Properties

Label 304.5.r.d.65.10
Level $304$
Weight $5$
Character 304.65
Analytic conductor $31.424$
Analytic rank $0$
Dimension $40$
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] = [304,5,Mod(65,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("304.65");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 304.r (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(31.4244687775\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 152)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 65.10
Character \(\chi\) \(=\) 304.65
Dual form 304.5.r.d.145.10

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.93870 - 1.11931i) q^{3} +(-15.6195 + 27.0538i) q^{5} +6.74912 q^{7} +(-37.9943 - 65.8080i) q^{9} +O(q^{10})\) \(q+(-1.93870 - 1.11931i) q^{3} +(-15.6195 + 27.0538i) q^{5} +6.74912 q^{7} +(-37.9943 - 65.8080i) q^{9} +124.903 q^{11} +(134.659 - 77.7455i) q^{13} +(60.5631 - 34.9661i) q^{15} +(127.427 - 220.710i) q^{17} +(-161.629 + 322.796i) q^{19} +(-13.0845 - 7.55435i) q^{21} +(193.260 + 334.736i) q^{23} +(-175.438 - 303.867i) q^{25} +351.438i q^{27} +(-985.611 + 569.043i) q^{29} +593.148i q^{31} +(-242.149 - 139.805i) q^{33} +(-105.418 + 182.589i) q^{35} +360.278i q^{37} -348.085 q^{39} +(-1647.63 - 951.257i) q^{41} +(-794.184 + 1375.57i) q^{43} +2373.81 q^{45} +(1347.57 + 2334.06i) q^{47} -2355.45 q^{49} +(-494.085 + 285.260i) q^{51} +(-1.97124 + 1.13810i) q^{53} +(-1950.92 + 3379.09i) q^{55} +(674.659 - 444.891i) q^{57} +(3490.96 + 2015.51i) q^{59} +(3481.53 + 6030.18i) q^{61} +(-256.428 - 444.146i) q^{63} +4857.38i q^{65} +(4546.94 - 2625.18i) q^{67} -865.270i q^{69} +(-2217.11 - 1280.05i) q^{71} +(110.303 - 191.050i) q^{73} +785.477i q^{75} +842.982 q^{77} +(-2249.78 - 1298.91i) q^{79} +(-2684.17 + 4649.12i) q^{81} -7437.65 q^{83} +(3980.68 + 6894.75i) q^{85} +2547.74 q^{87} +(-8825.30 + 5095.29i) q^{89} +(908.830 - 524.713i) q^{91} +(663.917 - 1149.94i) q^{93} +(-6208.27 - 9414.59i) q^{95} +(8257.62 + 4767.54i) q^{97} +(-4745.58 - 8219.59i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q - 12 q^{3} + 32 q^{7} + 624 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 40 q - 12 q^{3} + 32 q^{7} + 624 q^{9} - 24 q^{11} + 264 q^{13} - 624 q^{15} + 216 q^{17} + 652 q^{19} - 216 q^{21} + 1296 q^{23} - 3044 q^{25} + 288 q^{29} - 6660 q^{33} - 360 q^{35} - 3184 q^{39} + 1260 q^{41} - 632 q^{43} + 256 q^{45} - 1248 q^{47} + 16696 q^{49} + 8064 q^{51} - 3672 q^{53} + 3408 q^{55} - 4552 q^{57} - 12492 q^{59} + 2720 q^{61} - 12472 q^{63} - 16260 q^{67} + 504 q^{71} + 9220 q^{73} - 14688 q^{77} + 28944 q^{79} - 1660 q^{81} + 39192 q^{83} - 18632 q^{85} + 34400 q^{87} + 3456 q^{89} - 54432 q^{91} - 17208 q^{93} - 44520 q^{95} - 30540 q^{97} + 10096 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.93870 1.11931i −0.215411 0.124368i 0.388412 0.921486i \(-0.373024\pi\)
−0.603824 + 0.797118i \(0.706357\pi\)
\(4\) 0 0
\(5\) −15.6195 + 27.0538i −0.624780 + 1.08215i 0.363803 + 0.931476i \(0.381478\pi\)
−0.988583 + 0.150675i \(0.951855\pi\)
\(6\) 0 0
\(7\) 6.74912 0.137737 0.0688685 0.997626i \(-0.478061\pi\)
0.0688685 + 0.997626i \(0.478061\pi\)
\(8\) 0 0
\(9\) −37.9943 65.8080i −0.469065 0.812445i
\(10\) 0 0
\(11\) 124.903 1.03225 0.516126 0.856513i \(-0.327374\pi\)
0.516126 + 0.856513i \(0.327374\pi\)
\(12\) 0 0
\(13\) 134.659 77.7455i 0.796800 0.460032i −0.0455512 0.998962i \(-0.514504\pi\)
0.842351 + 0.538930i \(0.181171\pi\)
\(14\) 0 0
\(15\) 60.5631 34.9661i 0.269169 0.155405i
\(16\) 0 0
\(17\) 127.427 220.710i 0.440923 0.763701i −0.556835 0.830623i \(-0.687985\pi\)
0.997758 + 0.0669221i \(0.0213179\pi\)
\(18\) 0 0
\(19\) −161.629 + 322.796i −0.447726 + 0.894171i
\(20\) 0 0
\(21\) −13.0845 7.55435i −0.0296701 0.0171300i
\(22\) 0 0
\(23\) 193.260 + 334.736i 0.365331 + 0.632771i 0.988829 0.149053i \(-0.0476226\pi\)
−0.623499 + 0.781824i \(0.714289\pi\)
\(24\) 0 0
\(25\) −175.438 303.867i −0.280700 0.486187i
\(26\) 0 0
\(27\) 351.438i 0.482082i
\(28\) 0 0
\(29\) −985.611 + 569.043i −1.17195 + 0.676626i −0.954139 0.299364i \(-0.903225\pi\)
−0.217812 + 0.975991i \(0.569892\pi\)
\(30\) 0 0
\(31\) 593.148i 0.617220i 0.951189 + 0.308610i \(0.0998638\pi\)
−0.951189 + 0.308610i \(0.900136\pi\)
\(32\) 0 0
\(33\) −242.149 139.805i −0.222359 0.128379i
\(34\) 0 0
\(35\) −105.418 + 182.589i −0.0860554 + 0.149052i
\(36\) 0 0
\(37\) 360.278i 0.263169i 0.991305 + 0.131585i \(0.0420065\pi\)
−0.991305 + 0.131585i \(0.957994\pi\)
\(38\) 0 0
\(39\) −348.085 −0.228853
\(40\) 0 0
\(41\) −1647.63 951.257i −0.980146 0.565888i −0.0778321 0.996966i \(-0.524800\pi\)
−0.902314 + 0.431079i \(0.858133\pi\)
\(42\) 0 0
\(43\) −794.184 + 1375.57i −0.429521 + 0.743952i −0.996831 0.0795528i \(-0.974651\pi\)
0.567310 + 0.823504i \(0.307984\pi\)
\(44\) 0 0
\(45\) 2373.81 1.17225
\(46\) 0 0
\(47\) 1347.57 + 2334.06i 0.610035 + 1.05661i 0.991234 + 0.132119i \(0.0421781\pi\)
−0.381198 + 0.924493i \(0.624489\pi\)
\(48\) 0 0
\(49\) −2355.45 −0.981028
\(50\) 0 0
\(51\) −494.085 + 285.260i −0.189959 + 0.109673i
\(52\) 0 0
\(53\) −1.97124 + 1.13810i −0.000701759 + 0.000405161i −0.500351 0.865823i \(-0.666796\pi\)
0.499649 + 0.866228i \(0.333462\pi\)
\(54\) 0 0
\(55\) −1950.92 + 3379.09i −0.644931 + 1.11705i
\(56\) 0 0
