Properties

Label 99.5.c.c.10.6
Level $99$
Weight $5$
Character 99.10
Analytic conductor $10.234$
Analytic rank $0$
Dimension $8$
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] = [99,5,Mod(10,99)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(99, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.10");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 99.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: \(10.2336263453\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 102x^{6} + 2913x^{4} + 23292x^{2} + 41364 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 10.6
Root \(3.00247i\) of defining polynomial
Character \(\chi\) \(=\) 99.10
Dual form 99.5.c.c.10.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00247i q^{2} +6.98517 q^{4} -8.72578 q^{5} -1.45810i q^{7} +69.0123i q^{8} +O(q^{10})\) \(q+3.00247i q^{2} +6.98517 q^{4} -8.72578 q^{5} -1.45810i q^{7} +69.0123i q^{8} -26.1989i q^{10} +(62.2476 + 103.760i) q^{11} +162.221i q^{13} +4.37791 q^{14} -95.4447 q^{16} -189.734i q^{17} +590.443i q^{19} -60.9511 q^{20} +(-311.538 + 186.897i) q^{22} +12.8557 q^{23} -548.861 q^{25} -487.064 q^{26} -10.1851i q^{28} +282.359i q^{29} -304.206 q^{31} +817.627i q^{32} +569.671 q^{34} +12.7231i q^{35} +464.276 q^{37} -1772.79 q^{38} -602.186i q^{40} -1193.81i q^{41} -1591.11i q^{43} +(434.810 + 724.784i) q^{44} +38.5989i q^{46} +1825.87 q^{47} +2398.87 q^{49} -1647.94i q^{50} +1133.14i q^{52} +4023.28 q^{53} +(-543.159 - 905.391i) q^{55} +100.627 q^{56} -847.774 q^{58} +1489.12 q^{59} -356.601i q^{61} -913.368i q^{62} -3982.02 q^{64} -1415.51i q^{65} +8259.07 q^{67} -1325.32i q^{68} -38.2007 q^{70} -7971.63 q^{71} -5780.93i q^{73} +1393.98i q^{74} +4124.34i q^{76} +(151.293 - 90.7633i) q^{77} -11304.5i q^{79} +832.830 q^{80} +3584.37 q^{82} +5449.44i q^{83} +1655.58i q^{85} +4777.26 q^{86} +(-7160.75 + 4295.85i) q^{88} -7332.12 q^{89} +236.535 q^{91} +89.7992 q^{92} +5482.13i q^{94} -5152.08i q^{95} -11228.1 q^{97} +7202.55i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 76 q^{4} + 36 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 76 q^{4} + 36 q^{5} - 36 q^{11} + 1140 q^{14} + 1412 q^{16} - 2532 q^{20} - 780 q^{22} - 516 q^{23} - 2280 q^{25} + 1524 q^{26} + 2752 q^{31} - 4920 q^{34} + 5296 q^{37} - 696 q^{38} + 6540 q^{44} - 420 q^{47} - 6832 q^{49} - 3540 q^{53} + 3784 q^{55} - 17964 q^{56} + 21624 q^{58} + 16632 q^{59} - 27508 q^{64} - 3656 q^{67} + 3312 q^{70} + 13212 q^{71} - 23268 q^{77} + 4476 q^{80} + 17088 q^{82} - 19896 q^{86} - 12516 q^{88} - 15528 q^{89} - 19752 q^{91} + 81180 q^{92} + 7624 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.00247i 0.750618i 0.926900 + 0.375309i \(0.122463\pi\)
−0.926900 + 0.375309i \(0.877537\pi\)
\(3\) 0 0
\(4\) 6.98517 0.436573
\(5\) −8.72578 −0.349031 −0.174516 0.984654i \(-0.555836\pi\)
−0.174516 + 0.984654i \(0.555836\pi\)
\(6\) 0 0
\(7\) 1.45810i 0.0297572i −0.999889 0.0148786i \(-0.995264\pi\)
0.999889 0.0148786i \(-0.00473618\pi\)
\(8\) 69.0123i 1.07832i
\(9\) 0 0
\(10\) 26.1989i 0.261989i
\(11\) 62.2476 + 103.760i 0.514443 + 0.857525i
\(12\) 0 0
\(13\) 162.221i 0.959888i 0.877299 + 0.479944i \(0.159343\pi\)
−0.877299 + 0.479944i \(0.840657\pi\)
\(14\) 4.37791 0.0223363
\(15\) 0 0
\(16\) −95.4447 −0.372831
\(17\) 189.734i 0.656519i −0.944588 0.328259i \(-0.893538\pi\)
0.944588 0.328259i \(-0.106462\pi\)
\(18\) 0 0
\(19\) 590.443i 1.63558i 0.575520 + 0.817788i \(0.304800\pi\)
−0.575520 + 0.817788i \(0.695200\pi\)
\(20\) −60.9511 −0.152378
\(21\) 0 0
\(22\) −311.538 + 186.897i −0.643673 + 0.386150i
\(23\) 12.8557 0.0243019 0.0121509 0.999926i \(-0.496132\pi\)
0.0121509 + 0.999926i \(0.496132\pi\)
\(24\) 0 0
\(25\) −548.861 −0.878177
\(26\) −487.064 −0.720509
\(27\) 0 0
\(28\) 10.1851i 0.0129912i
\(29\) 282.359i 0.335742i 0.985809 + 0.167871i \(0.0536892\pi\)
−0.985809 + 0.167871i \(0.946311\pi\)
\(30\) 0 0
\(31\) −304.206 −0.316551 −0.158275 0.987395i \(-0.550593\pi\)
−0.158275 + 0.987395i \(0.550593\pi\)
\(32\) 817.627i 0.798464i
\(33\) 0 0
\(34\) 569.671 0.492795
\(35\) 12.7231i 0.0103862i
\(36\) 0 0
\(37\) 464.276 0.339135 0.169568 0.985519i \(-0.445763\pi\)
0.169568 + 0.985519i \(0.445763\pi\)
\(38\) −1772.79 −1.22769
\(39\) 0 0
\(40\) 602.186i 0.376366i
\(41\) 1193.81i 0.710177i −0.934833 0.355089i \(-0.884451\pi\)
0.934833 0.355089i \(-0.115549\pi\)
\(42\) 0 0
\(43\) 1591.11i 0.860525i −0.902704 0.430263i \(-0.858421\pi\)
0.902704 0.430263i \(-0.141579\pi\)
\(44\) 434.810 + 724.784i 0.224592 + 0.374372i
\(45\) 0 0
\(46\) 38.5989i 0.0182414i
\(47\) 1825.87 0.826561 0.413281 0.910604i \(-0.364383\pi\)
0.413281 + 0.910604i \(0.364383\pi\)
\(48\) 0 0
\(49\) 2398.87 0.999115
\(50\) 1647.94i 0.659175i
\(51\) 0 0
\(52\) 1133.14i 0.419061i
\(53\) 4023.28 1.43228 0.716141 0.697955i \(-0.245907\pi\)
0.716141 + 0.697955i \(0.245907\pi\)
