Properties

Label 99.5.c.c.10.7
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 10.7
Root \(5.58567i\) of defining polynomial
Character \(\chi\) \(=\) 99.10
Dual form 99.5.c.c.10.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.58567i q^{2} -15.1997 q^{4} +29.7487 q^{5} +12.9420i q^{7} +4.47031i q^{8} +O(q^{10})\) \(q+5.58567i q^{2} -15.1997 q^{4} +29.7487 q^{5} +12.9420i q^{7} +4.47031i q^{8} +166.166i q^{10} +(100.434 + 67.4829i) q^{11} +36.6685i q^{13} -72.2898 q^{14} -268.165 q^{16} +464.257i q^{17} -327.633i q^{19} -452.171 q^{20} +(-376.937 + 560.993i) q^{22} -396.812 q^{23} +259.987 q^{25} -204.818 q^{26} -196.715i q^{28} -1152.21i q^{29} +437.378 q^{31} -1426.35i q^{32} -2593.18 q^{34} +385.009i q^{35} +276.604 q^{37} +1830.05 q^{38} +132.986i q^{40} +2782.30i q^{41} +1663.84i q^{43} +(-1526.57 - 1025.72i) q^{44} -2216.46i q^{46} +1855.68 q^{47} +2233.50 q^{49} +1452.20i q^{50} -557.350i q^{52} -3745.57 q^{53} +(2987.79 + 2007.53i) q^{55} -57.8549 q^{56} +6435.88 q^{58} +6573.17 q^{59} -5140.56i q^{61} +2443.05i q^{62} +3676.50 q^{64} +1090.84i q^{65} -3464.58 q^{67} -7056.56i q^{68} -2150.53 q^{70} +6031.86 q^{71} -3580.96i q^{73} +1545.02i q^{74} +4979.92i q^{76} +(-873.365 + 1299.82i) q^{77} -9711.53i q^{79} -7977.55 q^{80} -15541.0 q^{82} -11837.3i q^{83} +13811.1i q^{85} -9293.66 q^{86} +(-301.670 + 448.973i) q^{88} -2584.72 q^{89} -474.565 q^{91} +6031.42 q^{92} +10365.2i q^{94} -9746.67i q^{95} +1551.68 q^{97} +12475.6i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 76 q^{4} + 36 q^{5}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.58567i 1.39642i 0.715895 + 0.698208i \(0.246019\pi\)
−0.715895 + 0.698208i \(0.753981\pi\)
\(3\) 0 0
\(4\) −15.1997 −0.949980
\(5\) 29.7487 1.18995 0.594974 0.803745i \(-0.297162\pi\)
0.594974 + 0.803745i \(0.297162\pi\)
\(6\) 0 0
\(7\) 12.9420i 0.264123i 0.991242 + 0.132061i \(0.0421596\pi\)
−0.991242 + 0.132061i \(0.957840\pi\)
\(8\) 4.47031i 0.0698486i
\(9\) 0 0
\(10\) 166.166i 1.66166i
\(11\) 100.434 + 67.4829i 0.830036 + 0.557710i
\(12\) 0 0
\(13\) 36.6685i 0.216973i 0.994098 + 0.108487i \(0.0346005\pi\)
−0.994098 + 0.108487i \(0.965399\pi\)
\(14\) −72.2898 −0.368826
\(15\) 0 0
\(16\) −268.165 −1.04752
\(17\) 464.257i 1.60643i 0.595692 + 0.803213i \(0.296878\pi\)
−0.595692 + 0.803213i \(0.703122\pi\)
\(18\) 0 0
\(19\) 327.633i 0.907571i −0.891111 0.453786i \(-0.850073\pi\)
0.891111 0.453786i \(-0.149927\pi\)
\(20\) −452.171 −1.13043
\(21\) 0 0
\(22\) −376.937 + 560.993i −0.778796 + 1.15908i
\(23\) −396.812 −0.750118 −0.375059 0.927001i \(-0.622378\pi\)
−0.375059 + 0.927001i \(0.622378\pi\)
\(24\) 0 0
\(25\) 259.987 0.415979
\(26\) −204.818 −0.302985
\(27\) 0 0
\(28\) 196.715i 0.250911i
\(29\) 1152.21i 1.37005i −0.728519 0.685025i \(-0.759791\pi\)
0.728519 0.685025i \(-0.240209\pi\)
\(30\) 0 0
\(31\) 437.378 0.455128 0.227564 0.973763i \(-0.426924\pi\)
0.227564 + 0.973763i \(0.426924\pi\)
\(32\) 1426.35i 1.39292i
\(33\) 0 0
\(34\) −2593.18 −2.24324
\(35\) 385.009i 0.314293i
\(36\) 0 0
\(37\) 276.604 0.202049 0.101024 0.994884i \(-0.467788\pi\)
0.101024 + 0.994884i \(0.467788\pi\)
\(38\) 1830.05 1.26735
\(39\) 0 0
\(40\) 132.986i 0.0831163i
\(41\) 2782.30i 1.65515i 0.561358 + 0.827573i \(0.310279\pi\)
−0.561358 + 0.827573i \(0.689721\pi\)
\(42\) 0 0
\(43\) 1663.84i 0.899860i 0.893064 + 0.449930i \(0.148551\pi\)
−0.893064 + 0.449930i \(0.851449\pi\)
\(44\) −1526.57 1025.72i −0.788518 0.529813i
\(45\) 0 0
\(46\) 2216.46i 1.04748i
\(47\) 1855.68 0.840053 0.420026 0.907512i \(-0.362021\pi\)
0.420026 + 0.907512i \(0.362021\pi\)
\(48\) 0 0
\(49\) 2233.50 0.930239
\(50\) 1452.20i 0.580880i
\(51\) 0 0
\(52\) 557.350i 0.206120i
\(53\) −3745.57 −1.33342 −0.666709 0.745318i \(-0.732298\pi\)
−0.666709 + 0.745318i \(0.732298\pi\)
\(54\) 0 0
\(55\) 2987.79 + 2007.53i 0.987700 + 0.663646i
\(56\) −57.8549 −0.0184486
\(57\) 0 0
\(58\) 6435.88 1.91316
\(59\) 6573.17 1.88830 0.944150 0.329516i \(-0.106886\pi\)
0.944150 + 0.329516i \(0.106886\pi\)
\(60\) 0 0
\(61\) 5140.56i 1.38150i −0.723094 0.690750i \(-0.757281\pi\)
0.723094 0.690750i \(-0.242719\pi\)
\(62\) 2443.05i 0.635549i
\(63\) 0 0
\(64\) 3676.50 0.897583
\(65\) 1090.84i 0.258187i
\(66\) 0 0
\(67\) −3464.58 −0.771793 −0.385897 0.922542i \(-0.626108\pi\)
−0.385897 + 0.922542i \(0.626108\pi\)
\(68\) 7056.56i 1.52607i
\(69\) 0 0
\(70\) −2150.53 −0.438884
\(71\) 6031.86 1.19656 0.598280 0.801287i \(-0.295851\pi\)
0.598280 + 0.801287i \(0.295851\pi\)
\(72\) 0 0
\(73\) 3580.96i 0.671976i −0.941866 0.335988i \(-0.890930\pi\)
0.941866 0.335988i \(-0.109070\pi\)
\(74\) 1545.02i 0.282144i
\(75\) 0 0
\(76\) 4979.92i 0.862174i
\(77\) −873.365 + 1299.82i −0.147304 + 0.219231i