\(57\) 674.659 444.891i 0.207651 0.136932i
\(58\) 0 0
\(59\) 3490.96 + 2015.51i 1.00286 + 0.579002i 0.909094 0.416592i \(-0.136775\pi\)
0.0937680 + 0.995594i \(0.470109\pi\)
\(60\) 0 0
\(61\) 3481.53 + 6030.18i 0.935643 + 1.62058i 0.773483 + 0.633817i \(0.218513\pi\)
0.162161 + 0.986764i \(0.448154\pi\)
\(62\) 0 0
\(63\) −256.428 444.146i −0.0646077 0.111904i
\(64\) 0 0
\(65\) 4857.38i 1.14968i
\(66\) 0 0
\(67\) 4546.94 2625.18i 1.01291 0.584802i 0.100866 0.994900i \(-0.467839\pi\)
0.912041 + 0.410098i \(0.134505\pi\)
\(68\) 0 0
\(69\) 865.270i 0.181741i
\(70\) 0 0
\(71\) −2217.11 1280.05i −0.439816 0.253928i 0.263704 0.964604i \(-0.415056\pi\)
−0.703519 + 0.710676i \(0.748389\pi\)
\(72\) 0 0
\(73\) 110.303 191.050i 0.0206986 0.0358511i −0.855491 0.517818i \(-0.826744\pi\)
0.876189 + 0.481967i \(0.160078\pi\)
\(74\) 0 0
\(75\) 785.477i 0.139640i
\(76\) 0 0
\(77\) 842.982 0.142179
\(78\) 0 0
\(79\) −2249.78 1298.91i −0.360485 0.208126i 0.308809 0.951124i \(-0.400070\pi\)
−0.669293 + 0.742998i \(0.733403\pi\)
\(80\) 0 0
\(81\) −2684.17 + 4649.12i −0.409110 + 0.708599i
\(82\) 0 0
\(83\) −7437.65 −1.07964 −0.539821 0.841780i \(-0.681508\pi\)
−0.539821 + 0.841780i \(0.681508\pi\)
\(84\) 0 0
\(85\) 3980.68 + 6894.75i 0.550960 + 0.954290i
\(86\) 0 0
\(87\) 2547.74 0.336602
\(88\) 0 0
\(89\) −8825.30 + 5095.29i −1.11417 + 0.643263i −0.939905 0.341436i \(-0.889087\pi\)
−0.174260 + 0.984700i \(0.555753\pi\)
\(90\) 0 0
\(91\) 908.830 524.713i 0.109749 0.0633635i
\(92\) 0 0
\(93\) 663.917 1149.94i 0.0767622 0.132956i
\(94\) 0 0
\(95\) −6208.27 9414.59i −0.687897 1.04317i
\(96\) 0 0
\(97\) 8257.62 + 4767.54i 0.877630 + 0.506700i 0.869876 0.493270i \(-0.164198\pi\)
0.00775402 + 0.999970i \(0.497532\pi\)
\(98\) 0 0
\(99\) −4745.58 8219.59i −0.484194 0.838648i
\(100\) 0 0
\(101\) −2086.44 3613.82i −0.204533 0.354262i 0.745451 0.666561i \(-0.232234\pi\)
−0.949984 + 0.312299i \(0.898901\pi\)
\(102\) 0 0
\(103\) 18149.7i 1.71079i 0.517978 + 0.855394i \(0.326685\pi\)
−0.517978 + 0.855394i \(0.673315\pi\)
\(104\) 0 0
\(105\) 408.747 235.990i 0.0370746 0.0214050i
\(106\) 0 0
\(107\) 7747.72i 0.676715i 0.941018 + 0.338358i \(0.109871\pi\)
−0.941018 + 0.338358i \(0.890129\pi\)
\(108\) 0 0
\(109\) 5236.87 + 3023.51i 0.440777 + 0.254482i 0.703927 0.710272i \(-0.251428\pi\)
−0.263150 + 0.964755i \(0.584762\pi\)
\(110\) 0 0
\(111\) 403.263 698.472i 0.0327297 0.0566896i
\(112\) 0 0
\(113\) 9479.00i 0.742345i 0.928564 + 0.371172i \(0.121044\pi\)
−0.928564 + 0.371172i \(0.878956\pi\)
\(114\) 0 0
\(115\) −12074.5 −0.913005
\(116\) 0 0
\(117\) −10232.6 5907.77i −0.747502 0.431571i
\(118\) 0 0
\(119\) 860.018 1489.59i 0.0607314 0.105190i
\(120\) 0 0
\(121\) 959.647 0.0655452
\(122\) 0 0
\(123\) 2129.50 + 3688.41i 0.140756 + 0.243797i
\(124\) 0 0
\(125\) −8563.37 −0.548056
\(126\) 0 0
\(127\) 17568.6 10143.2i 1.08926 0.628882i 0.155877 0.987776i \(-0.450180\pi\)
0.933378 + 0.358895i \(0.116846\pi\)
\(128\) 0 0
\(129\) 3079.37 1777.87i 0.185047 0.106837i
\(130\) 0 0
\(131\) 11345.0 19650.1i 0.661091 1.14504i −0.319239 0.947674i \(-0.603427\pi\)
0.980329 0.197369i \(-0.0632395\pi\)
\(132\) 0 0
\(133\) −1090.85 + 2178.59i −0.0616685 + 0.123160i
\(134\) 0 0
\(135\) −9507.71 5489.28i −0.521685 0.301195i
\(136\) 0 0
\(137\) 1696.68 + 2938.74i 0.0903980 + 0.156574i 0.907679 0.419666i \(-0.137853\pi\)
−0.817281 + 0.576240i \(0.804519\pi\)
\(138\) 0 0
\(139\) 6450.95 + 11173.4i 0.333883 + 0.578302i 0.983270 0.182156i \(-0.0583077\pi\)
−0.649387 + 0.760458i \(0.724974\pi\)
\(140\) 0 0
\(141\) 6033.38i 0.303475i
\(142\) 0 0
\(143\) 16819.3 9710.61i 0.822498 0.474870i
\(144\) 0 0
\(145\) 35552.7i 1.69097i
\(146\) 0 0
\(147\) 4566.51 + 2636.48i 0.211325 + 0.122008i
\(148\) 0 0
\(149\) 17701.7 30660.2i 0.797336 1.38103i −0.124010 0.992281i \(-0.539575\pi\)
0.921345 0.388745i \(-0.127091\pi\)
\(150\) 0 0
\(151\) 17092.2i 0.749626i 0.927100 + 0.374813i \(0.122293\pi\)
−0.927100 + 0.374813i \(0.877707\pi\)
\(152\) 0 0
\(153\) −19366.0 −0.827287
\(154\) 0 0
\(155\) −16046.9 9264.68i −0.667925 0.385627i
\(156\) 0 0
\(157\) 17919.8 31038.0i 0.726999 1.25920i −0.231147 0.972919i \(-0.574248\pi\)
0.958146 0.286280i \(-0.0924187\pi\)
\(158\) 0 0
\(159\) 5.09553 0.000201556
\(160\) 0 0
\(161\) 1304.33 + 2259.17i 0.0503196 + 0.0871560i
\(162\) 0 0
\(163\) 6266.05 0.235841 0.117920 0.993023i \(-0.462377\pi\)
0.117920 + 0.993023i \(0.462377\pi\)
\(164\) 0 0
\(165\) 7564.48 4367.36i 0.277851 0.160417i
\(166\) 0 0
\(167\) −20148.4 + 11632.7i −0.722448 + 0.417106i −0.815653 0.578541i \(-0.803622\pi\)
0.0932050 + 0.995647i \(0.470289\pi\)
\(168\) 0 0
\(169\) −2191.78 + 3796.27i −0.0767402 + 0.132918i
\(170\) 0 0
\(171\) 27383.5 1627.89i 0.936477 0.0556714i
\(172\) 0 0
\(173\) 7165.62 + 4137.07i 0.239421 + 0.138230i 0.614910 0.788597i \(-0.289192\pi\)
−0.375490 + 0.926827i \(0.622525\pi\)
\(174\) 0 0
\(175\) −1184.05 2050.84i −0.0386629 0.0669660i
\(176\) 0 0
\(177\) −4511.95 7814.93i −0.144018 0.249447i
\(178\) 0 0
\(179\) 43730.2i 1.36482i 0.730969 + 0.682410i \(0.239068\pi\)
−0.730969 + 0.682410i \(0.760932\pi\)
\(180\) 0 0
\(181\) 7670.94 4428.82i 0.234149 0.135186i −0.378336 0.925668i \(-0.623504\pi\)
0.612484 + 0.790483i \(0.290170\pi\)
\(182\) 0 0
\(183\) 15587.6i 0.465455i
\(184\) 0 0