\(54\) 0 0
\(55\) −543.159 905.391i −0.179557 0.299303i
\(56\) 100.627 0.0320877
\(57\) 0 0
\(58\) −847.774 −0.252014
\(59\) 1489.12 0.427785 0.213893 0.976857i \(-0.431386\pi\)
0.213893 + 0.976857i \(0.431386\pi\)
\(60\) 0 0
\(61\) 356.601i 0.0958346i −0.998851 0.0479173i \(-0.984742\pi\)
0.998851 0.0479173i \(-0.0152584\pi\)
\(62\) 913.368i 0.237609i
\(63\) 0 0
\(64\) −3982.02 −0.972172
\(65\) 1415.51i 0.335031i
\(66\) 0 0
\(67\) 8259.07 1.83985 0.919923 0.392098i \(-0.128251\pi\)
0.919923 + 0.392098i \(0.128251\pi\)
\(68\) 1325.32i 0.286618i
\(69\) 0 0
\(70\) −38.2007 −0.00779606
\(71\) −7971.63 −1.58136 −0.790680 0.612230i \(-0.790273\pi\)
−0.790680 + 0.612230i \(0.790273\pi\)
\(72\) 0 0
\(73\) 5780.93i 1.08481i −0.840118 0.542403i \(-0.817515\pi\)
0.840118 0.542403i \(-0.182485\pi\)
\(74\) 1393.98i 0.254561i
\(75\) 0 0
\(76\) 4124.34i 0.714048i
\(77\) 151.293 90.7633i 0.0255175 0.0153084i
\(78\) 0 0
\(79\) 11304.5i 1.81133i −0.423998 0.905663i \(-0.639374\pi\)
0.423998 0.905663i \(-0.360626\pi\)
\(80\) 832.830 0.130130
\(81\) 0 0
\(82\) 3584.37 0.533071
\(83\) 5449.44i 0.791036i 0.918458 + 0.395518i \(0.129435\pi\)
−0.918458 + 0.395518i \(0.870565\pi\)
\(84\) 0 0
\(85\) 1655.58i 0.229146i
\(86\) 4777.26 0.645925
\(87\) 0 0
\(88\) −7160.75 + 4295.85i −0.924684 + 0.554733i
\(89\) −7332.12 −0.925656 −0.462828 0.886448i \(-0.653165\pi\)
−0.462828 + 0.886448i \(0.653165\pi\)
\(90\) 0 0
\(91\) 236.535 0.0285636
\(92\) 89.7992 0.0106096
\(93\) 0 0
\(94\) 5482.13i 0.620432i
\(95\) 5152.08i 0.570867i
\(96\) 0 0
\(97\) −11228.1 −1.19334 −0.596669 0.802487i \(-0.703509\pi\)
−0.596669 + 0.802487i \(0.703509\pi\)
\(98\) 7202.55i 0.749953i
\(99\) 0 0
\(100\) −3833.88 −0.383388
\(101\) 12735.1i 1.24842i −0.781258 0.624209i \(-0.785422\pi\)
0.781258 0.624209i \(-0.214578\pi\)
\(102\) 0 0
\(103\) 4637.37 0.437117 0.218558 0.975824i \(-0.429865\pi\)
0.218558 + 0.975824i \(0.429865\pi\)
\(104\) −11195.2 −1.03506
\(105\) 0 0
\(106\) 12079.8i 1.07510i
\(107\) 15858.6i 1.38515i 0.721347 + 0.692574i \(0.243524\pi\)
−0.721347 + 0.692574i \(0.756476\pi\)
\(108\) 0 0
\(109\) 19516.7i 1.64268i 0.570438 + 0.821341i \(0.306774\pi\)
−0.570438 + 0.821341i \(0.693226\pi\)
\(110\) 2718.41 1630.82i 0.224662 0.134778i
\(111\) 0 0
\(112\) 139.168i 0.0110944i
\(113\) 15343.5 1.20162 0.600811 0.799391i \(-0.294844\pi\)
0.600811 + 0.799391i \(0.294844\pi\)
\(114\) 0 0
\(115\) −112.176 −0.00848212
\(116\) 1972.32i 0.146576i
\(117\) 0 0
\(118\) 4471.04i 0.321103i
\(119\) −276.652 −0.0195362
\(120\) 0 0
\(121\) −6891.47 + 12917.7i −0.470697 + 0.882295i
\(122\) 1070.68 0.0719351
\(123\) 0 0
\(124\) −2124.93 −0.138198
\(125\) 10242.9 0.655543
\(126\) 0 0
\(127\) 19359.4i 1.20028i −0.799893 0.600142i \(-0.795111\pi\)
0.799893 0.600142i \(-0.204889\pi\)
\(128\) 1126.14i 0.0687341i
\(129\) 0 0
\(130\) 4250.01 0.251480
\(131\) 13966.3i 0.813842i −0.913463 0.406921i \(-0.866602\pi\)
0.913463 0.406921i \(-0.133398\pi\)
\(132\) 0 0
\(133\) 860.926 0.0486701
\(134\) 24797.6i 1.38102i
\(135\) 0 0
\(136\) 13094.0 0.707936
\(137\) −10923.8 −0.582010 −0.291005 0.956721i \(-0.593990\pi\)
−0.291005 + 0.956721i \(0.593990\pi\)
\(138\) 0 0
\(139\) 27425.4i 1.41946i −0.704474 0.709730i \(-0.748817\pi\)
0.704474 0.709730i \(-0.251183\pi\)
\(140\) 88.8729i 0.00453433i
\(141\) 0 0
\(142\) 23934.6i 1.18700i
\(143\) −16832.1 + 10097.9i −0.823127 + 0.493807i
\(144\) 0 0
\(145\) 2463.80i 0.117184i
\(146\) 17357.1 0.814275
\(147\) 0 0
\(148\) 3243.05 0.148057
\(149\) 17646.6i 0.794856i −0.917633 0.397428i \(-0.869903\pi\)
0.917633 0.397428i \(-0.130097\pi\)
\(150\) 0 0
\(151\) 19482.1i 0.854441i 0.904147 + 0.427221i \(0.140507\pi\)
−0.904147 + 0.427221i \(0.859493\pi\)
\(152\) −40747.8 −1.76367
\(153\) 0 0
\(154\) 272.514 + 454.254i 0.0114907 + 0.0191539i
\(155\) 2654.43 0.110486
\(156\) 0 0
\(157\) 20642.6 0.837462 0.418731 0.908110i \(-0.362475\pi\)
0.418731 + 0.908110i \(0.362475\pi\)
\(158\) 33941.4 1.35961
\(159\) 0 0
\(160\) 7134.43i 0.278689i
\(161\) 18.7449i 0.000723156i
\(162\) 0 0
\(163\) 5888.63 0.221635 0.110818 0.993841i \(-0.464653\pi\)
0.110818 + 0.993841i \(0.464653\pi\)
\(164\) 8338.95i 0.310044i
\(165\) 0 0
\(166\) −16361.8 −0.593765
\(167\) 25996.3i 0.932135i −0.884749 0.466067i \(-0.845670\pi\)
0.884749 0.466067i \(-0.154330\pi\)
\(168\) 0 0
\(169\) 2245.35 0.0786159
\(170\) −4970.82 −0.172001
\(171\) 0 0
\(172\) 11114.2i 0.375682i
\(173\) 22355.9i 0.746964i −0.927637 0.373482i \(-0.878164\pi\)
0.927637 0.373482i \(-0.121836\pi\)
\(174\) 0 0
\(175\) 800.295i 0.0261321i
\(176\) −5941.21 9903.39i −0.191800 0.319712i
\(177\) 0 0
\(178\) 22014.5i 0.694814i
\(179\) −9227.98 −0.288005 −0.144003 0.989577i \(-0.545997\pi\)
−0.144003 + 0.989577i \(0.545997\pi\)
\(180\) 0 0
\(181\) −52256.0 −1.59507 −0.797534 0.603274i \(-0.793862\pi\)