\(78\) 0 0
\(79\) 9711.53i 1.55609i −0.628211 0.778043i \(-0.716213\pi\)
0.628211 0.778043i \(-0.283787\pi\)
\(80\) −7977.55 −1.24649
\(81\) 0 0
\(82\) −15541.0 −2.31127
\(83\) 11837.3i 1.71829i −0.511729 0.859147i \(-0.670995\pi\)
0.511729 0.859147i \(-0.329005\pi\)
\(84\) 0 0
\(85\) 13811.1i 1.91156i
\(86\) −9293.66 −1.25658
\(87\) 0 0
\(88\) −301.670 + 448.973i −0.0389553 + 0.0579769i
\(89\) −2584.72 −0.326312 −0.163156 0.986600i \(-0.552167\pi\)
−0.163156 + 0.986600i \(0.552167\pi\)
\(90\) 0 0
\(91\) −474.565 −0.0573077
\(92\) 6031.42 0.712597
\(93\) 0 0
\(94\) 10365.2i 1.17306i
\(95\) 9746.67i 1.07996i
\(96\) 0 0
\(97\) 1551.68 0.164915 0.0824574 0.996595i \(-0.473723\pi\)
0.0824574 + 0.996595i \(0.473723\pi\)
\(98\) 12475.6i 1.29900i
\(99\) 0 0
\(100\) −3951.71 −0.395171
\(101\) 3939.16i 0.386154i 0.981184 + 0.193077i \(0.0618468\pi\)
−0.981184 + 0.193077i \(0.938153\pi\)
\(102\) 0 0
\(103\) 12210.1 1.15092 0.575459 0.817830i \(-0.304823\pi\)
0.575459 + 0.817830i \(0.304823\pi\)
\(104\) −163.920 −0.0151553
\(105\) 0 0
\(106\) 20921.5i 1.86201i
\(107\) 3756.40i 0.328099i −0.986452 0.164049i \(-0.947544\pi\)
0.986452 0.164049i \(-0.0524556\pi\)
\(108\) 0 0
\(109\) 15338.3i 1.29099i −0.763764 0.645495i \(-0.776651\pi\)
0.763764 0.645495i \(-0.223349\pi\)
\(110\) −11213.4 + 16688.8i −0.926727 + 1.37924i
\(111\) 0 0
\(112\) 3470.59i 0.276673i
\(113\) 7205.76 0.564316 0.282158 0.959368i \(-0.408950\pi\)
0.282158 + 0.959368i \(0.408950\pi\)
\(114\) 0 0
\(115\) −11804.7 −0.892602
\(116\) 17513.3i 1.30152i
\(117\) 0 0
\(118\) 36715.6i 2.63685i
\(119\) −6008.42 −0.424294
\(120\) 0 0
\(121\) 5533.12 + 13555.2i 0.377919 + 0.925839i
\(122\) 28713.5 1.92915
\(123\) 0 0
\(124\) −6648.01 −0.432363
\(125\) −10858.7 −0.694956
\(126\) 0 0
\(127\) 10757.4i 0.666960i 0.942757 + 0.333480i \(0.108223\pi\)
−0.942757 + 0.333480i \(0.891777\pi\)
\(128\) 2285.94i 0.139523i
\(129\) 0 0
\(130\) −6093.08 −0.360537
\(131\) 14623.4i 0.852131i 0.904692 + 0.426066i \(0.140101\pi\)
−0.904692 + 0.426066i \(0.859899\pi\)
\(132\) 0 0
\(133\) 4240.24 0.239710
\(134\) 19352.0i 1.07775i
\(135\) 0 0
\(136\) −2075.37 −0.112207
\(137\) −25143.7 −1.33964 −0.669821 0.742523i \(-0.733629\pi\)
−0.669821 + 0.742523i \(0.733629\pi\)
\(138\) 0 0
\(139\) 4454.38i 0.230546i −0.993334 0.115273i \(-0.963226\pi\)
0.993334 0.115273i \(-0.0367743\pi\)
\(140\) 5852.01i 0.298572i
\(141\) 0 0
\(142\) 33692.0i 1.67090i
\(143\) −2474.50 + 3682.78i −0.121008 + 0.180096i
\(144\) 0 0
\(145\) 34276.9i 1.63029i
\(146\) 20002.0 0.938358
\(147\) 0 0
\(148\) −4204.30 −0.191942
\(149\) 21755.9i 0.979950i −0.871737 0.489975i \(-0.837006\pi\)
0.871737 0.489975i \(-0.162994\pi\)
\(150\) 0 0
\(151\) 20736.4i 0.909450i −0.890632 0.454725i \(-0.849738\pi\)
0.890632 0.454725i \(-0.150262\pi\)
\(152\) 1464.62 0.0633926
\(153\) 0 0
\(154\) −7260.38 4878.33i −0.306139 0.205698i
\(155\) 13011.5 0.541580
\(156\) 0 0
\(157\) −16991.5 −0.689340 −0.344670 0.938724i \(-0.612009\pi\)
−0.344670 + 0.938724i \(0.612009\pi\)
\(158\) 54245.4 2.17294
\(159\) 0 0
\(160\) 42432.2i 1.65751i
\(161\) 5135.55i 0.198123i
\(162\) 0 0
\(163\) 15654.2 0.589192 0.294596 0.955622i \(-0.404815\pi\)
0.294596 + 0.955622i \(0.404815\pi\)
\(164\) 42290.1i 1.57236i
\(165\) 0 0
\(166\) 66119.4 2.39945
\(167\) 13741.9i 0.492734i −0.969177 0.246367i \(-0.920763\pi\)
0.969177 0.246367i \(-0.0792369\pi\)
\(168\) 0 0
\(169\) 27216.4 0.952923
\(170\) −77143.9 −2.66934
\(171\) 0 0
\(172\) 25289.9i 0.854849i
\(173\) 41623.1i 1.39073i 0.718657 + 0.695365i \(0.244757\pi\)
−0.718657 + 0.695365i \(0.755243\pi\)
\(174\) 0 0
\(175\) 3364.75i 0.109869i
\(176\) −26932.9 18096.5i −0.869478 0.584211i
\(177\) 0 0
\(178\) 14437.4i 0.455668i
\(179\) 17574.8 0.548510 0.274255 0.961657i \(-0.411569\pi\)
0.274255 + 0.961657i \(0.411569\pi\)
\(180\) 0 0
\(181\) 10371.8 0.316588 0.158294 0.987392i \(-0.449401\pi\)
0.158294 + 0.987392i \(0.449401\pi\)
\(182\) 2650.76i 0.0800254i
\(183\) 0 0
\(184\) 1773.88i 0.0523947i
\(185\) 8228.63 0.240427
\(186\) 0 0
\(187\) −31329.4 + 46627.3i −0.895919 + 1.33339i
\(188\) −28205.7 −0.798034
\(189\) 0 0
\(190\) 54441.7 1.50808
\(191\) 2219.45 0.0608386 0.0304193 0.999537i \(-0.490316\pi\)
0.0304193 + 0.999537i \(0.490316\pi\)
\(192\) 0 0
\(193\) 18368.1i 0.493117i −0.969128 0.246559i \(-0.920700\pi\)
0.969128 0.246559i \(-0.0792998\pi\)
\(194\) 8667.19i 0.230290i
\(195\) 0 0
\(196\) −33948.6 −0.883709
\(197\) 23671.0i 0.609936i 0.952363 + 0.304968i \(0.0986458\pi\)
−0.952363 + 0.304968i \(0.901354\pi\)
\(198\) 0 0
\(199\) −27214.8 −0.687226 −0.343613 0.939111i \(-0.611651\pi\)
−0.343613 + 0.939111i \(0.611651\pi\)
