Properties

Label 490.4.c.f.99.12
Level $490$
Weight $4$
Character 490.99
Analytic conductor $28.911$
Analytic rank $0$
Dimension $12$
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] = [490,4,Mod(99,490)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(490, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("490.99");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 490.c (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: \(28.9109359028\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 185x^{10} + 12748x^{8} + 405460x^{6} + 5908496x^{4} + 33016000x^{2} + 60840000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.12
Root \(7.00242i\) of defining polynomial
Character \(\chi\) \(=\) 490.99
Dual form 490.4.c.f.99.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000i q^{2} +6.00242i q^{3} -4.00000 q^{4} +(3.14360 - 10.7293i) q^{5} -12.0048 q^{6} -8.00000i q^{8} -9.02909 q^{9} +O(q^{10})\) \(q+2.00000i q^{2} +6.00242i q^{3} -4.00000 q^{4} +(3.14360 - 10.7293i) q^{5} -12.0048 q^{6} -8.00000i q^{8} -9.02909 q^{9} +(21.4586 + 6.28719i) q^{10} +9.14868 q^{11} -24.0097i q^{12} -32.7036i q^{13} +(64.4018 + 18.8692i) q^{15} +16.0000 q^{16} +109.798i q^{17} -18.0582i q^{18} +22.2644 q^{19} +(-12.5744 + 42.9172i) q^{20} +18.2974i q^{22} +191.279i q^{23} +48.0194 q^{24} +(-105.236 - 67.4571i) q^{25} +65.4072 q^{26} +107.869i q^{27} -190.986 q^{29} +(-37.7384 + 128.804i) q^{30} +35.6111 q^{31} +32.0000i q^{32} +54.9143i q^{33} -219.595 q^{34} +36.1164 q^{36} +57.7420i q^{37} +44.5287i q^{38} +196.301 q^{39} +(-85.8344 - 25.1488i) q^{40} +39.7463 q^{41} +323.003i q^{43} -36.5947 q^{44} +(-28.3838 + 96.8758i) q^{45} -382.557 q^{46} +57.1037i q^{47} +96.0388i q^{48} +(134.914 - 210.471i) q^{50} -659.052 q^{51} +130.814i q^{52} +529.738i q^{53} -215.738 q^{54} +(28.7597 - 98.1589i) q^{55} +133.640i q^{57} -381.973i q^{58} +766.697 q^{59} +(-257.607 - 75.4768i) q^{60} +524.009 q^{61} +71.2221i q^{62} -64.0000 q^{64} +(-350.887 - 102.807i) q^{65} -109.829 q^{66} +370.893i q^{67} -439.191i q^{68} -1148.14 q^{69} -722.165 q^{71} +72.2327i q^{72} -829.047i q^{73} -115.484 q^{74} +(404.906 - 631.669i) q^{75} -89.0575 q^{76} +392.602i q^{78} -494.621 q^{79} +(50.2975 - 171.669i) q^{80} -891.261 q^{81} +79.4926i q^{82} -1241.34i q^{83} +(1178.05 + 345.159i) q^{85} -646.005 q^{86} -1146.38i q^{87} -73.1895i q^{88} -20.7580 q^{89} +(-193.752 - 56.7676i) q^{90} -765.115i q^{92} +213.753i q^{93} -114.207 q^{94} +(69.9902 - 238.881i) q^{95} -192.078 q^{96} +1010.36i q^{97} -82.6042 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 48 q^{4} + 8 q^{5} + 28 q^{6} - 62 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 48 q^{4} + 8 q^{5} + 28 q^{6} - 62 q^{9} - 12 q^{10} + 62 q^{11} + 86 q^{15} + 192 q^{16} - 186 q^{19} - 32 q^{20} - 112 q^{24} - 126 q^{25} + 236 q^{26} - 338 q^{29} - 28 q^{30} + 652 q^{31} - 272 q^{34} + 248 q^{36} + 868 q^{39} + 48 q^{40} + 396 q^{41} - 248 q^{44} - 664 q^{45} - 376 q^{46} + 160 q^{50} + 1448 q^{51} - 1540 q^{54} + 298 q^{55} - 1336 q^{59} - 344 q^{60} + 314 q^{61} - 768 q^{64} - 1862 q^{65} + 1600 q^{66} + 90 q^{69} + 2216 q^{71} + 1012 q^{74} + 4550 q^{75} + 744 q^{76} + 1772 q^{79} + 128 q^{80} - 1228 q^{81} + 2282 q^{85} - 396 q^{86} - 6094 q^{89} + 100 q^{90} - 3604 q^{94} - 1166 q^{95} + 448 q^{96} - 8546 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/490\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000i 0.707107i
\(3\) 6.00242i 1.15517i 0.816332 + 0.577583i \(0.196004\pi\)
−0.816332 + 0.577583i \(0.803996\pi\)
\(4\) −4.00000 −0.500000
\(5\) 3.14360 10.7293i 0.281172 0.959657i
\(6\) −12.0048 −0.816826
\(7\) 0 0
\(8\) 8.00000i 0.353553i
\(9\) −9.02909 −0.334411
\(10\) 21.4586 + 6.28719i 0.678580 + 0.198818i
\(11\) 9.14868 0.250766 0.125383 0.992108i \(-0.459984\pi\)
0.125383 + 0.992108i \(0.459984\pi\)
\(12\) 24.0097i 0.577583i
\(13\) 32.7036i 0.697719i −0.937175 0.348859i \(-0.886569\pi\)
0.937175 0.348859i \(-0.113431\pi\)
\(14\) 0 0
\(15\) 64.4018 + 18.8692i 1.10856 + 0.324800i
\(16\) 16.0000 0.250000
\(17\) 109.798i 1.56646i 0.621731 + 0.783231i \(0.286430\pi\)
−0.621731 + 0.783231i \(0.713570\pi\)
\(18\) 18.0582i 0.236464i
\(19\) 22.2644 0.268831 0.134416 0.990925i \(-0.457084\pi\)
0.134416 + 0.990925i \(0.457084\pi\)
\(20\) −12.5744 + 42.9172i −0.140586 + 0.479829i
\(21\) 0 0
\(22\) 18.2974i 0.177319i
\(23\) 191.279i 1.73410i 0.498219 + 0.867051i \(0.333988\pi\)
−0.498219 + 0.867051i \(0.666012\pi\)
\(24\) 48.0194 0.408413
\(25\) −105.236 67.4571i −0.841885 0.539657i
\(26\) 65.4072 0.493362
\(27\) 107.869i 0.768867i
\(28\) 0 0
\(29\) −190.986 −1.22294 −0.611470 0.791267i \(-0.709422\pi\)
−0.611470 + 0.791267i \(0.709422\pi\)
\(30\) −37.7384 + 128.804i −0.229668 + 0.783874i
\(31\) 35.6111 0.206320 0.103160 0.994665i \(-0.467105\pi\)
0.103160 + 0.994665i \(0.467105\pi\)
\(32\) 32.0000i 0.176777i
\(33\) 54.9143i 0.289677i
\(34\) −219.595 −1.10766
\(35\) 0 0
\(36\) 36.1164 0.167205
\(37\) 57.7420i 0.256560i 0.991738 + 0.128280i \(0.0409457\pi\)
−0.991738 + 0.128280i \(0.959054\pi\)
\(38\) 44.5287i 0.190092i
\(39\) 196.301 0.805982
\(40\) −85.8344 25.1488i −0.339290 0.0994092i
\(41\) 39.7463 0.151398 0.0756991 0.997131i \(-0.475881\pi\)
0.0756991 + 0.997131i \(0.475881\pi\)
\(42\) 0 0
\(43\) 323.003i 1.14552i 0.819722 + 0.572761i \(0.194128\pi\)
−0.819722 + 0.572761i \(0.805872\pi\)
\(44\) −36.5947 −0.125383
\(45\) −28.3838 + 96.8758i −0.0940268 + 0.320920i
\(46\) −382.557 −1.22620
\(47\) 57.1037i 0.177222i 0.996066 + 0.0886110i \(0.0282428\pi\)
−0.996066 + 0.0886110i \(0.971757\pi\)
\(48\) 96.0388i 0.288792i
