Properties

Label 441.4.p.d.80.9
Level $441$
Weight $4$
Character 441.80
Analytic conductor $26.020$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,4,Mod(80,441)]
 
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([3, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.80");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.p (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: \(26.0198423125\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 80.9
Character \(\chi\) \(=\) 441.80
Dual form 441.4.p.d.215.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.40358 + 0.810356i) q^{2} +(-2.68665 + 4.65341i) q^{4} +(-2.35993 - 4.08752i) q^{5} -21.6743i q^{8} +O(q^{10})\) \(q+(-1.40358 + 0.810356i) q^{2} +(-2.68665 + 4.65341i) q^{4} +(-2.35993 - 4.08752i) q^{5} -21.6743i q^{8} +(6.62469 + 3.82477i) q^{10} +(25.9801 + 14.9996i) q^{11} +27.1586i q^{13} +(-3.92932 - 6.80578i) q^{16} +(-8.92464 + 15.4579i) q^{17} +(-107.297 + 61.9477i) q^{19} +25.3612 q^{20} -48.6201 q^{22} +(-71.0802 + 41.0382i) q^{23} +(51.3615 - 88.9606i) q^{25} +(-22.0081 - 38.1191i) q^{26} -88.2194i q^{29} +(-220.389 - 127.242i) q^{31} +(161.194 + 93.0653i) q^{32} -28.9285i q^{34} +(-107.695 - 186.533i) q^{37} +(100.399 - 173.897i) q^{38} +(-88.5939 + 51.1497i) q^{40} +427.760 q^{41} +62.4602 q^{43} +(-139.599 + 80.5974i) q^{44} +(66.5110 - 115.200i) q^{46} +(-211.787 - 366.825i) q^{47} +166.484i q^{50} +(-126.380 - 72.9654i) q^{52} +(587.234 + 339.039i) q^{53} -141.592i q^{55} +(71.4891 + 123.823i) q^{58} +(383.719 - 664.621i) q^{59} +(-313.156 + 180.801i) q^{61} +412.444 q^{62} -238.795 q^{64} +(111.011 - 64.0923i) q^{65} +(323.947 - 561.093i) q^{67} +(-47.9547 - 83.0600i) q^{68} -536.663i q^{71} +(-547.293 - 315.980i) q^{73} +(302.316 + 174.542i) q^{74} -665.726i q^{76} +(298.709 + 517.379i) q^{79} +(-18.5458 + 32.1223i) q^{80} +(-600.395 + 346.638i) q^{82} -591.773 q^{83} +84.2461 q^{85} +(-87.6677 + 50.6150i) q^{86} +(325.106 - 563.100i) q^{88} +(-769.502 - 1332.82i) q^{89} -441.020i q^{92} +(594.517 + 343.245i) q^{94} +(506.425 + 292.385i) q^{95} -654.102i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 96 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 48 q + 96 q^{4} - 144 q^{16} + 1248 q^{22} - 312 q^{25} + 864 q^{37} + 2496 q^{43} + 3888 q^{46} + 7440 q^{58} - 6720 q^{64} + 2688 q^{67} - 480 q^{79} + 26496 q^{85} + 7248 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(199\) \(344\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.40358 + 0.810356i −0.496240 + 0.286504i −0.727159 0.686469i \(-0.759160\pi\)
0.230920 + 0.972973i \(0.425827\pi\)
\(3\) 0 0
\(4\) −2.68665 + 4.65341i −0.335831 + 0.581676i
\(5\) −2.35993 4.08752i −0.211079 0.365599i 0.740974 0.671534i \(-0.234364\pi\)
−0.952052 + 0.305935i \(0.901031\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 21.6743i 0.957876i
\(9\) 0 0
\(10\) 6.62469 + 3.82477i 0.209491 + 0.120950i
\(11\) 25.9801 + 14.9996i 0.712118 + 0.411142i 0.811845 0.583873i \(-0.198464\pi\)