\(200\) 1162.22i 0.0290555i
\(201\) 0 0
\(202\) −22002.8 −0.539232
\(203\) 14912.0 0.361862
\(204\) 0 0
\(205\) 82769.9i 1.96954i
\(206\) 68201.5i 1.60716i
\(207\) 0 0
\(208\) 9833.20i 0.227284i
\(209\) 22109.6 32905.6i 0.506161 0.753317i
\(210\) 0 0
\(211\) 7465.71i 0.167690i −0.996479 0.0838449i \(-0.973280\pi\)
0.996479 0.0838449i \(-0.0267200\pi\)
\(212\) 56931.5 1.26672
\(213\) 0 0
\(214\) 20982.0 0.458162
\(215\) 49497.1i 1.07079i
\(216\) 0 0
\(217\) 5660.56i 0.120210i
\(218\) 85674.4 1.80276
\(219\) 0 0
\(220\) −45413.5 30513.8i −0.938296 0.630451i
\(221\) −17023.6 −0.348552
\(222\) 0 0
\(223\) 13654.6 0.274581 0.137291 0.990531i \(-0.456161\pi\)
0.137291 + 0.990531i \(0.456161\pi\)
\(224\) 18459.9 0.367903
\(225\) 0 0
\(226\) 40249.0i 0.788021i
\(227\) 15860.1i 0.307790i −0.988087 0.153895i \(-0.950818\pi\)
0.988087 0.153895i \(-0.0491817\pi\)
\(228\) 0 0
\(229\) −99238.3 −1.89238 −0.946190 0.323612i \(-0.895103\pi\)
−0.946190 + 0.323612i \(0.895103\pi\)
\(230\) 65936.9i 1.24644i
\(231\) 0 0
\(232\) 5150.75 0.0956962
\(233\) 44066.2i 0.811697i 0.913940 + 0.405849i \(0.133024\pi\)
−0.913940 + 0.405849i \(0.866976\pi\)
\(234\) 0 0
\(235\) 55204.0 0.999620
\(236\) −99910.1 −1.79385
\(237\) 0 0
\(238\) 33561.1i 0.592491i
\(239\) 57559.5i 1.00768i −0.863798 0.503838i \(-0.831921\pi\)
0.863798 0.503838i \(-0.168079\pi\)
\(240\) 0 0
\(241\) 86315.0i 1.48611i 0.669228 + 0.743057i \(0.266625\pi\)
−0.669228 + 0.743057i \(0.733375\pi\)
\(242\) −75714.9 + 30906.2i −1.29286 + 0.527733i
\(243\) 0 0
\(244\) 78134.9i 1.31240i
\(245\) 66443.9 1.10694
\(246\) 0 0
\(247\) 12013.8 0.196919
\(248\) 1955.22i 0.0317901i
\(249\) 0 0
\(250\) 60653.0i 0.970448i
\(251\) −93826.2 −1.48928 −0.744641 0.667465i \(-0.767379\pi\)
−0.744641 + 0.667465i \(0.767379\pi\)
\(252\) 0 0
\(253\) −39853.6 26778.1i −0.622625 0.418348i
\(254\) −60087.2 −0.931354
\(255\) 0 0
\(256\) 71592.5 1.09241
\(257\) 24489.3 0.370775 0.185387 0.982666i \(-0.440646\pi\)
0.185387 + 0.982666i \(0.440646\pi\)
\(258\) 0 0
\(259\) 3579.82i 0.0533656i
\(260\) 16580.4i 0.245273i
\(261\) 0 0
\(262\) −81681.6 −1.18993
\(263\) 74432.3i 1.07609i 0.842915 + 0.538047i \(0.180838\pi\)
−0.842915 + 0.538047i \(0.819162\pi\)
\(264\) 0 0
\(265\) −111426. −1.58670
\(266\) 23684.5i 0.334736i
\(267\) 0 0
\(268\) 52660.5 0.733188
\(269\) −114719. −1.58537 −0.792684 0.609633i \(-0.791317\pi\)
−0.792684 + 0.609633i \(0.791317\pi\)
\(270\) 0 0
\(271\) 18016.5i 0.245319i −0.992449 0.122660i \(-0.960858\pi\)
0.992449 0.122660i \(-0.0391423\pi\)
\(272\) 124497.i 1.68276i
\(273\) 0 0
\(274\) 140445.i 1.87070i
\(275\) 26111.6 + 17544.7i 0.345277 + 0.231995i
\(276\) 0 0
\(277\) 82563.5i 1.07604i 0.842932 + 0.538020i \(0.180828\pi\)
−0.842932 + 0.538020i \(0.819172\pi\)
\(278\) 24880.7 0.321938
\(279\) 0 0
\(280\) −1721.11 −0.0219529
\(281\) 7495.15i 0.0949222i 0.998873 + 0.0474611i \(0.0151130\pi\)
−0.998873 + 0.0474611i \(0.984887\pi\)
\(282\) 0 0
\(283\) 104220.i 1.30130i −0.759376 0.650652i \(-0.774495\pi\)
0.759376 0.650652i \(-0.225505\pi\)
\(284\) −91682.3 −1.13671
\(285\) 0 0
\(286\) −20570.8 13821.7i −0.251489 0.168978i
\(287\) −36008.6 −0.437162
\(288\) 0 0
\(289\) −132014. −1.58060
\(290\) 191459. 2.27656
\(291\) 0 0
\(292\) 54429.4i 0.638363i
\(293\) 54033.8i 0.629405i 0.949190 + 0.314703i \(0.101905\pi\)
−0.949190 + 0.314703i \(0.898095\pi\)
\(294\) 0 0
\(295\) 195543. 2.24698
\(296\) 1236.51i 0.0141128i
\(297\) 0 0
\(298\) 121521. 1.36842
\(299\) 14550.5i 0.162756i
\(300\) 0 0
\(301\) −21533.5 −0.237674
\(302\) 115826. 1.26997
\(303\) 0 0
\(304\) 87859.6i 0.950697i
\(305\) 152925.i 1.64391i
\(306\) 0 0
\(307\) 69862.2i 0.741251i −0.928782 0.370625i \(-0.879143\pi\)
0.928782 0.370625i \(-0.120857\pi\)
\(308\) 13274.9 19756.9i 0.139936 0.208266i
\(309\) 0 0
\(310\) 72677.6i 0.756271i
\(311\) 34406.8 0.355732 0.177866 0.984055i \(-0.443081\pi\)
0.177866 + 0.984055i \(0.443081\pi\)
\(312\) 0 0
\(313\) 34145.2 0.348531 0.174266 0.984699i \(-0.444245\pi\)
0.174266 + 0.984699i \(0.444245\pi\)
\(314\) 94909.0i 0.962605i
\(315\) 0 0
\(316\) 147612.i 1.47825i
\(317\) −67080.7 −0.667543 −0.333771 0.942654i \(-0.608321\pi\)
−0.333771 + 0.942654i \(0.608321\pi\)
\(318\) 0 0
\(319\) 77754.6 115722.i 0.764091 1.13719i
\(320\) 109371. 1.06808
\(321\) 0 0
\(322\) 28685.5 0.276663
\(323\) 152106. 1.45795
\(324\) 0 0
\(325\) 9533.32i 0.0902563i
\(326\) 87439.4i 0.822757i
\(327\) 0 0
\(328\) −12437.8 −0.115610
\(329\) 24016.2i 0.221877i
\(330\) 0 0
\(331\) −22562.2 −0.205933 −0.102967 0.994685i \(-0.532833\pi\)
−0.102967 + 0.994685i \(0.532833\pi\)
\(332\) 179924.i 1.63235i
\(333\) 0 0
\(334\) 76757.5 0.688062
\(335\) −103067. −0.918395
\(336\) 0 0
