Properties

Label 441.4.e.o.226.1
Level $441$
Weight $4$
Character 441.226
Analytic conductor $26.020$
Analytic rank $0$
Dimension $2$
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] = [441,4,Mod(226,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.226");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.e (of order \(3\), 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: \(26.0198423125\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 147)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 226.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 441.226
Dual form 441.4.e.o.361.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.00000 - 3.46410i) q^{2} +(-4.00000 - 6.92820i) q^{4} +(9.00000 - 15.5885i) q^{5} +O(q^{10})\) \(q+(2.00000 - 3.46410i) q^{2} +(-4.00000 - 6.92820i) q^{4} +(9.00000 - 15.5885i) q^{5} +(-36.0000 - 62.3538i) q^{10} +(-25.0000 - 43.3013i) q^{11} -36.0000 q^{13} +(32.0000 - 55.4256i) q^{16} +(63.0000 + 109.119i) q^{17} +(36.0000 - 62.3538i) q^{19} -144.000 q^{20} -200.000 q^{22} +(7.00000 - 12.1244i) q^{23} +(-99.5000 - 172.339i) q^{25} +(-72.0000 + 124.708i) q^{26} -158.000 q^{29} +(18.0000 + 31.1769i) q^{31} +(-128.000 - 221.703i) q^{32} +504.000 q^{34} +(81.0000 - 140.296i) q^{37} +(-144.000 - 249.415i) q^{38} +270.000 q^{41} -324.000 q^{43} +(-200.000 + 346.410i) q^{44} +(-28.0000 - 48.4974i) q^{46} +(-36.0000 + 62.3538i) q^{47} -796.000 q^{50} +(144.000 + 249.415i) q^{52} +(-11.0000 - 19.0526i) q^{53} -900.000 q^{55} +(-316.000 + 547.328i) q^{58} +(234.000 + 405.300i) q^{59} +(-396.000 + 685.892i) q^{61} +144.000 q^{62} -512.000 q^{64} +(-324.000 + 561.184i) q^{65} +(-116.000 - 200.918i) q^{67} +(504.000 - 872.954i) q^{68} +734.000 q^{71} +(-90.0000 - 155.885i) q^{73} +(-324.000 - 561.184i) q^{74} -576.000 q^{76} +(-118.000 + 204.382i) q^{79} +(-576.000 - 997.661i) q^{80} +(540.000 - 935.307i) q^{82} -36.0000 q^{83} +2268.00 q^{85} +(-648.000 + 1122.37i) q^{86} +(117.000 - 202.650i) q^{89} -112.000 q^{92} +(144.000 + 249.415i) q^{94} +(-648.000 - 1122.37i) q^{95} +468.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} - 8 q^{4} + 18 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} - 8 q^{4} + 18 q^{5} - 72 q^{10} - 50 q^{11} - 72 q^{13} + 64 q^{16} + 126 q^{17} + 72 q^{19} - 288 q^{20} - 400 q^{22} + 14 q^{23} - 199 q^{25} - 144 q^{26} - 316 q^{29} + 36 q^{31} - 256 q^{32} + 1008 q^{34} + 162 q^{37} - 288 q^{38} + 540 q^{41} - 648 q^{43} - 400 q^{44} - 56 q^{46} - 72 q^{47} - 1592 q^{50} + 288 q^{52} - 22 q^{53} - 1800 q^{55} - 632 q^{58} + 468 q^{59} - 792 q^{61} + 288 q^{62} - 1024 q^{64} - 648 q^{65} - 232 q^{67} + 1008 q^{68} + 1468 q^{71} - 180 q^{73} - 648 q^{74} - 1152 q^{76} - 236 q^{79} - 1152 q^{80} + 1080 q^{82} - 72 q^{83} + 4536 q^{85} - 1296 q^{86} + 234 q^{89} - 224 q^{92} + 288 q^{94} - 1296 q^{95} + 936 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 3.46410i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(3\) 0 0
\(4\) −4.00000 6.92820i −0.500000 0.866025i
\(5\) 9.00000 15.5885i 0.804984 1.39427i −0.111317 0.993785i \(-0.535507\pi\)
0.916302 0.400489i \(-0.131160\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) −36.0000 62.3538i −1.13842 1.97180i
\(11\) −25.0000 43.3013i −0.685253 1.18689i −0.973357 0.229294i \(-0.926358\pi\)
0.288104 0.957599i \(-0.406975\pi\)
\(12\) 0 0
\(13\) −36.0000 −0.768046 −0.384023 0.923323i \(-0.625462\pi\)
−0.384023 + 0.923323i \(0.625462\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 32.0000 55.4256i 0.500000 0.866025i
\(17\) 63.0000 + 109.119i 0.898808 + 1.55678i 0.829019 + 0.559220i \(0.188899\pi\)
0.0697893 + 0.997562i \(0.477767\pi\)
\(18\) 0 0
\(19\) 36.0000 62.3538i 0.434682 0.752892i −0.562587 0.826738i \(-0.690194\pi\)
0.997270 + 0.0738459i \(0.0235273\pi\)
\(20\) −144.000 −1.60997
\(21\) 0 0
\(22\) −200.000 −1.93819
\(23\) 7.00000 12.1244i 0.0634609 0.109918i −0.832549 0.553951i \(-0.813120\pi\)
0.896010 + 0.444033i \(0.146453\pi\)
\(24\) 0 0
\(25\) −99.5000 172.339i −0.796000 1.37871i
\(26\) −72.0000 + 124.708i −0.543091 + 0.940661i
\(27\) 0 0
\(28\) 0 0
\(29\) −158.000 −1.01172 −0.505860 0.862616i \(-0.668825\pi\)
−0.505860 + 0.862616i \(0.668825\pi\)
\(30\) 0 0
\(31\) 18.0000 + 31.1769i 0.104287 + 0.180630i 0.913447 0.406958i \(-0.133411\pi\)
−0.809160 + 0.587589i \(0.800077\pi\)
\(32\) −128.000 221.703i −0.707107 1.22474i
\(33\) 0 0
\(34\) 504.000 2.54221
\(35\) 0 0
\(36\) 0 0
\(37\) 81.0000 140.296i 0.359900 0.623366i −0.628043 0.778178i \(-0.716144\pi\)
0.987944 + 0.154812i \(0.0494773\pi\)
\(38\) −144.000 249.415i −0.614734 1.06475i
\(39\) 0 0
\(40\) 0 0
\(41\) 270.000 1.02846 0.514231 0.857652i \(-0.328078\pi\)
0.514231 + 0.857652i \(0.328078\pi\)
\(42\) 0 0
\(43\) −324.000 −1.14906 −0.574529 0.818484i \(-0.694815\pi\)
−0.574529 + 0.818484i \(0.694815\pi\)
\(44\) −200.000 + 346.410i −0.685253 + 1.18689i
\(45\) 0 0
\(46\) −28.0000 48.4974i −0.0897473 0.155447i
\(47\) −36.0000 + 62.3538i −0.111726 + 0.193516i −0.916466 0.400112i \(-0.868971\pi\)
0.804740 + 0.593627i \(0.202305\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −796.000 −2.25143
\(51\) 0 0
\(52\) 144.000 + 249.415i 0.384023 + 0.665148i
\(53\) −11.0000 19.0526i −0.0285088 0.0493787i 0.851419 0.524486i \(-0.175743\pi\)
−0.879928 + 0.475107i \(0.842409\pi\)
\(54\) 0 0
\(55\) −900.000 −2.20647
\(56\) 0 0
\(57\) 0 0
\(58\) −316.000 + 547.328i −0.715394 + 1.23910i