−0.0997267 + 0.995015i \(0.531797\pi\)
\(12\) 0 0
\(13\) 27.1586i 0.579417i 0.957115 + 0.289709i \(0.0935584\pi\)
−0.957115 + 0.289709i \(0.906442\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −3.92932 6.80578i −0.0613956 0.106340i
\(17\) −8.92464 + 15.4579i −0.127326 + 0.220535i −0.922640 0.385663i \(-0.873973\pi\)
0.795314 + 0.606198i \(0.207306\pi\)
\(18\) 0 0
\(19\) −107.297 + 61.9477i −1.29555 + 0.747988i −0.979633 0.200797i \(-0.935647\pi\)
−0.315921 + 0.948786i \(0.602313\pi\)
\(20\) 25.3612 0.283547
\(21\) 0 0
\(22\) −48.6201 −0.471175
\(23\) −71.0802 + 41.0382i −0.644402 + 0.372046i −0.786308 0.617835i \(-0.788010\pi\)
0.141906 + 0.989880i \(0.454677\pi\)
\(24\) 0 0
\(25\) 51.3615 88.9606i 0.410892 0.711685i
\(26\) −22.0081 38.1191i −0.166005 0.287530i
\(27\) 0 0
\(28\) 0 0
\(29\) 88.2194i 0.564894i −0.959283 0.282447i \(-0.908854\pi\)
0.959283 0.282447i \(-0.0911462\pi\)
\(30\) 0 0
\(31\) −220.389 127.242i −1.27687 0.737203i −0.300600 0.953750i \(-0.597187\pi\)
−0.976272 + 0.216548i \(0.930520\pi\)
\(32\) 161.194 + 93.0653i 0.890479 + 0.514118i
\(33\) 0 0
\(34\) 28.9285i 0.145918i
\(35\) 0 0
\(36\) 0 0
\(37\) −107.695 186.533i −0.478511 0.828805i 0.521186 0.853443i \(-0.325490\pi\)
−0.999696 + 0.0246385i \(0.992157\pi\)
\(38\) 100.399 173.897i 0.428603 0.742363i
\(39\) 0 0
\(40\) −88.5939 + 51.1497i −0.350198 + 0.202187i
\(41\) 427.760 1.62939 0.814695 0.579890i \(-0.196905\pi\)
0.814695 + 0.579890i \(0.196905\pi\)
\(42\) 0 0
\(43\) 62.4602 0.221514 0.110757 0.993848i \(-0.464673\pi\)
0.110757 + 0.993848i \(0.464673\pi\)
\(44\) −139.599 + 80.5974i −0.478303 + 0.276148i
\(45\) 0 0
\(46\) 66.5110 115.200i 0.213185 0.369247i
\(47\) −211.787 366.825i −0.657282 1.13845i −0.981317 0.192400i \(-0.938373\pi\)
0.324035 0.946045i \(-0.394960\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 166.484i 0.470888i
\(51\) 0 0
\(52\) −126.380 72.9654i −0.337033 0.194586i
\(53\) 587.234 + 339.039i 1.52194 + 0.878692i 0.999664 + 0.0259140i \(0.00824962\pi\)
0.522274 + 0.852778i \(0.325084\pi\)
\(54\) 0 0
\(55\) 141.592i 0.347133i
\(56\) 0 0
\(57\) 0 0
\(58\) 71.4891 + 123.823i 0.161844 + 0.280323i
\(59\) 383.719 664.621i 0.846711 1.46655i −0.0374163 0.999300i \(-0.511913\pi\)
0.884127 0.467246i \(-0.154754\pi\)
\(60\) 0 0
\(61\) −313.156 + 180.801i −0.657303 + 0.379494i −0.791249 0.611495i \(-0.790569\pi\)
0.133946 + 0.990989i \(0.457235\pi\)
\(62\) 412.444 0.844846
\(63\) 0 0
\(64\) −238.795 −0.466396
\(65\) 111.011 64.0923i 0.211834 0.122303i
\(66\) 0 0
\(67\) 323.947 561.093i 0.590694 1.02311i −0.403446 0.915004i \(-0.632188\pi\)
0.994139 0.108108i \(-0.0344791\pi\)
\(68\) −47.9547 83.0600i −0.0855200 0.148125i
\(69\) 0 0
\(70\) 0 0
\(71\) 536.663i 0.897045i −0.893772 0.448522i \(-0.851950\pi\)
0.893772 0.448522i \(-0.148050\pi\)
\(72\) 0 0
\(73\) −547.293 315.980i −0.877477 0.506612i −0.00765130 0.999971i \(-0.502436\pi\)
−0.869826 + 0.493359i \(0.835769\pi\)
\(74\) 302.316 + 174.542i 0.474912 + 0.274190i
\(75\) 0 0
\(76\) 665.726i 1.00479i
\(77\) 0 0