\(49\) 0 0
\(50\) 134.914 210.471i 0.381595 0.595303i
\(51\) −659.052 −1.80952
\(52\) 130.814i 0.348859i
\(53\) 529.738i 1.37293i 0.727165 + 0.686463i \(0.240838\pi\)
−0.727165 + 0.686463i \(0.759162\pi\)
\(54\) −215.738 −0.543671
\(55\) 28.7597 98.1589i 0.0705084 0.240650i
\(56\) 0 0
\(57\) 133.640i 0.310545i
\(58\) 381.973i 0.864750i
\(59\) 766.697 1.69179 0.845894 0.533352i \(-0.179068\pi\)
0.845894 + 0.533352i \(0.179068\pi\)
\(60\) −257.607 75.4768i −0.554282 0.162400i
\(61\) 524.009 1.09988 0.549939 0.835205i \(-0.314651\pi\)
0.549939 + 0.835205i \(0.314651\pi\)
\(62\) 71.2221i 0.145891i
\(63\) 0 0
\(64\) −64.0000 −0.125000
\(65\) −350.887 102.807i −0.669571 0.196179i
\(66\) −109.829 −0.204833
\(67\) 370.893i 0.676295i 0.941093 + 0.338148i \(0.109800\pi\)
−0.941093 + 0.338148i \(0.890200\pi\)
\(68\) 439.191i 0.783231i
\(69\) −1148.14 −2.00318
\(70\) 0 0
\(71\) −722.165 −1.20712 −0.603558 0.797319i \(-0.706251\pi\)
−0.603558 + 0.797319i \(0.706251\pi\)
\(72\) 72.2327i 0.118232i
\(73\) 829.047i 1.32921i −0.747193 0.664607i \(-0.768599\pi\)
0.747193 0.664607i \(-0.231401\pi\)
\(74\) −115.484 −0.181416
\(75\) 404.906 631.669i 0.623394 0.972518i
\(76\) −89.0575 −0.134416
\(77\) 0 0
\(78\) 392.602i 0.569915i
\(79\) −494.621 −0.704421 −0.352210 0.935921i \(-0.614570\pi\)
−0.352210 + 0.935921i \(0.614570\pi\)
\(80\) 50.2975 171.669i 0.0702929 0.239914i
\(81\) −891.261 −1.22258
\(82\) 79.4926i 0.107055i
\(83\) 1241.34i 1.64162i −0.571201 0.820810i \(-0.693522\pi\)
0.571201 0.820810i \(-0.306478\pi\)
\(84\) 0 0
\(85\) 1178.05 + 345.159i 1.50327 + 0.440445i
\(86\) −646.005 −0.810007
\(87\) 1146.38i 1.41270i
\(88\) 73.1895i 0.0886593i
\(89\) −20.7580 −0.0247229 −0.0123615 0.999924i \(-0.503935\pi\)
−0.0123615 + 0.999924i \(0.503935\pi\)
\(90\) −193.752 56.7676i −0.226924 0.0664870i
\(91\) 0 0
\(92\) 765.115i 0.867051i
\(93\) 213.753i 0.238335i
\(94\) −114.207 −0.125315
\(95\) 69.9902 238.881i 0.0755878 0.257986i
\(96\) −192.078 −0.204207
\(97\) 1010.36i 1.05759i 0.848748 + 0.528797i \(0.177357\pi\)
−0.848748 + 0.528797i \(0.822643\pi\)
\(98\) 0 0
\(99\) −82.6042 −0.0838590
\(100\) 420.942 + 269.829i 0.420942 + 0.269829i
\(101\) −1286.97 −1.26790 −0.633950 0.773374i \(-0.718567\pi\)
−0.633950 + 0.773374i \(0.718567\pi\)
\(102\) 1318.10i 1.27953i
\(103\) 1648.16i 1.57668i 0.615242 + 0.788338i \(0.289058\pi\)
−0.615242 + 0.788338i \(0.710942\pi\)
\(104\) −261.629 −0.246681
\(105\) 0 0
\(106\) −1059.48 −0.970805
\(107\) 68.3883i 0.0617883i 0.999523 + 0.0308941i \(0.00983548\pi\)
−0.999523 + 0.0308941i \(0.990165\pi\)
\(108\) 431.476i 0.384433i
\(109\) −187.622 −0.164871 −0.0824353 0.996596i \(-0.526270\pi\)
−0.0824353 + 0.996596i \(0.526270\pi\)
\(110\) 196.318 + 57.5195i 0.170165 + 0.0498570i
\(111\) −346.592 −0.296370
\(112\) 0 0
\(113\) 979.328i 0.815287i −0.913141 0.407643i \(-0.866351\pi\)
0.913141 0.407643i \(-0.133649\pi\)
\(114\) −267.280 −0.219589
\(115\) 2052.29 + 601.303i 1.66414 + 0.487580i
\(116\) 763.946 0.611470
\(117\) 295.284i 0.233325i
\(118\) 1533.39i 1.19627i
\(119\) 0 0
\(120\) 150.954 515.214i 0.114834 0.391937i
\(121\) −1247.30 −0.937116
\(122\) 1048.02i 0.777731i
\(123\) 238.574i 0.174890i
\(124\) −142.444 −0.103160
\(125\) −1054.59 + 917.046i −0.754600 + 0.656185i
\(126\) 0 0
\(127\) 58.6732i 0.0409953i −0.999790 0.0204977i \(-0.993475\pi\)
0.999790 0.0204977i \(-0.00652507\pi\)
\(128\) 128.000i 0.0883883i
\(129\) −1938.80 −1.32327
\(130\) 205.614 701.773i 0.138719 0.473458i
\(131\) 1530.15 1.02053 0.510265 0.860017i \(-0.329547\pi\)
0.510265 + 0.860017i \(0.329547\pi\)
\(132\) 219.657i 0.144839i
\(133\) 0 0
\(134\) −741.786 −0.478213
\(135\) 1157.36 + 339.097i 0.737849 + 0.216184i
\(136\) 878.381 0.553828
\(137\) 3042.88i 1.89760i 0.315882 + 0.948798i \(0.397699\pi\)
−0.315882 + 0.948798i \(0.602301\pi\)
\(138\) 2296.27i 1.41646i
\(139\) 1793.28 1.09427 0.547136 0.837044i \(-0.315718\pi\)
0.547136 + 0.837044i \(0.315718\pi\)
\(140\) 0 0
\(141\) −342.761 −0.204721
\(142\) 1444.33i 0.853559i
\(143\) 299.195i 0.174964i
\(144\) −144.465 −0.0836027
\(145\) −600.384 + 2049.15i −0.343856 + 1.17360i
\(146\) 1658.09 0.939897
\(147\) 0 0
\(148\) 230.968i 0.128280i
\(149\) 2008.16 1.10413 0.552064 0.833802i \(-0.313841\pi\)
0.552064 + 0.833802i \(0.313841\pi\)
\(150\) 1263.34 + 809.813i 0.687674 + 0.440806i
\(151\) −59.1933 −0.0319012 −0.0159506 0.999873i \(-0.505077\pi\)
−0.0159506 + 0.999873i \(0.505077\pi\)
\(152\) 178.115i 0.0950462i
\(153\) 991.373i 0.523841i
\(154\) 0 0
\(155\) 111.947 382.082i 0.0580115 0.197997i
\(156\) −785.203 −0.402991
\(157\) 2671.04i 1.35778i 0.734238 + 0.678892i \(0.237540\pi\)
−0.734238 + 0.678892i \(0.762460\pi\)
\(158\) 989.242i 0.498101i
\(159\) −3179.71 −1.58596
\(160\) 343.337 + 100.595i 0.169645 + 0.0497046i
\(161\) 0 0
\(162\) 1782.52i 0.864495i
\(163\) 2767.06i 1.32965i −0.747000 0.664824i \(-0.768507\pi\)
0.747000 0.664824i \(-0.231493\pi\)
\(164\) −158.985 −0.0756991
\(165\) 589.191 + 172.628i 0.277991 + 0.0814490i
\(166\) 2482.68 1.16080
\(167\) 1603.11i 0.742828i 0.928467 + 0.371414i \(0.121127\pi\)
−0.928467 + 0.371414i \(0.878873\pi\)
\(168\) 0 0
\(169\) 1127.48 0.513189
\(170\) −690.319 + 2356.10i −0.311441 + 1.06297i
\(171\) −201.027 −0.0899001
\(172\) 1292.01i 0.572761i
\(173\) 592.913i 0.260568i −0.991477 0.130284i \(-0.958411\pi\)
0.991477 0.130284i \(-0.0415890\pi\)
\(174\) 2292.76 0.998930
\(175\) 0 0
\(176\) 146.379 0.0626916
\(177\) 4602.04i 1.95430i
\(178\) 41.5159i 0.0174817i
\(179\) 499.392 0.208527 0.104263 0.994550i \(-0.466751\pi\)
0.104263 + 0.994550i \(0.466751\pi\)
\(180\) 113.535 387.503i 0.0470134 0.160460i