\(59\) 234.000 + 405.300i 0.516342 + 0.894331i 0.999820 + 0.0189746i \(0.00604016\pi\)
−0.483478 + 0.875357i \(0.660627\pi\)
\(60\) 0 0
\(61\) −396.000 + 685.892i −0.831190 + 1.43966i 0.0659047 + 0.997826i \(0.479007\pi\)
−0.897095 + 0.441838i \(0.854327\pi\)
\(62\) 144.000 0.294968
\(63\) 0 0
\(64\) −512.000 −1.00000
\(65\) −324.000 + 561.184i −0.618265 + 1.07087i
\(66\) 0 0
\(67\) −116.000 200.918i −0.211517 0.366359i 0.740672 0.671866i \(-0.234507\pi\)
−0.952190 + 0.305508i \(0.901174\pi\)
\(68\) 504.000 872.954i 0.898808 1.55678i
\(69\) 0 0
\(70\) 0 0
\(71\) 734.000 1.22690 0.613449 0.789734i \(-0.289782\pi\)
0.613449 + 0.789734i \(0.289782\pi\)
\(72\) 0 0
\(73\) −90.0000 155.885i −0.144297 0.249930i 0.784813 0.619732i \(-0.212759\pi\)
−0.929111 + 0.369802i \(0.879425\pi\)
\(74\) −324.000 561.184i −0.508976 0.881573i
\(75\) 0 0
\(76\) −576.000 −0.869365
\(77\) 0 0
\(78\) 0 0
\(79\) −118.000 + 204.382i −0.168051 + 0.291073i −0.937735 0.347353i \(-0.887081\pi\)
0.769683 + 0.638426i \(0.220414\pi\)
\(80\) −576.000 997.661i −0.804984 1.39427i
\(81\) 0 0
\(82\) 540.000 935.307i 0.727232 1.25960i
\(83\) −36.0000 −0.0476086 −0.0238043 0.999717i \(-0.507578\pi\)
−0.0238043 + 0.999717i \(0.507578\pi\)
\(84\) 0 0
\(85\) 2268.00 2.89411
\(86\) −648.000 + 1122.37i −0.812507 + 1.40730i
\(87\) 0 0
\(88\) 0 0
\(89\) 117.000 202.650i 0.139348 0.241358i −0.787902 0.615801i \(-0.788833\pi\)
0.927250 + 0.374443i \(0.122166\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −112.000 −0.126922
\(93\) 0 0
\(94\) 144.000 + 249.415i 0.158005 + 0.273673i
\(95\) −648.000 1122.37i −0.699825 1.21213i
\(96\) 0 0
\(97\) 468.000 0.489878 0.244939 0.969538i \(-0.421232\pi\)
0.244939 + 0.969538i \(0.421232\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −796.000 + 1378.71i −0.796000 + 1.37871i
\(101\) −333.000 576.773i −0.328067 0.568228i 0.654061 0.756441i \(-0.273064\pi\)
−0.982128 + 0.188213i \(0.939730\pi\)
\(102\) 0 0
\(103\) 126.000 218.238i 0.120535 0.208773i −0.799444 0.600741i \(-0.794872\pi\)
0.919979 + 0.391968i \(0.128206\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −88.0000 −0.0806351
\(107\) 335.000 580.237i 0.302670 0.524240i −0.674070 0.738668i \(-0.735455\pi\)
0.976740 + 0.214428i \(0.0687887\pi\)
\(108\) 0 0
\(109\) −81.0000 140.296i −0.0711779 0.123284i 0.828240 0.560374i \(-0.189342\pi\)
−0.899418 + 0.437090i \(0.856009\pi\)
\(110\) −1800.00 + 3117.69i −1.56021 + 2.70237i
\(111\) 0 0
\(112\) 0 0
\(113\) 1390.00 1.15717 0.578585 0.815622i \(-0.303605\pi\)
0.578585 + 0.815622i \(0.303605\pi\)
\(114\) 0 0
\(115\) −126.000 218.238i −0.102170 0.176964i
\(116\) 632.000 + 1094.66i 0.505860 + 0.876175i
\(117\) 0 0
\(118\) 1872.00 1.46044
\(119\) 0 0
\(120\) 0 0
\(121\) −584.500 + 1012.38i −0.439144 + 0.760619i
\(122\) 1584.00 + 2743.57i 1.17548 + 2.03599i
\(123\) 0 0
\(124\) 144.000 249.415i 0.104287 0.180630i
\(125\) −1332.00 −0.953102
\(126\) 0 0
\(127\) 916.000 0.640015 0.320007 0.947415i \(-0.396315\pi\)
0.320007 + 0.947415i \(0.396315\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 1296.00 + 2244.74i 0.874359 + 1.51443i
\(131\) 1134.00 1964.15i 0.756321 1.30999i −0.188394 0.982094i \(-0.560328\pi\)
0.944715 0.327893i \(-0.106338\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −928.000 −0.598261
\(135\) 0 0
\(136\) 0 0
\(137\) 403.000 + 698.016i 0.251318 + 0.435296i 0.963889 0.266304i \(-0.0858026\pi\)
−0.712571 + 0.701600i \(0.752469\pi\)
\(138\) 0 0
\(139\) 2628.00 1.60363 0.801813 0.597575i \(-0.203869\pi\)
0.801813 + 0.597575i \(0.203869\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1468.00 2542.65i 0.867548 1.50264i
\(143\) 900.000 + 1558.85i 0.526306 + 0.911589i
\(144\) 0 0
\(145\) −1422.00 + 2462.98i −0.814418 + 1.41061i
\(146\) −720.000 −0.408134
\(147\) 0 0
\(148\) −1296.00 −0.719801
\(149\) −1195.00 + 2069.80i −0.657035 + 1.13802i 0.324344 + 0.945939i \(0.394856\pi\)
−0.981379 + 0.192079i \(0.938477\pi\)
\(150\) 0 0
\(151\) −1620.00 2805.92i −0.873071 1.51220i −0.858803 0.512305i \(-0.828792\pi\)
−0.0142676 0.999898i \(-0.504542\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 648.000 0.335798
\(156\) 0 0
\(157\) −1512.00 2618.86i −0.768603 1.33126i −0.938320 0.345767i \(-0.887619\pi\)
0.169717 0.985493i \(-0.445715\pi\)
\(158\) 472.000 + 817.528i 0.237660 + 0.411639i
\(159\) 0 0
\(160\) −4608.00 −2.27684
\(161\) 0 0
\(162\) 0 0
\(163\) 892.000 1544.99i 0.428631 0.742410i −0.568121 0.822945i \(-0.692329\pi\)
0.996752 + 0.0805346i \(0.0256628\pi\)
\(164\) −1080.00 1870.61i −0.514231 0.890674i
\(165\) 0 0
\(166\) −72.0000 + 124.708i −0.0336644 + 0.0583084i
\(167\) 3024.00 1.40122 0.700611 0.713543i \(-0.252911\pi\)
0.700611 + 0.713543i \(0.252911\pi\)
\(168\) 0 0
\(169\) −901.000 −0.410105
\(170\) 4536.00 7856.58i 2.04644 3.54454i
\(171\) 0 0
\(172\) 1296.00 + 2244.74i 0.574529 + 0.995114i
\(173\) 783.000 1356.20i 0.344106 0.596010i −0.641085 0.767470i \(-0.721515\pi\)
0.985191 + 0.171461i \(0.0548486\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3200.00 −1.37051
\(177\) 0 0
\(178\) −468.000 810.600i −0.197068 0.341332i
\(179\) 1901.00 + 3292.63i 0.793784 + 1.37487i 0.923608 + 0.383338i \(0.125226\pi\)
−0.129824 + 0.991537i \(0.541441\pi\)
\(180\) 0 0
\(181\) −468.000 −0.192189 −0.0960944 0.995372i \(-0.530635\pi\)
−0.0960944 + 0.995372i \(0.530635\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1458.00 2525.33i −0.579429 1.00360i
\(186\) 0 0
\(187\) 3150.00 5455.96i 1.23182 2.13358i
\(188\) 576.000 0.223453