\(78\) 0 0
\(79\) 298.709 + 517.379i 0.425410 + 0.736832i 0.996459 0.0840844i \(-0.0267965\pi\)
−0.571049 + 0.820916i \(0.693463\pi\)
\(80\) −18.5458 + 32.1223i −0.0259186 + 0.0448923i
\(81\) 0 0
\(82\) −600.395 + 346.638i −0.808567 + 0.466827i
\(83\) −591.773 −0.782597 −0.391299 0.920264i \(-0.627974\pi\)
−0.391299 + 0.920264i \(0.627974\pi\)
\(84\) 0 0
\(85\) 84.2461 0.107503
\(86\) −87.6677 + 50.6150i −0.109924 + 0.0634646i
\(87\) 0 0
\(88\) 325.106 563.100i 0.393823 0.682121i
\(89\) −769.502 1332.82i −0.916484 1.58740i −0.804714 0.593662i \(-0.797682\pi\)
−0.111769 0.993734i \(-0.535652\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 441.020i 0.499778i
\(93\) 0 0
\(94\) 594.517 + 343.245i 0.652338 + 0.376628i
\(95\) 506.425 + 292.385i 0.546927 + 0.315769i
\(96\) 0 0
\(97\) 654.102i 0.684680i −0.939576 0.342340i \(-0.888781\pi\)
0.939576 0.342340i \(-0.111219\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 275.980 + 478.012i 0.275980 + 0.478012i
\(101\) 148.389 257.018i 0.146191 0.253210i −0.783626 0.621233i \(-0.786632\pi\)
0.929817 + 0.368023i \(0.119965\pi\)
\(102\) 0 0
\(103\) 387.255 223.582i 0.370460 0.213885i −0.303199 0.952927i \(-0.598055\pi\)
0.673660 + 0.739042i \(0.264721\pi\)
\(104\) 588.641 0.555010
\(105\) 0 0
\(106\) −1098.97 −1.00699
\(107\) 655.418 378.406i 0.592165 0.341887i −0.173788 0.984783i \(-0.555601\pi\)
0.765953 + 0.642896i \(0.222267\pi\)
\(108\) 0 0
\(109\) −59.2503 + 102.625i −0.0520656 + 0.0901803i −0.890884 0.454232i \(-0.849914\pi\)
0.838818 + 0.544412i \(0.183247\pi\)
\(110\) 114.740 + 198.736i 0.0994550 + 0.172261i
\(111\) 0 0
\(112\) 0 0
\(113\) 90.4761i 0.0753210i −0.999291 0.0376605i \(-0.988009\pi\)
0.999291 0.0376605i \(-0.0119905\pi\)
\(114\) 0 0
\(115\) 335.489 + 193.694i 0.272039 + 0.157062i
\(116\) 410.521 + 237.014i 0.328586 + 0.189709i
\(117\) 0 0
\(118\) 1243.80i 0.970344i
\(119\) 0 0
\(120\) 0 0
\(121\) −215.522 373.296i −0.161925 0.280463i
\(122\) 293.026 507.535i 0.217453 0.376640i
\(123\) 0 0
\(124\) 1184.22 683.707i 0.857626 0.495151i
\(125\) −1074.82 −0.769079
\(126\) 0 0
\(127\) 324.991 0.227073 0.113536 0.993534i \(-0.463782\pi\)
0.113536 + 0.993534i \(0.463782\pi\)
\(128\) −954.384 + 551.014i −0.659034 + 0.380494i
\(129\) 0 0
\(130\) −103.875 + 179.917i −0.0700804 + 0.121383i
\(131\) −338.925 587.035i −0.226046 0.391523i 0.730587 0.682820i \(-0.239247\pi\)
−0.956633 + 0.291297i \(0.905913\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 1050.05i 0.676944i
\(135\) 0 0
\(136\) 335.039 + 193.435i 0.211245 + 0.121963i
\(137\) −1866.23 1077.47i −1.16381 0.671928i −0.211598 0.977357i \(-0.567867\pi\)
−0.952215 + 0.305429i \(0.901200\pi\)
\(138\) 0 0
\(139\) 719.518i 0.439055i −0.975606 0.219528i \(-0.929548\pi\)
0.975606 0.219528i \(-0.0704516\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 434.888 + 753.248i 0.257007 + 0.445149i
\(143\) −407.368 + 705.582i −0.238223 + 0.412614i
\(144\) 0 0
\(145\) −360.599 + 208.192i −0.206525 + 0.119237i
\(146\) 1024.22 0.580585