\(181\) −1133.42 −0.465450 −0.232725 0.972543i \(-0.574764\pi\)
−0.232725 + 0.972543i \(0.574764\pi\)
\(182\) 0 0
\(183\) 3145.33i 1.27054i
\(184\) 1530.23 0.613098
\(185\) 619.531 + 181.518i 0.246210 + 0.0721375i
\(186\) −427.505 −0.168528
\(187\) 1004.50i 0.392816i
\(188\) 228.415i 0.0886110i
\(189\) 0 0
\(190\) 477.762 + 139.980i 0.182424 + 0.0534486i
\(191\) 2389.57 0.905251 0.452625 0.891701i \(-0.350487\pi\)
0.452625 + 0.891701i \(0.350487\pi\)
\(192\) 384.155i 0.144396i
\(193\) 608.230i 0.226846i −0.993547 0.113423i \(-0.963818\pi\)
0.993547 0.113423i \(-0.0361816\pi\)
\(194\) −2020.72 −0.747832
\(195\) 617.090 2106.17i 0.226619 0.773466i
\(196\) 0 0
\(197\) 2976.07i 1.07633i 0.842841 + 0.538163i \(0.180882\pi\)
−0.842841 + 0.538163i \(0.819118\pi\)
\(198\) 165.208i 0.0592973i
\(199\) −1731.83 −0.616916 −0.308458 0.951238i \(-0.599813\pi\)
−0.308458 + 0.951238i \(0.599813\pi\)
\(200\) −539.657 + 841.885i −0.190798 + 0.297651i
\(201\) −2226.26 −0.781234
\(202\) 2573.93i 0.896540i
\(203\) 0 0
\(204\) 2636.21 0.904762
\(205\) 124.946 426.450i 0.0425689 0.145291i
\(206\) −3296.31 −1.11488
\(207\) 1727.07i 0.579902i
\(208\) 523.257i 0.174430i
\(209\) 203.690 0.0674139
\(210\) 0 0
\(211\) −785.955 −0.256433 −0.128217 0.991746i \(-0.540925\pi\)
−0.128217 + 0.991746i \(0.540925\pi\)
\(212\) 2118.95i 0.686463i
\(213\) 4334.74i 1.39442i
\(214\) −136.777 −0.0436909
\(215\) 3465.59 + 1015.39i 1.09931 + 0.322088i
\(216\) 862.952 0.271835
\(217\) 0 0
\(218\) 375.243i 0.116581i
\(219\) 4976.29 1.53546
\(220\) −115.039 + 392.636i −0.0352542 + 0.120325i
\(221\) 3590.78 1.09295
\(222\) 693.184i 0.209565i
\(223\) 6112.17i 1.83543i −0.397237 0.917716i \(-0.630031\pi\)
0.397237 0.917716i \(-0.369969\pi\)
\(224\) 0 0
\(225\) 950.182 + 609.076i 0.281535 + 0.180467i
\(226\) 1958.66 0.576495
\(227\) 2133.14i 0.623706i −0.950130 0.311853i \(-0.899050\pi\)
0.950130 0.311853i \(-0.100950\pi\)
\(228\) 534.561i 0.155273i
\(229\) −2452.04 −0.707578 −0.353789 0.935325i \(-0.615107\pi\)
−0.353789 + 0.935325i \(0.615107\pi\)
\(230\) −1202.61 + 4104.57i −0.344771 + 1.17673i
\(231\) 0 0
\(232\) 1527.89i 0.432375i
\(233\) 4067.42i 1.14363i −0.820383 0.571814i \(-0.806240\pi\)
0.820383 0.571814i \(-0.193760\pi\)
\(234\) −590.567 −0.164985
\(235\) 612.683 + 179.511i 0.170072 + 0.0498298i
\(236\) −3066.79 −0.845894
\(237\) 2968.93i 0.813723i
\(238\) 0 0
\(239\) −860.208 −0.232813 −0.116406 0.993202i \(-0.537137\pi\)
−0.116406 + 0.993202i \(0.537137\pi\)
\(240\) 1030.43 + 301.907i 0.277141 + 0.0812001i
\(241\) 60.9641 0.0162948 0.00814739 0.999967i \(-0.497407\pi\)
0.00814739 + 0.999967i \(0.497407\pi\)
\(242\) 2494.60i 0.662641i
\(243\) 2437.26i 0.643417i
\(244\) −2096.04 −0.549939
\(245\) 0 0
\(246\) −477.148 −0.123666
\(247\) 728.125i 0.187569i
\(248\) 284.888i 0.0729453i
\(249\) 7451.03 1.89635
\(250\) −1834.09 2109.17i −0.463993 0.533583i
\(251\) 4683.77 1.17784 0.588919 0.808192i \(-0.299554\pi\)
0.588919 + 0.808192i \(0.299554\pi\)
\(252\) 0 0
\(253\) 1749.95i 0.434855i
\(254\) 117.346 0.0289881
\(255\) −2071.79 + 7071.17i −0.508787 + 1.73652i
\(256\) 256.000 0.0625000
\(257\) 930.412i 0.225827i −0.993605 0.112913i \(-0.963982\pi\)
0.993605 0.112913i \(-0.0360183\pi\)
\(258\) 3877.60i 0.935693i
\(259\) 0 0
\(260\) 1403.55 + 411.227i 0.334785 + 0.0980894i
\(261\) 1724.43 0.408964
\(262\) 3060.29i 0.721624i
\(263\) 5682.86i 1.33240i 0.745775 + 0.666198i \(0.232079\pi\)
−0.745775 + 0.666198i \(0.767921\pi\)
\(264\) 439.314 0.102416
\(265\) 5683.71 + 1665.28i 1.31754 + 0.386028i
\(266\) 0 0
\(267\) 124.598i 0.0285591i
\(268\) 1483.57i 0.338148i
\(269\) −2418.73 −0.548224 −0.274112 0.961698i \(-0.588384\pi\)
−0.274112 + 0.961698i \(0.588384\pi\)
\(270\) −678.193 + 2314.72i −0.152865 + 0.521738i
\(271\) 327.762 0.0734692 0.0367346 0.999325i \(-0.488304\pi\)
0.0367346 + 0.999325i \(0.488304\pi\)
\(272\) 1756.76i 0.391615i
\(273\) 0 0
\(274\) −6085.76 −1.34180
\(275\) −962.767 617.144i −0.211117 0.135328i
\(276\) 4592.54 1.00159
\(277\) 1530.62i 0.332008i 0.986125 + 0.166004i \(0.0530864\pi\)
−0.986125 + 0.166004i \(0.946914\pi\)
\(278\) 3586.55i 0.773767i
\(279\) −321.535 −0.0689958
\(280\) 0 0
\(281\) 9173.67 1.94753 0.973764 0.227561i \(-0.0730751\pi\)
0.973764 + 0.227561i \(0.0730751\pi\)
\(282\) 685.522i 0.144760i
\(283\) 3450.42i 0.724756i −0.932031 0.362378i \(-0.881965\pi\)
0.932031 0.362378i \(-0.118035\pi\)
\(284\) 2888.66 0.603558
\(285\) 1433.87 + 420.111i 0.298017 + 0.0873165i
\(286\) 598.389 0.123719
\(287\) 0 0
\(288\) 288.931i 0.0591160i
\(289\) −7142.53 −1.45380
\(290\) −4098.30 1200.77i −0.829864 0.243143i
\(291\) −6064.62 −1.22170
\(292\) 3316.19i 0.664607i
\(293\) 5618.66i 1.12029i 0.828394 + 0.560146i \(0.189255\pi\)
−0.828394 + 0.560146i \(0.810745\pi\)
\(294\) 0 0
\(295\) 2410.18 8226.12i 0.475683 1.62354i
\(296\) 461.936 0.0907078
\(297\) 986.859i 0.192806i
\(298\) 4016.32i 0.780736i
\(299\) 6255.50 1.20992
\(300\) −1619.63 + 2526.68i −0.311697 + 0.486259i
\(301\) 0 0
\(302\) 118.387i 0.0225576i
\(303\) 7724.91i 1.46464i
\(304\) 356.230 0.0672078
\(305\) 1647.27 5622.25i 0.309254 1.05551i
\(306\) 1982.75 0.370412
\(307\) 3600.31i 0.669317i 0.942339 + 0.334658i \(0.108621\pi\)
−0.942339 + 0.334658i \(0.891379\pi\)
\(308\) 0 0
\(309\) −9892.94 −1.82132
\(310\) 764.163 + 223.894i 0.140005 + 0.0410203i
\(311\) 4567.54 0.832803 0.416401 0.909181i \(-0.363291\pi\)
0.416401 + 0.909181i \(0.363291\pi\)
\(312\) 1570.41i 0.284958i
\(313\) 2299.84i 0.415318i −0.978201 0.207659i \(-0.933416\pi\)
0.978201 0.207659i \(-0.0665844\pi\)
\(314\) −5342.08 −0.960099
\(315\) 0 0
\(316\) 1978.48 0.352210