\(185\) −9746.89 5627.37i −0.284789 0.164423i
\(186\) 0 0
\(187\) 15915.9 27567.2i 0.455144 0.788332i
\(188\) 0 0
\(189\) 2371.89i 0.0664005i
\(190\) 0 0
\(191\) −7936.99 −0.217565 −0.108783 0.994066i \(-0.534695\pi\)
−0.108783 + 0.994066i \(0.534695\pi\)
\(192\) 0 0
\(193\) −43557.3 25147.8i −1.16936 0.675128i −0.215827 0.976432i \(-0.569245\pi\)
−0.953528 + 0.301304i \(0.902578\pi\)
\(194\) 0 0
\(195\) 5436.92 9417.01i 0.142983 0.247653i
\(196\) 0 0
\(197\) −19563.5 −0.504096 −0.252048 0.967715i \(-0.581104\pi\)
−0.252048 + 0.967715i \(0.581104\pi\)
\(198\) 0 0
\(199\) 12415.8 + 21504.9i 0.313523 + 0.543038i 0.979122 0.203271i \(-0.0651573\pi\)
−0.665599 + 0.746309i \(0.731824\pi\)
\(200\) 0 0
\(201\) −11753.5 −0.290922
\(202\) 0 0
\(203\) −6652.00 + 3840.54i −0.161421 + 0.0931965i
\(204\) 0 0
\(205\) 51470.2 29716.3i 1.22475 0.707111i
\(206\) 0 0
\(207\) 14685.5 25436.1i 0.342728 0.593622i
\(208\) 0 0
\(209\) −20187.9 + 40318.0i −0.462167 + 0.923010i
\(210\) 0 0
\(211\) −18973.0 10954.1i −0.426158 0.246043i 0.271550 0.962424i \(-0.412464\pi\)
−0.697709 + 0.716382i \(0.745797\pi\)
\(212\) 0 0
\(213\) 2865.54 + 4963.27i 0.0631608 + 0.109398i
\(214\) 0 0
\(215\) −24809.5 42971.3i −0.536712 0.929612i
\(216\) 0 0
\(217\) 4003.23i 0.0850141i
\(218\) 0 0
\(219\) −427.689 + 246.926i −0.00891743 + 0.00514848i
\(220\) 0 0
\(221\) 39627.4i 0.811355i
\(222\) 0 0
\(223\) 3246.29 + 1874.25i 0.0652796 + 0.0376892i 0.532285 0.846565i \(-0.321334\pi\)
−0.467005 + 0.884255i \(0.654667\pi\)
\(224\) 0 0
\(225\) −13331.3 + 23090.4i −0.263334 + 0.456107i
\(226\) 0 0
\(227\) 69169.5i 1.34234i −0.741303 0.671170i \(-0.765792\pi\)
0.741303 0.671170i \(-0.234208\pi\)
\(228\) 0 0
\(229\) −4633.97 −0.0883655 −0.0441827 0.999023i \(-0.514068\pi\)
−0.0441827 + 0.999023i \(0.514068\pi\)
\(230\) 0 0
\(231\) −1634.29 943.558i −0.0306270 0.0176825i
\(232\) 0 0
\(233\) −8090.94 + 14013.9i −0.149035 + 0.258136i −0.930871 0.365348i \(-0.880950\pi\)
0.781836 + 0.623484i \(0.214283\pi\)
\(234\) 0 0
\(235\) −84193.4 −1.52455
\(236\) 0 0
\(237\) 2907.77 + 5036.41i 0.0517683 + 0.0896653i
\(238\) 0 0
\(239\) −9138.88 −0.159992 −0.0799958 0.996795i \(-0.525491\pi\)
−0.0799958 + 0.996795i \(0.525491\pi\)
\(240\) 0 0
\(241\) 33766.2 19494.9i 0.581363 0.335650i −0.180312 0.983610i \(-0.557711\pi\)
0.761675 + 0.647959i \(0.224377\pi\)
\(242\) 0 0
\(243\) 35060.3 20242.1i 0.593749 0.342801i
\(244\) 0 0
\(245\) 36791.0 63723.8i 0.612927 1.06162i
\(246\) 0 0
\(247\) 3331.05 + 56033.3i 0.0545993 + 0.918443i
\(248\) 0 0
\(249\) 14419.4 + 8325.03i 0.232567 + 0.134273i
\(250\) 0 0
\(251\) 21264.0 + 36830.3i 0.337519 + 0.584599i 0.983965 0.178360i \(-0.0570791\pi\)
−0.646447 + 0.762959i \(0.723746\pi\)
\(252\) 0 0
\(253\) 24138.7 + 41809.4i 0.377113 + 0.653180i
\(254\) 0 0
\(255\) 17822.5i 0.274086i
\(256\) 0 0
\(257\) −75878.9 + 43808.7i −1.14883 + 0.663276i −0.948601 0.316474i \(-0.897501\pi\)
−0.200226 + 0.979750i \(0.564168\pi\)
\(258\) 0 0
\(259\) 2431.56i 0.0362481i
\(260\) 0 0
\(261\) 74895.2 + 43240.7i 1.09944 + 0.634764i
\(262\) 0 0
\(263\) 30776.8 53306.9i 0.444950 0.770677i −0.553098 0.833116i \(-0.686555\pi\)
0.998049 + 0.0624394i \(0.0198880\pi\)
\(264\) 0 0
\(265\) 71.1060i 0.00101255i
\(266\) 0 0
\(267\) 22812.8 0.320005
\(268\) 0 0
\(269\) 67084.3 + 38731.2i 0.927078 + 0.535249i 0.885886 0.463902i \(-0.153551\pi\)
0.0411921 + 0.999151i \(0.486884\pi\)
\(270\) 0 0
\(271\) 56026.7 97041.1i 0.762880 1.32135i −0.178480 0.983944i \(-0.557118\pi\)
0.941360 0.337404i \(-0.109549\pi\)
\(272\) 0 0
\(273\) −2349.27 −0.0315215
\(274\) 0 0
\(275\) −21912.6 37953.8i −0.289754 0.501868i
\(276\) 0 0
\(277\) 128472. 1.67436 0.837179 0.546929i \(-0.184203\pi\)
0.837179 + 0.546929i \(0.184203\pi\)
\(278\) 0 0
\(279\) 39033.9 22536.3i 0.501457 0.289516i
\(280\) 0 0
\(281\) −111706. + 64493.4i −1.41470 + 0.816776i −0.995826 0.0912700i \(-0.970907\pi\)
−0.418871 + 0.908046i \(0.637574\pi\)
\(282\) 0 0
\(283\) −72402.9 + 125405.i −0.904030 + 1.56583i −0.0818162 + 0.996647i \(0.526072\pi\)
−0.822214 + 0.569179i \(0.807261\pi\)
\(284\) 0 0
\(285\) 1498.14 + 25201.0i 0.0184444 + 0.310262i
\(286\) 0 0
\(287\) −11120.0 6420.15i −0.135003 0.0779437i
\(288\) 0 0
\(289\) 9285.36 + 16082.7i 0.111174 + 0.192559i
\(290\) 0 0
\(291\) −10672.7 18485.7i −0.126034 0.218298i
\(292\) 0 0
\(293\) 142973.i 1.66541i 0.553720 + 0.832703i \(0.313208\pi\)
−0.553720 + 0.832703i \(0.686792\pi\)
\(294\) 0 0
\(295\) −109054. + 62962.5i −1.25314 + 0.723499i
\(296\) 0 0
\(297\) 43895.5i 0.497630i
\(298\) 0 0
\(299\) 52048.4 + 30050.2i 0.582191 + 0.336128i
\(300\) 0 0
\(301\) −5360.04 + 9283.86i −0.0591609 + 0.102470i
\(302\) 0 0
\(303\) 9341.50i 0.101749i
\(304\) 0 0
\(305\) −217519. −2.33829
\(306\) 0 0
\(307\) −93661.1 54075.2i −0.993762 0.573749i −0.0873652 0.996176i \(-0.527845\pi\)
−0.906397 + 0.422428i \(0.861178\pi\)
\(308\) 0 0
\(309\) 20315.2 35186.9i 0.212767 0.368523i
\(310\) 0 0
\(311\) 115895. 1.19824 0.599121 0.800659i \(-0.295517\pi\)
0.599121 + 0.800659i \(0.295517\pi\)
\(312\) 0 0
\(313\) −58215.1 100831.i −0.594219 1.02922i −0.993657 0.112457i \(-0.964128\pi\)
0.399437 0.916760i \(-0.369206\pi\)
\(314\) 0 0
\(315\) 16021.1 0.161462
\(316\) 0 0