\(189\) 0 0
\(190\) −5184.00 −1.97940
\(191\) 241.000 417.424i 0.0912992 0.158135i −0.816759 0.576979i \(-0.804231\pi\)
0.908058 + 0.418844i \(0.137565\pi\)
\(192\) 0 0
\(193\) 405.000 + 701.481i 0.151049 + 0.261625i 0.931614 0.363450i \(-0.118401\pi\)
−0.780564 + 0.625076i \(0.785068\pi\)
\(194\) 936.000 1621.20i 0.346396 0.599976i
\(195\) 0 0
\(196\) 0 0
\(197\) 2462.00 0.890407 0.445204 0.895429i \(-0.353131\pi\)
0.445204 + 0.895429i \(0.353131\pi\)
\(198\) 0 0
\(199\) 2268.00 + 3928.29i 0.807911 + 1.39934i 0.914308 + 0.405019i \(0.132735\pi\)
−0.106397 + 0.994324i \(0.533932\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −2664.00 −0.927913
\(203\) 0 0
\(204\) 0 0
\(205\) 2430.00 4208.88i 0.827895 1.43396i
\(206\) −504.000 872.954i −0.170463 0.295250i
\(207\) 0 0
\(208\) −1152.00 + 1995.32i −0.384023 + 0.665148i
\(209\) −3600.00 −1.19147
\(210\) 0 0
\(211\) 2916.00 0.951402 0.475701 0.879607i \(-0.342195\pi\)
0.475701 + 0.879607i \(0.342195\pi\)
\(212\) −88.0000 + 152.420i −0.0285088 + 0.0493787i
\(213\) 0 0
\(214\) −1340.00 2320.95i −0.428040 0.741387i
\(215\) −2916.00 + 5050.66i −0.924975 + 1.60210i
\(216\) 0 0
\(217\) 0 0
\(218\) −648.000 −0.201322
\(219\) 0 0
\(220\) 3600.00 + 6235.38i 1.10324 + 1.91086i
\(221\) −2268.00 3928.29i −0.690327 1.19568i
\(222\) 0 0
\(223\) −1080.00 −0.324315 −0.162157 0.986765i \(-0.551845\pi\)
−0.162157 + 0.986765i \(0.551845\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 2780.00 4815.10i 0.818243 1.41724i
\(227\) −666.000 1153.55i −0.194731 0.337284i 0.752081 0.659070i \(-0.229050\pi\)
−0.946812 + 0.321786i \(0.895717\pi\)
\(228\) 0 0
\(229\) −810.000 + 1402.96i −0.233739 + 0.404848i −0.958906 0.283725i \(-0.908430\pi\)
0.725166 + 0.688574i \(0.241763\pi\)
\(230\) −1008.00 −0.288981
\(231\) 0 0
\(232\) 0 0
\(233\) 3359.00 5817.96i 0.944444 1.63582i 0.187583 0.982249i \(-0.439935\pi\)
0.756861 0.653576i \(-0.226732\pi\)
\(234\) 0 0
\(235\) 648.000 + 1122.37i 0.179876 + 0.311554i
\(236\) 1872.00 3242.40i 0.516342 0.894331i
\(237\) 0 0
\(238\) 0 0
\(239\) 3578.00 0.968375 0.484187 0.874964i \(-0.339115\pi\)
0.484187 + 0.874964i \(0.339115\pi\)
\(240\) 0 0
\(241\) 378.000 + 654.715i 0.101034 + 0.174995i 0.912111 0.409944i \(-0.134452\pi\)
−0.811077 + 0.584939i \(0.801118\pi\)
\(242\) 2338.00 + 4049.53i 0.621043 + 1.07568i
\(243\) 0 0
\(244\) 6336.00 1.66238
\(245\) 0 0
\(246\) 0 0
\(247\) −1296.00 + 2244.74i −0.333856 + 0.578256i
\(248\) 0 0
\(249\) 0 0
\(250\) −2664.00 + 4614.18i −0.673945 + 1.16731i
\(251\) 6516.00 1.63859 0.819295 0.573372i \(-0.194365\pi\)
0.819295 + 0.573372i \(0.194365\pi\)
\(252\) 0 0
\(253\) −700.000 −0.173947
\(254\) 1832.00 3173.12i 0.452559 0.783855i
\(255\) 0 0
\(256\) −2048.00 3547.24i −0.500000 0.866025i
\(257\) −3015.00 + 5222.13i −0.731792 + 1.26750i 0.224325 + 0.974514i \(0.427982\pi\)
−0.956117 + 0.292986i \(0.905351\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 5184.00 1.23653
\(261\) 0 0
\(262\) −4536.00 7856.58i −1.06960 1.85260i
\(263\) 295.000 + 510.955i 0.0691653 + 0.119798i 0.898534 0.438904i \(-0.144633\pi\)
−0.829369 + 0.558702i \(0.811300\pi\)
\(264\) 0 0
\(265\) −396.000 −0.0917966
\(266\) 0 0
\(267\) 0 0
\(268\) −928.000 + 1607.34i −0.211517 + 0.366359i
\(269\) −495.000 857.365i −0.112196 0.194329i 0.804459 0.594008i \(-0.202455\pi\)
−0.916655 + 0.399679i \(0.869122\pi\)
\(270\) 0 0
\(271\) 1710.00 2961.81i 0.383303 0.663900i −0.608229 0.793761i \(-0.708120\pi\)
0.991532 + 0.129861i \(0.0414532\pi\)
\(272\) 8064.00 1.79762
\(273\) 0 0
\(274\) 3224.00 0.710836
\(275\) −4975.00 + 8616.95i −1.09092 + 1.88953i
\(276\) 0 0
\(277\) 1367.00 + 2367.71i 0.296516 + 0.513582i 0.975337 0.220723i \(-0.0708417\pi\)
−0.678820 + 0.734305i \(0.737508\pi\)
\(278\) 5256.00 9103.66i 1.13394 1.96403i
\(279\) 0 0
\(280\) 0 0
\(281\) −598.000 −0.126953 −0.0634763 0.997983i \(-0.520219\pi\)
−0.0634763 + 0.997983i \(0.520219\pi\)
\(282\) 0 0
\(283\) −1800.00 3117.69i −0.378088 0.654868i 0.612696 0.790319i \(-0.290085\pi\)
−0.990784 + 0.135451i \(0.956752\pi\)
\(284\) −2936.00 5085.30i −0.613449 1.06253i
\(285\) 0 0
\(286\) 7200.00 1.48862
\(287\) 0 0
\(288\) 0 0
\(289\) −5481.50 + 9494.24i −1.11571 + 1.93247i
\(290\) 5688.00 + 9851.90i 1.15176 + 1.99491i
\(291\) 0 0
\(292\) −720.000 + 1247.08i −0.144297 + 0.249930i
\(293\) −7902.00 −1.57556 −0.787781 0.615955i \(-0.788770\pi\)
−0.787781 + 0.615955i \(0.788770\pi\)
\(294\) 0 0
\(295\) 8424.00 1.66259
\(296\) 0 0
\(297\) 0 0
\(298\) 4780.00 + 8279.20i 0.929188 + 1.60940i
\(299\) −252.000 + 436.477i −0.0487409 + 0.0844218i
\(300\) 0 0
\(301\) 0 0
\(302\) −12960.0 −2.46942
\(303\) 0 0
\(304\) −2304.00 3990.65i −0.434682 0.752892i
\(305\) 7128.00 + 12346.1i 1.33819 + 2.31781i
\(306\) 0 0
\(307\) −10224.0 −1.90070 −0.950349 0.311185i \(-0.899274\pi\)
−0.950349 + 0.311185i \(0.899274\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 1296.00 2244.74i 0.237445 0.411266i
\(311\) 1944.00 + 3367.11i 0.354451 + 0.613926i 0.987024 0.160574i \(-0.0513346\pi\)
−0.632573 + 0.774501i \(0.718001\pi\)
\(312\) 0 0
\(313\) −2556.00 + 4427.12i −0.461577 + 0.799475i −0.999040 0.0438124i \(-0.986050\pi\)
0.537463 + 0.843288i \(0.319383\pi\)
\(314\) −12096.0 −2.17394
\(315\) 0 0
\(316\) 1888.00 0.336102
\(317\) −5051.00 + 8748.59i −0.894929 + 1.55006i −0.0610361 + 0.998136i \(0.519440\pi\)
−0.833893 + 0.551927i \(0.813893\pi\)
\(318\) 0 0
\(319\) 3950.00 + 6841.60i 0.693284 + 1.20080i