−0.797534 + 0.603274i \(0.793862\pi\)
\(182\) 710.189i 0.0214403i
\(183\) 0 0
\(184\) 887.202i 0.0262052i
\(185\) −4051.17 −0.118369
\(186\) 0 0
\(187\) 19686.9 11810.5i 0.562981 0.337742i
\(188\) 12754.0 0.360854
\(189\) 0 0
\(190\) 15469.0 0.428503
\(191\) 70655.7 1.93678 0.968390 0.249442i \(-0.0802471\pi\)
0.968390 + 0.249442i \(0.0802471\pi\)
\(192\) 0 0
\(193\) 21752.7i 0.583981i 0.956421 + 0.291991i \(0.0943176\pi\)
−0.956421 + 0.291991i \(0.905682\pi\)
\(194\) 33712.1i 0.895741i
\(195\) 0 0
\(196\) 16756.5 0.436186
\(197\) 67346.7i 1.73534i 0.497144 + 0.867668i \(0.334382\pi\)
−0.497144 + 0.867668i \(0.665618\pi\)
\(198\) 0 0
\(199\) 4956.39 0.125158 0.0625791 0.998040i \(-0.480067\pi\)
0.0625791 + 0.998040i \(0.480067\pi\)
\(200\) 37878.1i 0.946954i
\(201\) 0 0
\(202\) 38236.8 0.937084
\(203\) 411.708 0.00999073
\(204\) 0 0
\(205\) 10416.9i 0.247874i
\(206\) 13923.6i 0.328108i
\(207\) 0 0
\(208\) 15483.1i 0.357876i
\(209\) −61264.6 + 36753.6i −1.40255 + 0.841410i
\(210\) 0 0
\(211\) 5230.33i 0.117480i 0.998273 + 0.0587400i \(0.0187083\pi\)
−0.998273 + 0.0587400i \(0.981292\pi\)
\(212\) 28103.3 0.625296
\(213\) 0 0
\(214\) −47614.9 −1.03972
\(215\) 13883.7i 0.300350i
\(216\) 0 0
\(217\) 443.563i 0.00941967i
\(218\) −58598.3 −1.23303
\(219\) 0 0
\(220\) −3794.06 6324.31i −0.0783896 0.130668i
\(221\) 30778.8 0.630184
\(222\) 0 0
\(223\) 70116.0 1.40996 0.704981 0.709226i \(-0.250956\pi\)
0.704981 + 0.709226i \(0.250956\pi\)
\(224\) 1192.18 0.0237600
\(225\) 0 0
\(226\) 46068.5i 0.901960i
\(227\) 69181.4i 1.34257i 0.741198 + 0.671287i \(0.234258\pi\)
−0.741198 + 0.671287i \(0.765742\pi\)
\(228\) 0 0
\(229\) −27970.4 −0.533369 −0.266685 0.963784i \(-0.585928\pi\)
−0.266685 + 0.963784i \(0.585928\pi\)
\(230\) 336.805i 0.00636683i
\(231\) 0 0
\(232\) −19486.2 −0.362036
\(233\) 67591.9i 1.24504i 0.782604 + 0.622519i \(0.213891\pi\)
−0.782604 + 0.622519i \(0.786109\pi\)
\(234\) 0 0
\(235\) −15932.2 −0.288496
\(236\) 10401.8 0.186759
\(237\) 0 0
\(238\) 830.638i 0.0146642i
\(239\) 31589.5i 0.553028i −0.961010 0.276514i \(-0.910821\pi\)
0.961010 0.276514i \(-0.0891792\pi\)
\(240\) 0 0
\(241\) 64415.3i 1.10906i 0.832164 + 0.554530i \(0.187102\pi\)
−0.832164 + 0.554530i \(0.812898\pi\)
\(242\) −38785.0 20691.4i −0.662266 0.353313i
\(243\) 0 0
\(244\) 2490.91i 0.0418388i
\(245\) −20932.1 −0.348722
\(246\) 0 0
\(247\) −95782.2 −1.56997
\(248\) 20993.9i 0.341342i
\(249\) 0 0
\(250\) 30753.9i 0.492062i
\(251\) −34822.1 −0.552722 −0.276361 0.961054i \(-0.589129\pi\)
−0.276361 + 0.961054i \(0.589129\pi\)
\(252\) 0 0
\(253\) 800.237 + 1333.91i 0.0125019 + 0.0208395i
\(254\) 58126.0 0.900955
\(255\) 0 0
\(256\) −67093.5 −1.02376
\(257\) 73746.1 1.11654 0.558268 0.829661i \(-0.311466\pi\)
0.558268 + 0.829661i \(0.311466\pi\)
\(258\) 0 0
\(259\) 676.962i 0.0100917i
\(260\) 9887.54i 0.146265i
\(261\) 0 0
\(262\) 41933.5 0.610884
\(263\) 103570.i 1.49734i −0.662942 0.748671i \(-0.730692\pi\)
0.662942 0.748671i \(-0.269308\pi\)
\(264\) 0 0
\(265\) −35106.3 −0.499911
\(266\) 2584.91i 0.0365327i
\(267\) 0 0
\(268\) 57691.0 0.803227
\(269\) −118870. −1.64274 −0.821371 0.570395i \(-0.806790\pi\)
−0.821371 + 0.570395i \(0.806790\pi\)
\(270\) 0 0
\(271\) 102742.i 1.39897i 0.714646 + 0.699486i \(0.246588\pi\)
−0.714646 + 0.699486i \(0.753412\pi\)
\(272\) 18109.1i 0.244771i
\(273\) 0 0
\(274\) 32798.3i 0.436867i
\(275\) −34165.3 56950.1i −0.451772 0.753059i
\(276\) 0 0
\(277\) 103336.i 1.34677i 0.739292 + 0.673385i \(0.235161\pi\)
−0.739292 + 0.673385i \(0.764839\pi\)
\(278\) 82343.9 1.06547
\(279\) 0 0
\(280\) −878.049 −0.0111996
\(281\) 37257.8i 0.471850i 0.971771 + 0.235925i \(0.0758120\pi\)
−0.971771 + 0.235925i \(0.924188\pi\)
\(282\) 0 0
\(283\) 126640.i 1.58124i −0.612310 0.790618i \(-0.709759\pi\)
0.612310 0.790618i \(-0.290241\pi\)
\(284\) −55683.2 −0.690379
\(285\) 0 0
\(286\) −30318.6 50538.0i −0.370661 0.617854i
\(287\) −1740.69 −0.0211329
\(288\) 0 0
\(289\) 47522.0 0.568983
\(290\) 7397.49 0.0879607
\(291\) 0 0
\(292\) 40380.8i 0.473597i
\(293\) 1337.08i 0.0155748i −0.999970 0.00778741i \(-0.997521\pi\)
0.999970 0.00778741i \(-0.00247883\pi\)
\(294\) 0 0
\(295\) −12993.7 −0.149310
\(296\) 32040.8i 0.365695i
\(297\) 0 0
\(298\) 52983.4 0.596633
\(299\) 2085.46i 0.0233271i
\(300\) 0 0
\(301\) −2320.00 −0.0256068
\(302\) −58494.5 −0.641359
\(303\) 0 0
\(304\) 56354.7i 0.609793i
\(305\) 3111.62i 0.0334493i
\(306\) 0 0
\(307\) 70825.1i 0.751468i −0.926728 0.375734i \(-0.877391\pi\)
0.926728 0.375734i \(-0.122609\pi\)
\(308\) 1056.81 633.997i 0.0111403 0.00668322i
\(309\) 0 0
\(310\) 7969.85i 0.0829329i
\(311\) 79979.1 0.826905 0.413453 0.910526i \(-0.364323\pi\)
0.413453 + 0.910526i \(0.364323\pi\)
\(312\) 0 0
\(313\) 45118.3 0.460537 0.230268 0.973127i \(-0.426040\pi\)
0.230268 + 0.973127i \(0.426040\pi\)