\(317\) 24219.2 13983.0i 0.241014 0.139149i −0.374629 0.927175i \(-0.622230\pi\)
0.615643 + 0.788025i \(0.288896\pi\)
\(318\) 0 0
\(319\) −123105. + 71074.9i −1.20975 + 0.698449i
\(320\) 0 0
\(321\) 8672.09 15020.5i 0.0841616 0.145772i
\(322\) 0 0
\(323\) 50648.2 + 76805.9i 0.485466 + 0.736189i
\(324\) 0 0
\(325\) −47248.6 27279.0i −0.447324 0.258263i
\(326\) 0 0
\(327\) −6768.48 11723.3i −0.0632988 0.109637i
\(328\) 0 0
\(329\) 9094.89 + 15752.8i 0.0840245 + 0.145535i
\(330\) 0 0
\(331\) 101916.i 0.930224i 0.885252 + 0.465112i \(0.153986\pi\)
−0.885252 + 0.465112i \(0.846014\pi\)
\(332\) 0 0
\(333\) 23709.2 13688.5i 0.213810 0.123443i
\(334\) 0 0
\(335\) 164016.i 1.46149i
\(336\) 0 0
\(337\) −7232.06 4175.43i −0.0636799 0.0367656i 0.467822 0.883823i \(-0.345039\pi\)
−0.531502 + 0.847057i \(0.678372\pi\)
\(338\) 0 0
\(339\) 10609.9 18376.9i 0.0923237 0.159909i
\(340\) 0 0
\(341\) 74085.7i 0.637127i
\(342\) 0 0
\(343\) −32101.8 −0.272861
\(344\) 0 0
\(345\) 23408.8 + 13515.1i 0.196672 + 0.113548i
\(346\) 0 0
\(347\) −45577.0 + 78941.6i −0.378518 + 0.655612i −0.990847 0.134991i \(-0.956899\pi\)
0.612329 + 0.790603i \(0.290233\pi\)
\(348\) 0 0
\(349\) −29079.9 −0.238750 −0.119375 0.992849i \(-0.538089\pi\)
−0.119375 + 0.992849i \(0.538089\pi\)
\(350\) 0 0
\(351\) 27322.7 + 47324.3i 0.221773 + 0.384123i
\(352\) 0 0
\(353\) 134292. 1.07771 0.538854 0.842399i \(-0.318857\pi\)
0.538854 + 0.842399i \(0.318857\pi\)
\(354\) 0 0
\(355\) 69260.4 39987.5i 0.549576 0.317298i
\(356\) 0 0
\(357\) −3334.63 + 1925.25i −0.0261645 + 0.0151061i
\(358\) 0 0
\(359\) −69994.3 + 121234.i −0.543093 + 0.940664i 0.455632 + 0.890168i \(0.349413\pi\)
−0.998724 + 0.0504957i \(0.983920\pi\)
\(360\) 0 0
\(361\) −78073.0 104346.i −0.599082 0.800688i
\(362\) 0 0
\(363\) −1860.47 1074.14i −0.0141192 0.00815170i
\(364\) 0 0
\(365\) 3445.76 + 5968.22i 0.0258642 + 0.0447981i
\(366\) 0 0
\(367\) 84019.8 + 145527.i 0.623806 + 1.08046i 0.988770 + 0.149442i \(0.0477478\pi\)
−0.364965 + 0.931021i \(0.618919\pi\)
\(368\) 0 0
\(369\) 144569.i 1.06175i
\(370\) 0 0
\(371\) −13.3041 + 7.68115i −9.66582e−5 + 5.58057e-5i
\(372\) 0 0
\(373\) 88343.1i 0.634973i −0.948263 0.317486i \(-0.897161\pi\)
0.948263 0.317486i \(-0.102839\pi\)
\(374\) 0 0
\(375\) 16601.8 + 9585.07i 0.118057 + 0.0681605i
\(376\) 0 0
\(377\) −88481.0 + 153254.i −0.622540 + 1.07827i
\(378\) 0 0
\(379\) 2419.75i 0.0168458i 0.999965 + 0.00842292i \(0.00268113\pi\)
−0.999965 + 0.00842292i \(0.997319\pi\)
\(380\) 0 0
\(381\) −45413.7 −0.312850
\(382\) 0 0
\(383\) −104816. 60515.8i −0.714549 0.412545i 0.0981944 0.995167i \(-0.468693\pi\)
−0.812743 + 0.582622i \(0.802027\pi\)
\(384\) 0 0
\(385\) −13167.0 + 22805.8i −0.0888309 + 0.153860i
\(386\) 0 0
\(387\) 120698. 0.805893
\(388\) 0 0
\(389\) −101360. 175561.i −0.669834 1.16019i −0.977950 0.208838i \(-0.933032\pi\)
0.308116 0.951349i \(-0.400301\pi\)
\(390\) 0 0
\(391\) 98505.9 0.644330
\(392\) 0 0
\(393\) −43989.0 + 25397.1i −0.284813 + 0.164437i
\(394\) 0 0
\(395\) 70281.0 40576.8i 0.450447 0.260066i
\(396\) 0 0
\(397\) 57516.2 99621.1i 0.364930 0.632077i −0.623835 0.781556i \(-0.714426\pi\)
0.988765 + 0.149479i \(0.0477596\pi\)
\(398\) 0 0
\(399\) 4553.35 3002.62i 0.0286013 0.0188606i
\(400\) 0 0
\(401\) 129215. + 74602.5i 0.803573 + 0.463943i 0.844719 0.535210i \(-0.179768\pi\)
−0.0411460 + 0.999153i \(0.513101\pi\)
\(402\) 0 0
\(403\) 46114.6 + 79872.8i 0.283941 + 0.491801i
\(404\) 0 0
\(405\) −83850.8 145234.i −0.511208 0.885437i
\(406\) 0 0
\(407\) 44999.7i 0.271657i
\(408\) 0 0
\(409\) 146230. 84426.2i 0.874160 0.504697i 0.00543191 0.999985i \(-0.498271\pi\)
0.868729 + 0.495288i \(0.164938\pi\)
\(410\) 0 0
\(411\) 7596.44i 0.0449704i
\(412\) 0 0
\(413\) 23560.9 + 13602.9i 0.138131 + 0.0797501i
\(414\) 0 0
\(415\) 116172. 201216.i 0.674538 1.16833i
\(416\) 0 0
\(417\) 28882.4i 0.166097i
\(418\) 0 0
\(419\) 61370.4 0.349567 0.174784 0.984607i \(-0.444077\pi\)
0.174784 + 0.984607i \(0.444077\pi\)
\(420\) 0 0
\(421\) 203447. + 117460.i 1.14786 + 0.662716i 0.948364 0.317183i \(-0.102737\pi\)
0.199493 + 0.979899i \(0.436070\pi\)
\(422\) 0 0
\(423\) 102400. 177362.i 0.572293 0.991240i
\(424\) 0 0
\(425\) −89421.9 −0.495069
\(426\) 0 0
\(427\) 23497.2 + 40698.4i 0.128873 + 0.223214i
\(428\) 0 0
\(429\) −43476.7 −0.236234
\(430\) 0 0
\(431\) −199518. + 115192.i −1.07406 + 0.620106i −0.929287 0.369359i \(-0.879577\pi\)
−0.144769 + 0.989466i \(0.546244\pi\)
\(432\) 0 0
\(433\) 144263. 83290.1i 0.769447 0.444240i −0.0632306 0.997999i \(-0.520140\pi\)
0.832677 + 0.553759i \(0.186807\pi\)
\(434\) 0 0
\(435\) −39794.4 + 68926.0i −0.210302 + 0.364254i
\(436\) 0 0
\(437\) −139288. + 8280.33i −0.729373 + 0.0433595i
\(438\) 0 0
\(439\) −226379. 130700.i −1.17465 0.678183i −0.219877 0.975528i \(-0.570566\pi\)
−0.954770 + 0.297345i \(0.903899\pi\)
\(440\) 0 0
\(441\) 89493.6 + 155008.i 0.460166 + 0.797032i
\(442\) 0 0
\(443\) −104145. 180384.i −0.530676 0.919157i −0.999359 0.0357911i \(-0.988605\pi\)
0.468684 0.883366i \(-0.344728\pi\)
\(444\) 0 0
\(445\) 318344.i 1.60759i
\(446\) 0 0
\(447\) −68636.4 + 39627.3i −0.343510 + 0.198326i
\(448\) 0 0
\(449\) 110756.i 0.549381i −0.961533 0.274691i \(-0.911425\pi\)
0.961533 0.274691i \(-0.0885754\pi\)
\(450\) 0 0
\(451\) −205793. 118814.i −1.01176 0.584139i