\(317\) 10423.8i 1.84688i 0.383743 + 0.923440i \(0.374635\pi\)
−0.383743 + 0.923440i \(0.625365\pi\)
\(318\) 6359.42i 1.12144i
\(319\) −1747.27 −0.306673
\(320\) −201.190 + 686.675i −0.0351465 + 0.119957i
\(321\) −410.496 −0.0713758
\(322\) 0 0
\(323\) 2444.58i 0.421114i
\(324\) 3565.04 0.611290
\(325\) −2206.09 + 3441.58i −0.376529 + 0.587399i
\(326\) 5534.11 0.940203
\(327\) 1126.18i 0.190453i
\(328\) 317.970i 0.0535274i
\(329\) 0 0
\(330\) −345.256 + 1178.38i −0.0575931 + 0.196569i
\(331\) 6217.79 1.03251 0.516255 0.856435i \(-0.327326\pi\)
0.516255 + 0.856435i \(0.327326\pi\)
\(332\) 4965.35i 0.820810i
\(333\) 521.358i 0.0857965i
\(334\) −3206.22 −0.525259
\(335\) 3979.42 + 1165.94i 0.649012 + 0.190155i
\(336\) 0 0
\(337\) 11329.0i 1.83125i −0.402034 0.915625i \(-0.631697\pi\)
0.402034 0.915625i \(-0.368303\pi\)
\(338\) 2254.95i 0.362879i
\(339\) 5878.34 0.941792
\(340\) −4712.21 1380.64i −0.751633 0.220222i
\(341\) 325.794 0.0517383
\(342\) 402.054i 0.0635690i
\(343\) 0 0
\(344\) 2584.02 0.405003
\(345\) −3609.27 + 12318.7i −0.563237 + 1.92236i
\(346\) 1185.83 0.184250
\(347\) 10969.1i 1.69698i −0.529211 0.848490i \(-0.677512\pi\)
0.529211 0.848490i \(-0.322488\pi\)
\(348\) 4585.53i 0.706350i
\(349\) 3925.09 0.602021 0.301010 0.953621i \(-0.402676\pi\)
0.301010 + 0.953621i \(0.402676\pi\)
\(350\) 0 0
\(351\) 3527.70 0.536453
\(352\) 292.758i 0.0443297i
\(353\) 8409.37i 1.26795i −0.773354 0.633974i \(-0.781422\pi\)
0.773354 0.633974i \(-0.218578\pi\)
\(354\) −9204.08 −1.38190
\(355\) −2270.19 + 7748.32i −0.339407 + 1.15842i
\(356\) 83.0319 0.0123615
\(357\) 0 0
\(358\) 998.784i 0.147451i
\(359\) −5024.14 −0.738618 −0.369309 0.929307i \(-0.620406\pi\)
−0.369309 + 0.929307i \(0.620406\pi\)
\(360\) 775.006 + 227.070i 0.113462 + 0.0332435i
\(361\) −6363.30 −0.927730
\(362\) 2266.84i 0.329123i
\(363\) 7486.83i 1.08253i
\(364\) 0 0
\(365\) −8895.10 2606.19i −1.27559 0.373738i
\(366\) −6290.65 −0.898409
\(367\) 2700.78i 0.384140i 0.981381 + 0.192070i \(0.0615201\pi\)
−0.981381 + 0.192070i \(0.938480\pi\)
\(368\) 3060.46i 0.433526i
\(369\) −358.873 −0.0506292
\(370\) −363.035 + 1239.06i −0.0510089 + 0.174097i
\(371\) 0 0
\(372\) 855.011i 0.119167i
\(373\) 7738.90i 1.07428i −0.843494 0.537138i \(-0.819505\pi\)
0.843494 0.537138i \(-0.180495\pi\)
\(374\) −2009.01 −0.277763
\(375\) −5504.50 6330.07i −0.758003 0.871689i
\(376\) 456.830 0.0626574
\(377\) 6245.94i 0.853269i
\(378\) 0 0
\(379\) −5512.60 −0.747132 −0.373566 0.927604i \(-0.621865\pi\)
−0.373566 + 0.927604i \(0.621865\pi\)
\(380\) −279.961 + 955.524i −0.0377939 + 0.128993i
\(381\) 352.182 0.0473565
\(382\) 4779.13i 0.640109i
\(383\) 921.050i 0.122881i −0.998111 0.0614405i \(-0.980431\pi\)
0.998111 0.0614405i \(-0.0195695\pi\)
\(384\) 768.310 0.102103
\(385\) 0 0
\(386\) 1216.46 0.160405
\(387\) 2916.42i 0.383075i
\(388\) 4041.45i 0.528797i
\(389\) −734.690 −0.0957590 −0.0478795 0.998853i \(-0.515246\pi\)
−0.0478795 + 0.998853i \(0.515246\pi\)
\(390\) 4212.34 + 1234.18i 0.546923 + 0.160244i
\(391\) −21002.0 −2.71640
\(392\) 0 0
\(393\) 9184.58i 1.17888i
\(394\) −5952.14 −0.761078
\(395\) −1554.89 + 5306.94i −0.198063 + 0.676002i
\(396\) 330.417 0.0419295
\(397\) 6234.68i 0.788185i −0.919071 0.394093i \(-0.871059\pi\)
0.919071 0.394093i \(-0.128941\pi\)
\(398\) 3463.67i 0.436226i
\(399\) 0 0
\(400\) −1683.77 1079.31i −0.210471 0.134914i
\(401\) 1147.33 0.142880 0.0714399 0.997445i \(-0.477241\pi\)
0.0714399 + 0.997445i \(0.477241\pi\)
\(402\) 4452.51i 0.552416i
\(403\) 1164.61i 0.143954i
\(404\) 5147.86 0.633950
\(405\) −2801.76 + 9562.60i −0.343755 + 1.17326i
\(406\) 0 0
\(407\) 528.263i 0.0643367i
\(408\) 5272.42i 0.639764i
\(409\) 5333.98 0.644862 0.322431 0.946593i \(-0.395500\pi\)
0.322431 + 0.946593i \(0.395500\pi\)
\(410\) 852.900 + 249.893i 0.102736 + 0.0301008i
\(411\) −18264.7 −2.19204
\(412\) 6592.63i 0.788338i
\(413\) 0 0
\(414\) 3454.14 0.410053
\(415\) −13318.7 3902.26i −1.57539 0.461577i
\(416\) 1046.51 0.123340
\(417\) 10764.0i 1.26407i
\(418\) 407.379i 0.0476688i
\(419\) −11602.6 −1.35280 −0.676401 0.736533i \(-0.736462\pi\)
−0.676401 + 0.736533i \(0.736462\pi\)
\(420\) 0 0
\(421\) −5948.07 −0.688578 −0.344289 0.938864i \(-0.611880\pi\)
−0.344289 + 0.938864i \(0.611880\pi\)
\(422\) 1571.91i 0.181326i
\(423\) 515.595i 0.0592649i
\(424\) 4237.90 0.485403
\(425\) 7406.64 11554.6i 0.845352 1.31878i
\(426\) 8669.48 0.986004
\(427\) 0 0
\(428\) 273.553i 0.0308941i
\(429\) 1795.89 0.202113
\(430\) −2030.78 + 6931.18i −0.227751 + 0.777329i
\(431\) 3708.25 0.414432 0.207216 0.978295i \(-0.433560\pi\)
0.207216 + 0.978295i \(0.433560\pi\)
\(432\) 1725.90i 0.192217i
\(433\) 4775.00i 0.529958i −0.964254 0.264979i \(-0.914635\pi\)
0.964254 0.264979i \(-0.0853651\pi\)
\(434\) 0 0
\(435\) −12299.9 3603.76i −1.35571 0.397211i
\(436\) 750.487 0.0824353
\(437\) 4258.70i 0.466181i
\(438\) 9952.59i 1.08574i
\(439\) 3933.57 0.427652 0.213826 0.976872i \(-0.431407\pi\)
0.213826 + 0.976872i \(0.431407\pi\)
\(440\) −785.271 230.078i −0.0850826 0.0249285i
\(441\) 0 0
\(442\) 7181.56i 0.772832i
\(443\) 8730.40i 0.936329i −0.883641 0.468165i \(-0.844915\pi\)
0.883641 0.468165i \(-0.155085\pi\)
\(444\) 1386.37 0.148185
\(445\) −65.2547 + 222.718i −0.00695139 + 0.0237255i
\(446\) 12224.3 1.29785
\(447\) 12053.8i 1.27545i
\(448\) 0 0
\(449\) 18724.5 1.96807 0.984035 0.177977i \(-0.0569553\pi\)
0.984035 + 0.177977i \(0.0569553\pi\)
\(450\) −1218.15 + 1900.36i −0.127609 + 0.199076i
\(451\) 363.626 0.0379656
\(452\) 3917.31i 0.407643i
\(453\) 355.303i 0.0368512i
\(454\) 4266.28 0.441027
\(455\) 0 0
\(456\) 1069.12 0.109794