\(314\) 61978.8i 0.628614i
\(315\) 0 0
\(316\) 78963.7i 0.790776i
\(317\) 53392.1 0.531322 0.265661 0.964066i \(-0.414410\pi\)
0.265661 + 0.964066i \(0.414410\pi\)
\(318\) 0 0
\(319\) −29297.7 + 17576.2i −0.287907 + 0.172720i
\(320\) 34746.2 0.339318
\(321\) 0 0
\(322\) 56.2811 0.000542814
\(323\) 112027. 1.07379
\(324\) 0 0
\(325\) 89036.7i 0.842951i
\(326\) 17680.4i 0.166363i
\(327\) 0 0
\(328\) 82387.4 0.765796
\(329\) 2662.31i 0.0245961i
\(330\) 0 0
\(331\) 97744.9 0.892151 0.446075 0.894995i \(-0.352821\pi\)
0.446075 + 0.894995i \(0.352821\pi\)
\(332\) 38065.3i 0.345345i
\(333\) 0 0
\(334\) 78053.2 0.699677
\(335\) −72066.9 −0.642164
\(336\) 0 0
\(337\) 75385.5i 0.663786i 0.943317 + 0.331893i \(0.107687\pi\)
−0.943317 + 0.331893i \(0.892313\pi\)
\(338\) 6741.59i 0.0590105i
\(339\) 0 0
\(340\) 11564.5i 0.100039i
\(341\) −18936.1 31564.5i −0.162847 0.271450i
\(342\) 0 0
\(343\) 6998.71i 0.0594880i
\(344\) 109806. 0.927919
\(345\) 0 0
\(346\) 67122.9 0.560684
\(347\) 95151.0i 0.790232i −0.918631 0.395116i \(-0.870704\pi\)
0.918631 0.395116i \(-0.129296\pi\)
\(348\) 0 0
\(349\) 106717.i 0.876156i −0.898937 0.438078i \(-0.855659\pi\)
0.898937 0.438078i \(-0.144341\pi\)
\(350\) −2402.86 −0.0196152
\(351\) 0 0
\(352\) −84837.3 + 50895.3i −0.684702 + 0.410764i
\(353\) −62919.0 −0.504932 −0.252466 0.967606i \(-0.581242\pi\)
−0.252466 + 0.967606i \(0.581242\pi\)
\(354\) 0 0
\(355\) 69558.7 0.551944
\(356\) −51216.1 −0.404116
\(357\) 0 0
\(358\) 27706.7i 0.216182i
\(359\) 147077.i 1.14119i −0.821232 0.570594i \(-0.806713\pi\)
0.821232 0.570594i \(-0.193287\pi\)
\(360\) 0 0
\(361\) −218302. −1.67511
\(362\) 156897.i 1.19729i
\(363\) 0 0
\(364\) 1652.24 0.0124701
\(365\) 50443.2i 0.378631i
\(366\) 0 0
\(367\) −50293.7 −0.373406 −0.186703 0.982416i \(-0.559780\pi\)
−0.186703 + 0.982416i \(0.559780\pi\)
\(368\) −1227.01 −0.00906050
\(369\) 0 0
\(370\) 12163.5i 0.0888497i
\(371\) 5866.36i 0.0426207i
\(372\) 0 0
\(373\) 29786.2i 0.214090i −0.994254 0.107045i \(-0.965861\pi\)
0.994254 0.107045i \(-0.0341389\pi\)
\(374\) 35460.6 + 59109.3i 0.253515 + 0.422584i
\(375\) 0 0
\(376\) 126008.i 0.891295i
\(377\) −45804.5 −0.322274
\(378\) 0 0
\(379\) 80930.7 0.563423 0.281712 0.959499i \(-0.409098\pi\)
0.281712 + 0.959499i \(0.409098\pi\)
\(380\) 35988.1i 0.249225i
\(381\) 0 0
\(382\) 212142.i 1.45378i
\(383\) 266517. 1.81688 0.908441 0.418012i \(-0.137273\pi\)
0.908441 + 0.418012i \(0.137273\pi\)
\(384\) 0 0
\(385\) −1320.15 + 791.981i −0.00890641 + 0.00534310i
\(386\) −65311.9 −0.438347
\(387\) 0 0
\(388\) −78430.3 −0.520979
\(389\) −4358.49 −0.0288029 −0.0144015 0.999896i \(-0.504584\pi\)
−0.0144015 + 0.999896i \(0.504584\pi\)
\(390\) 0 0
\(391\) 2439.16i 0.0159547i
\(392\) 165552.i 1.07736i
\(393\) 0 0
\(394\) −202206. −1.30257
\(395\) 98640.5i 0.632209i
\(396\) 0 0
\(397\) −11014.6 −0.0698859 −0.0349429 0.999389i \(-0.511125\pi\)
−0.0349429 + 0.999389i \(0.511125\pi\)
\(398\) 14881.4i 0.0939459i
\(399\) 0 0
\(400\) 52385.9 0.327412
\(401\) −244214. −1.51873 −0.759366 0.650664i \(-0.774491\pi\)
−0.759366 + 0.650664i \(0.774491\pi\)
\(402\) 0 0
\(403\) 49348.5i 0.303853i
\(404\) 88956.8i 0.545025i
\(405\) 0 0
\(406\) 1236.14i 0.00749922i
\(407\) 28900.1 + 48173.5i 0.174466 + 0.290817i
\(408\) 0 0
\(409\) 9069.88i 0.0542194i 0.999632 + 0.0271097i \(0.00863035\pi\)
−0.999632 + 0.0271097i \(0.991370\pi\)
\(410\) −31276.5 −0.186059
\(411\) 0 0
\(412\) 32392.8 0.190833
\(413\) 2171.29i 0.0127297i
\(414\) 0 0
\(415\) 47550.7i 0.276096i
\(416\) −132636. −0.766435
\(417\) 0 0
\(418\) −110352. 183945.i −0.631578 1.05278i
\(419\) −317776. −1.81006 −0.905029 0.425350i \(-0.860151\pi\)
−0.905029 + 0.425350i \(0.860151\pi\)
\(420\) 0 0
\(421\) 89267.7 0.503651 0.251826 0.967773i \(-0.418969\pi\)
0.251826 + 0.967773i \(0.418969\pi\)
\(422\) −15703.9 −0.0881826
\(423\) 0 0
\(424\) 277656.i 1.54445i
\(425\) 104138.i 0.576540i
\(426\) 0 0
\(427\) −519.960 −0.00285177
\(428\) 110775.i 0.604718i
\(429\) 0 0
\(430\) −41685.4 −0.225448
\(431\) 80855.4i 0.435266i −0.976031 0.217633i \(-0.930166\pi\)
0.976031 0.217633i \(-0.0698335\pi\)
\(432\) 0 0
\(433\) 17118.8 0.0913057 0.0456529 0.998957i \(-0.485463\pi\)
0.0456529 + 0.998957i \(0.485463\pi\)
\(434\) −1331.78 −0.00707057
\(435\) 0 0
\(436\) 136327.i 0.717150i
\(437\) 7590.56i 0.0397476i
\(438\) 0 0
\(439\) 32476.8i 0.168517i −0.996444 0.0842585i \(-0.973148\pi\)
0.996444 0.0842585i \(-0.0268522\pi\)
\(440\) 62483.1 37484.6i 0.322743 0.193619i
\(441\) 0 0
\(442\) 92412.6i 0.473028i
\(443\) −4735.01 −0.0241276 −0.0120638 0.999927i \(-0.503840\pi\)
−0.0120638 + 0.999927i \(0.503840\pi\)
\(444\) 0 0
\(445\) 63978.5 0.323083
\(446\) 210521.i 1.05834i
\(447\) 0 0
\(448\) 5806.19i 0.0289291i
\(449\) 217096. 1.07686 0.538430 0.842670i \(-0.319018\pi\)
0.538430 + 0.842670i \(0.319018\pi\)
\(450\) 0 0