\(147\) 0 0
\(148\) 1157.35 0.642795
\(149\) 1034.65 597.357i 0.568873 0.328439i −0.187826 0.982202i \(-0.560144\pi\)
0.756699 + 0.653763i \(0.226811\pi\)
\(150\) 0 0
\(151\) −536.066 + 928.494i −0.288904 + 0.500396i −0.973548 0.228481i \(-0.926624\pi\)
0.684645 + 0.728877i \(0.259957\pi\)
\(152\) 1342.67 + 2325.57i 0.716480 + 1.24098i
\(153\) 0 0
\(154\) 0 0
\(155\) 1201.13i 0.622431i
\(156\) 0 0
\(157\) −2022.12 1167.47i −1.02792 0.593468i −0.111530 0.993761i \(-0.535575\pi\)
−0.916387 + 0.400293i \(0.868908\pi\)
\(158\) −838.523 484.121i −0.422211 0.243763i
\(159\) 0 0
\(160\) 878.511i 0.434077i
\(161\) 0 0
\(162\) 0 0
\(163\) −1803.42 3123.61i −0.866592 1.50098i −0.865458 0.500982i \(-0.832972\pi\)
−0.00113428 0.999999i \(-0.500361\pi\)
\(164\) −1149.24 + 1990.54i −0.547199 + 0.947777i
\(165\) 0 0
\(166\) 830.600 479.547i 0.388356 0.224217i
\(167\) −2249.44 −1.04232 −0.521159 0.853460i \(-0.674500\pi\)
−0.521159 + 0.853460i \(0.674500\pi\)
\(168\) 0 0
\(169\) 1459.41 0.664275
\(170\) −118.246 + 68.2693i −0.0533473 + 0.0308001i
\(171\) 0 0
\(172\) −167.809 + 290.653i −0.0743912 + 0.128849i
\(173\) 2022.40 + 3502.90i 0.888788 + 1.53943i 0.841310 + 0.540553i \(0.181785\pi\)
0.0474778 + 0.998872i \(0.484882\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 235.753i 0.100969i
\(177\) 0 0
\(178\) 2160.11 + 1247.14i 0.909591 + 0.525153i
\(179\) −1843.22 1064.18i −0.769658 0.444362i 0.0630946 0.998008i \(-0.479903\pi\)
−0.832753 + 0.553645i \(0.813236\pi\)
\(180\) 0 0
\(181\) 4272.10i 1.75438i 0.480143 + 0.877190i \(0.340585\pi\)
−0.480143 + 0.877190i \(0.659415\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 889.472 + 1540.61i 0.356373 + 0.617257i
\(185\) −508.304 + 880.408i −0.202007 + 0.349886i
\(186\) 0 0
\(187\) −463.726 + 267.732i −0.181342 + 0.104698i
\(188\) 2275.98 0.882942
\(189\) 0 0
\(190\) −947.742 −0.361876
\(191\) 670.762 387.265i 0.254108 0.146709i −0.367536 0.930009i \(-0.619798\pi\)
0.621644 + 0.783300i \(0.286465\pi\)
\(192\) 0 0
\(193\) −2456.61 + 4254.97i −0.916221 + 1.58694i −0.111117 + 0.993807i \(0.535443\pi\)
−0.805104 + 0.593134i \(0.797890\pi\)
\(194\) 530.055 + 918.082i 0.196164 + 0.339765i
\(195\) 0 0
\(196\) 0 0
\(197\) 1004.44i 0.363265i 0.983366 + 0.181632i \(0.0581381\pi\)
−0.983366 + 0.181632i \(0.941862\pi\)
\(198\) 0 0
\(199\) 1734.53 + 1001.43i 0.617878 + 0.356732i 0.776042 0.630681i \(-0.217224\pi\)
−0.158164 + 0.987413i \(0.550558\pi\)
\(200\) −1928.16 1113.22i −0.681706 0.393583i
\(201\) 0 0
\(202\) 480.992i 0.167537i
\(203\) 0 0
\(204\) 0 0
\(205\) −1009.49 1748.48i −0.343929 0.595703i
\(206\) −362.362 + 627.629i −0.122558 + 0.212277i
\(207\) 0 0
\(208\) 184.835 106.715i 0.0616154 0.0355737i
\(209\) −3716.77 −1.23012
\(210\) 0 0
\(211\) 2630.90 0.858383 0.429191 0.903214i \(-0.358799\pi\)
0.429191 + 0.903214i \(0.358799\pi\)
\(212\) −3155.38 + 1821.76i −1.02223 + 0.590184i
\(213\) 0 0
\(214\) −613.287 + 1062.24i −0.195904 + 0.339315i
\(215\) −147.402 255.307i −0.0467568 0.0809852i
\(216\) 0 0