\(457\) 4384.10i 0.448752i 0.974503 + 0.224376i \(0.0720343\pi\)
−0.974503 + 0.224376i \(0.927966\pi\)
\(458\) 4904.08i 0.500333i
\(459\) −11843.8 −1.20440
\(460\) −8209.14 2405.21i −0.832072 0.243790i
\(461\) −12719.7 −1.28507 −0.642533 0.766258i \(-0.722117\pi\)
−0.642533 + 0.766258i \(0.722117\pi\)
\(462\) 0 0
\(463\) 4806.34i 0.482440i −0.970470 0.241220i \(-0.922452\pi\)
0.970470 0.241220i \(-0.0775476\pi\)
\(464\) −3055.78 −0.305735
\(465\) 2293.42 + 671.952i 0.228720 + 0.0670129i
\(466\) 8134.83 0.808667
\(467\) 1985.62i 0.196753i 0.995149 + 0.0983763i \(0.0313649\pi\)
−0.995149 + 0.0983763i \(0.968635\pi\)
\(468\) 1181.13i 0.116662i
\(469\) 0 0
\(470\) −359.022 + 1225.37i −0.0352350 + 0.120259i
\(471\) −16032.7 −1.56847
\(472\) 6133.58i 0.598137i
\(473\) 2955.05i 0.287259i
\(474\) 5937.85 0.575389
\(475\) −2343.00 1501.89i −0.226325 0.145077i
\(476\) 0 0
\(477\) 4783.05i 0.459121i
\(478\) 1720.42i 0.164623i
\(479\) −6897.05 −0.657900 −0.328950 0.944347i \(-0.606695\pi\)
−0.328950 + 0.944347i \(0.606695\pi\)
\(480\) −603.814 + 2060.86i −0.0574171 + 0.195968i
\(481\) 1888.37 0.179007
\(482\) 121.928i 0.0115222i
\(483\) 0 0
\(484\) 4989.21 0.468558
\(485\) 10840.5 + 3176.17i 1.01493 + 0.297366i
\(486\) 4874.52 0.454965
\(487\) 11260.0i 1.04772i −0.851805 0.523859i \(-0.824492\pi\)
0.851805 0.523859i \(-0.175508\pi\)
\(488\) 4192.07i 0.388865i
\(489\) 16609.0 1.53596
\(490\) 0 0
\(491\) 13161.5 1.20972 0.604858 0.796333i \(-0.293230\pi\)
0.604858 + 0.796333i \(0.293230\pi\)
\(492\) 954.297i 0.0874452i
\(493\) 20969.9i 1.91569i
\(494\) 1456.25 0.132631
\(495\) −259.674 + 886.285i −0.0235788 + 0.0804759i
\(496\) 569.777 0.0515801
\(497\) 0 0
\(498\) 14902.1i 1.34092i
\(499\) −12404.0 −1.11278 −0.556392 0.830920i \(-0.687815\pi\)
−0.556392 + 0.830920i \(0.687815\pi\)
\(500\) 4218.34 3668.19i 0.377300 0.328092i
\(501\) −9622.54 −0.858091
\(502\) 9367.55i 0.832857i
\(503\) 10548.2i 0.935027i −0.883986 0.467514i \(-0.845150\pi\)
0.883986 0.467514i \(-0.154850\pi\)
\(504\) 0 0
\(505\) −4045.70 + 13808.2i −0.356497 + 1.21675i
\(506\) −3499.89 −0.307489
\(507\) 6767.58i 0.592818i
\(508\) 234.693i 0.0204977i
\(509\) 10749.6 0.936088 0.468044 0.883705i \(-0.344959\pi\)
0.468044 + 0.883705i \(0.344959\pi\)
\(510\) −14142.3 4143.59i −1.22791 0.359767i
\(511\) 0 0
\(512\) 512.000i 0.0441942i
\(513\) 2401.64i 0.206696i
\(514\) 1860.82 0.159684
\(515\) 17683.6 + 5181.14i 1.51307 + 0.443317i
\(516\) 7755.20 0.661635
\(517\) 522.424i 0.0444413i
\(518\) 0 0
\(519\) 3558.91 0.301000
\(520\) −822.455 + 2807.09i −0.0693597 + 0.236729i
\(521\) 13734.3 1.15492 0.577459 0.816420i \(-0.304044\pi\)
0.577459 + 0.816420i \(0.304044\pi\)
\(522\) 3448.87i 0.289182i
\(523\) 13966.9i 1.16775i −0.811845 0.583873i \(-0.801537\pi\)
0.811845 0.583873i \(-0.198463\pi\)
\(524\) −6120.58 −0.510265
\(525\) 0 0
\(526\) −11365.7 −0.942146
\(527\) 3910.01i 0.323193i
\(528\) 878.628i 0.0724193i
\(529\) −24420.5 −2.00711
\(530\) −3330.56 + 11367.4i −0.272963 + 0.931641i
\(531\) −6922.57 −0.565752
\(532\) 0 0
\(533\) 1299.85i 0.105633i
\(534\) 249.196 0.0201943
\(535\) 733.759 + 214.985i 0.0592956 + 0.0173731i
\(536\) 2967.14 0.239106
\(537\) 2997.56i 0.240883i
\(538\) 4837.45i 0.387653i
\(539\) 0 0
\(540\) −4629.44 1356.39i −0.368924 0.108092i
\(541\) 19473.7 1.54758 0.773790 0.633443i \(-0.218359\pi\)
0.773790 + 0.633443i \(0.218359\pi\)
\(542\) 655.525i 0.0519505i
\(543\) 6803.27i 0.537672i
\(544\) −3513.53 −0.276914
\(545\) −589.806 + 2013.05i −0.0463569 + 0.158219i
\(546\) 0 0
\(547\) 14279.9i 1.11620i 0.829773 + 0.558101i \(0.188470\pi\)
−0.829773 + 0.558101i \(0.811530\pi\)
\(548\) 12171.5i 0.948798i
\(549\) −4731.33 −0.367811
\(550\) 1234.29 1925.53i 0.0956913 0.149282i
\(551\) −4252.19 −0.328765
\(552\) 9185.08i 0.708230i
\(553\) 0 0
\(554\) −3061.24 −0.234765
\(555\) −1089.55 + 3718.69i −0.0833308 + 0.284414i
\(556\) −7173.11 −0.547136
\(557\) 8.99517i 0.000684269i 1.00000 0.000342134i \(0.000108905\pi\)
−1.00000 0.000342134i \(0.999891\pi\)
\(558\) 643.071i 0.0487874i
\(559\) 10563.3 0.799252
\(560\) 0 0
\(561\) −6029.46 −0.453768
\(562\) 18347.3i 1.37711i
\(563\) 14998.9i 1.12279i 0.827548 + 0.561394i \(0.189735\pi\)
−0.827548 + 0.561394i \(0.810265\pi\)
\(564\) 1371.04 0.102361
\(565\) −10507.5 3078.61i −0.782396 0.229236i
\(566\) 6900.83 0.512480
\(567\) 0 0
\(568\) 5777.32i 0.426780i
\(569\) −995.051 −0.0733123 −0.0366561 0.999328i \(-0.511671\pi\)
−0.0366561 + 0.999328i \(0.511671\pi\)
\(570\) −840.221 + 2867.73i −0.0617421 + 0.210730i
\(571\) −10830.2 −0.793750 −0.396875 0.917873i \(-0.629905\pi\)
−0.396875 + 0.917873i \(0.629905\pi\)
\(572\) 1196.78i 0.0874822i
\(573\) 14343.2i 1.04572i
\(574\) 0 0
\(575\) 12903.1 20129.3i 0.935821 1.45991i
\(576\) 577.862 0.0418013
\(577\) 7969.91i 0.575029i 0.957776 + 0.287515i \(0.0928290\pi\)
−0.957776 + 0.287515i \(0.907171\pi\)
\(578\) 14285.1i 1.02799i
\(579\) 3650.86 0.262046
\(580\) 2401.54 8196.60i 0.171928 0.586802i
\(581\) 0 0
\(582\) 12129.2i 0.863871i
\(583\) 4846.40i 0.344284i
\(584\) −6632.38 −0.469948
\(585\) 3168.19 + 928.252i 0.223912 + 0.0656043i
\(586\) −11237.3 −0.792166
\(587\) 17898.0i 1.25849i −0.777209 0.629243i \(-0.783365\pi\)
0.777209 0.629243i \(-0.216635\pi\)
\(588\) 0 0
\(589\) 792.858 0.0554654
\(590\) 16452.2 + 4820.37i 1.14801 + 0.336358i
\(591\) −17863.6 −1.24334
\(592\) 923.872i 0.0641401i
\(593\) 10769.4i 0.745781i −0.927875 0.372890i \(-0.878367\pi\)
0.927875 0.372890i \(-0.121633\pi\)
\(594\) −1973.72 −0.136334
\(595\) 0 0
\(596\) −8032.65 −0.552064
\(597\) 10395.2i 0.712642i
\(598\) 12511.0i 0.855540i
\(599\) 17177.4 1.17170 0.585852 0.810418i \(-0.300760\pi\)