\(320\) −4608.00 + 7981.29i −0.804984 + 1.39427i
\(321\) 0 0
\(322\) 0 0
\(323\) 9072.00 1.56279
\(324\) 0 0
\(325\) 3582.00 + 6204.21i 0.611365 + 1.05892i
\(326\) −3568.00 6179.96i −0.606176 1.04993i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −2754.00 + 4770.07i −0.457322 + 0.792105i −0.998818 0.0485983i \(-0.984525\pi\)
0.541497 + 0.840703i \(0.317858\pi\)
\(332\) 144.000 + 249.415i 0.0238043 + 0.0412303i
\(333\) 0 0
\(334\) 6048.00 10475.4i 0.990814 1.71614i
\(335\) −4176.00 −0.681072
\(336\) 0 0
\(337\) −9234.00 −1.49261 −0.746303 0.665607i \(-0.768173\pi\)
−0.746303 + 0.665607i \(0.768173\pi\)
\(338\) −1802.00 + 3121.16i −0.289988 + 0.502274i
\(339\) 0 0
\(340\) −9072.00 15713.2i −1.44705 2.50637i
\(341\) 900.000 1558.85i 0.142926 0.247555i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −3132.00 5424.78i −0.486640 0.842885i
\(347\) −3247.00 5623.97i −0.502329 0.870059i −0.999996 0.00269115i \(-0.999143\pi\)
0.497668 0.867368i \(-0.334190\pi\)
\(348\) 0 0
\(349\) 10080.0 1.54605 0.773023 0.634378i \(-0.218744\pi\)
0.773023 + 0.634378i \(0.218744\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −6400.00 + 11085.1i −0.969094 + 1.67852i
\(353\) −369.000 639.127i −0.0556371 0.0963662i 0.836865 0.547409i \(-0.184386\pi\)
−0.892503 + 0.451042i \(0.851052\pi\)
\(354\) 0 0
\(355\) 6606.00 11441.9i 0.987634 1.71063i
\(356\) −1872.00 −0.278696
\(357\) 0 0
\(358\) 15208.0 2.24516
\(359\) 97.0000 168.009i 0.0142603 0.0246996i −0.858807 0.512299i \(-0.828794\pi\)
0.873068 + 0.487599i \(0.162127\pi\)
\(360\) 0 0
\(361\) 837.500 + 1450.59i 0.122102 + 0.211487i
\(362\) −936.000 + 1621.20i −0.135898 + 0.235382i
\(363\) 0 0
\(364\) 0 0
\(365\) −3240.00 −0.464628
\(366\) 0 0
\(367\) 2376.00 + 4115.35i 0.337946 + 0.585340i 0.984046 0.177913i \(-0.0569345\pi\)
−0.646100 + 0.763253i \(0.723601\pi\)
\(368\) −448.000 775.959i −0.0634609 0.109918i
\(369\) 0 0
\(370\) −11664.0 −1.63887
\(371\) 0 0
\(372\) 0 0
\(373\) 1153.00 1997.05i 0.160054 0.277221i −0.774834 0.632165i \(-0.782167\pi\)
0.934888 + 0.354943i \(0.115500\pi\)
\(374\) −12600.0 21823.8i −1.74206 3.01734i
\(375\) 0 0
\(376\) 0 0
\(377\) 5688.00 0.777047
\(378\) 0 0
\(379\) −7452.00 −1.00998 −0.504991 0.863124i \(-0.668504\pi\)
−0.504991 + 0.863124i \(0.668504\pi\)
\(380\) −5184.00 + 8978.95i −0.699825 + 1.21213i
\(381\) 0 0
\(382\) −964.000 1669.70i −0.129117 0.223636i
\(383\) −576.000 + 997.661i −0.0768465 + 0.133102i −0.901888 0.431970i \(-0.857818\pi\)
0.825041 + 0.565073i \(0.191152\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 3240.00 0.427232
\(387\) 0 0
\(388\) −1872.00 3242.40i −0.244939 0.424247i
\(389\) 947.000 + 1640.25i 0.123431 + 0.213789i 0.921119 0.389282i \(-0.127277\pi\)
−0.797687 + 0.603071i \(0.793943\pi\)
\(390\) 0 0
\(391\) 1764.00 0.228157
\(392\) 0 0
\(393\) 0 0
\(394\) 4924.00 8528.62i 0.629613 1.09052i
\(395\) 2124.00 + 3678.88i 0.270557 + 0.468619i
\(396\) 0 0
\(397\) −4608.00 + 7981.29i −0.582541 + 1.00899i 0.412636 + 0.910896i \(0.364608\pi\)
−0.995177 + 0.0980950i \(0.968725\pi\)
\(398\) 18144.0 2.28512
\(399\) 0 0
\(400\) −12736.0 −1.59200
\(401\) −5825.00 + 10089.2i −0.725403 + 1.25643i 0.233405 + 0.972380i \(0.425013\pi\)
−0.958808 + 0.284055i \(0.908320\pi\)
\(402\) 0 0
\(403\) −648.000 1122.37i −0.0800972 0.138732i
\(404\) −2664.00 + 4614.18i −0.328067 + 0.568228i
\(405\) 0 0
\(406\) 0 0
\(407\) −8100.00 −0.986492
\(408\) 0 0
\(409\) 3762.00 + 6515.98i 0.454814 + 0.787761i 0.998677 0.0514127i \(-0.0163724\pi\)
−0.543863 + 0.839174i \(0.683039\pi\)
\(410\) −9720.00 16835.5i −1.17082 2.02792i
\(411\) 0 0
\(412\) −2016.00 −0.241071
\(413\) 0 0
\(414\) 0 0
\(415\) −324.000 + 561.184i −0.0383242 + 0.0663794i
\(416\) 4608.00 + 7981.29i 0.543091 + 0.940661i
\(417\) 0 0
\(418\) −7200.00 + 12470.8i −0.842496 + 1.45925i
\(419\) 3852.00 0.449123 0.224561 0.974460i \(-0.427905\pi\)
0.224561 + 0.974460i \(0.427905\pi\)
\(420\) 0 0
\(421\) 10402.0 1.20419 0.602093 0.798426i \(-0.294334\pi\)
0.602093 + 0.798426i \(0.294334\pi\)
\(422\) 5832.00 10101.3i 0.672742 1.16522i
\(423\) 0 0
\(424\) 0 0
\(425\) 12537.0 21714.7i 1.43090 2.47840i
\(426\) 0 0
\(427\) 0 0
\(428\) −5360.00 −0.605340
\(429\) 0 0
\(430\) 11664.0 + 20202.6i 1.30811 + 2.26572i
\(431\) −5195.00 8998.00i −0.580590 1.00561i −0.995409 0.0957078i \(-0.969489\pi\)
0.414819 0.909904i \(-0.363845\pi\)
\(432\) 0 0
\(433\) 11232.0 1.24659 0.623297 0.781985i \(-0.285793\pi\)
0.623297 + 0.781985i \(0.285793\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −648.000 + 1122.37i −0.0711779 + 0.123284i
\(437\) −504.000 872.954i −0.0551707 0.0955584i
\(438\) 0 0
\(439\) −7308.00 + 12657.8i −0.794514 + 1.37614i 0.128633 + 0.991692i \(0.458941\pi\)
−0.923147 + 0.384447i \(0.874392\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −18144.0 −1.95254
\(443\) 5969.00 10338.6i 0.640171 1.10881i −0.345223 0.938521i \(-0.612197\pi\)
0.985394 0.170288i \(-0.0544698\pi\)
\(444\) 0 0
\(445\) −2106.00 3647.70i −0.224346 0.388579i
\(446\) −2160.00 + 3741.23i −0.229325 + 0.397203i
\(447\) 0 0
\(448\) 0 0
\(449\) −8186.00 −0.860404 −0.430202 0.902733i \(-0.641558\pi\)
−0.430202 + 0.902733i \(0.641558\pi\)
\(450\) 0 0
\(451\) −6750.00 11691.3i −0.704756 1.22067i
\(452\) −5560.00 9630.20i −0.578585 1.00214i
\(453\) 0 0
\(454\) −5328.00 −0.550783
\(455\) 0 0
\(456\) 0 0
\(457\) −1053.00 + 1823.85i −0.107784 + 0.186687i −0.914872 0.403744i \(-0.867709\pi\)
0.807088 + 0.590431i \(0.201042\pi\)