\(337\) 54658.6i 0.481281i 0.970614 + 0.240641i \(0.0773575\pi\)
−0.970614 + 0.240641i \(0.922642\pi\)
\(338\) 152022.i 1.33068i
\(339\) 0 0
\(340\) 209924.i 1.81595i
\(341\) 43927.8 + 29515.6i 0.377773 + 0.253830i
\(342\) 0 0
\(343\) 59979.9i 0.509820i
\(344\) −7437.89 −0.0628540
\(345\) 0 0
\(346\) −232493. −1.94204
\(347\) 116149.i 0.964618i −0.876001 0.482309i \(-0.839798\pi\)
0.876001 0.482309i \(-0.160202\pi\)
\(348\) 0 0
\(349\) 131862.i 1.08260i −0.840829 0.541301i \(-0.817932\pi\)
0.840829 0.541301i \(-0.182068\pi\)
\(350\) −18794.4 −0.153424
\(351\) 0 0
\(352\) 96254.4 143255.i 0.776847 1.15618i
\(353\) −27814.5 −0.223215 −0.111607 0.993752i \(-0.535600\pi\)
−0.111607 + 0.993752i \(0.535600\pi\)
\(354\) 0 0
\(355\) 179440. 1.42385
\(356\) 39286.9 0.309990
\(357\) 0 0
\(358\) 98167.0i 0.765948i
\(359\) 75277.7i 0.584087i 0.956405 + 0.292044i \(0.0943352\pi\)
−0.956405 + 0.292044i \(0.905665\pi\)
\(360\) 0 0
\(361\) 22977.5 0.176315
\(362\) 57933.2i 0.442089i
\(363\) 0 0
\(364\) 7213.23 0.0544411
\(365\) 106529.i 0.799617i
\(366\) 0 0
\(367\) 133254. 0.989346 0.494673 0.869079i \(-0.335288\pi\)
0.494673 + 0.869079i \(0.335288\pi\)
\(368\) 106411. 0.785762
\(369\) 0 0
\(370\) 45962.4i 0.335737i
\(371\) 48475.3i 0.352186i
\(372\) 0 0
\(373\) 125160.i 0.899600i 0.893129 + 0.449800i \(0.148505\pi\)
−0.893129 + 0.449800i \(0.851495\pi\)
\(374\) −260445. 174996.i −1.86197 1.25108i
\(375\) 0 0
\(376\) 8295.46i 0.0586766i
\(377\) 42249.9 0.297265
\(378\) 0 0
\(379\) −138734. −0.965841 −0.482920 0.875664i \(-0.660424\pi\)
−0.482920 + 0.875664i \(0.660424\pi\)
\(380\) 148146.i 1.02594i
\(381\) 0 0
\(382\) 12397.1i 0.0849561i
\(383\) −121841. −0.830611 −0.415305 0.909682i \(-0.636325\pi\)
−0.415305 + 0.909682i \(0.636325\pi\)
\(384\) 0 0
\(385\) −25981.5 + 38668.1i −0.175284 + 0.260874i
\(386\) 102598. 0.688597
\(387\) 0 0
\(388\) −23585.1 −0.156666
\(389\) −275759. −1.82234 −0.911172 0.412026i \(-0.864821\pi\)
−0.911172 + 0.412026i \(0.864821\pi\)
\(390\) 0 0
\(391\) 184223.i 1.20501i
\(392\) 9984.46i 0.0649759i
\(393\) 0 0
\(394\) −132218. −0.851725
\(395\) 288906.i 1.85166i
\(396\) 0 0
\(397\) −1004.21 −0.00637151 −0.00318576 0.999995i \(-0.501014\pi\)
−0.00318576 + 0.999995i \(0.501014\pi\)
\(398\) 152013.i 0.959654i
\(399\) 0 0
\(400\) −69719.2 −0.435745
\(401\) 127748. 0.794447 0.397224 0.917722i \(-0.369974\pi\)
0.397224 + 0.917722i \(0.369974\pi\)
\(402\) 0 0
\(403\) 16038.0i 0.0987508i
\(404\) 59874.0i 0.366839i
\(405\) 0 0
\(406\) 83293.2i 0.505310i
\(407\) 27780.6 + 18666.1i 0.167708 + 0.112684i
\(408\) 0 0
\(409\) 147696.i 0.882920i 0.897281 + 0.441460i \(0.145539\pi\)
−0.897281 + 0.441460i \(0.854461\pi\)
\(410\) −462325. −2.75030
\(411\) 0 0
\(412\) −185590. −1.09335
\(413\) 85070.1i 0.498743i
\(414\) 0 0
\(415\) 352145.i 2.04468i
\(416\) 52302.3 0.302227
\(417\) 0 0
\(418\) 183800. + 123497.i 1.05194 + 0.706812i
\(419\) 319997. 1.82271 0.911355 0.411621i \(-0.135037\pi\)
0.911355 + 0.411621i \(0.135037\pi\)
\(420\) 0 0
\(421\) −244090. −1.37716 −0.688582 0.725159i \(-0.741766\pi\)
−0.688582 + 0.725159i \(0.741766\pi\)
\(422\) 41701.0 0.234165
\(423\) 0 0
\(424\) 16743.9i 0.0931375i
\(425\) 120701.i 0.668239i
\(426\) 0 0
\(427\) 66529.2 0.364886
\(428\) 57096.1i 0.311687i
\(429\) 0 0
\(430\) −276475. −1.49527
\(431\) 144764.i 0.779303i 0.920962 + 0.389651i \(0.127404\pi\)
−0.920962 + 0.389651i \(0.872596\pi\)
\(432\) 0 0
\(433\) −98589.6 −0.525842 −0.262921 0.964817i \(-0.584686\pi\)
−0.262921 + 0.964817i \(0.584686\pi\)
\(434\) −31618.0 −0.167863
\(435\) 0 0
\(436\) 233137.i 1.22642i
\(437\) 130009.i 0.680785i
\(438\) 0 0
\(439\) 241475.i 1.25298i 0.779430 + 0.626490i \(0.215509\pi\)
−0.779430 + 0.626490i \(0.784491\pi\)
\(440\) −8974.29 + 13356.4i −0.0463548 + 0.0689895i
\(441\) 0 0
\(442\) 95088.3i 0.486724i
\(443\) 106301. 0.541666 0.270833 0.962626i \(-0.412701\pi\)
0.270833 + 0.962626i \(0.412701\pi\)
\(444\) 0 0
\(445\) −76892.1 −0.388295
\(446\) 76270.3i 0.383430i
\(447\) 0 0
\(448\) 47581.4i 0.237072i
\(449\) 265627. 1.31759 0.658794 0.752323i \(-0.271067\pi\)
0.658794 + 0.752323i \(0.271067\pi\)
\(450\) 0 0
\(451\) −187758. + 279439.i −0.923091 + 1.37383i
\(452\) −109525. −0.536089
\(453\) 0 0
\(454\) 88589.2 0.429803
\(455\) −14117.7 −0.0681932
\(456\) 0 0
\(457\) 116157.i 0.556177i −0.960556 0.278088i \(-0.910299\pi\)
0.960556 0.278088i \(-0.0897008\pi\)
\(458\) 554312.i 2.64255i
\(459\) 0 0
\(460\) 179427. 0.847954
\(461\) 121516.i 0.571784i −0.958262 0.285892i \(-0.907710\pi\)
0.958262 0.285892i \(-0.0922899\pi\)
\(462\) 0 0
\(463\) 370675. 1.72914 0.864572 0.502509i \(-0.167590\pi\)
0.864572 + 0.502509i \(0.167590\pi\)