0.585852 + 0.810418i \(0.300760\pi\)
\(600\) −5053.35 3239.25i −0.343837 0.220403i
\(601\) 13125.5 0.890849 0.445424 0.895320i \(-0.353053\pi\)
0.445424 + 0.895320i \(0.353053\pi\)
\(602\) 0 0
\(603\) 3348.82i 0.226160i
\(604\) 236.773 0.0159506
\(605\) −3921.01 + 13382.7i −0.263491 + 0.899311i
\(606\) 15449.8 1.03565
\(607\) 5942.07i 0.397333i 0.980067 + 0.198667i \(0.0636611\pi\)
−0.980067 + 0.198667i \(0.936339\pi\)
\(608\) 712.460i 0.0475231i
\(609\) 0 0
\(610\) 11244.5 + 3294.55i 0.746355 + 0.218676i
\(611\) 1867.50 0.123651
\(612\) 3965.49i 0.261921i
\(613\) 4731.24i 0.311734i 0.987778 + 0.155867i \(0.0498172\pi\)
−0.987778 + 0.155867i \(0.950183\pi\)
\(614\) −7200.61 −0.473279
\(615\) 2559.73 + 749.981i 0.167835 + 0.0491742i
\(616\) 0 0
\(617\) 4234.66i 0.276306i −0.990411 0.138153i \(-0.955883\pi\)
0.990411 0.138153i \(-0.0441166\pi\)
\(618\) 19785.9i 1.28787i
\(619\) 23698.8 1.53883 0.769414 0.638751i \(-0.220548\pi\)
0.769414 + 0.638751i \(0.220548\pi\)
\(620\) −447.787 + 1528.33i −0.0290057 + 0.0989985i
\(621\) −20633.0 −1.33329
\(622\) 9135.09i 0.588880i
\(623\) 0 0
\(624\) 3140.81 0.201495
\(625\) 6524.07 + 14197.8i 0.417541 + 0.908658i
\(626\) 4599.67 0.293674
\(627\) 1222.63i 0.0778743i
\(628\) 10684.2i 0.678892i
\(629\) −6339.94 −0.401892
\(630\) 0 0
\(631\) −21626.1 −1.36438 −0.682188 0.731177i \(-0.738971\pi\)
−0.682188 + 0.731177i \(0.738971\pi\)
\(632\) 3956.97i 0.249050i
\(633\) 4717.64i 0.296223i
\(634\) −20847.7 −1.30594
\(635\) −629.523 184.445i −0.0393415 0.0115267i
\(636\) 12718.8 0.792979
\(637\) 0 0
\(638\) 3494.55i 0.216850i
\(639\) 6520.49 0.403672
\(640\) −1373.35 402.380i −0.0848225 0.0248523i
\(641\) 334.627 0.0206193 0.0103096 0.999947i \(-0.496718\pi\)
0.0103096 + 0.999947i \(0.496718\pi\)
\(642\) 820.991i 0.0504703i
\(643\) 9514.06i 0.583512i −0.956493 0.291756i \(-0.905761\pi\)
0.956493 0.291756i \(-0.0942395\pi\)
\(644\) 0 0
\(645\) −6094.80 + 20802.0i −0.372066 + 1.26989i
\(646\) −4889.15 −0.297773
\(647\) 3009.69i 0.182880i 0.995811 + 0.0914398i \(0.0291469\pi\)
−0.995811 + 0.0914398i \(0.970853\pi\)
\(648\) 7130.09i 0.432247i
\(649\) 7014.27 0.424244
\(650\) −6883.16 4412.18i −0.415354 0.266246i
\(651\) 0 0
\(652\) 11068.2i 0.664824i
\(653\) 24188.9i 1.44960i 0.688962 + 0.724798i \(0.258067\pi\)
−0.688962 + 0.724798i \(0.741933\pi\)
\(654\) 2252.37 0.134671
\(655\) 4810.16 16417.4i 0.286944 0.979359i
\(656\) 635.941 0.0378496
\(657\) 7485.54i 0.444504i
\(658\) 0 0
\(659\) 9983.62 0.590147 0.295073 0.955475i \(-0.404656\pi\)
0.295073 + 0.955475i \(0.404656\pi\)
\(660\) −2356.77 690.513i −0.138995 0.0407245i
\(661\) −20564.0 −1.21006 −0.605028 0.796204i \(-0.706838\pi\)
−0.605028 + 0.796204i \(0.706838\pi\)
\(662\) 12435.6i 0.730095i
\(663\) 21553.4i 1.26254i
\(664\) −9930.70 −0.580401
\(665\) 0 0
\(666\) 1042.72 0.0606673
\(667\) 36531.6i 2.12070i
\(668\) 6412.44i 0.371414i
\(669\) 36687.9 2.12023
\(670\) −2331.87 + 7958.84i −0.134460 + 0.458921i
\(671\) 4793.99 0.275812
\(672\) 0 0
\(673\) 28158.5i 1.61282i −0.591355 0.806411i \(-0.701407\pi\)
0.591355 0.806411i \(-0.298593\pi\)
\(674\) 22658.0 1.29489
\(675\) 7276.53 11351.7i 0.414924 0.647297i
\(676\) −4509.90 −0.256594
\(677\) 4810.13i 0.273070i −0.990635 0.136535i \(-0.956403\pi\)
0.990635 0.136535i \(-0.0435966\pi\)
\(678\) 11756.7i 0.665948i
\(679\) 0 0
\(680\) 2761.28 9424.42i 0.155721 0.531485i
\(681\) 12804.0 0.720485
\(682\) 651.589i 0.0365845i
\(683\) 22559.8i 1.26387i −0.775019 0.631937i \(-0.782260\pi\)
0.775019 0.631937i \(-0.217740\pi\)
\(684\) 804.108 0.0449500
\(685\) 32648.0 + 9565.58i 1.82104 + 0.533550i
\(686\) 0 0
\(687\) 14718.2i 0.817370i
\(688\) 5168.04i 0.286381i
\(689\) 17324.3 0.957916
\(690\) −24637.4 7218.55i −1.35932 0.398269i
\(691\) 22147.2 1.21928 0.609638 0.792680i \(-0.291315\pi\)
0.609638 + 0.792680i \(0.291315\pi\)
\(692\) 2371.65i 0.130284i
\(693\) 0 0
\(694\) 21938.2 1.19995
\(695\) 5637.34 19240.6i 0.307678 1.05013i
\(696\) −9171.05 −0.499465
\(697\) 4364.05i 0.237160i
\(698\) 7850.18i 0.425693i
\(699\) 24414.4 1.32108
\(700\) 0 0
\(701\) −3360.84 −0.181080 −0.0905400 0.995893i \(-0.528859\pi\)
−0.0905400 + 0.995893i \(0.528859\pi\)
\(702\) 7055.41i 0.379329i
\(703\) 1285.59i 0.0689715i
\(704\) −585.516 −0.0313458
\(705\) −1077.50 + 3677.58i −0.0575618 + 0.196462i
\(706\) 16818.7 0.896575
\(707\) 0 0
\(708\) 18408.2i 0.977148i
\(709\) 15432.0 0.817437 0.408718 0.912661i \(-0.365976\pi\)
0.408718 + 0.912661i \(0.365976\pi\)
\(710\) −15496.6 4540.39i −0.819125 0.239997i
\(711\) 4465.98 0.235566
\(712\) 166.064i 0.00874087i
\(713\) 6811.64i 0.357781i
\(714\) 0 0
\(715\) −3210.15 940.547i −0.167906 0.0491951i
\(716\) −1997.57 −0.104263
\(717\) 5163.33i 0.268937i
\(718\) 10048.3i 0.522282i
\(719\) 11650.6 0.604301 0.302151 0.953260i \(-0.402295\pi\)
0.302151 + 0.953260i \(0.402295\pi\)
\(720\) −454.141 + 1550.01i −0.0235067 + 0.0802299i
\(721\) 0 0
\(722\) 12726.6i 0.656004i
\(723\) 365.932i 0.0188232i
\(724\) 4533.68 0.232725
\(725\) 20098.6 + 12883.4i 1.02958 + 0.659969i
\(726\) 14973.7 0.765461
\(727\) 6809.99i 0.347412i −0.984798 0.173706i \(-0.944426\pi\)
0.984798 0.173706i \(-0.0555743\pi\)
\(728\) 0 0
\(729\) −9434.57 −0.479326
\(730\) 5212.38 17790.2i 0.264272 0.901979i
\(731\) −35465.0 −1.79442
\(732\) 12581.3i 0.635271i
\(733\) 31889.5i 1.60691i 0.595365 + 0.803456i \(0.297008\pi\)
−0.595365 + 0.803456i \(0.702992\pi\)
\(734\) −5401.55 −0.271628
\(735\) 0 0
\(736\) −6120.92 −0.306549
\(737\) 3393.18i 0.169592i
\(738\) 717.746i 0.0358003i
\(739\) 20503.7 1.02062 0.510312 0.859989i \(-0.329530\pi\)
0.510312 + 0.859989i \(0.329530\pi\)