\(458\) 3240.00 + 5611.84i 0.330557 + 0.572542i
\(459\) 0 0
\(460\) −1008.00 + 1745.91i −0.102170 + 0.176964i
\(461\) −9486.00 −0.958367 −0.479183 0.877715i \(-0.659067\pi\)
−0.479183 + 0.877715i \(0.659067\pi\)
\(462\) 0 0
\(463\) −12652.0 −1.26995 −0.634977 0.772531i \(-0.718990\pi\)
−0.634977 + 0.772531i \(0.718990\pi\)
\(464\) −5056.00 + 8757.25i −0.505860 + 0.876175i
\(465\) 0 0
\(466\) −13436.0 23271.8i −1.33565 2.31341i
\(467\) −1854.00 + 3211.22i −0.183711 + 0.318196i −0.943141 0.332392i \(-0.892144\pi\)
0.759431 + 0.650588i \(0.225478\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 5184.00 0.508766
\(471\) 0 0
\(472\) 0 0
\(473\) 8100.00 + 14029.6i 0.787396 + 1.36381i
\(474\) 0 0
\(475\) −14328.0 −1.38403
\(476\) 0 0
\(477\) 0 0
\(478\) 7156.00 12394.6i 0.684744 1.18601i
\(479\) −4032.00 6983.63i −0.384607 0.666159i 0.607108 0.794620i \(-0.292330\pi\)
−0.991715 + 0.128461i \(0.958996\pi\)
\(480\) 0 0
\(481\) −2916.00 + 5050.66i −0.276420 + 0.478774i
\(482\) 3024.00 0.285766
\(483\) 0 0
\(484\) 9352.00 0.878287
\(485\) 4212.00 7295.40i 0.394344 0.683025i
\(486\) 0 0
\(487\) 5832.00 + 10101.3i 0.542655 + 0.939907i 0.998750 + 0.0499756i \(0.0159143\pi\)
−0.456095 + 0.889931i \(0.650752\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 9814.00 0.902036 0.451018 0.892515i \(-0.351061\pi\)
0.451018 + 0.892515i \(0.351061\pi\)
\(492\) 0 0
\(493\) −9954.00 17240.8i −0.909342 1.57503i
\(494\) 5184.00 + 8978.95i 0.472144 + 0.817778i
\(495\) 0 0
\(496\) 2304.00 0.208574
\(497\) 0 0
\(498\) 0 0
\(499\) 7614.00 13187.8i 0.683065 1.18310i −0.290976 0.956730i \(-0.593980\pi\)
0.974041 0.226373i \(-0.0726868\pi\)
\(500\) 5328.00 + 9228.37i 0.476551 + 0.825410i
\(501\) 0 0
\(502\) 13032.0 22572.1i 1.15866 2.00686i
\(503\) 11088.0 0.982882 0.491441 0.870911i \(-0.336470\pi\)
0.491441 + 0.870911i \(0.336470\pi\)
\(504\) 0 0
\(505\) −11988.0 −1.05635
\(506\) −1400.00 + 2424.87i −0.122999 + 0.213041i
\(507\) 0 0
\(508\) −3664.00 6346.23i −0.320007 0.554269i
\(509\) −2907.00 + 5035.07i −0.253144 + 0.438459i −0.964390 0.264485i \(-0.914798\pi\)
0.711245 + 0.702944i \(0.248131\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −16384.0 −1.41421
\(513\) 0 0
\(514\) 12060.0 + 20888.5i 1.03491 + 1.79252i
\(515\) −2268.00 3928.29i −0.194058 0.336119i
\(516\) 0 0
\(517\) 3600.00 0.306243
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −5841.00 10116.9i −0.491169 0.850729i 0.508780 0.860897i \(-0.330097\pi\)
−0.999948 + 0.0101677i \(0.996763\pi\)
\(522\) 0 0
\(523\) 1494.00 2587.68i 0.124910 0.216351i −0.796788 0.604259i \(-0.793469\pi\)
0.921698 + 0.387909i \(0.126802\pi\)
\(524\) −18144.0 −1.51264
\(525\) 0 0
\(526\) 2360.00 0.195629
\(527\) −2268.00 + 3928.29i −0.187468 + 0.324704i
\(528\) 0 0
\(529\) 5985.50 + 10367.2i 0.491945 + 0.852074i
\(530\) −792.000 + 1371.78i −0.0649100 + 0.112427i
\(531\) 0 0
\(532\) 0 0
\(533\) −9720.00 −0.789906
\(534\) 0 0
\(535\) −6030.00 10444.3i −0.487289 0.844009i
\(536\) 0 0
\(537\) 0 0
\(538\) −3960.00 −0.317338
\(539\) 0 0
\(540\) 0 0
\(541\) −3565.00 + 6174.76i −0.283311 + 0.490709i −0.972198 0.234159i \(-0.924766\pi\)
0.688887 + 0.724869i \(0.258100\pi\)
\(542\) −6840.00 11847.2i −0.542072 0.938897i
\(543\) 0 0
\(544\) 16128.0 27934.5i 1.27111 2.20162i
\(545\) −2916.00 −0.229188
\(546\) 0 0
\(547\) −5488.00 −0.428976 −0.214488 0.976727i \(-0.568808\pi\)
−0.214488 + 0.976727i \(0.568808\pi\)
\(548\) 3224.00 5584.13i 0.251318 0.435296i
\(549\) 0 0
\(550\) 19900.0 + 34467.8i 1.54280 + 2.67220i
\(551\) −5688.00 + 9851.90i −0.439777 + 0.761716i
\(552\) 0 0
\(553\) 0 0
\(554\) 10936.0 0.838675
\(555\) 0 0
\(556\) −10512.0 18207.3i −0.801813 1.38878i
\(557\) 2873.00 + 4976.18i 0.218551 + 0.378541i 0.954365 0.298642i \(-0.0965337\pi\)
−0.735814 + 0.677183i \(0.763200\pi\)
\(558\) 0 0
\(559\) 11664.0 0.882531
\(560\) 0 0
\(561\) 0 0
\(562\) −1196.00 + 2071.53i −0.0897691 + 0.155485i
\(563\) 6534.00 + 11317.2i 0.489121 + 0.847183i 0.999922 0.0125165i \(-0.00398422\pi\)
−0.510800 + 0.859699i \(0.670651\pi\)
\(564\) 0 0
\(565\) 12510.0 21668.0i 0.931504 1.61341i
\(566\) −14400.0 −1.06939
\(567\) 0 0
\(568\) 0 0
\(569\) −565.000 + 978.609i −0.0416275 + 0.0721009i −0.886088 0.463516i \(-0.846588\pi\)
0.844461 + 0.535617i \(0.179921\pi\)
\(570\) 0 0
\(571\) −8432.00 14604.7i −0.617983 1.07038i −0.989853 0.142093i \(-0.954617\pi\)
0.371870 0.928285i \(-0.378717\pi\)
\(572\) 7200.00 12470.8i 0.526306 0.911589i
\(573\) 0 0
\(574\) 0 0
\(575\) −2786.00 −0.202060
\(576\) 0 0
\(577\) 1044.00 + 1808.26i 0.0753246 + 0.130466i 0.901227 0.433347i \(-0.142667\pi\)
−0.825903 + 0.563813i \(0.809334\pi\)
\(578\) 21926.0 + 37976.9i 1.57786 + 2.73293i
\(579\) 0 0
\(580\) 22752.0 1.62884
\(581\) 0 0
\(582\) 0 0
\(583\) −550.000 + 952.628i −0.0390715 + 0.0676738i
\(584\) 0 0
\(585\) 0 0
\(586\) −15804.0 + 27373.3i −1.11409 + 1.92966i
\(587\) −10260.0 −0.721423 −0.360712 0.932677i \(-0.617466\pi\)
−0.360712 + 0.932677i \(0.617466\pi\)
\(588\) 0 0
\(589\) 2592.00 0.181327
\(590\) 16848.0 29181.6i 1.17563 2.03625i
\(591\) 0 0
\(592\) −5184.00 8978.95i −0.359900 0.623366i
\(593\) 1791.00 3102.10i 0.124026 0.214820i −0.797326 0.603549i \(-0.793753\pi\)
0.921352 + 0.388730i \(0.127086\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 19120.0 1.31407
\(597\) 0 0
\(598\) 1008.00 + 1745.91i 0.0689301 + 0.119390i
\(599\) 3517.00 + 6091.62i 0.239901 + 0.415521i 0.960686 0.277638i \(-0.0895517\pi\)
−0.720785 + 0.693159i \(0.756218\pi\)