\(451\) 123870. 74311.7i 0.608994 0.365346i
\(452\) 107177. 0.524596
\(453\) 0 0
\(454\) −207715. −1.00776
\(455\) −2063.95 −0.00996957
\(456\) 0 0
\(457\) 226266.i 1.08340i 0.840573 + 0.541698i \(0.182218\pi\)
−0.840573 + 0.541698i \(0.817782\pi\)
\(458\) 83980.4i 0.400356i
\(459\) 0 0
\(460\) −783.569 −0.00370307
\(461\) 280319.i 1.31902i 0.751697 + 0.659509i \(0.229236\pi\)
−0.751697 + 0.659509i \(0.770764\pi\)
\(462\) 0 0
\(463\) 176739. 0.824463 0.412231 0.911079i \(-0.364750\pi\)
0.412231 + 0.911079i \(0.364750\pi\)
\(464\) 26949.7i 0.125175i
\(465\) 0 0
\(466\) −202943. −0.934548
\(467\) 211683. 0.970625 0.485312 0.874341i \(-0.338706\pi\)
0.485312 + 0.874341i \(0.338706\pi\)
\(468\) 0 0
\(469\) 12042.6i 0.0547487i
\(470\) 47835.9i 0.216550i
\(471\) 0 0
\(472\) 102768.i 0.461288i
\(473\) 165094. 99042.8i 0.737921 0.442691i
\(474\) 0 0
\(475\) 324071.i 1.43633i
\(476\) −1932.46 −0.00852896
\(477\) 0 0
\(478\) 94846.5 0.415112
\(479\) 80152.2i 0.349337i −0.984627 0.174668i \(-0.944115\pi\)
0.984627 0.174668i \(-0.0558854\pi\)
\(480\) 0 0
\(481\) 75315.3i 0.325532i
\(482\) −193405. −0.832480
\(483\) 0 0
\(484\) −48138.1 + 90232.2i −0.205494 + 0.385186i
\(485\) 97974.1 0.416512
\(486\) 0 0
\(487\) −173605. −0.731988 −0.365994 0.930617i \(-0.619271\pi\)
−0.365994 + 0.930617i \(0.619271\pi\)
\(488\) 24609.8 0.103340
\(489\) 0 0
\(490\) 62847.9i 0.261757i
\(491\) 152879.i 0.634138i 0.948402 + 0.317069i \(0.102699\pi\)
−0.948402 + 0.317069i \(0.897301\pi\)
\(492\) 0 0
\(493\) 53573.1 0.220421
\(494\) 287583.i 1.17845i
\(495\) 0 0
\(496\) 29034.8 0.118020
\(497\) 11623.5i 0.0470568i
\(498\) 0 0
\(499\) 122586. 0.492312 0.246156 0.969230i \(-0.420832\pi\)
0.246156 + 0.969230i \(0.420832\pi\)
\(500\) 71548.0 0.286192
\(501\) 0 0
\(502\) 104552.i 0.414883i
\(503\) 355305.i 1.40432i −0.712020 0.702159i \(-0.752220\pi\)
0.712020 0.702159i \(-0.247780\pi\)
\(504\) 0 0
\(505\) 111124.i 0.435737i
\(506\) −4005.04 + 2402.69i −0.0156425 + 0.00938418i
\(507\) 0 0
\(508\) 135229.i 0.524012i
\(509\) −34844.7 −0.134493 −0.0672467 0.997736i \(-0.521421\pi\)
−0.0672467 + 0.997736i \(0.521421\pi\)
\(510\) 0 0
\(511\) −8429.19 −0.0322808
\(512\) 183428.i 0.699722i
\(513\) 0 0
\(514\) 221421.i 0.838092i
\(515\) −40464.7 −0.152567
\(516\) 0 0
\(517\) 113656. + 189454.i 0.425219 + 0.708797i
\(518\) 2032.56 0.00757501
\(519\) 0 0
\(520\) 97687.3 0.361269
\(521\) −130516. −0.480825 −0.240412 0.970671i \(-0.577283\pi\)
−0.240412 + 0.970671i \(0.577283\pi\)
\(522\) 0 0
\(523\) 436331.i 1.59519i −0.603192 0.797596i \(-0.706105\pi\)
0.603192 0.797596i \(-0.293895\pi\)
\(524\) 97557.3i 0.355301i
\(525\) 0 0
\(526\) 310965. 1.12393
\(527\) 57718.1i 0.207822i
\(528\) 0 0
\(529\) −279676. −0.999409
\(530\) 105406.i 0.375242i
\(531\) 0 0
\(532\) 6013.71 0.0212481
\(533\) 193661. 0.681690
\(534\) 0 0
\(535\) 138378.i 0.483460i
\(536\) 569978.i 1.98394i
\(537\) 0 0
\(538\) 356905.i 1.23307i
\(539\) 149324. + 248908.i 0.513987 + 0.856765i
\(540\) 0 0
\(541\) 80892.7i 0.276385i −0.990405 0.138193i \(-0.955871\pi\)
0.990405 0.138193i \(-0.0441293\pi\)
\(542\) −308480. −1.05009
\(543\) 0 0
\(544\) 155132. 0.524206
\(545\) 170298.i 0.573347i
\(546\) 0 0
\(547\) 231848.i 0.774870i −0.921897 0.387435i \(-0.873361\pi\)
0.921897 0.387435i \(-0.126639\pi\)
\(548\) −76304.2 −0.254090
\(549\) 0 0
\(550\) 170991. 102580.i 0.565259 0.339108i
\(551\) −166717. −0.549131
\(552\) 0 0
\(553\) −16483.1 −0.0539000
\(554\) −310264. −1.01091
\(555\) 0 0
\(556\) 191571.i 0.619698i
\(557\) 461600.i 1.48784i −0.668270 0.743919i \(-0.732965\pi\)
0.668270 0.743919i \(-0.267035\pi\)
\(558\) 0 0
\(559\) 258112. 0.826007
\(560\) 1214.35i 0.00387229i
\(561\) 0 0
\(562\) −111865. −0.354179
\(563\) 35282.9i 0.111313i −0.998450 0.0556566i \(-0.982275\pi\)
0.998450 0.0556566i \(-0.0177252\pi\)
\(564\) 0 0
\(565\) −133884. −0.419404
\(566\) 380232. 1.18690
\(567\) 0 0
\(568\) 550141.i 1.70521i
\(569\) 296349.i 0.915332i 0.889124 + 0.457666i \(0.151314\pi\)
−0.889124 + 0.457666i \(0.848686\pi\)
\(570\) 0 0
\(571\) 241258.i 0.739963i 0.929039 + 0.369982i \(0.120636\pi\)
−0.929039 + 0.369982i \(0.879364\pi\)
\(572\) −117575. + 70535.3i −0.359355 + 0.215583i
\(573\) 0 0
\(574\) 5226.38i 0.0158627i
\(575\) −7055.99 −0.0213414
\(576\) 0 0
\(577\) −457944. −1.37550 −0.687750 0.725948i \(-0.741401\pi\)
−0.687750 + 0.725948i \(0.741401\pi\)
\(578\) 142683.i 0.427089i
\(579\) 0 0
\(580\) 17210.1i 0.0511595i
\(581\) 7945.85 0.0235390
\(582\) 0 0
\(583\) 250440. + 417458.i 0.736828 + 1.22822i
\(584\) 398956. 1.16977
\(585\) 0 0
\(586\) 4014.55 0.0116907
\(587\) −481837. −1.39838 −0.699188 0.714938i \(-0.746455\pi\)
−0.699188 + 0.714938i \(0.746455\pi\)
\(588\) 0 0
\(589\) 179616.i 0.517743i
\(590\) 39013.3i 0.112075i
\(591\) 0 0
\(592\) −44312.7 −0.126440