\(464\) 308983.i 1.43515i
\(465\) 0 0
\(466\) −246139. −1.13347
\(467\) −17468.5 −0.0800980 −0.0400490 0.999198i \(-0.512751\pi\)
−0.0400490 + 0.999198i \(0.512751\pi\)
\(468\) 0 0
\(469\) 44838.7i 0.203848i
\(470\) 308351.i 1.39589i
\(471\) 0 0
\(472\) 29384.1i 0.131895i
\(473\) −112281. + 167107.i −0.501861 + 0.746916i
\(474\) 0 0
\(475\) 85180.2i 0.377530i
\(476\) 91326.1 0.403071
\(477\) 0 0
\(478\) 321508. 1.40714
\(479\) 37311.4i 0.162619i 0.996689 + 0.0813094i \(0.0259102\pi\)
−0.996689 + 0.0813094i \(0.974090\pi\)
\(480\) 0 0
\(481\) 10142.7i 0.0438392i
\(482\) −482127. −2.07523
\(483\) 0 0
\(484\) −84101.6 206035.i −0.359016 0.879528i
\(485\) 46160.6 0.196240
\(486\) 0 0
\(487\) 177099. 0.746721 0.373360 0.927686i \(-0.378205\pi\)
0.373360 + 0.927686i \(0.378205\pi\)
\(488\) 22979.9 0.0964958
\(489\) 0 0
\(490\) 371134.i 1.54575i
\(491\) 127053.i 0.527013i 0.964658 + 0.263507i \(0.0848791\pi\)
−0.964658 + 0.263507i \(0.915121\pi\)
\(492\) 0 0
\(493\) 534923. 2.20088
\(494\) 67105.2i 0.274981i
\(495\) 0 0
\(496\) −117289. −0.476755
\(497\) 78064.5i 0.316039i
\(498\) 0 0
\(499\) −166524. −0.668767 −0.334383 0.942437i \(-0.608528\pi\)
−0.334383 + 0.942437i \(0.608528\pi\)
\(500\) 165049. 0.660194
\(501\) 0 0
\(502\) 524082.i 2.07966i
\(503\) 163251.i 0.645236i −0.946529 0.322618i \(-0.895437\pi\)
0.946529 0.322618i \(-0.104563\pi\)
\(504\) 0 0
\(505\) 117185.i 0.459504i
\(506\) 149573. 222609.i 0.584189 0.869444i
\(507\) 0 0
\(508\) 163509.i 0.633599i
\(509\) 206271. 0.796164 0.398082 0.917350i \(-0.369676\pi\)
0.398082 + 0.917350i \(0.369676\pi\)
\(510\) 0 0
\(511\) 46344.8 0.177484
\(512\) 363317.i 1.38594i
\(513\) 0 0
\(514\) 136789.i 0.517756i
\(515\) 363235. 1.36953
\(516\) 0 0
\(517\) 186374. + 125226.i 0.697274 + 0.468506i
\(518\) −19995.7 −0.0745207
\(519\) 0 0
\(520\) −4876.40 −0.0180340
\(521\) −442001. −1.62835 −0.814174 0.580620i \(-0.802810\pi\)
−0.814174 + 0.580620i \(0.802810\pi\)
\(522\) 0 0
\(523\) 373577.i 1.36577i −0.730527 0.682884i \(-0.760726\pi\)
0.730527 0.682884i \(-0.239274\pi\)
\(524\) 222271.i 0.809508i
\(525\) 0 0
\(526\) −415754. −1.50268
\(527\) 203056.i 0.731130i
\(528\) 0 0
\(529\) −122381. −0.437323
\(530\) 622389.i 2.21569i
\(531\) 0 0
\(532\) −64450.2 −0.227720
\(533\) −102023. −0.359123
\(534\) 0 0
\(535\) 111748.i 0.390421i
\(536\) 15487.8i 0.0539087i
\(537\) 0 0
\(538\) 640781.i 2.21383i
\(539\) 224321. + 150723.i 0.772132 + 0.518804i
\(540\) 0 0
\(541\) 339954.i 1.16152i −0.814075 0.580759i \(-0.802756\pi\)
0.814075 0.580759i \(-0.197244\pi\)
\(542\) 100634. 0.342568
\(543\) 0 0
\(544\) 662194. 2.23763
\(545\) 456294.i 1.53621i
\(546\) 0 0
\(547\) 568245.i 1.89916i 0.313531 + 0.949578i \(0.398488\pi\)
−0.313531 + 0.949578i \(0.601512\pi\)
\(548\) 382177. 1.27263
\(549\) 0 0
\(550\) −97998.6 + 145851.i −0.323962 + 0.482151i
\(551\) −377503. −1.24342
\(552\) 0 0
\(553\) 125687. 0.410998
\(554\) −461172. −1.50260
\(555\) 0 0
\(556\) 67705.2i 0.219014i
\(557\) 211216.i 0.680795i −0.940282 0.340398i \(-0.889438\pi\)
0.940282 0.340398i \(-0.110562\pi\)
\(558\) 0 0
\(559\) −61010.6 −0.195246
\(560\) 103246.i 0.329227i
\(561\) 0 0
\(562\) −41865.4 −0.132551
\(563\) 595910.i 1.88003i −0.341140 0.940013i \(-0.610813\pi\)
0.341140 0.940013i \(-0.389187\pi\)
\(564\) 0 0
\(565\) 214362. 0.671508
\(566\) 582139. 1.81716
\(567\) 0 0
\(568\) 26964.3i 0.0835781i
\(569\) 15875.4i 0.0490342i 0.999699 + 0.0245171i \(0.00780482\pi\)
−0.999699 + 0.0245171i \(0.992195\pi\)
\(570\) 0 0
\(571\) 413970.i 1.26969i 0.772641 + 0.634843i \(0.218935\pi\)
−0.772641 + 0.634843i \(0.781065\pi\)
\(572\) 37611.6 55977.1i 0.114955 0.171087i
\(573\) 0 0
\(574\) 201132.i 0.610460i
\(575\) −103166. −0.312033
\(576\) 0 0
\(577\) −102654. −0.308337 −0.154169 0.988045i \(-0.549270\pi\)
−0.154169 + 0.988045i \(0.549270\pi\)
\(578\) 737384.i 2.20718i
\(579\) 0 0
\(580\) 520997.i 1.54874i
\(581\) 153199. 0.453841
\(582\) 0 0
\(583\) −376184. 252762.i −1.10679 0.743661i
\(584\) 16008.0 0.0469366
\(585\) 0 0
\(586\) −301815. −0.878912
\(587\) −127414. −0.369779 −0.184889 0.982759i \(-0.559193\pi\)
−0.184889 + 0.982759i \(0.559193\pi\)
\(588\) 0 0
\(589\) 143300.i 0.413061i
\(590\) 1.09224e6i 3.13772i
\(591\) 0 0
\(592\) −74175.5 −0.211649
\(593\) 477628.i 1.35825i 0.734022 + 0.679126i \(0.237641\pi\)
−0.734022 + 0.679126i \(0.762359\pi\)
\(594\) 0 0
\(595\) −178743. −0.504888
\(596\) 330682.i 0.930933i
\(597\) 0 0
\(598\) 81274.4 0.227275
\(599\) −301738. −0.840961 −0.420480 0.907302i \(-0.638138\pi\)
−0.420480 + 0.907302i \(0.638138\pi\)
\(600\) 0 0
\(601\) 294665.i 0.815793i −0.913028 0.407896i \(-0.866262\pi\)
0.913028 0.407896i \(-0.133738\pi\)