\(600\) 0 0
\(601\) −18072.0 −1.22658 −0.613288 0.789859i \(-0.710154\pi\)
−0.613288 + 0.789859i \(0.710154\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −12960.0 + 22447.4i −0.873071 + 1.51220i
\(605\) 10521.0 + 18222.9i 0.707007 + 1.22457i
\(606\) 0 0
\(607\) −14292.0 + 24754.5i −0.955674 + 1.65528i −0.222857 + 0.974851i \(0.571538\pi\)
−0.732817 + 0.680426i \(0.761795\pi\)
\(608\) −18432.0 −1.22947
\(609\) 0 0
\(610\) 57024.0 3.78497
\(611\) 1296.00 2244.74i 0.0858110 0.148629i
\(612\) 0 0
\(613\) 5455.00 + 9448.34i 0.359421 + 0.622536i 0.987864 0.155320i \(-0.0496407\pi\)
−0.628443 + 0.777856i \(0.716307\pi\)
\(614\) −20448.0 + 35417.0i −1.34400 + 2.32787i
\(615\) 0 0
\(616\) 0 0
\(617\) 5522.00 0.360304 0.180152 0.983639i \(-0.442341\pi\)
0.180152 + 0.983639i \(0.442341\pi\)
\(618\) 0 0
\(619\) −1206.00 2088.85i −0.0783089 0.135635i 0.824211 0.566282i \(-0.191619\pi\)
−0.902520 + 0.430647i \(0.858285\pi\)
\(620\) −2592.00 4489.48i −0.167899 0.290809i
\(621\) 0 0
\(622\) 15552.0 1.00254
\(623\) 0 0
\(624\) 0 0
\(625\) 449.500 778.557i 0.0287680 0.0498276i
\(626\) 10224.0 + 17708.5i 0.652769 + 1.13063i
\(627\) 0 0
\(628\) −12096.0 + 20950.9i −0.768603 + 1.33126i
\(629\) 20412.0 1.29393
\(630\) 0 0
\(631\) 24676.0 1.55679 0.778396 0.627773i \(-0.216034\pi\)
0.778396 + 0.627773i \(0.216034\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 20204.0 + 34994.4i 1.26562 + 2.19212i
\(635\) 8244.00 14279.0i 0.515202 0.892356i
\(636\) 0 0
\(637\) 0 0
\(638\) 31600.0 1.96090
\(639\) 0 0
\(640\) 0 0
\(641\) −13741.0 23800.1i −0.846703 1.46653i −0.884134 0.467234i \(-0.845251\pi\)
0.0374303 0.999299i \(-0.488083\pi\)
\(642\) 0 0
\(643\) −22752.0 −1.39541 −0.697707 0.716383i \(-0.745796\pi\)
−0.697707 + 0.716383i \(0.745796\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 18144.0 31426.3i 1.10506 1.91401i
\(647\) 7416.00 + 12844.9i 0.450623 + 0.780502i 0.998425 0.0561063i \(-0.0178686\pi\)
−0.547802 + 0.836608i \(0.684535\pi\)
\(648\) 0 0
\(649\) 11700.0 20265.0i 0.707650 1.22569i
\(650\) 28656.0 1.72920
\(651\) 0 0
\(652\) −14272.0 −0.857262
\(653\) 1411.00 2443.92i 0.0845585 0.146460i −0.820645 0.571439i \(-0.806385\pi\)
0.905203 + 0.424979i \(0.139719\pi\)
\(654\) 0 0
\(655\) −20412.0 35354.6i −1.21765 2.10904i
\(656\) 8640.00 14964.9i 0.514231 0.890674i
\(657\) 0 0
\(658\) 0 0
\(659\) 15826.0 0.935498 0.467749 0.883861i \(-0.345065\pi\)
0.467749 + 0.883861i \(0.345065\pi\)
\(660\) 0 0
\(661\) 11916.0 + 20639.1i 0.701178 + 1.21448i 0.968053 + 0.250745i \(0.0806755\pi\)
−0.266875 + 0.963731i \(0.585991\pi\)
\(662\) 11016.0 + 19080.3i 0.646751 + 1.12021i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1106.00 + 1915.65i −0.0642046 + 0.111206i
\(668\) −12096.0 20950.9i −0.700611 1.21349i
\(669\) 0 0
\(670\) −8352.00 + 14466.1i −0.481591 + 0.834140i
\(671\) 39600.0 2.27830
\(672\) 0 0
\(673\) 13770.0 0.788699 0.394350 0.918961i \(-0.370970\pi\)
0.394350 + 0.918961i \(0.370970\pi\)
\(674\) −18468.0 + 31987.5i −1.05543 + 1.82806i
\(675\) 0 0
\(676\) 3604.00 + 6242.31i 0.205052 + 0.355161i
\(677\) 4167.00 7217.46i 0.236560 0.409733i −0.723165 0.690675i \(-0.757313\pi\)
0.959725 + 0.280942i \(0.0906468\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) −3600.00 6235.38i −0.202128 0.350096i
\(683\) −9299.00 16106.3i −0.520961 0.902331i −0.999703 0.0243752i \(-0.992240\pi\)
0.478742 0.877956i \(-0.341093\pi\)
\(684\) 0 0
\(685\) 14508.0 0.809229
\(686\) 0 0
\(687\) 0 0
\(688\) −10368.0 + 17957.9i −0.574529 + 0.995114i
\(689\) 396.000 + 685.892i 0.0218961 + 0.0379251i
\(690\) 0 0
\(691\) −4482.00 + 7763.05i −0.246749 + 0.427381i −0.962622 0.270849i \(-0.912696\pi\)
0.715873 + 0.698230i \(0.246029\pi\)
\(692\) −12528.0 −0.688213
\(693\) 0 0
\(694\) −25976.0 −1.42080
\(695\) 23652.0 40966.5i 1.29089 2.23589i
\(696\) 0 0
\(697\) 17010.0 + 29462.2i 0.924390 + 1.60109i
\(698\) 20160.0 34918.1i 1.09322 1.89351i
\(699\) 0 0
\(700\) 0 0
\(701\) −3542.00 −0.190841 −0.0954205 0.995437i \(-0.530420\pi\)
−0.0954205 + 0.995437i \(0.530420\pi\)
\(702\) 0 0
\(703\) −5832.00 10101.3i −0.312885 0.541932i
\(704\) 12800.0 + 22170.3i 0.685253 + 1.18689i
\(705\) 0 0
\(706\) −2952.00 −0.157365
\(707\) 0 0
\(708\) 0 0
\(709\) 243.000 420.888i 0.0128717 0.0222945i −0.859518 0.511106i \(-0.829236\pi\)
0.872390 + 0.488811i \(0.162569\pi\)
\(710\) −26424.0 45767.7i −1.39673 2.41920i
\(711\) 0 0
\(712\) 0 0
\(713\) 504.000 0.0264726
\(714\) 0 0
\(715\) 32400.0 1.69467
\(716\) 15208.0 26341.0i 0.793784 1.37487i
\(717\) 0 0
\(718\) −388.000 672.036i −0.0201672 0.0349306i
\(719\) −13464.0 + 23320.3i −0.698362 + 1.20960i 0.270672 + 0.962672i \(0.412754\pi\)
−0.969034 + 0.246927i \(0.920579\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 6700.00 0.345358
\(723\) 0 0
\(724\) 1872.00 + 3242.40i 0.0960944 + 0.166440i
\(725\) 15721.0 + 27229.6i 0.805329 + 1.39487i
\(726\) 0 0
\(727\) 20628.0 1.05234 0.526169 0.850380i \(-0.323628\pi\)
0.526169 + 0.850380i \(0.323628\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −6480.00 + 11223.7i −0.328542 + 0.569051i
\(731\) −20412.0 35354.6i −1.03278 1.78883i
\(732\) 0 0
\(733\) 4878.00 8448.94i 0.245802 0.425742i −0.716555 0.697531i \(-0.754282\pi\)
0.962357 + 0.271789i \(0.0876153\pi\)
\(734\) 19008.0 0.955856
\(735\) 0 0
\(736\) −3584.00 −0.179495
\(737\) −5800.00 + 10045.9i −0.289886 + 0.502097i
\(738\) 0 0
\(739\) −9532.00 16509.9i −0.474479 0.821822i 0.525094 0.851045i \(-0.324030\pi\)