\(593\) 495129.i 1.40802i −0.710190 0.704010i \(-0.751391\pi\)
0.710190 0.704010i \(-0.248609\pi\)
\(594\) 0 0
\(595\) 2414.00 0.00681873
\(596\) 123264.i 0.347013i
\(597\) 0 0
\(598\) −6261.55 −0.0175097
\(599\) 341737. 0.952442 0.476221 0.879326i \(-0.342006\pi\)
0.476221 + 0.879326i \(0.342006\pi\)
\(600\) 0 0
\(601\) 404022.i 1.11855i 0.828982 + 0.559275i \(0.188920\pi\)
−0.828982 + 0.559275i \(0.811080\pi\)
\(602\) 6965.74i 0.0192209i
\(603\) 0 0
\(604\) 136086.i 0.373026i
\(605\) 60133.5 112717.i 0.164288 0.307949i
\(606\) 0 0
\(607\) 85001.6i 0.230701i 0.993325 + 0.115351i \(0.0367991\pi\)
−0.993325 + 0.115351i \(0.963201\pi\)
\(608\) −482762. −1.30595
\(609\) 0 0
\(610\) −9342.54 −0.0251076
\(611\) 296195.i 0.793406i
\(612\) 0 0
\(613\) 296743.i 0.789695i 0.918747 + 0.394847i \(0.129203\pi\)
−0.918747 + 0.394847i \(0.870797\pi\)
\(614\) 212650. 0.564065
\(615\) 0 0
\(616\) 6263.79 + 10441.1i 0.0165073 + 0.0275160i
\(617\) 205712. 0.540368 0.270184 0.962809i \(-0.412915\pi\)
0.270184 + 0.962809i \(0.412915\pi\)
\(618\) 0 0
\(619\) −23798.6 −0.0621112 −0.0310556 0.999518i \(-0.509887\pi\)
−0.0310556 + 0.999518i \(0.509887\pi\)
\(620\) 18541.6 0.0482353
\(621\) 0 0
\(622\) 240135.i 0.620690i
\(623\) 10691.0i 0.0275449i
\(624\) 0 0
\(625\) 253661. 0.649372
\(626\) 135466.i 0.345687i
\(627\) 0 0
\(628\) 144192. 0.365613
\(629\) 88088.9i 0.222649i
\(630\) 0 0
\(631\) −464381. −1.16631 −0.583157 0.812359i \(-0.698183\pi\)
−0.583157 + 0.812359i \(0.698183\pi\)
\(632\) 780149. 1.95318
\(633\) 0 0
\(634\) 160308.i 0.398820i
\(635\) 168926.i 0.418937i
\(636\) 0 0
\(637\) 389148.i 0.959038i
\(638\) −52771.9 87965.4i −0.129647 0.216108i
\(639\) 0 0
\(640\) 9826.44i 0.0239903i
\(641\) −433460. −1.05495 −0.527476 0.849570i \(-0.676862\pi\)
−0.527476 + 0.849570i \(0.676862\pi\)
\(642\) 0 0
\(643\) −228934. −0.553718 −0.276859 0.960910i \(-0.589294\pi\)
−0.276859 + 0.960910i \(0.589294\pi\)
\(644\) 130.936i 0.000315710i
\(645\) 0 0
\(646\) 336358.i 0.806003i
\(647\) 83987.4 0.200634 0.100317 0.994956i \(-0.468014\pi\)
0.100317 + 0.994956i \(0.468014\pi\)
\(648\) 0 0
\(649\) 92694.1 + 154512.i 0.220071 + 0.366836i
\(650\) 267330. 0.632734
\(651\) 0 0
\(652\) 41133.1 0.0967600
\(653\) −337789. −0.792172 −0.396086 0.918213i \(-0.629632\pi\)
−0.396086 + 0.918213i \(0.629632\pi\)
\(654\) 0 0
\(655\) 121867.i 0.284056i
\(656\) 113943.i 0.264776i
\(657\) 0 0
\(658\) 7993.51 0.0184623
\(659\) 586098.i 1.34958i 0.738008 + 0.674791i \(0.235766\pi\)
−0.738008 + 0.674791i \(0.764234\pi\)
\(660\) 0 0
\(661\) −660271. −1.51119 −0.755596 0.655038i \(-0.772652\pi\)
−0.755596 + 0.655038i \(0.772652\pi\)
\(662\) 293476.i 0.669664i
\(663\) 0 0
\(664\) −376079. −0.852987
\(665\) −7512.25 −0.0169874
\(666\) 0 0
\(667\) 3629.92i 0.00815916i
\(668\) 181589.i 0.406945i
\(669\) 0 0
\(670\) 216379.i 0.482020i
\(671\) 37001.0 22197.5i 0.0821805 0.0493014i
\(672\) 0 0
\(673\) 362391.i 0.800105i −0.916492 0.400052i \(-0.868992\pi\)
0.916492 0.400052i \(-0.131008\pi\)
\(674\) −226343. −0.498249
\(675\) 0 0
\(676\) 15684.1 0.0343216
\(677\) 92662.7i 0.202175i 0.994878 + 0.101088i \(0.0322322\pi\)
−0.994878 + 0.101088i \(0.967768\pi\)
\(678\) 0 0
\(679\) 16371.7i 0.0355104i
\(680\) −114255. −0.247092
\(681\) 0 0
\(682\) 94771.5 56855.0i 0.203755 0.122236i
\(683\) −448358. −0.961132 −0.480566 0.876958i \(-0.659569\pi\)
−0.480566 + 0.876958i \(0.659569\pi\)
\(684\) 0 0
\(685\) 95318.3 0.203140
\(686\) 21013.4 0.0446528
\(687\) 0 0
\(688\) 151863.i 0.320830i
\(689\) 652661.i 1.37483i
\(690\) 0 0
\(691\) 358700. 0.751234 0.375617 0.926775i \(-0.377431\pi\)
0.375617 + 0.926775i \(0.377431\pi\)
\(692\) 156160.i 0.326104i
\(693\) 0 0
\(694\) 285688. 0.593162
\(695\) 239308.i 0.495436i
\(696\) 0 0
\(697\) −226506. −0.466245
\(698\) 320414. 0.657659
\(699\) 0 0
\(700\) 5590.19i 0.0114086i
\(701\) 159779.i 0.325150i −0.986696 0.162575i \(-0.948020\pi\)
0.986696 0.162575i \(-0.0519799\pi\)
\(702\) 0 0
\(703\) 274128.i 0.554681i
\(704\) −247871. 413176.i −0.500127 0.833661i
\(705\) 0 0
\(706\) 188913.i 0.379011i
\(707\) −18569.1 −0.0371494
\(708\) 0 0
\(709\) 455705. 0.906550 0.453275 0.891371i \(-0.350256\pi\)
0.453275 + 0.891371i \(0.350256\pi\)
\(710\) 208848.i 0.414299i
\(711\) 0 0
\(712\) 506006.i 0.998151i
\(713\) −3910.78 −0.00769279
\(714\) 0 0
\(715\) 146873. 88111.8i 0.287297 0.172354i
\(716\) −64459.0 −0.125735
\(717\) 0 0
\(718\) 441596. 0.856596
\(719\) −640858. −1.23966 −0.619832 0.784735i \(-0.712799\pi\)
−0.619832 + 0.784735i \(0.712799\pi\)
\(720\) 0 0
\(721\) 6761.76i 0.0130074i
\(722\) 655445.i 1.25737i
\(723\) 0 0
\(724\) −365017. −0.696363
\(725\) 154976.i 0.294841i
\(726\) 0 0
\(727\) −324283. −0.613558 −0.306779 0.951781i \(-0.599251\pi\)
−0.306779 + 0.951781i \(0.599251\pi\)
\(728\) 16323.8i 0.0308006i
\(729\) 0 0
\(730\) −151454. −0.284207
\(731\) −301888. −0.564951