\(602\) 120279.i 0.331891i
\(603\) 0 0
\(604\) 315186.i 0.863959i
\(605\) 164603. + 403250.i 0.449705 + 1.10170i
\(606\) 0 0
\(607\) 308433.i 0.837113i −0.908191 0.418556i \(-0.862536\pi\)
0.908191 0.418556i \(-0.137464\pi\)
\(608\) −467321. −1.26418
\(609\) 0 0
\(610\) 854189. 2.29559
\(611\) 68044.9i 0.182269i
\(612\) 0 0
\(613\) 222563.i 0.592287i −0.955143 0.296144i \(-0.904299\pi\)
0.955143 0.296144i \(-0.0957007\pi\)
\(614\) 390227. 1.03510
\(615\) 0 0
\(616\) −5810.62 3904.22i −0.0153130 0.0102890i
\(617\) −478612. −1.25723 −0.628613 0.777718i \(-0.716377\pi\)
−0.628613 + 0.777718i \(0.716377\pi\)
\(618\) 0 0
\(619\) −110719. −0.288963 −0.144482 0.989507i \(-0.546151\pi\)
−0.144482 + 0.989507i \(0.546151\pi\)
\(620\) −197770. −0.514490
\(621\) 0 0
\(622\) 192185.i 0.496750i
\(623\) 33451.5i 0.0861866i
\(624\) 0 0
\(625\) −485524. −1.24294
\(626\) 190724.i 0.486695i
\(627\) 0 0
\(628\) 258266. 0.654859
\(629\) 128416.i 0.324576i
\(630\) 0 0
\(631\) −77313.0 −0.194175 −0.0970876 0.995276i \(-0.530953\pi\)
−0.0970876 + 0.995276i \(0.530953\pi\)
\(632\) 43413.6 0.108690
\(633\) 0 0
\(634\) 374690.i 0.932168i
\(635\) 320019.i 0.793648i
\(636\) 0 0
\(637\) 81899.3i 0.201837i
\(638\) 646383. + 434312.i 1.58799 + 1.06699i
\(639\) 0 0
\(640\) 68003.7i 0.166025i
\(641\) −382285. −0.930404 −0.465202 0.885205i \(-0.654018\pi\)
−0.465202 + 0.885205i \(0.654018\pi\)
\(642\) 0 0
\(643\) −216575. −0.523825 −0.261912 0.965092i \(-0.584353\pi\)
−0.261912 + 0.965092i \(0.584353\pi\)
\(644\) 78058.8i 0.188213i
\(645\) 0 0
\(646\) 849613.i 2.03590i
\(647\) 151109. 0.360979 0.180489 0.983577i \(-0.442232\pi\)
0.180489 + 0.983577i \(0.442232\pi\)
\(648\) 0 0
\(649\) 660172. + 443577.i 1.56736 + 1.05312i
\(650\) −53250.0 −0.126035
\(651\) 0 0
\(652\) −237939. −0.559721
\(653\) 191077. 0.448107 0.224053 0.974577i \(-0.428071\pi\)
0.224053 + 0.974577i \(0.428071\pi\)
\(654\) 0 0
\(655\) 435028.i 1.01399i
\(656\) 746114.i 1.73380i
\(657\) 0 0
\(658\) −134147. −0.309833
\(659\) 452930.i 1.04294i 0.853269 + 0.521471i \(0.174616\pi\)
−0.853269 + 0.521471i \(0.825384\pi\)
\(660\) 0 0
\(661\) 307053. 0.702767 0.351383 0.936232i \(-0.385711\pi\)
0.351383 + 0.936232i \(0.385711\pi\)
\(662\) 126025.i 0.287568i
\(663\) 0 0
\(664\) 52916.6 0.120020
\(665\) 126142. 0.285243
\(666\) 0 0
\(667\) 457212.i 1.02770i
\(668\) 208872.i 0.468088i
\(669\) 0 0
\(670\) 575697.i 1.28246i
\(671\) 346900. 516289.i 0.770476 1.14669i
\(672\) 0 0
\(673\) 216281.i 0.477517i 0.971079 + 0.238758i \(0.0767404\pi\)
−0.971079 + 0.238758i \(0.923260\pi\)
\(674\) −305305. −0.672069
\(675\) 0 0
\(676\) −413681. −0.905257
\(677\) 23957.4i 0.0522712i 0.999658 + 0.0261356i \(0.00832017\pi\)
−0.999658 + 0.0261356i \(0.991680\pi\)
\(678\) 0 0
\(679\) 20081.9i 0.0435578i
\(680\) −61739.7 −0.133520
\(681\) 0 0
\(682\) −164864. + 245366.i −0.354452 + 0.527529i
\(683\) −547099. −1.17280 −0.586401 0.810021i \(-0.699456\pi\)
−0.586401 + 0.810021i \(0.699456\pi\)
\(684\) 0 0
\(685\) −747994. −1.59411
\(686\) −335028. −0.711922
\(687\) 0 0
\(688\) 446183.i 0.942619i
\(689\) 137345.i 0.289316i
\(690\) 0 0
\(691\) −299138. −0.626493 −0.313246 0.949672i \(-0.601417\pi\)
−0.313246 + 0.949672i \(0.601417\pi\)
\(692\) 632658.i 1.32117i
\(693\) 0 0
\(694\) 648768. 1.34701
\(695\) 132512.i 0.274338i
\(696\) 0 0
\(697\) −1.29170e6 −2.65887
\(698\) 736538. 1.51176
\(699\) 0 0
\(700\) 51143.2i 0.104374i
\(701\) 876409.i 1.78349i −0.452538 0.891745i \(-0.649481\pi\)
0.452538 0.891745i \(-0.350519\pi\)
\(702\) 0 0
\(703\) 90624.8i 0.183373i
\(704\) 369247. + 248101.i 0.745026 + 0.500591i
\(705\) 0 0
\(706\) 155363.i 0.311701i
\(707\) −50980.7 −0.101992
\(708\) 0 0
\(709\) 266386. 0.529931 0.264965 0.964258i \(-0.414639\pi\)
0.264965 + 0.964258i \(0.414639\pi\)
\(710\) 1.00229e6i 1.98828i
\(711\) 0 0
\(712\) 11554.5i 0.0227925i
\(713\) −173557. −0.341400
\(714\) 0 0
\(715\) −73613.2 + 109558.i −0.143994 + 0.214305i
\(716\) −267131. −0.521073
\(717\) 0 0
\(718\) −420476. −0.815629
\(719\) 747175. 1.44532 0.722661 0.691203i \(-0.242919\pi\)
0.722661 + 0.691203i \(0.242919\pi\)
\(720\) 0 0
\(721\) 158023.i 0.303984i
\(722\) 128345.i 0.246209i
\(723\) 0 0
\(724\) −157647. −0.300753
\(725\) 299560.i 0.569912i
\(726\) 0 0
\(727\) −685597. −1.29718 −0.648590 0.761138i \(-0.724641\pi\)
−0.648590 + 0.761138i \(0.724641\pi\)
\(728\) 2121.45i 0.00400286i
\(729\) 0 0
\(730\) 595035. 1.11660
\(731\) −772450. −1.44556
\(732\) 0 0
\(733\) 64133.6i 0.119365i −0.998217 0.0596826i \(-0.980991\pi\)
0.998217 0.0596826i \(-0.0190089\pi\)
\(734\) 744313.i 1.38154i
\(735\) 0 0
\(736\) 565995.i 1.04486i
\(737\) −347963. 233800.i −0.640616 0.430437i
\(738\) 0 0