\(452\) 0 0
\(453\) 19131.5 33136.7i 0.0932293 0.161478i
\(454\) 0 0
\(455\) 32783.1i 0.158353i
\(456\) 0 0
\(457\) −128375. −0.614680 −0.307340 0.951600i \(-0.599439\pi\)
−0.307340 + 0.951600i \(0.599439\pi\)
\(458\) 0 0
\(459\) 77565.6 + 44782.5i 0.368166 + 0.212561i
\(460\) 0 0
\(461\) −1280.81 + 2218.42i −0.00602674 + 0.0104386i −0.869023 0.494772i \(-0.835252\pi\)
0.862996 + 0.505210i \(0.168585\pi\)
\(462\) 0 0
\(463\) 318274. 1.48470 0.742350 0.670012i \(-0.233711\pi\)
0.742350 + 0.670012i \(0.233711\pi\)
\(464\) 0 0
\(465\) 20740.1 + 35922.9i 0.0959190 + 0.166137i
\(466\) 0 0
\(467\) −262467. −1.20349 −0.601743 0.798690i \(-0.705527\pi\)
−0.601743 + 0.798690i \(0.705527\pi\)
\(468\) 0 0
\(469\) 30687.8 17717.6i 0.139515 0.0805489i
\(470\) 0 0
\(471\) −69482.2 + 40115.6i −0.313207 + 0.180830i
\(472\) 0 0
\(473\) −99195.6 + 171812.i −0.443374 + 0.767946i
\(474\) 0 0
\(475\) 126443. 7516.73i 0.560412 0.0333151i
\(476\) 0 0
\(477\) 149.792 + 86.4824i 0.000658342 + 0.000380094i
\(478\) 0 0
\(479\) −102699. 177879.i −0.447604 0.775273i 0.550626 0.834752i \(-0.314389\pi\)
−0.998230 + 0.0594795i \(0.981056\pi\)
\(480\) 0 0
\(481\) 28010.0 + 48514.8i 0.121066 + 0.209693i
\(482\) 0 0
\(483\) 5839.81i 0.0250325i
\(484\) 0 0
\(485\) −257960. + 148933.i −1.09665 + 0.633152i
\(486\) 0 0
\(487\) 24740.3i 0.104315i −0.998639 0.0521576i \(-0.983390\pi\)
0.998639 0.0521576i \(-0.0166098\pi\)
\(488\) 0 0
\(489\) −12148.0 7013.65i −0.0508028 0.0293310i
\(490\) 0 0
\(491\) 162868. 282096.i 0.675574 1.17013i −0.300727 0.953710i \(-0.597229\pi\)
0.976301 0.216418i \(-0.0694374\pi\)
\(492\) 0 0
\(493\) 290045.i 1.19336i
\(494\) 0 0
\(495\) 296495. 1.21006
\(496\) 0 0
\(497\) −14963.5 8639.21i −0.0605789 0.0349753i
\(498\) 0 0
\(499\) −42795.3 + 74123.6i −0.171868 + 0.297684i −0.939073 0.343718i \(-0.888314\pi\)
0.767205 + 0.641402i \(0.221647\pi\)
\(500\) 0 0
\(501\) 52082.2 0.207498
\(502\) 0 0
\(503\) −102109. 176858.i −0.403578 0.699017i 0.590577 0.806981i \(-0.298900\pi\)
−0.994155 + 0.107964i \(0.965567\pi\)
\(504\) 0 0
\(505\) 130357. 0.511153
\(506\) 0 0
\(507\) 8498.40 4906.56i 0.0330614 0.0190880i
\(508\) 0 0
\(509\) 309500. 178690.i 1.19461 0.689707i 0.235259 0.971933i \(-0.424406\pi\)
0.959348 + 0.282226i \(0.0910728\pi\)
\(510\) 0 0
\(511\) 744.448 1289.42i 0.00285097 0.00493802i
\(512\) 0 0
\(513\) −113443. 56802.6i −0.431063 0.215841i
\(514\) 0 0
\(515\) −491019. 283490.i −1.85133 1.06887i
\(516\) 0 0
\(517\) 168315. + 291530.i 0.629711 + 1.09069i
\(518\) 0 0
\(519\) −9261.33 16041.1i −0.0343826 0.0595524i
\(520\) 0 0
\(521\) 279875.i 1.03107i −0.856868 0.515536i \(-0.827593\pi\)
0.856868 0.515536i \(-0.172407\pi\)
\(522\) 0 0
\(523\) 472250. 272654.i 1.72651 0.996800i 0.823288 0.567625i \(-0.192137\pi\)
0.903221 0.429176i \(-0.141196\pi\)
\(524\) 0 0
\(525\) 5301.27i 0.0192336i
\(526\) 0 0
\(527\) 130914. + 75583.0i 0.471371 + 0.272146i
\(528\) 0 0
\(529\) 65221.7 112967.i 0.233067 0.403684i
\(530\) 0 0
\(531\) 306311.i 1.08636i
\(532\) 0 0
\(533\) −295824. −1.04131
\(534\) 0 0
\(535\) −209605. 121015.i −0.732308 0.422798i
\(536\) 0 0
\(537\) 48947.7 84779.8i 0.169740 0.293998i
\(538\) 0 0
\(539\) −294202. −1.01267
\(540\) 0 0
\(541\) 111400. + 192950.i 0.380618 + 0.659249i 0.991151 0.132742i \(-0.0423780\pi\)
−0.610533 + 0.791991i \(0.709045\pi\)
\(542\) 0 0
\(543\) −19828.9 −0.0672510
\(544\) 0 0
\(545\) −163594. + 94451.3i −0.550777 + 0.317991i
\(546\) 0 0
\(547\) 28990.1 16737.5i 0.0968893 0.0559390i −0.450772 0.892639i \(-0.648851\pi\)
0.547662 + 0.836700i \(0.315518\pi\)
\(548\) 0 0
\(549\) 264556. 458225.i 0.877756 1.52032i
\(550\) 0 0
\(551\) −24381.0 410125.i −0.0803059 1.35087i
\(552\) 0 0
\(553\) −15184.1 8766.52i −0.0496521 0.0286667i
\(554\) 0 0
\(555\) 12597.5 + 21819.6i 0.0408978 + 0.0708370i
\(556\) 0 0
\(557\) 175167. + 303398.i 0.564601 + 0.977918i 0.997087 + 0.0762774i \(0.0243034\pi\)
−0.432485 + 0.901641i \(0.642363\pi\)
\(558\) 0 0
\(559\) 246977.i 0.790374i
\(560\) 0 0
\(561\) −61712.4 + 35629.7i −0.196086 + 0.113210i
\(562\) 0 0
\(563\) 510897.i 1.61182i 0.592038 + 0.805910i \(0.298324\pi\)
−0.592038 + 0.805910i \(0.701676\pi\)
\(564\) 0 0
\(565\) −256443. 148057.i −0.803329 0.463802i
\(566\) 0 0
\(567\) −18115.8 + 31377.4i −0.0563496 + 0.0976004i
\(568\) 0 0
\(569\) 298435.i 0.921775i −0.887459 0.460887i \(-0.847531\pi\)
0.887459 0.460887i \(-0.152469\pi\)
\(570\) 0 0
\(571\) 411772. 1.26294 0.631472 0.775399i \(-0.282451\pi\)
0.631472 + 0.775399i \(0.282451\pi\)
\(572\) 0 0
\(573\) 15387.4 + 8883.95i 0.0468659 + 0.0270581i
\(574\) 0 0
\(575\) 67810.2 117451.i 0.205097 0.355238i
\(576\) 0 0
\(577\) 454898. 1.36635 0.683176 0.730254i \(-0.260598\pi\)
0.683176 + 0.730254i \(0.260598\pi\)
\(578\) 0 0
\(579\) 56296.4 + 97508.3i 0.167928 + 0.290860i
\(580\) 0 0
\(581\) −50197.6 −0.148707
\(582\) 0 0
\(583\) −246.213 + 142.151i −0.000724393 + 0.000418228i
\(584\) 0 0
\(585\) 319655. 184553.i 0.934049 0.539273i
\(586\) 0 0
\(587\) −18802.2 + 32566.4i −0.0545673 + 0.0945133i −0.892019 0.451998i \(-0.850711\pi\)
0.837451 + 0.546512i \(0.184045\pi\)
\(588\) 0 0
\(589\) −191466. 95870.1i −0.551900 0.276346i
\(590\) 0 0
\(591\) 37927.7 + 21897.6i 0.108588 + 0.0626933i
\(592\) 0 0