\(732\) 0 0
\(733\) 633078.i 1.17828i −0.808030 0.589141i \(-0.799466\pi\)
0.808030 0.589141i \(-0.200534\pi\)
\(734\) 151005.i 0.280285i
\(735\) 0 0
\(736\) 10511.2i 0.0194042i
\(737\) 514107. + 856965.i 0.946496 + 1.57771i
\(738\) 0 0
\(739\) 218657.i 0.400382i 0.979757 + 0.200191i \(0.0641562\pi\)
−0.979757 + 0.200191i \(0.935844\pi\)
\(740\) −28298.1 −0.0516766
\(741\) 0 0
\(742\) 17613.6 0.0319919
\(743\) 740582.i 1.34152i −0.741677 0.670758i \(-0.765969\pi\)
0.741677 0.670758i \(-0.234031\pi\)
\(744\) 0 0
\(745\) 153980.i 0.277430i
\(746\) 89432.1 0.160700
\(747\) 0 0
\(748\) 137516. 82498.2i 0.245782 0.147449i
\(749\) 23123.4 0.0412181
\(750\) 0 0
\(751\) −1.08089e6 −1.91647 −0.958235 0.285982i \(-0.907680\pi\)
−0.958235 + 0.285982i \(0.907680\pi\)
\(752\) −174270. −0.308168
\(753\) 0 0
\(754\) 137527.i 0.241905i
\(755\) 169997.i 0.298227i
\(756\) 0 0
\(757\) 604976. 1.05571 0.527857 0.849333i \(-0.322996\pi\)
0.527857 + 0.849333i \(0.322996\pi\)
\(758\) 242992.i 0.422916i
\(759\) 0 0
\(760\) 355557. 0.615576
\(761\) 1.06160e6i 1.83312i −0.399901 0.916558i \(-0.630955\pi\)
0.399901 0.916558i \(-0.369045\pi\)
\(762\) 0 0
\(763\) 28457.3 0.0488816
\(764\) 493542. 0.845546
\(765\) 0 0
\(766\) 800209.i 1.36378i
\(767\) 241567.i 0.410626i
\(768\) 0 0
\(769\) 405621.i 0.685912i −0.939352 0.342956i \(-0.888572\pi\)
0.939352 0.342956i \(-0.111428\pi\)
\(770\) −2377.90 3963.72i −0.00401063 0.00668531i
\(771\) 0 0
\(772\) 151946.i 0.254950i
\(773\) 279272. 0.467377 0.233689 0.972311i \(-0.424920\pi\)
0.233689 + 0.972311i \(0.424920\pi\)
\(774\) 0 0
\(775\) 166966. 0.277988
\(776\) 774878.i 1.28680i
\(777\) 0 0
\(778\) 13086.2i 0.0216200i
\(779\) 704875. 1.16155
\(780\) 0 0
\(781\) −496215. 827141.i −0.813519 1.35605i
\(782\) 7323.52 0.0119758
\(783\) 0 0
\(784\) −228960. −0.372501
\(785\) −180123. −0.292301
\(786\) 0 0
\(787\) 637120.i 1.02866i 0.857593 + 0.514330i \(0.171959\pi\)
−0.857593 + 0.514330i \(0.828041\pi\)
\(788\) 470428.i 0.757601i
\(789\) 0 0
\(790\) −296165. −0.474548
\(791\) 22372.4i 0.0357569i
\(792\) 0 0
\(793\) 57848.1 0.0919904
\(794\) 33071.1i 0.0524576i
\(795\) 0 0
\(796\) 34621.2 0.0546407
\(797\) −275712. −0.434049 −0.217024 0.976166i \(-0.569635\pi\)
−0.217024 + 0.976166i \(0.569635\pi\)
\(798\) 0 0
\(799\) 346430.i 0.542653i
\(800\) 448763.i 0.701193i
\(801\) 0 0
\(802\) 733244.i 1.13999i
\(803\) 599832. 359849.i 0.930248 0.558071i
\(804\) 0 0
\(805\) 163.564i 0.000252404i
\(806\) 148168. 0.228078
\(807\) 0 0
\(808\) 878879. 1.34619
\(809\) 974096.i 1.48835i 0.667986 + 0.744174i \(0.267157\pi\)
−0.667986 + 0.744174i \(0.732843\pi\)
\(810\) 0 0
\(811\) 379198.i 0.576534i 0.957550 + 0.288267i \(0.0930790\pi\)
−0.957550 + 0.288267i \(0.906921\pi\)
\(812\) 2875.85 0.00436168
\(813\) 0 0
\(814\) −144640. + 86771.6i −0.218292 + 0.130957i
\(815\) −51382.9 −0.0773576
\(816\) 0 0
\(817\) 939460. 1.40745
\(818\) −27232.1 −0.0406981
\(819\) 0 0
\(820\) 72763.8i 0.108215i
\(821\) 87100.3i 0.129221i 0.997911 + 0.0646106i \(0.0205805\pi\)
−0.997911 + 0.0646106i \(0.979419\pi\)
\(822\) 0 0
\(823\) 177783. 0.262477 0.131239 0.991351i \(-0.458105\pi\)
0.131239 + 0.991351i \(0.458105\pi\)
\(824\) 320036.i 0.471350i
\(825\) 0 0
\(826\) 6519.23 0.00955513
\(827\) 369448.i 0.540185i 0.962834 + 0.270093i \(0.0870543\pi\)
−0.962834 + 0.270093i \(0.912946\pi\)
\(828\) 0 0
\(829\) 805060. 1.17144 0.585718 0.810515i \(-0.300812\pi\)
0.585718 + 0.810515i \(0.300812\pi\)
\(830\) 142769. 0.207243
\(831\) 0 0
\(832\) 645967.i 0.933176i
\(833\) 455148.i 0.655938i
\(834\) 0 0
\(835\) 226838.i 0.325344i
\(836\) −427944. + 256730.i −0.612314 + 0.367337i
\(837\) 0 0
\(838\) 954112.i 1.35866i
\(839\) 762552. 1.08329 0.541646 0.840606i \(-0.317801\pi\)
0.541646 + 0.840606i \(0.317801\pi\)
\(840\) 0 0
\(841\) 627555. 0.887278
\(842\) 268024.i 0.378050i
\(843\) 0 0
\(844\) 36534.7i 0.0512886i
\(845\) −19592.4 −0.0274394
\(846\) 0 0
\(847\) 18835.3 + 10048.5i 0.0262546 + 0.0140066i
\(848\) −384001. −0.533999
\(849\) 0 0
\(850\) −312670. −0.432761
\(851\) 5968.59 0.00824163
\(852\) 0 0
\(853\) 602505.i 0.828061i −0.910263 0.414031i \(-0.864121\pi\)
0.910263 0.414031i \(-0.135879\pi\)
\(854\) 1561.16i 0.00214059i
\(855\) 0 0
\(856\) −1.09444e6 −1.49363
\(857\) 119350.i 0.162502i −0.996694 0.0812512i \(-0.974108\pi\)
0.996694 0.0812512i \(-0.0258916\pi\)
\(858\) 0 0
\(859\) 437992. 0.593580 0.296790 0.954943i \(-0.404084\pi\)
0.296790 + 0.954943i \(0.404084\pi\)
\(860\) 96979.9i 0.131125i
\(861\) 0 0
\(862\) 242766. 0.326718
\(863\) −776719. −1.04290 −0.521450 0.853282i \(-0.674609\pi\)
−0.521450 + 0.853282i \(0.674609\pi\)
\(864\) 0 0
\(865\) 195073.i 0.260714i
\(866\) 51398.7i 0.0685357i
\(867\) 0 0
\(868\) 3098.36i 0.00411237i
\(869\) 1.17296e6 703677.i 1.55326 0.931824i
\(870\) 0 0
\(871\) 1.33979e6i 1.76605i