\(739\) 350880.i 0.642495i −0.946995 0.321248i \(-0.895898\pi\)
0.946995 0.321248i \(-0.104102\pi\)
\(740\) −125073. −0.228401
\(741\) 0 0
\(742\) 270767. 0.491799
\(743\) 214737.i 0.388981i −0.980904 0.194491i \(-0.937695\pi\)
0.980904 0.194491i \(-0.0623054\pi\)
\(744\) 0 0
\(745\) 647209.i 1.16609i
\(746\) −699105. −1.25622
\(747\) 0 0
\(748\) 476197. 708721.i 0.851106 1.26669i
\(749\) 48615.4 0.0866583
\(750\) 0 0
\(751\) 656340. 1.16372 0.581861 0.813288i \(-0.302325\pi\)
0.581861 + 0.813288i \(0.302325\pi\)
\(752\) −497627. −0.879970
\(753\) 0 0
\(754\) 235994.i 0.415105i
\(755\) 616880.i 1.08220i
\(756\) 0 0
\(757\) 667153. 1.16422 0.582108 0.813111i \(-0.302228\pi\)
0.582108 + 0.813111i \(0.302228\pi\)
\(758\) 774924.i 1.34872i
\(759\) 0 0
\(760\) 43570.7 0.0754340
\(761\) 125398.i 0.216532i 0.994122 + 0.108266i \(0.0345298\pi\)
−0.994122 + 0.108266i \(0.965470\pi\)
\(762\) 0 0
\(763\) 198508. 0.340980
\(764\) −33735.0 −0.0577955
\(765\) 0 0
\(766\) 680566.i 1.15988i
\(767\) 241028.i 0.409711i
\(768\) 0 0
\(769\) 232947.i 0.393916i 0.980412 + 0.196958i \(0.0631063\pi\)
−0.980412 + 0.196958i \(0.936894\pi\)
\(770\) −215987. 145124.i −0.364289 0.244770i
\(771\) 0 0
\(772\) 279190.i 0.468452i
\(773\) 283047. 0.473697 0.236848 0.971547i \(-0.423886\pi\)
0.236848 + 0.971547i \(0.423886\pi\)
\(774\) 0 0
\(775\) 113713. 0.189324
\(776\) 6936.51i 0.0115191i
\(777\) 0 0
\(778\) 1.54030e6i 2.54475i
\(779\) 911574. 1.50216
\(780\) 0 0
\(781\) 605806. + 407047.i 0.993188 + 0.667333i
\(782\) 1.02901e6 1.68269
\(783\) 0 0
\(784\) −598947. −0.974442
\(785\) −505476. −0.820279
\(786\) 0 0
\(787\) 299296.i 0.483227i −0.970373 0.241613i \(-0.922323\pi\)
0.970373 0.241613i \(-0.0776766\pi\)
\(788\) 359792.i 0.579427i
\(789\) 0 0
\(790\) 1.61373e6 2.58569
\(791\) 93257.0i 0.149049i
\(792\) 0 0
\(793\) 188497. 0.299749
\(794\) 5609.17i 0.00889729i
\(795\) 0 0
\(796\) 413657. 0.652851
\(797\) 768258. 1.20946 0.604729 0.796432i \(-0.293282\pi\)
0.604729 + 0.796432i \(0.293282\pi\)
\(798\) 0 0
\(799\) 861511.i 1.34948i
\(800\) 370833.i 0.579426i
\(801\) 0 0
\(802\) 713557.i 1.10938i
\(803\) 241653. 359651.i 0.374767 0.557764i
\(804\) 0 0
\(805\) 152776.i 0.235757i
\(806\) −89583.1 −0.137897
\(807\) 0 0
\(808\) −17609.3 −0.0269723
\(809\) 982802.i 1.50165i 0.660500 + 0.750826i \(0.270344\pi\)
−0.660500 + 0.750826i \(0.729656\pi\)
\(810\) 0 0
\(811\) 322384.i 0.490153i −0.969504 0.245077i \(-0.921187\pi\)
0.969504 0.245077i \(-0.0788131\pi\)
\(812\) −226657. −0.343761
\(813\) 0 0
\(814\) −104262. + 155173.i −0.157355 + 0.234190i
\(815\) 465694. 0.701108
\(816\) 0 0
\(817\) 545129. 0.816687
\(818\) −824979. −1.23292
\(819\) 0 0
\(820\) 1.25808e6i 1.87102i
\(821\) 862907.i 1.28020i 0.768292 + 0.640100i \(0.221107\pi\)
−0.768292 + 0.640100i \(0.778893\pi\)
\(822\) 0 0
\(823\) −457793. −0.675879 −0.337940 0.941168i \(-0.609730\pi\)
−0.337940 + 0.941168i \(0.609730\pi\)
\(824\) 54583.0i 0.0803901i
\(825\) 0 0
\(826\) −475173. −0.696453
\(827\) 1.26608e6i 1.85119i 0.378516 + 0.925595i \(0.376435\pi\)
−0.378516 + 0.925595i \(0.623565\pi\)
\(828\) 0 0
\(829\) 336361. 0.489437 0.244718 0.969594i \(-0.421305\pi\)
0.244718 + 0.969594i \(0.421305\pi\)
\(830\) 1.96697e6 2.85523
\(831\) 0 0
\(832\) 134812.i 0.194752i
\(833\) 1.03692e6i 1.49436i
\(834\) 0 0
\(835\) 408803.i 0.586329i
\(836\) −336059. + 500155.i −0.480843 + 0.715636i
\(837\) 0 0
\(838\) 1.78740e6i 2.54526i
\(839\) −673978. −0.957463 −0.478732 0.877961i \(-0.658903\pi\)
−0.478732 + 0.877961i \(0.658903\pi\)
\(840\) 0 0
\(841\) −620313. −0.877039
\(842\) 1.36341e6i 1.92309i
\(843\) 0 0
\(844\) 113476.i 0.159302i
\(845\) 809654. 1.13393
\(846\) 0 0
\(847\) −175432. + 71609.7i −0.244535 + 0.0998171i
\(848\) 1.00443e6 1.39678
\(849\) 0 0
\(850\) −674193. −0.933140
\(851\) −109760. −0.151560
\(852\) 0 0
\(853\) 293271.i 0.403062i −0.979482 0.201531i \(-0.935408\pi\)
0.979482 0.201531i \(-0.0645916\pi\)
\(854\) 371610.i 0.509532i
\(855\) 0 0
\(856\) 16792.3 0.0229172
\(857\) 404960.i 0.551379i 0.961247 + 0.275690i \(0.0889062\pi\)
−0.961247 + 0.275690i \(0.911094\pi\)
\(858\) 0 0
\(859\) −354890. −0.480958 −0.240479 0.970654i \(-0.577305\pi\)
−0.240479 + 0.970654i \(0.577305\pi\)
\(860\) 752341.i 1.01723i
\(861\) 0 0
\(862\) −808604. −1.08823
\(863\) −1.17452e6 −1.57703 −0.788514 0.615017i \(-0.789149\pi\)
−0.788514 + 0.615017i \(0.789149\pi\)
\(864\) 0 0
\(865\) 1.23824e6i 1.65490i
\(866\) 550689.i 0.734294i
\(867\) 0 0
\(868\) 86038.7i 0.114197i
\(869\) 655362. 975371.i 0.867845 1.29161i
\(870\) 0 0
\(871\) 127041.i 0.167459i
\(872\) 68566.8 0.0901739
\(873\) 0 0
\(874\) −726187. −0.950660
\(875\) 140533.i 0.183554i
\(876\) 0 0