−0.999573 + 0.0292221i \(0.990697\pi\)
\(740\) −11664.0 + 20202.6i −0.579429 + 1.00360i
\(741\) 0 0
\(742\) 0 0
\(743\) 3766.00 0.185950 0.0929752 0.995668i \(-0.470362\pi\)
0.0929752 + 0.995668i \(0.470362\pi\)
\(744\) 0 0
\(745\) 21510.0 + 37256.4i 1.05781 + 1.83217i
\(746\) −4612.00 7988.22i −0.226350 0.392050i
\(747\) 0 0
\(748\) −50400.0 −2.46365
\(749\) 0 0
\(750\) 0 0
\(751\) 5832.00 10101.3i 0.283372 0.490815i −0.688841 0.724913i \(-0.741880\pi\)
0.972213 + 0.234097i \(0.0752134\pi\)
\(752\) 2304.00 + 3990.65i 0.111726 + 0.193516i
\(753\) 0 0
\(754\) 11376.0 19703.8i 0.549456 0.951685i
\(755\) −58320.0 −2.81123
\(756\) 0 0
\(757\) −34182.0 −1.64117 −0.820585 0.571524i \(-0.806352\pi\)
−0.820585 + 0.571524i \(0.806352\pi\)
\(758\) −14904.0 + 25814.5i −0.714166 + 1.23697i
\(759\) 0 0
\(760\) 0 0
\(761\) −2367.00 + 4099.76i −0.112751 + 0.195291i −0.916879 0.399166i \(-0.869300\pi\)
0.804127 + 0.594457i \(0.202633\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −3856.00 −0.182598
\(765\) 0 0
\(766\) 2304.00 + 3990.65i 0.108677 + 0.188235i
\(767\) −8424.00 14590.8i −0.396575 0.686888i
\(768\) 0 0
\(769\) −30240.0 −1.41805 −0.709026 0.705182i \(-0.750865\pi\)
−0.709026 + 0.705182i \(0.750865\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 3240.00 5611.84i 0.151049 0.261625i
\(773\) 13851.0 + 23990.6i 0.644484 + 1.11628i 0.984421 + 0.175830i \(0.0562609\pi\)
−0.339937 + 0.940448i \(0.610406\pi\)
\(774\) 0 0
\(775\) 3582.00 6204.21i 0.166025 0.287563i
\(776\) 0 0
\(777\) 0 0
\(778\) 7576.00 0.349117
\(779\) 9720.00 16835.5i 0.447054 0.774320i
\(780\) 0 0
\(781\) −18350.0 31783.1i −0.840736 1.45620i
\(782\) 3528.00 6110.68i 0.161331 0.279434i
\(783\) 0 0
\(784\) 0 0
\(785\) −54432.0 −2.47486
\(786\) 0 0
\(787\) 11322.0 + 19610.3i 0.512815 + 0.888222i 0.999890 + 0.0148617i \(0.00473080\pi\)
−0.487074 + 0.873361i \(0.661936\pi\)
\(788\) −9848.00 17057.2i −0.445204 0.771115i
\(789\) 0 0
\(790\) 16992.0 0.765251
\(791\) 0 0
\(792\) 0 0
\(793\) 14256.0 24692.1i 0.638393 1.10573i
\(794\) 18432.0 + 31925.2i 0.823838 + 1.42693i
\(795\) 0 0
\(796\) 18144.0 31426.3i 0.807911 1.39934i
\(797\) −30150.0 −1.33998 −0.669992 0.742368i \(-0.733703\pi\)
−0.669992 + 0.742368i \(0.733703\pi\)
\(798\) 0 0
\(799\) −9072.00 −0.401682
\(800\) −25472.0 + 44118.8i −1.12571 + 1.94979i
\(801\) 0 0
\(802\) 23300.0 + 40356.8i 1.02587 + 1.77687i
\(803\) −4500.00 + 7794.23i −0.197760 + 0.342531i
\(804\) 0 0
\(805\) 0 0
\(806\) −5184.00 −0.226549
\(807\) 0 0
\(808\) 0 0
\(809\) −5659.00 9801.68i −0.245933 0.425969i 0.716460 0.697628i \(-0.245761\pi\)
−0.962394 + 0.271659i \(0.912428\pi\)
\(810\) 0 0
\(811\) −29628.0 −1.28284 −0.641418 0.767192i \(-0.721653\pi\)
−0.641418 + 0.767192i \(0.721653\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −16200.0 + 28059.2i −0.697555 + 1.20820i
\(815\) −16056.0 27809.8i −0.690082 1.19526i
\(816\) 0 0
\(817\) −11664.0 + 20202.6i −0.499476 + 0.865117i
\(818\) 30096.0 1.28641
\(819\) 0 0
\(820\) −38880.0 −1.65579
\(821\) 8885.00 15389.3i 0.377696 0.654189i −0.613030 0.790059i \(-0.710050\pi\)
0.990727 + 0.135870i \(0.0433830\pi\)
\(822\) 0 0
\(823\) −3934.00 6813.89i −0.166623 0.288599i 0.770608 0.637310i \(-0.219953\pi\)
−0.937230 + 0.348711i \(0.886620\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −35726.0 −1.50219 −0.751097 0.660192i \(-0.770475\pi\)
−0.751097 + 0.660192i \(0.770475\pi\)
\(828\) 0 0
\(829\) −13554.0 23476.2i −0.567853 0.983550i −0.996778 0.0802098i \(-0.974441\pi\)
0.428925 0.903340i \(-0.358892\pi\)
\(830\) 1296.00 + 2244.74i 0.0541986 + 0.0938747i
\(831\) 0 0
\(832\) 18432.0 0.768046
\(833\) 0 0
\(834\) 0 0
\(835\) 27216.0 47139.5i 1.12796 1.95369i
\(836\) 14400.0 + 24941.5i 0.595735 + 1.03184i
\(837\) 0 0
\(838\) 7704.00 13343.7i 0.317578 0.550061i
\(839\) −23256.0 −0.956956 −0.478478 0.878099i \(-0.658811\pi\)
−0.478478 + 0.878099i \(0.658811\pi\)
\(840\) 0 0
\(841\) 575.000 0.0235762
\(842\) 20804.0 36033.6i 0.851488 1.47482i
\(843\) 0 0
\(844\) −11664.0 20202.6i −0.475701 0.823938i
\(845\) −8109.00 + 14045.2i −0.330128 + 0.571798i
\(846\) 0 0
\(847\) 0 0
\(848\) −1408.00 −0.0570176
\(849\) 0 0
\(850\) −50148.0 86858.9i −2.02360 3.50498i
\(851\) −1134.00 1964.15i −0.0456792 0.0791187i
\(852\) 0 0
\(853\) 35280.0 1.41614 0.708068 0.706144i \(-0.249567\pi\)
0.708068 + 0.706144i \(0.249567\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −9855.00 17069.4i −0.392813 0.680371i 0.600007 0.799995i \(-0.295165\pi\)
−0.992819 + 0.119624i \(0.961831\pi\)
\(858\) 0 0
\(859\) −1944.00 + 3367.11i −0.0772159 + 0.133742i −0.902048 0.431636i \(-0.857936\pi\)
0.824832 + 0.565378i \(0.191270\pi\)
\(860\) 46656.0 1.84995
\(861\) 0 0
\(862\) −41560.0 −1.64216
\(863\) −18317.0 + 31726.0i −0.722500 + 1.25141i 0.237494 + 0.971389i \(0.423674\pi\)
−0.959995 + 0.280019i \(0.909659\pi\)
\(864\) 0 0
\(865\) −14094.0 24411.5i −0.554000 0.959557i
\(866\) 22464.0 38908.8i 0.881476 1.52676i
\(867\) 0 0
\(868\) 0 0
\(869\) 11800.0 0.460630
\(870\) 0 0
\(871\) 4176.00 + 7233.04i 0.162455 + 0.281380i
\(872\) 0 0
\(873\) 0 0
\(874\) −4032.00 −0.156046
\(875\) 0 0
\(876\) 0 0
\(877\) −613.000 + 1061.75i −0.0236027 + 0.0408810i −0.877585 0.479420i \(-0.840847\pi\)
0.853983 + 0.520301i \(0.174180\pi\)
\(878\) 29232.0 + 50631.3i 1.12361 + 1.94615i
\(879\) 0 0
\(880\) −28800.0 + 49883.1i −1.10324 + 1.91086i
\(881\) 38538.0 1.47376 0.736878 0.676026i \(-0.236299\pi\)