\(593\) 123237. + 213453.i 0.350454 + 0.607004i 0.986329 0.164788i \(-0.0526939\pi\)
−0.635875 + 0.771792i \(0.719361\pi\)
\(594\) 0 0
\(595\) 26866.1 + 46533.5i 0.0758876 + 0.131441i
\(596\) 0 0
\(597\) 55588.6i 0.155969i
\(598\) 0 0
\(599\) −70920.3 + 40945.9i −0.197659 + 0.114119i −0.595563 0.803308i \(-0.703071\pi\)
0.397904 + 0.917427i \(0.369738\pi\)
\(600\) 0 0
\(601\) 552301.i 1.52907i −0.644583 0.764534i \(-0.722969\pi\)
0.644583 0.764534i \(-0.277031\pi\)
\(602\) 0 0
\(603\) −345516. 199483.i −0.950239 0.548621i
\(604\) 0 0
\(605\) −14989.2 + 25962.1i −0.0409513 + 0.0709298i
\(606\) 0 0
\(607\) 123618.i 0.335509i 0.985829 + 0.167755i \(0.0536516\pi\)
−0.985829 + 0.167755i \(0.946348\pi\)
\(608\) 0 0
\(609\) 17195.0 0.0463626
\(610\) 0 0
\(611\) 362925. + 209535.i 0.972152 + 0.561272i
\(612\) 0 0
\(613\) −333223. + 577160.i −0.886777 + 1.53594i −0.0431137 + 0.999070i \(0.513728\pi\)
−0.843663 + 0.536873i \(0.819606\pi\)
\(614\) 0 0
\(615\) −133047. −0.351767
\(616\) 0 0
\(617\) −29382.7 50892.4i −0.0771831 0.133685i 0.824850 0.565351i \(-0.191259\pi\)
−0.902034 + 0.431666i \(0.857926\pi\)
\(618\) 0 0
\(619\) 466387. 1.21721 0.608605 0.793473i \(-0.291729\pi\)
0.608605 + 0.793473i \(0.291729\pi\)
\(620\) 0 0
\(621\) −117639. + 67918.8i −0.305047 + 0.176119i
\(622\) 0 0
\(623\) −59563.0 + 34388.7i −0.153462 + 0.0886012i
\(624\) 0 0
\(625\) 243404. 421589.i 0.623115 1.07927i
\(626\) 0 0
\(627\) 84266.6 55568.0i 0.214348 0.141348i
\(628\) 0 0
\(629\) 79516.9 + 45909.1i 0.200982 + 0.116037i
\(630\) 0 0
\(631\) −332333. 575618.i −0.834671 1.44569i −0.894298 0.447471i \(-0.852325\pi\)
0.0596278 0.998221i \(-0.481009\pi\)
\(632\) 0 0
\(633\) 24522.0 + 42473.3i 0.0611995 + 0.106001i
\(634\) 0 0
\(635\) 633729.i 1.57165i
\(636\) 0 0
\(637\) −317183. + 183126.i −0.781683 + 0.451305i
\(638\) 0 0
\(639\) 194538.i 0.476435i
\(640\) 0 0
\(641\) −643227. 371367.i −1.56548 0.903831i −0.996685 0.0813527i \(-0.974076\pi\)
−0.568796 0.822478i \(-0.692591\pi\)
\(642\) 0 0
\(643\) 93473.9 161902.i 0.226083 0.391588i −0.730561 0.682848i \(-0.760741\pi\)
0.956644 + 0.291260i \(0.0940745\pi\)
\(644\) 0 0
\(645\) 111078.i 0.266999i
\(646\) 0 0
\(647\) 476329. 1.13789 0.568943 0.822377i \(-0.307353\pi\)
0.568943 + 0.822377i \(0.307353\pi\)
\(648\) 0 0
\(649\) 436030. + 251742.i 1.03521 + 0.597677i
\(650\) 0 0
\(651\) 4480.85 7761.06i 0.0105730 0.0183130i
\(652\) 0 0
\(653\) −740783. −1.73726 −0.868630 0.495462i \(-0.834999\pi\)
−0.868630 + 0.495462i \(0.834999\pi\)
\(654\) 0 0
\(655\) 354406. + 613849.i 0.826073 + 1.43080i
\(656\) 0 0
\(657\) −16763.5 −0.0388360
\(658\) 0 0
\(659\) −686805. + 396527.i −1.58148 + 0.913066i −0.586833 + 0.809708i \(0.699625\pi\)
−0.994644 + 0.103358i \(0.967041\pi\)
\(660\) 0 0
\(661\) 279193. 161192.i 0.639001 0.368927i −0.145229 0.989398i \(-0.546392\pi\)
0.784230 + 0.620471i \(0.213058\pi\)
\(662\) 0 0
\(663\) −44355.3 + 76825.7i −0.100906 + 0.174775i
\(664\) 0 0
\(665\) −41900.3 63540.1i −0.0947489 0.143683i
\(666\) 0 0
\(667\) −380958. 219946.i −0.856299 0.494385i
\(668\) 0 0
\(669\) −4195.72 7267.21i −0.00937464 0.0162374i
\(670\) 0 0
\(671\) 434852. + 753185.i 0.965820 + 1.67285i
\(672\) 0 0
\(673\) 66308.4i 0.146399i −0.997317 0.0731996i \(-0.976679\pi\)
0.997317 0.0731996i \(-0.0233210\pi\)
\(674\) 0 0
\(675\) 106790. 61655.4i 0.234382 0.135321i
\(676\) 0 0
\(677\) 614850.i 1.34150i −0.741682 0.670752i \(-0.765972\pi\)
0.741682 0.670752i \(-0.234028\pi\)
\(678\) 0 0
\(679\) 55731.7 + 32176.7i 0.120882 + 0.0697914i
\(680\) 0 0
\(681\) −77422.0 + 134099.i −0.166944 + 0.289155i
\(682\) 0 0
\(683\) 386557.i 0.828652i −0.910129 0.414326i \(-0.864017\pi\)
0.910129 0.414326i \(-0.135983\pi\)
\(684\) 0 0
\(685\) −106005. −0.225915
\(686\) 0 0
\(687\) 8983.89 + 5186.85i 0.0190349 + 0.0109898i
\(688\) 0 0
\(689\) −176.964 + 306.510i −0.000372774 + 0.000645664i
\(690\) 0 0
\(691\) −267742. −0.560739 −0.280369 0.959892i \(-0.590457\pi\)
−0.280369 + 0.959892i \(0.590457\pi\)
\(692\) 0 0
\(693\) −32028.5 55475.0i −0.0666914 0.115513i
\(694\) 0 0
\(695\) −403042. −0.834413
\(696\) 0 0
\(697\) −419903. + 242431.i −0.864338 + 0.499026i
\(698\) 0 0
\(699\) 31371.8 18112.5i 0.0642075 0.0370702i
\(700\) 0 0
\(701\) −187934. + 325511.i −0.382445 + 0.662415i −0.991411 0.130782i \(-0.958251\pi\)
0.608966 + 0.793196i \(0.291585\pi\)
\(702\) 0 0
\(703\) −116296. 58231.5i −0.235318 0.117828i
\(704\) 0 0
\(705\) 163226. + 94238.4i 0.328406 + 0.189605i
\(706\) 0 0
\(707\) −14081.6 24390.1i −0.0281718 0.0487950i
\(708\) 0 0
\(709\) 37666.3 + 65239.9i 0.0749308 + 0.129784i 0.901056 0.433702i \(-0.142793\pi\)
−0.826125 + 0.563486i \(0.809460\pi\)
\(710\) 0 0
\(711\) 197405.i 0.390499i
\(712\) 0 0
\(713\) −198548. + 114632.i −0.390559 + 0.225489i
\(714\) 0 0
\(715\) 606700.i 1.18676i
\(716\) 0 0
\(717\) 17717.6 + 10229.2i 0.0344640 + 0.0198978i
\(718\) 0 0
\(719\) −347929. + 602631.i −0.673028 + 1.16572i 0.304013 + 0.952668i \(0.401673\pi\)
−0.977041 + 0.213051i \(0.931660\pi\)
\(720\) 0 0
\(721\) 122495.i 0.235639i
\(722\) 0 0
\(723\) −87283.3 −0.166976
\(724\) 0 0
\(725\) 345827. + 199663.i 0.657934 + 0.379859i
\(726\) 0 0
\(727\) 341985. 592336.i 0.647051 1.12072i −0.336773 0.941586i \(-0.609336\pi\)
0.983824 0.179139i \(-0.0573311\pi\)