\(740\) −2478.13 726.070i −0.123105 0.0360687i
\(741\) 4370.51 0.216673
\(742\) 0 0
\(743\) 27013.1i 1.33380i 0.745146 + 0.666902i \(0.232380\pi\)
−0.745146 + 0.666902i \(0.767620\pi\)
\(744\) 1710.02 0.0842640
\(745\) 6312.85 21546.2i 0.310449 1.05958i
\(746\) 15477.8 0.759628
\(747\) 11208.1i 0.548975i
\(748\) 4018.02i 0.196408i
\(749\) 0 0
\(750\) 12660.1 11009.0i 0.616377 0.535989i
\(751\) −2926.38 −0.142191 −0.0710953 0.997470i \(-0.522649\pi\)
−0.0710953 + 0.997470i \(0.522649\pi\)
\(752\) 913.660i 0.0443055i
\(753\) 28114.0i 1.36060i
\(754\) −12491.9 −0.603352
\(755\) −186.080 + 635.102i −0.00896972 + 0.0306142i
\(756\) 0 0
\(757\) 8929.57i 0.428733i 0.976753 + 0.214366i \(0.0687687\pi\)
−0.976753 + 0.214366i \(0.931231\pi\)
\(758\) 11025.2i 0.528302i
\(759\) −10503.9 −0.502330
\(760\) −1911.05 559.921i −0.0912118 0.0267243i
\(761\) 12222.6 0.582218 0.291109 0.956690i \(-0.405976\pi\)
0.291109 + 0.956690i \(0.405976\pi\)
\(762\) 704.363i 0.0334861i
\(763\) 0 0
\(764\) −9558.26 −0.452625
\(765\) −10636.7 3116.47i −0.502708 0.147289i
\(766\) 1842.10 0.0868900
\(767\) 25073.7i 1.18039i
\(768\) 1536.62i 0.0721979i
\(769\) −19856.8 −0.931148 −0.465574 0.885009i \(-0.654152\pi\)
−0.465574 + 0.885009i \(0.654152\pi\)
\(770\) 0 0
\(771\) 5584.72 0.260868
\(772\) 2432.92i 0.113423i
\(773\) 23058.2i 1.07289i −0.843934 0.536447i \(-0.819766\pi\)
0.843934 0.536447i \(-0.180234\pi\)
\(774\) 5832.84 0.270875
\(775\) −3747.55 2402.22i −0.173698 0.111342i
\(776\) 8082.89 0.373916
\(777\) 0 0
\(778\) 1469.38i 0.0677119i
\(779\) 884.926 0.0407006
\(780\) −2468.36 + 8424.68i −0.113310 + 0.386733i
\(781\) −6606.86 −0.302704
\(782\) 42003.9i 1.92079i
\(783\) 20601.5i 0.940279i
\(784\) 0 0
\(785\) 28658.4 + 8396.67i 1.30301 + 0.381771i
\(786\) −18369.2 −0.833596
\(787\) 12766.9i 0.578260i −0.957290 0.289130i \(-0.906634\pi\)
0.957290 0.289130i \(-0.0933660\pi\)
\(788\) 11904.3i 0.538163i
\(789\) −34110.9 −1.53914
\(790\) −10613.9 3109.78i −0.478006 0.140052i
\(791\) 0 0
\(792\) 660.834i 0.0296486i
\(793\) 17137.0i 0.767405i
\(794\) 12469.4 0.557331
\(795\) −9995.72 + 34116.1i −0.445927 + 1.52198i
\(796\) 6927.33 0.308458
\(797\) 23674.8i 1.05220i −0.850423 0.526100i \(-0.823654\pi\)
0.850423 0.526100i \(-0.176346\pi\)
\(798\) 0 0
\(799\) −6269.86 −0.277611
\(800\) 2158.63 3367.54i 0.0953988 0.148826i
\(801\) 187.426 0.00826761
\(802\) 2294.66i 0.101031i
\(803\) 7584.69i 0.333322i
\(804\) 8905.02 0.390617
\(805\) 0 0
\(806\) 2329.22 0.101791
\(807\) 14518.2i 0.633291i
\(808\) 10295.7i 0.448270i
\(809\) 18134.3 0.788093 0.394046 0.919091i \(-0.371075\pi\)
0.394046 + 0.919091i \(0.371075\pi\)
\(810\) −19125.2 5603.53i −0.829619 0.243071i
\(811\) −1527.66 −0.0661449 −0.0330724 0.999453i \(-0.510529\pi\)
−0.0330724 + 0.999453i \(0.510529\pi\)
\(812\) 0 0
\(813\) 1967.37i 0.0848691i
\(814\) −1056.53 −0.0454929
\(815\) −29688.6 8698.50i −1.27601 0.373859i
\(816\) −10544.8 −0.452381
\(817\) 7191.45i 0.307952i
\(818\) 10668.0i 0.455986i
\(819\) 0 0
\(820\) −499.785 + 1705.80i −0.0212845 + 0.0726453i
\(821\) −734.066 −0.0312047 −0.0156024 0.999878i \(-0.504967\pi\)
−0.0156024 + 0.999878i \(0.504967\pi\)
\(822\) 36529.3i 1.55001i
\(823\) 20888.0i 0.884700i 0.896842 + 0.442350i \(0.145855\pi\)
−0.896842 + 0.442350i \(0.854145\pi\)
\(824\) 13185.3 0.557439
\(825\) 3704.36 5778.94i 0.156326 0.243875i
\(826\) 0 0
\(827\) 3587.86i 0.150861i −0.997151 0.0754305i \(-0.975967\pi\)
0.997151 0.0754305i \(-0.0240331\pi\)
\(828\) 6908.29i 0.289951i
\(829\) −16536.6 −0.692812 −0.346406 0.938085i \(-0.612598\pi\)
−0.346406 + 0.938085i \(0.612598\pi\)
\(830\) 7804.53 26637.4i 0.326384 1.11397i
\(831\) −9187.44 −0.383524
\(832\) 2093.03i 0.0872148i
\(833\) 0 0
\(834\) −21528.0 −0.893830
\(835\) 17200.2 + 5039.53i 0.712861 + 0.208862i
\(836\) −814.758 −0.0337069
\(837\) 3841.33i 0.158633i
\(838\) 23205.2i 0.956576i
\(839\) −6813.22 −0.280356 −0.140178 0.990126i \(-0.544767\pi\)
−0.140178 + 0.990126i \(0.544767\pi\)
\(840\) 0 0
\(841\) 12086.8 0.495584
\(842\) 11896.1i 0.486898i
\(843\) 55064.3i 2.24972i
\(844\) 3143.82 0.128217
\(845\) 3544.33 12097.0i 0.144294 0.492485i
\(846\) 1031.19 0.0419066
\(847\) 0 0
\(848\) 8475.81i 0.343232i
\(849\) 20710.9 0.837214
\(850\) 23109.3 + 14813.3i 0.932519 + 0.597754i
\(851\) −11044.8 −0.444902
\(852\) 17339.0i 0.697210i
\(853\) 13905.1i 0.558149i 0.960269 + 0.279074i \(0.0900276\pi\)
−0.960269 + 0.279074i \(0.909972\pi\)
\(854\) 0 0
\(855\) −631.947 + 2156.88i −0.0252774 + 0.0862733i
\(856\) 547.107 0.0218455
\(857\) 10419.5i 0.415311i −0.978202 0.207656i \(-0.933417\pi\)
0.978202 0.207656i \(-0.0665833\pi\)
\(858\) 3591.79i 0.142916i
\(859\) −8686.75 −0.345039 −0.172519 0.985006i \(-0.555191\pi\)
−0.172519 + 0.985006i \(0.555191\pi\)
\(860\) −13862.4 4061.56i −0.549654 0.161044i
\(861\) 0 0
\(862\) 7416.51i 0.293048i
\(863\) 1991.10i 0.0785376i −0.999229 0.0392688i \(-0.987497\pi\)
0.999229 0.0392688i \(-0.0125029\pi\)
\(864\) −3451.81 −0.135918
\(865\) −6361.54 1863.88i −0.250056 0.0732644i
\(866\) 9550.00 0.374737
\(867\) 42872.5i 1.67938i
\(868\) 0 0
\(869\) −4525.13 −0.176645
\(870\) 7207.52 24599.7i 0.280871 0.958631i
\(871\) 12129.5 0.471864
\(872\) 1500.97i 0.0582906i
\(873\) 9122.64i 0.353671i
\(874\) −8517.40 −0.329640
\(875\) 0 0
\(876\) −19905.2 −0.767732
\(877\) 23847.3i 0.918205i 0.888383 + 0.459102i \(0.151829\pi\)
−0.888383 + 0.459102i \(0.848171\pi\)
\(878\) 7867.15i 0.302396i
\(879\) −33725.6 −1.29412
\(880\) 460.156 1570.54i 0.0176271 0.0601625i
\(881\) −18359.1 −0.702082 −0.351041 0.936360i \(-0.614172\pi\)
−0.351041 + 0.936360i \(0.614172\pi\)