\(877\) 199151.i 0.258931i 0.991584 + 0.129466i \(0.0413262\pi\)
−0.991584 + 0.129466i \(0.958674\pi\)
\(878\) −1.34880e6 −1.74968
\(879\) 0 0
\(880\) −801220. 538348.i −1.03463 0.695181i
\(881\) −538123. −0.693314 −0.346657 0.937992i \(-0.612683\pi\)
−0.346657 + 0.937992i \(0.612683\pi\)
\(882\) 0 0
\(883\) 1.41933e6 1.82038 0.910192 0.414187i \(-0.135934\pi\)
0.910192 + 0.414187i \(0.135934\pi\)
\(884\) 258754. 0.331117
\(885\) 0 0
\(886\) 593765.i 0.756392i
\(887\) 1.08483e6i 1.37884i 0.724360 + 0.689422i \(0.242135\pi\)
−0.724360 + 0.689422i \(0.757865\pi\)
\(888\) 0 0
\(889\) −139222. −0.176159
\(890\) 429494.i 0.542222i
\(891\) 0 0
\(892\) −207546. −0.260847
\(893\) 607981.i 0.762408i
\(894\) 0 0
\(895\) 522828. 0.652699
\(896\) 29584.6 0.0368511
\(897\) 0 0
\(898\) 1.48371e6i 1.83990i
\(899\) 503953.i 0.623549i
\(900\) 0 0
\(901\) 1.73891e6i 2.14204i
\(902\) −1.56085e6 1.04875e6i −1.91844 1.28902i
\(903\) 0 0
\(904\) 32212.0i 0.0394167i
\(905\) 308546. 0.376724
\(906\) 0 0
\(907\) −396615. −0.482119 −0.241060 0.970510i \(-0.577495\pi\)
−0.241060 + 0.970510i \(0.577495\pi\)
\(908\) 241068.i 0.292394i
\(909\) 0 0
\(910\) 78856.8i 0.0952261i
\(911\) 5038.05 0.00607052 0.00303526 0.999995i \(-0.499034\pi\)
0.00303526 + 0.999995i \(0.499034\pi\)
\(912\) 0 0
\(913\) 798817. 1.18887e6i 0.958310 1.42625i
\(914\) 648814. 0.776654
\(915\) 0 0
\(916\) 1.50839e6 1.79772
\(917\) −189257. −0.225067
\(918\) 0 0
\(919\) 1.61726e6i 1.91491i 0.288574 + 0.957457i \(0.406819\pi\)
−0.288574 + 0.957457i \(0.593181\pi\)
\(920\) 52770.5i 0.0623470i
\(921\) 0 0
\(922\) 678749. 0.798449
\(923\) 221179.i 0.259622i
\(924\) 0 0
\(925\) 71913.5 0.0840479
\(926\) 2.07047e6i 2.41461i
\(927\) 0 0
\(928\) −1.64346e6 −1.90838
\(929\) 433742. 0.502575 0.251287 0.967913i \(-0.419146\pi\)
0.251287 + 0.967913i \(0.419146\pi\)
\(930\) 0 0
\(931\) 731770.i 0.844258i
\(932\) 669793.i 0.771096i
\(933\) 0 0
\(934\) 97573.1i 0.111850i
\(935\) −932010. + 1.38710e6i −1.06610 + 1.58667i
\(936\) 0 0
\(937\) 428901.i 0.488515i 0.969710 + 0.244257i \(0.0785442\pi\)
−0.969710 + 0.244257i \(0.921456\pi\)
\(938\) 250454. 0.284657
\(939\) 0 0
\(940\) −839084. −0.949619
\(941\) 1.50772e6i 1.70272i −0.524583 0.851359i \(-0.675779\pi\)
0.524583 0.851359i \(-0.324221\pi\)
\(942\) 0 0
\(943\) 1.10405e6i 1.24155i
\(944\) −1.76269e6 −1.97803
\(945\) 0 0
\(946\) −933403. 627163.i −1.04301 0.700807i
\(947\) −130017. −0.144977 −0.0724887 0.997369i \(-0.523094\pi\)
−0.0724887 + 0.997369i \(0.523094\pi\)
\(948\) 0 0
\(949\) 131308. 0.145801
\(950\) 475788. 0.527189
\(951\) 0 0
\(952\) 26859.5i 0.0296363i
\(953\) 1.78038e6i 1.96032i −0.198209 0.980160i \(-0.563512\pi\)
0.198209 0.980160i \(-0.436488\pi\)
\(954\) 0 0
\(955\) 66025.9 0.0723949
\(956\) 874886.i 0.957273i
\(957\) 0 0
\(958\) −208409. −0.227084
\(959\) 325411.i 0.353830i
\(960\) 0 0
\(961\) −732221. −0.792858
\(962\) −56653.6 −0.0612178
\(963\) 0 0
\(964\) 1.31196e6i 1.41178i
\(965\) 546428.i 0.586785i
\(966\) 0 0
\(967\) 445739.i 0.476680i −0.971182 0.238340i \(-0.923397\pi\)
0.971182 0.238340i \(-0.0766033\pi\)
\(968\) −60596.0 + 24734.8i −0.0646686 + 0.0263971i
\(969\) 0 0
\(970\) 257838.i 0.274033i
\(971\) −1.44428e6 −1.53183 −0.765917 0.642939i \(-0.777715\pi\)
−0.765917 + 0.642939i \(0.777715\pi\)
\(972\) 0 0
\(973\) 57648.7 0.0608925
\(974\) 989216.i 1.04273i
\(975\) 0 0
\(976\) 1.37852e6i 1.44715i
\(977\) 1.79884e6 1.88453 0.942265 0.334868i \(-0.108692\pi\)
0.942265 + 0.334868i \(0.108692\pi\)
\(978\) 0 0
\(979\) −259595. 174424.i −0.270851 0.181988i
\(980\) −1.00993e6 −1.05157
\(981\) 0 0
\(982\) −709675. −0.735931
\(983\) 562567. 0.582194 0.291097 0.956694i \(-0.405980\pi\)
0.291097 + 0.956694i \(0.405980\pi\)
\(984\) 0 0
\(985\) 704182.i 0.725793i
\(986\) 2.98790e6i 3.07335i
\(987\) 0 0
\(988\) −182606. −0.187069
\(989\) 660233.i 0.675001i
\(990\) 0 0
\(991\) −679908. −0.692313 −0.346157 0.938177i \(-0.612513\pi\)
−0.346157 + 0.938177i \(0.612513\pi\)
\(992\) 623856.i 0.633959i
\(993\) 0 0
\(994\) −436042. −0.441322
\(995\) −809607. −0.817764
\(996\) 0 0
\(997\) 723578.i 0.727939i −0.931411 0.363970i \(-0.881421\pi\)
0.931411 0.363970i \(-0.118579\pi\)
\(998\) 930145.i 0.933877i
\(999\) 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.10.7 8
3.2 odd 2 33.5.c.a.10.2 8
11.10 odd 2 inner 99.5.c.c.10.2 8
12.11 even 2 528.5.j.a.241.5 8
33.32 even 2 33.5.c.a.10.7 yes 8
132.131 odd 2 528.5.j.a.241.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.5.c.a.10.2 8 3.2 odd 2
33.5.c.a.10.7 yes 8 33.32 even 2
99.5.c.c.10.2 8 11.10 odd 2 inner
99.5.c.c.10.7 8 1.1 even 1 trivial
528.5.j.a.241.5 8 12.11 even 2
528.5.j.a.241.6 8 132.131 odd 2