\(728\) 0 0
\(729\) 344207. 0.647686
\(730\) 0 0
\(731\) 202400. + 350568.i 0.378771 + 0.656051i
\(732\) 0 0
\(733\) −408184. −0.759709 −0.379855 0.925046i \(-0.624026\pi\)
−0.379855 + 0.925046i \(0.624026\pi\)
\(734\) 0 0
\(735\) −142653. + 82360.9i −0.264063 + 0.152457i
\(736\) 0 0
\(737\) 567924. 327891.i 1.04558 0.603663i
\(738\) 0 0
\(739\) −522561. + 905102.i −0.956859 + 1.65733i −0.226802 + 0.973941i \(0.572827\pi\)
−0.730056 + 0.683387i \(0.760506\pi\)
\(740\) 0 0
\(741\) 56260.7 112360.i 0.102463 0.204633i
\(742\) 0 0
\(743\) −680267. 392752.i −1.23226 0.711444i −0.264758 0.964315i \(-0.585292\pi\)
−0.967500 + 0.252870i \(0.918625\pi\)
\(744\) 0 0
\(745\) 552982. + 957793.i 0.996319 + 1.72568i
\(746\) 0 0
\(747\) 282588. + 489457.i 0.506422 + 0.877149i
\(748\) 0 0
\(749\) 52290.2i 0.0932088i
\(750\) 0 0
\(751\) 411795. 237750.i 0.730132 0.421542i −0.0883387 0.996090i \(-0.528156\pi\)
0.818470 + 0.574549i \(0.194822\pi\)
\(752\) 0 0
\(753\) 95204.0i 0.167906i
\(754\) 0 0
\(755\) −462409. 266972.i −0.811209 0.468352i
\(756\) 0 0
\(757\) −496513. + 859985.i −0.866440 + 1.50072i −0.000829615 1.00000i \(0.500264\pi\)
−0.865610 + 0.500718i \(0.833069\pi\)
\(758\) 0 0
\(759\) 108074.i 0.187603i
\(760\) 0 0
\(761\) −81817.3 −0.141278 −0.0706392 0.997502i \(-0.522504\pi\)
−0.0706392 + 0.997502i \(0.522504\pi\)
\(762\) 0 0
\(763\) 35344.2 + 20406.0i 0.0607113 + 0.0350517i
\(764\) 0 0
\(765\) 302487. 523922.i 0.516872 0.895249i
\(766\) 0 0
\(767\) 626787. 1.06544
\(768\) 0 0
\(769\) −98144.6 169991.i −0.165964 0.287458i 0.771033 0.636795i \(-0.219740\pi\)
−0.936997 + 0.349337i \(0.886407\pi\)
\(770\) 0 0
\(771\) 196142. 0.329960
\(772\) 0 0
\(773\) −821675. + 474394.i −1.37512 + 0.793927i −0.991567 0.129592i \(-0.958633\pi\)
−0.383554 + 0.923518i \(0.625300\pi\)
\(774\) 0 0
\(775\) 180238. 104061.i 0.300085 0.173254i
\(776\) 0 0
\(777\) 2721.67 4714.07i 0.00450810 0.00780826i
\(778\) 0 0
\(779\) 573366. 378095.i 0.944838 0.623055i
\(780\) 0 0
\(781\) −276923. 159881.i −0.454001 0.262118i
\(782\) 0 0
\(783\) −199983. 346381.i −0.326189 0.564976i
\(784\) 0 0
\(785\) 559797. + 969596.i 0.908429 + 1.57344i
\(786\) 0 0
\(787\) 994511.i 1.60568i −0.596192 0.802842i \(-0.703320\pi\)
0.596192 0.802842i \(-0.296680\pi\)
\(788\) 0 0
\(789\) −119334. + 68897.5i −0.191695 + 0.110675i
\(790\) 0 0
\(791\) 63974.9i 0.102248i
\(792\) 0 0
\(793\) 937639. + 541346.i 1.49104 + 0.860853i
\(794\) 0 0
\(795\) −79.5896 + 137.853i −0.000125928 + 0.000218114i
\(796\) 0 0
\(797\) 272658.i 0.429242i −0.976697 0.214621i \(-0.931148\pi\)
0.976697 0.214621i \(-0.0688516\pi\)
\(798\) 0 0
\(799\) 686865. 1.07591
\(800\) 0 0
\(801\) 670622. + 387184.i 1.04523 + 0.603465i
\(802\) 0 0
\(803\) 13777.1 23862.7i 0.0213662 0.0370074i
\(804\) 0 0
\(805\) −81492.2 −0.125755
\(806\) 0 0
\(807\) −86704.3 150176.i −0.133135 0.230597i
\(808\) 0 0
\(809\) 588970. 0.899903 0.449952 0.893053i \(-0.351441\pi\)
0.449952 + 0.893053i \(0.351441\pi\)
\(810\) 0 0
\(811\) 699080. 403614.i 1.06288 0.613655i 0.136654 0.990619i \(-0.456365\pi\)
0.926228 + 0.376964i \(0.123032\pi\)
\(812\) 0 0
\(813\) −217238. + 125422.i −0.328666 + 0.189755i
\(814\) 0 0
\(815\) −97872.7 + 169520.i −0.147349 + 0.255215i
\(816\) 0 0
\(817\) −315663. 478691.i −0.472912 0.717151i
\(818\) 0 0
\(819\) −69060.7 39872.2i −0.102959 0.0594433i
\(820\) 0 0
\(821\) 225213. + 390081.i 0.334124 + 0.578719i 0.983316 0.181905i \(-0.0582263\pi\)
−0.649192 + 0.760624i \(0.724893\pi\)
\(822\) 0 0
\(823\) 212992. + 368912.i 0.314458 + 0.544657i 0.979322 0.202307i \(-0.0648439\pi\)
−0.664864 + 0.746964i \(0.731511\pi\)
\(824\) 0 0
\(825\) 98108.0i 0.144144i
\(826\) 0 0
\(827\) −1.00410e6 + 579720.i −1.46814 + 0.847632i −0.999363 0.0356845i \(-0.988639\pi\)
−0.468778 + 0.883316i \(0.655306\pi\)
\(828\) 0 0
\(829\) 119777.i 0.174287i 0.996196 + 0.0871436i \(0.0277739\pi\)
−0.996196 + 0.0871436i \(0.972226\pi\)
\(830\) 0 0
\(831\) −249068. 143800.i −0.360675 0.208236i
\(832\) 0 0
\(833\) −300147. + 519870.i −0.432558 + 0.749212i
\(834\) 0 0
\(835\) 726785.i 1.04240i
\(836\) 0 0
\(837\) −208455. −0.297550
\(838\) 0 0
\(839\) −1.17823e6 680253.i −1.67381 0.966377i −0.965472 0.260508i \(-0.916110\pi\)
−0.708342 0.705869i \(-0.750557\pi\)
\(840\) 0 0
\(841\) 293979. 509186.i 0.415646 0.719920i
\(842\) 0 0
\(843\) 288752. 0.406322
\(844\) 0 0
\(845\) −68469.0 118592.i −0.0958915 0.166089i
\(846\) 0 0
\(847\) 6476.77 0.00902800
\(848\) 0 0
\(849\) 280735. 162082.i 0.389476 0.224864i
\(850\) 0 0
\(851\) −120598. + 69627.4i −0.166526 + 0.0961437i
\(852\) 0 0
\(853\) −406987. + 704922.i −0.559349 + 0.968820i 0.438202 + 0.898876i \(0.355615\pi\)
−0.997551 + 0.0699438i \(0.977718\pi\)
\(854\) 0 0
\(855\) −383677. + 766255.i −0.524848 + 1.04819i
\(856\) 0 0
\(857\) 201512. + 116343.i 0.274372 + 0.158409i 0.630873 0.775886i \(-0.282697\pi\)
−0.356501 + 0.934295i \(0.616030\pi\)
\(858\) 0 0
\(859\) 522786. + 905492.i 0.708497 + 1.22715i 0.965415 + 0.260719i \(0.0839597\pi\)
−0.256918 + 0.966433i \(0.582707\pi\)
\(860\) 0 0
\(861\) 14372.3 + 24893.5i 0.0193874 + 0.0335799i
\(862\) 0 0
\(863\) 1.15469e6i 1.55040i 0.631715 + 0.775200i \(0.282351\pi\)
−0.631715 + 0.775200i \(0.717649\pi\)
\(864\) 0 0
\(865\) −223847. + 129238.i −0.299171 + 0.172726i