\(882\) 0 0
\(883\) 32592.8i 1.24217i 0.783743 + 0.621085i \(0.213308\pi\)
−0.783743 + 0.621085i \(0.786692\pi\)
\(884\) −14363.1 −0.546475
\(885\) 49376.7 + 14467.0i 1.87546 + 0.549493i
\(886\) 17460.8 0.662085
\(887\) 3241.53i 0.122706i 0.998116 + 0.0613528i \(0.0195415\pi\)
−0.998116 + 0.0613528i \(0.980459\pi\)
\(888\) 2772.74i 0.104783i
\(889\) 0 0
\(890\) −445.437 130.509i −0.0167765 0.00491537i
\(891\) −8153.86 −0.306582
\(892\) 24448.7i 0.917716i
\(893\) 1271.38i 0.0476428i
\(894\) −24107.7 −0.901880
\(895\) 1569.89 5358.13i 0.0586319 0.200114i
\(896\) 0 0
\(897\) 37548.1i 1.39765i
\(898\) 37449.0i 1.39164i
\(899\) −6801.23 −0.252318
\(900\) −3800.73 2436.31i −0.140768 0.0902335i
\(901\) −58164.0 −2.15064
\(902\) 727.253i 0.0268457i
\(903\) 0 0
\(904\) −7834.62 −0.288247
\(905\) −3563.01 + 12160.8i −0.130871 + 0.446673i
\(906\) 710.606 0.0260577
\(907\) 25884.4i 0.947606i 0.880631 + 0.473803i \(0.157119\pi\)
−0.880631 + 0.473803i \(0.842881\pi\)
\(908\) 8532.55i 0.311853i
\(909\) 11620.1 0.423999
\(910\) 0 0
\(911\) 1292.86 0.0470189 0.0235095 0.999724i \(-0.492516\pi\)
0.0235095 + 0.999724i \(0.492516\pi\)
\(912\) 2138.24i 0.0776363i
\(913\) 11356.6i 0.411663i
\(914\) −8768.20 −0.317315
\(915\) 33747.1 + 9887.63i 1.21928 + 0.357240i
\(916\) 9808.16 0.353789
\(917\) 0 0
\(918\) 23687.5i 0.851640i
\(919\) −20310.5 −0.729032 −0.364516 0.931197i \(-0.618765\pi\)
−0.364516 + 0.931197i \(0.618765\pi\)
\(920\) 4810.42 16418.3i 0.172386 0.588364i
\(921\) −21610.6 −0.773173
\(922\) 25439.4i 0.908679i
\(923\) 23617.4i 0.842227i
\(924\) 0 0
\(925\) 3895.11 6076.52i 0.138455 0.215994i
\(926\) 9612.69 0.341137
\(927\) 14881.4i 0.527258i
\(928\) 6111.56i 0.216187i
\(929\) 49155.2 1.73599 0.867993 0.496577i \(-0.165410\pi\)
0.867993 + 0.496577i \(0.165410\pi\)
\(930\) −1343.90 + 4586.83i −0.0473853 + 0.161729i
\(931\) 0 0
\(932\) 16269.7i 0.571814i
\(933\) 27416.3i 0.962026i
\(934\) −3971.24 −0.139125
\(935\) 10777.6 + 3157.75i 0.376969 + 0.110449i
\(936\) 2362.27 0.0824927
\(937\) 16877.6i 0.588440i 0.955738 + 0.294220i \(0.0950599\pi\)
−0.955738 + 0.294220i \(0.904940\pi\)
\(938\) 0 0
\(939\) 13804.6 0.479761
\(940\) −2450.73 718.044i −0.0850362 0.0249149i
\(941\) 782.274 0.0271003 0.0135502 0.999908i \(-0.495687\pi\)
0.0135502 + 0.999908i \(0.495687\pi\)
\(942\) 32065.4i 1.10907i
\(943\) 7602.62i 0.262540i
\(944\) 12267.2 0.422947
\(945\) 0 0
\(946\) −5910.10 −0.203122
\(947\) 10355.1i 0.355329i 0.984091 + 0.177664i \(0.0568541\pi\)
−0.984091 + 0.177664i \(0.943146\pi\)
\(948\) 11875.7i 0.406862i
\(949\) −27112.8 −0.927418
\(950\) 3003.78 4686.01i 0.102585 0.160036i
\(951\) −62568.3 −2.13345
\(952\) 0 0
\(953\) 21907.4i 0.744647i −0.928103 0.372324i \(-0.878561\pi\)
0.928103 0.372324i \(-0.121439\pi\)
\(954\) 9566.10 0.324648
\(955\) 7511.83 25638.4i 0.254531 0.868731i
\(956\) 3440.83 0.116406
\(957\) 10487.9i 0.354258i
\(958\) 13794.1i 0.465206i
\(959\) 0 0
\(960\) −4121.71 1207.63i −0.138571 0.0406000i
\(961\) −28522.9 −0.957432
\(962\) 3776.74i 0.126577i
\(963\) 617.484i 0.0206627i
\(964\) −243.856 −0.00814739
\(965\) −6525.88 1912.03i −0.217695 0.0637828i
\(966\) 0 0
\(967\) 41853.5i 1.39185i −0.718116 0.695924i \(-0.754995\pi\)
0.718116 0.695924i \(-0.245005\pi\)
\(968\) 9978.41i 0.331321i
\(969\) −14673.4 −0.486457
\(970\) −6352.34 + 21680.9i −0.210269 + 0.717663i
\(971\) 853.827 0.0282190 0.0141095 0.999900i \(-0.495509\pi\)
0.0141095 + 0.999900i \(0.495509\pi\)
\(972\) 9749.05i 0.321709i
\(973\) 0 0
\(974\) 22520.0 0.740849
\(975\) −20657.8 13241.9i −0.678544 0.434954i
\(976\) 8384.15 0.274969
\(977\) 20962.6i 0.686443i −0.939255 0.343221i \(-0.888482\pi\)
0.939255 0.343221i \(-0.111518\pi\)
\(978\) 33218.1i 1.08609i
\(979\) −189.908 −0.00619968
\(980\) 0 0
\(981\) 1694.05 0.0551345
\(982\) 26323.0i 0.855398i
\(983\) 46290.9i 1.50198i 0.660311 + 0.750992i \(0.270424\pi\)
−0.660311 + 0.750992i \(0.729576\pi\)
\(984\) 1908.59 0.0618331
\(985\) 31931.2 + 9355.57i 1.03291 + 0.302633i
\(986\) 41939.7 1.35460
\(987\) 0 0
\(988\) 2912.50i 0.0937843i
\(989\) −61783.5 −1.98645
\(990\) −1772.57 519.349i −0.0569051 0.0166727i
\(991\) −29867.9 −0.957402 −0.478701 0.877978i \(-0.658892\pi\)
−0.478701 + 0.877978i \(0.658892\pi\)
\(992\) 1139.55i 0.0364727i
\(993\) 37321.8i 1.19272i
\(994\) 0 0
\(995\) −5444.18 + 18581.3i −0.173459 + 0.592029i
\(996\) −29804.1 −0.948173
\(997\) 21989.4i 0.698506i 0.937029 + 0.349253i \(0.113565\pi\)
−0.937029 + 0.349253i \(0.886435\pi\)
\(998\) 24808.0i 0.786857i
\(999\) −6228.58 −0.197261
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 490.4.c.f.99.12 12
5.2 odd 4 2450.4.a.cv.1.6 6
5.3 odd 4 2450.4.a.cy.1.1 6
5.4 even 2 inner 490.4.c.f.99.1 12
7.2 even 3 70.4.i.a.39.7 yes 24
7.4 even 3 70.4.i.a.9.6 24
7.6 odd 2 490.4.c.e.99.7 12
35.2 odd 12 350.4.e.o.151.1 12
35.4 even 6 70.4.i.a.9.7 yes 24
35.9 even 6 70.4.i.a.39.6 yes 24
35.13 even 4 2450.4.a.cx.1.6 6
35.18 odd 12 350.4.e.n.51.6 12
35.23 odd 12 350.4.e.n.151.6 12
35.27 even 4 2450.4.a.cw.1.1 6
35.32 odd 12 350.4.e.o.51.1 12
35.34 odd 2 490.4.c.e.99.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.4.i.a.9.6 24 7.4 even 3
70.4.i.a.9.7 yes 24 35.4 even 6
70.4.i.a.39.6 yes 24 35.9 even 6
70.4.i.a.39.7 yes 24 7.2 even 3
350.4.e.n.51.6 12 35.18 odd 12
350.4.e.n.151.6 12 35.23 odd 12
350.4.e.o.51.1 12 35.32 odd 12
350.4.e.o.151.1 12 35.2 odd 12
490.4.c.e.99.6 12 35.34 odd 2
490.4.c.e.99.7 12 7.6 odd 2
490.4.c.f.99.1 12 5.4 even 2 inner
490.4.c.f.99.12 12 1.1 even 1 trivial
2450.4.a.cv.1.6 6 5.2 odd 4
2450.4.a.cw.1.1 6 35.27 even 4
2450.4.a.cx.1.6 6 35.13 even 4
2450.4.a.cy.1.1 6 5.3 odd 4