\(872\) −1.34689e6 −1.77133
\(873\) 0 0
\(874\) −22790.4 −0.0298352
\(875\) 14935.1i 0.0195071i
\(876\) 0 0
\(877\) 1.15156e6i 1.49722i −0.663009 0.748612i \(-0.730721\pi\)
0.663009 0.748612i \(-0.269279\pi\)
\(878\) 97510.6 0.126492
\(879\) 0 0
\(880\) 51841.7 + 86414.8i 0.0669443 + 0.111589i
\(881\) −228842. −0.294839 −0.147419 0.989074i \(-0.547097\pi\)
−0.147419 + 0.989074i \(0.547097\pi\)
\(882\) 0 0
\(883\) −1.22059e6 −1.56549 −0.782743 0.622345i \(-0.786180\pi\)
−0.782743 + 0.622345i \(0.786180\pi\)
\(884\) 214995. 0.275121
\(885\) 0 0
\(886\) 14216.7i 0.0181106i
\(887\) 11201.7i 0.0142375i −0.999975 0.00711877i \(-0.997734\pi\)
0.999975 0.00711877i \(-0.00226599\pi\)
\(888\) 0 0
\(889\) −28228.0 −0.0357171
\(890\) 192094.i 0.242512i
\(891\) 0 0
\(892\) 489772. 0.615551
\(893\) 1.07807e6i 1.35190i
\(894\) 0 0
\(895\) 80521.3 0.100523
\(896\) 1642.03 0.00204533
\(897\) 0 0
\(898\) 651825.i 0.808310i
\(899\) 85895.1i 0.106279i
\(900\) 0 0
\(901\) 763353.i 0.940321i
\(902\) 223119. + 371916.i 0.274235 + 0.457122i
\(903\) 0 0
\(904\) 1.05889e6i 1.29573i
\(905\) 455975. 0.556729
\(906\) 0 0
\(907\) −612933. −0.745072 −0.372536 0.928018i \(-0.621512\pi\)
−0.372536 + 0.928018i \(0.621512\pi\)
\(908\) 483244.i 0.586131i
\(909\) 0 0
\(910\) 6196.95i 0.00748334i
\(911\) 606369. 0.730634 0.365317 0.930883i \(-0.380961\pi\)
0.365317 + 0.930883i \(0.380961\pi\)
\(912\) 0 0
\(913\) −565437. + 339215.i −0.678332 + 0.406943i
\(914\) −679358. −0.813216
\(915\) 0 0
\(916\) −195378. −0.232855
\(917\) −20364.4 −0.0242176
\(918\) 0 0
\(919\) 19670.4i 0.0232906i 0.999932 + 0.0116453i \(0.00370690\pi\)
−0.999932 + 0.0116453i \(0.996293\pi\)
\(920\) 7741.53i 0.00914642i
\(921\) 0 0
\(922\) −841649. −0.990078
\(923\) 1.29317e6i 1.51793i
\(924\) 0 0
\(925\) −254823. −0.297821
\(926\) 530654.i 0.618856i
\(927\) 0 0
\(928\) −230864. −0.268078
\(929\) −1.51101e6 −1.75080 −0.875399 0.483402i \(-0.839401\pi\)
−0.875399 + 0.483402i \(0.839401\pi\)
\(930\) 0 0
\(931\) 1.41640e6i 1.63413i
\(932\) 472141.i 0.543550i
\(933\) 0 0
\(934\) 635571.i 0.728568i
\(935\) −171783. + 103056.i −0.196498 + 0.117882i
\(936\) 0 0
\(937\) 998435.i 1.13721i 0.822611 + 0.568605i \(0.192517\pi\)
−0.822611 + 0.568605i \(0.807483\pi\)
\(938\) 36157.5 0.0410953
\(939\) 0 0
\(940\) −111289. −0.125949
\(941\) 1.51328e6i 1.70900i −0.519455 0.854498i \(-0.673865\pi\)
0.519455 0.854498i \(-0.326135\pi\)
\(942\) 0 0
\(943\) 15347.2i 0.0172586i
\(944\) −142129. −0.159492
\(945\) 0 0
\(946\) 297373. + 495691.i 0.332292 + 0.553897i
\(947\) −873603. −0.974124 −0.487062 0.873367i \(-0.661931\pi\)
−0.487062 + 0.873367i \(0.661931\pi\)
\(948\) 0 0
\(949\) 937789. 1.04129
\(950\) 973013. 1.07813
\(951\) 0 0
\(952\) 19092.4i 0.0210662i
\(953\) 140477.i 0.154675i 0.997005 + 0.0773374i \(0.0246419\pi\)
−0.997005 + 0.0773374i \(0.975358\pi\)
\(954\) 0 0
\(955\) −616526. −0.675997
\(956\) 220658.i 0.241437i
\(957\) 0 0
\(958\) 240655. 0.262218
\(959\) 15927.9i 0.0173190i
\(960\) 0 0
\(961\) −830980. −0.899795
\(962\) −226132. −0.244350
\(963\) 0 0
\(964\) 449952.i 0.484185i
\(965\) 189809.i 0.203828i
\(966\) 0 0
\(967\) 7957.10i 0.00850946i 0.999991 + 0.00425473i \(0.00135433\pi\)
−0.999991 + 0.00425473i \(0.998646\pi\)
\(968\) −891479. 475596.i −0.951394 0.507561i
\(969\) 0 0
\(970\) 294164.i 0.312642i
\(971\) 1.09461e6 1.16097 0.580487 0.814270i \(-0.302862\pi\)
0.580487 + 0.814270i \(0.302862\pi\)
\(972\) 0 0
\(973\) −39989.0 −0.0422391
\(974\) 521244.i 0.549443i
\(975\) 0 0
\(976\) 34035.6i 0.0357301i
\(977\) −45038.9 −0.0471844 −0.0235922 0.999722i \(-0.507510\pi\)
−0.0235922 + 0.999722i \(0.507510\pi\)
\(978\) 0 0
\(979\) −456407. 760784.i −0.476197 0.793773i
\(980\) −146214. −0.152243
\(981\) 0 0
\(982\) −459014. −0.475995
\(983\) −113557. −0.117518 −0.0587591 0.998272i \(-0.518714\pi\)
−0.0587591 + 0.998272i \(0.518714\pi\)
\(984\) 0 0
\(985\) 587652.i 0.605687i
\(986\) 160852.i 0.165452i
\(987\) 0 0
\(988\) −669055. −0.685406
\(989\) 20454.8i 0.0209124i
\(990\) 0 0
\(991\) 657037. 0.669025 0.334512 0.942391i \(-0.391428\pi\)
0.334512 + 0.942391i \(0.391428\pi\)
\(992\) 248727.i 0.252754i
\(993\) 0 0
\(994\) −34899.1 −0.0353217
\(995\) −43248.4 −0.0436841
\(996\) 0 0
\(997\) 349112.i 0.351216i −0.984460 0.175608i \(-0.943811\pi\)
0.984460 0.175608i \(-0.0561892\pi\)
\(998\) 368061.i 0.369538i
\(999\) 0 0
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 99.5.c.c.10.6 8
3.2 odd 2 33.5.c.a.10.3 8
11.10 odd 2 inner 99.5.c.c.10.3 8
12.11 even 2 528.5.j.a.241.4 8
33.32 even 2 33.5.c.a.10.6 yes 8
132.131 odd 2 528.5.j.a.241.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.5.c.a.10.3 8 3.2 odd 2
33.5.c.a.10.6 yes 8 33.32 even 2
99.5.c.c.10.3 8 11.10 odd 2 inner
99.5.c.c.10.6 8 1.1 even 1 trivial
528.5.j.a.241.3 8 132.131 odd 2
528.5.j.a.241.4 8 12.11 even 2