\(866\) 0 0
\(867\) 41572.8i 0.0553058i
\(868\) 0 0
\(869\) −281004. 162238.i −0.372111 0.214838i
\(870\) 0 0
\(871\) 408191. 707008.i 0.538056 0.931940i
\(872\) 0 0
\(873\) 724558.i 0.950702i
\(874\) 0 0
\(875\) −57795.2 −0.0754876
\(876\) 0 0
\(877\) 108941. + 62897.4i 0.141643 + 0.0817774i 0.569147 0.822236i \(-0.307274\pi\)
−0.427504 + 0.904013i \(0.640607\pi\)
\(878\) 0 0
\(879\) 160032. 277183.i 0.207123 0.358747i
\(880\) 0 0
\(881\) −170516. −0.219692 −0.109846 0.993949i \(-0.535036\pi\)
−0.109846 + 0.993949i \(0.535036\pi\)
\(882\) 0 0
\(883\) −418950. 725643.i −0.537330 0.930683i −0.999047 0.0436555i \(-0.986100\pi\)
0.461717 0.887028i \(-0.347234\pi\)
\(884\) 0 0
\(885\) 281898. 0.359919
\(886\) 0 0
\(887\) −1.16240e6 + 671114.i −1.47744 + 0.853000i −0.999675 0.0254881i \(-0.991886\pi\)
−0.477764 + 0.878488i \(0.658553\pi\)
\(888\) 0 0
\(889\) 118573. 68457.9i 0.150031 0.0866203i
\(890\) 0 0
\(891\) −335260. + 580687.i −0.422305 + 0.731453i
\(892\) 0 0
\(893\) −971230. + 57737.3i −1.21792 + 0.0724025i
\(894\) 0 0
\(895\) −1.18307e6 683044.i −1.47694 0.852713i
\(896\) 0 0
\(897\) −67270.9 116517.i −0.0836069 0.144811i
\(898\) 0 0
\(899\) −337527. 584613.i −0.417627 0.723352i
\(900\) 0 0
\(901\) 580.096i 0.000714579i
\(902\) 0 0
\(903\) 20783.0 11999.1i 0.0254878 0.0147154i
\(904\) 0 0
\(905\) 276704.i 0.337846i
\(906\) 0 0
\(907\) 72775.9 + 42017.2i 0.0884653 + 0.0510755i 0.543580 0.839357i \(-0.317068\pi\)
−0.455115 + 0.890433i \(0.650402\pi\)
\(908\) 0 0
\(909\) −158546. + 274609.i −0.191879 + 0.332344i
\(910\) 0 0
\(911\) 429356.i 0.517345i −0.965965 0.258673i \(-0.916715\pi\)
0.965965 0.258673i \(-0.0832851\pi\)
\(912\) 0 0
\(913\) −928981. −1.11446
\(914\) 0 0
\(915\) 421704. + 243471.i 0.503693 + 0.290807i
\(916\) 0 0
\(917\) 76568.6 132621.i 0.0910567 0.157715i
\(918\) 0 0
\(919\) 826582. 0.978711 0.489356 0.872084i \(-0.337232\pi\)
0.489356 + 0.872084i \(0.337232\pi\)
\(920\) 0 0
\(921\) 121054. + 209671.i 0.142712 + 0.247184i
\(922\) 0 0
\(923\) −398072. −0.467260
\(924\) 0 0
\(925\) 109477. 63206.5i 0.127950 0.0738717i
\(926\) 0 0
\(927\) 1.19440e6 689587.i 1.38992 0.802471i
\(928\) 0 0
\(929\) 755950. 1.30934e6i 0.875915 1.51713i 0.0201297 0.999797i \(-0.493592\pi\)
0.855785 0.517332i \(-0.173075\pi\)
\(930\) 0 0
\(931\) 380709. 760329.i 0.439232 0.877207i
\(932\) 0 0
\(933\) −224686. 129722.i −0.258115 0.149023i
\(934\) 0 0
\(935\) 497198. + 861172.i 0.568730 + 0.985069i
\(936\) 0 0
\(937\) −278250. 481942.i −0.316924 0.548929i 0.662920 0.748690i \(-0.269317\pi\)
−0.979845 + 0.199761i \(0.935983\pi\)
\(938\) 0 0
\(939\) 260643.i 0.295607i
\(940\) 0 0
\(941\) 127927. 73858.6i 0.144471 0.0834107i −0.426022 0.904713i \(-0.640085\pi\)
0.570493 + 0.821302i \(0.306752\pi\)
\(942\) 0 0
\(943\) 735360.i 0.826945i
\(944\) 0 0
\(945\) −64168.7 37047.8i −0.0718554 0.0414857i
\(946\) 0 0
\(947\) 805825. 1.39573e6i 0.898547 1.55633i 0.0691937 0.997603i \(-0.477957\pi\)
0.829353 0.558725i \(-0.188709\pi\)
\(948\) 0 0
\(949\) 34302.2i 0.0380882i
\(950\) 0 0
\(951\) −62605.2 −0.0692228
\(952\) 0 0
\(953\) −468828. 270678.i −0.516211 0.298035i 0.219172 0.975686i \(-0.429664\pi\)
−0.735383 + 0.677651i \(0.762998\pi\)
\(954\) 0 0
\(955\) 123972. 214726.i 0.135930 0.235438i
\(956\) 0 0
\(957\) 318219. 0.347458
\(958\) 0 0
\(959\) 11451.1 + 19833.9i 0.0124512 + 0.0215660i
\(960\) 0 0
\(961\) 571696. 0.619040
\(962\) 0 0
\(963\) 509862. 294369.i 0.549794 0.317424i
\(964\) 0 0
\(965\) 1.36069e6 785593.i 1.46118 0.843613i
\(966\) 0 0
\(967\) −300748. + 520910.i −0.321624 + 0.557070i −0.980823 0.194899i \(-0.937562\pi\)
0.659199 + 0.751969i \(0.270896\pi\)
\(968\) 0 0
\(969\) −12222.1 205595.i −0.0130166 0.218960i
\(970\) 0 0
\(971\) 330430. + 190774.i 0.350463 + 0.202340i 0.664889 0.746942i \(-0.268479\pi\)
−0.314426 + 0.949282i \(0.601812\pi\)
\(972\) 0 0
\(973\) 43538.2 + 75410.4i 0.0459880 + 0.0796536i
\(974\) 0 0
\(975\) 61067.3 + 105772.i 0.0642391 + 0.111265i
\(976\) 0 0
\(977\) 445014.i 0.466213i −0.972451 0.233106i \(-0.925111\pi\)
0.972451 0.233106i \(-0.0748890\pi\)
\(978\) 0 0
\(979\) −1.10230e6 + 636415.i −1.15010 + 0.664010i
\(980\) 0 0
\(981\) 459504.i 0.477476i
\(982\) 0 0
\(983\) 470900. + 271874.i 0.487329 + 0.281359i 0.723466 0.690360i \(-0.242548\pi\)
−0.236137 + 0.971720i \(0.575881\pi\)
\(984\) 0 0
\(985\) 305572. 529266.i 0.314949 0.545508i
\(986\) 0 0
\(987\) 40720.0i 0.0417997i
\(988\) 0 0
\(989\) −613935. −0.627668
\(990\) 0 0
\(991\) 527372. + 304478.i 0.536994 + 0.310034i 0.743860 0.668336i \(-0.232993\pi\)
−0.206866 + 0.978369i \(0.566326\pi\)
\(992\) 0 0
\(993\) 114076. 197585.i 0.115690 0.200381i
\(994\) 0 0
\(995\) −775717. −0.783532
\(996\) 0 0
\(997\) −563173. 975445.i −0.566568 0.981324i −0.996902 0.0786545i \(-0.974938\pi\)
0.430334 0.902670i \(-0.358396\pi\)
\(998\) 0 0
\(999\) −126615. −0.126869
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 304.5.r.d.65.10 40
4.3 odd 2 152.5.n.a.65.11 40
19.12 odd 6 inner 304.5.r.d.145.10 40
76.31 even 6 152.5.n.a.145.11 yes 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.5.n.a.65.11 40 4.3 odd 2
152.5.n.a.145.11 yes 40 76.31 even 6
304.5.r.d.65.10 40 1.1 even 1 trivial
304.5.r.d.145.10 40 19.12 odd 6 inner