0.736878 + 0.676026i \(0.236299\pi\)
\(882\) 0 0
\(883\) −37260.0 −1.42004 −0.710022 0.704180i \(-0.751315\pi\)
−0.710022 + 0.704180i \(0.751315\pi\)
\(884\) −18144.0 + 31426.3i −0.690327 + 1.19568i
\(885\) 0 0
\(886\) −23876.0 41354.4i −0.905338 1.56809i
\(887\) 13320.0 23070.9i 0.504219 0.873332i −0.495770 0.868454i \(-0.665114\pi\)
0.999988 0.00487800i \(-0.00155272\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −16848.0 −0.634546
\(891\) 0 0
\(892\) 4320.00 + 7482.46i 0.162157 + 0.280865i
\(893\) 2592.00 + 4489.48i 0.0971310 + 0.168236i
\(894\) 0 0
\(895\) 68436.0 2.55594
\(896\) 0 0
\(897\) 0 0
\(898\) −16372.0 + 28357.1i −0.608397 + 1.05377i
\(899\) −2844.00 4925.95i −0.105509 0.182747i
\(900\) 0 0
\(901\) 1386.00 2400.62i 0.0512479 0.0887640i
\(902\) −54000.0 −1.99335
\(903\) 0 0
\(904\) 0 0
\(905\) −4212.00 + 7295.40i −0.154709 + 0.267964i
\(906\) 0 0
\(907\) 6318.00 + 10943.1i 0.231296 + 0.400617i 0.958190 0.286133i \(-0.0923700\pi\)
−0.726894 + 0.686750i \(0.759037\pi\)
\(908\) −5328.00 + 9228.37i −0.194731 + 0.337284i
\(909\) 0 0
\(910\) 0 0
\(911\) 33638.0 1.22336 0.611678 0.791107i \(-0.290495\pi\)
0.611678 + 0.791107i \(0.290495\pi\)
\(912\) 0 0
\(913\) 900.000 + 1558.85i 0.0326239 + 0.0565063i
\(914\) 4212.00 + 7295.40i 0.152430 + 0.264016i
\(915\) 0 0
\(916\) 12960.0 0.467479
\(917\) 0 0
\(918\) 0 0
\(919\) −18468.0 + 31987.5i −0.662898 + 1.14817i 0.316953 + 0.948441i \(0.397340\pi\)
−0.979851 + 0.199731i \(0.935993\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −18972.0 + 32860.5i −0.677668 + 1.17375i
\(923\) −26424.0 −0.942315
\(924\) 0 0
\(925\) −32238.0 −1.14592
\(926\) −25304.0 + 43827.8i −0.897992 + 1.55537i
\(927\) 0 0
\(928\) 20224.0 + 35029.0i 0.715394 + 1.23910i
\(929\) −11151.0 + 19314.1i −0.393813 + 0.682104i −0.992949 0.118543i \(-0.962178\pi\)
0.599136 + 0.800647i \(0.295511\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −53744.0 −1.88889
\(933\) 0 0
\(934\) 7416.00 + 12844.9i 0.259806 + 0.449997i
\(935\) −56700.0 98207.3i −1.98320 3.43500i
\(936\) 0 0
\(937\) 13824.0 0.481975 0.240987 0.970528i \(-0.422529\pi\)
0.240987 + 0.970528i \(0.422529\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 5184.00 8978.95i 0.179876 0.311554i
\(941\) 6777.00 + 11738.1i 0.234776 + 0.406643i 0.959207 0.282703i \(-0.0912311\pi\)
−0.724432 + 0.689346i \(0.757898\pi\)
\(942\) 0 0
\(943\) 1890.00 3273.58i 0.0652671 0.113046i
\(944\) 29952.0 1.03268
\(945\) 0 0
\(946\) 64800.0 2.22709
\(947\) 22439.0 38865.5i 0.769978 1.33364i −0.167596 0.985856i \(-0.553600\pi\)
0.937574 0.347786i \(-0.113066\pi\)
\(948\) 0 0
\(949\) 3240.00 + 5611.84i 0.110827 + 0.191958i
\(950\) −28656.0 + 49633.6i −0.978656 + 1.69508i
\(951\) 0 0
\(952\) 0 0
\(953\) −38362.0 −1.30395 −0.651976 0.758239i \(-0.726060\pi\)
−0.651976 + 0.758239i \(0.726060\pi\)
\(954\) 0 0
\(955\) −4338.00 7513.64i −0.146989 0.254592i
\(956\) −14312.0 24789.1i −0.484187 0.838637i
\(957\) 0 0
\(958\) −32256.0 −1.08783
\(959\) 0 0
\(960\) 0 0
\(961\) 14247.5 24677.4i 0.478248 0.828351i
\(962\) 11664.0 + 20202.6i 0.390917 + 0.677089i
\(963\) 0 0
\(964\) 3024.00 5237.72i 0.101034 0.174995i
\(965\) 14580.0 0.486370
\(966\) 0 0
\(967\) 26444.0 0.879402 0.439701 0.898144i \(-0.355084\pi\)
0.439701 + 0.898144i \(0.355084\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −16848.0 29181.6i −0.557687 0.965943i
\(971\) 8910.00 15432.6i 0.294475 0.510046i −0.680387 0.732853i \(-0.738188\pi\)
0.974863 + 0.222806i \(0.0715218\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 46656.0 1.53486
\(975\) 0 0
\(976\) 25344.0 + 43897.1i 0.831190 + 1.43966i
\(977\) 17219.0 + 29824.2i 0.563853 + 0.976622i 0.997155 + 0.0753737i \(0.0240150\pi\)
−0.433302 + 0.901249i \(0.642652\pi\)
\(978\) 0 0
\(979\) −11700.0 −0.381955
\(980\) 0 0
\(981\) 0 0
\(982\) 19628.0 33996.7i 0.637836 1.10476i
\(983\) 13032.0 + 22572.1i 0.422845 + 0.732388i 0.996216 0.0869069i \(-0.0276983\pi\)
−0.573372 + 0.819295i \(0.694365\pi\)
\(984\) 0 0
\(985\) 22158.0 38378.8i 0.716764 1.24147i
\(986\) −79632.0 −2.57201
\(987\) 0 0
\(988\) 20736.0 0.667713
\(989\) −2268.00 + 3928.29i −0.0729203 + 0.126302i
\(990\) 0 0
\(991\) −16848.0 29181.6i −0.540055 0.935402i −0.998900 0.0468863i \(-0.985070\pi\)
0.458845 0.888516i \(-0.348263\pi\)
\(992\) 4608.00 7981.29i 0.147484 0.255450i
\(993\) 0 0
\(994\) 0 0
\(995\) 81648.0 2.60142
\(996\) 0 0
\(997\) −18036.0 31239.3i −0.572925 0.992335i −0.996264 0.0863632i \(-0.972475\pi\)
0.423339 0.905971i \(-0.360858\pi\)
\(998\) −30456.0 52751.3i −0.966000 1.67316i
\(999\) 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 441.4.e.o.226.1 2
3.2 odd 2 147.4.e.a.79.1 2
7.2 even 3 441.4.a.a.1.1 1
7.3 odd 6 441.4.e.l.361.1 2
7.4 even 3 inner 441.4.e.o.361.1 2
7.5 odd 6 441.4.a.c.1.1 1
7.6 odd 2 441.4.e.l.226.1 2
21.2 odd 6 147.4.a.h.1.1 yes 1
21.5 even 6 147.4.a.f.1.1 1
21.11 odd 6 147.4.e.a.67.1 2
21.17 even 6 147.4.e.d.67.1 2
21.20 even 2 147.4.e.d.79.1 2
84.23 even 6 2352.4.a.s.1.1 1
84.47 odd 6 2352.4.a.t.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.4.a.f.1.1 1 21.5 even 6
147.4.a.h.1.1 yes 1 21.2 odd 6
147.4.e.a.67.1 2 21.11 odd 6
147.4.e.a.79.1 2 3.2 odd 2
147.4.e.d.67.1 2 21.17 even 6
147.4.e.d.79.1 2 21.20 even 2
441.4.a.a.1.1 1 7.2 even 3
441.4.a.c.1.1 1 7.5 odd 6
441.4.e.l.226.1 2 7.6 odd 2
441.4.e.l.361.1 2 7.3 odd 6
441.4.e.o.226.1 2 1.1 even 1 trivial
441.4.e.o.361.1 2 7.4 even 3 inner
2352.4.a.s.1.1 1 84.23 even 6
2352.4.a.t.1.1 1 84.47 odd 6