\(217\) 0 0
\(218\) 192.055i 0.0596680i
\(219\) 0 0
\(220\) 658.887 + 380.409i 0.201919 + 0.116578i
\(221\) −419.815 242.380i −0.127782 0.0737749i
\(222\) 0 0
\(223\) 4972.20i 1.49311i 0.665324 + 0.746555i \(0.268293\pi\)
−0.665324 + 0.746555i \(0.731707\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 73.3178 + 126.990i 0.0215798 + 0.0373773i
\(227\) −1710.26 + 2962.26i −0.500062 + 0.866133i 0.499938 + 0.866061i \(0.333356\pi\)
−1.00000 7.18991e-5i \(0.999977\pi\)
\(228\) 0 0
\(229\) 2770.81 1599.73i 0.799566 0.461629i −0.0437537 0.999042i \(-0.513932\pi\)
0.843319 + 0.537413i \(0.180598\pi\)
\(230\) −627.846 −0.179995
\(231\) 0 0
\(232\) −1912.09 −0.541099
\(233\) 1859.97 1073.85i 0.522964 0.301933i −0.215183 0.976574i \(-0.569035\pi\)
0.738146 + 0.674641i \(0.235701\pi\)
\(234\) 0 0
\(235\) −999.603 + 1731.36i −0.277476 + 0.480603i
\(236\) 2061.83 + 3571.20i 0.568703 + 0.985023i
\(237\) 0 0
\(238\) 0 0
\(239\) 2853.29i 0.772236i −0.922449 0.386118i \(-0.873816\pi\)
0.922449 0.386118i \(-0.126184\pi\)
\(240\) 0 0
\(241\) 340.988 + 196.869i 0.0911409 + 0.0526202i 0.544878 0.838516i \(-0.316576\pi\)
−0.453737 + 0.891136i \(0.649909\pi\)
\(242\) 605.005 + 349.300i 0.160707 + 0.0927844i
\(243\) 0 0
\(244\) 1942.99i 0.509783i
\(245\) 0 0
\(246\) 0 0
\(247\) −1682.41 2914.02i −0.433397 0.750666i
\(248\) −2757.87 + 4776.77i −0.706149 + 1.22309i
\(249\) 0 0
\(250\) 1508.59 870.987i 0.381647 0.220344i
\(251\) −2520.34 −0.633794 −0.316897 0.948460i \(-0.602641\pi\)
−0.316897 + 0.948460i \(0.602641\pi\)
\(252\) 0 0
\(253\) −2462.23 −0.611854
\(254\) −456.150 + 263.358i −0.112683 + 0.0650573i
\(255\) 0 0
\(256\) 1848.21 3201.20i 0.451224 0.781543i
\(257\) −2125.96 3682.26i −0.516006 0.893748i −0.999827 0.0185815i \(-0.994085\pi\)
0.483822 0.875167i \(-0.339248\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 688.774i 0.164292i
\(261\) 0 0
\(262\) 951.414 + 549.299i 0.224346 + 0.129526i
\(263\) −6145.87 3548.32i −1.44095 0.831935i −0.443040 0.896502i \(-0.646100\pi\)
−0.997913 + 0.0645666i \(0.979434\pi\)
\(264\) 0 0
\(265\) 3200.44i 0.741892i
\(266\) 0 0
\(267\) 0 0
\(268\) 1740.66 + 3014.92i 0.396746 + 0.687185i
\(269\) 2180.43 3776.61i 0.494212 0.856000i −0.505766 0.862671i \(-0.668790\pi\)
0.999978 + 0.00667106i \(0.00212348\pi\)
\(270\) 0 0
\(271\) −3737.79 + 2158.02i −0.837841 + 0.483727i −0.856530 0.516098i \(-0.827384\pi\)
0.0186891 + 0.999825i \(0.494051\pi\)
\(272\) 140.271 0.0312690
\(273\) 0 0
\(274\) 3492.52 0.770040
\(275\) 2668.75 1540.81i 0.585207 0.337869i
\(276\) 0 0
\(277\) 1319.46 2285.38i 0.286205 0.495722i −0.686696 0.726945i \(-0.740939\pi\)
0.972901 + 0.231223i \(0.0742728\pi\)
\(278\) 583.065 + 1009.90i 0.125791 + 0.217877i
\(279\) 0 0
\(280\) 0 0
\(281\) 5925.56i 1.25797i 0.777417 + 0.628985i \(0.216529\pi\)
−0.777417 + 0.628985i \(0.783471\pi\)
\(282\) 0 0
\(283\) −2663.30 1537.65i −0.559422 0.322983i 0.193491 0.981102i \(-0.438019\pi\)
−0.752914 + 0.658119i \(0.771352\pi\)
\(284\) 2497.31 + 1441.82i 0.521789 + 0.301255i
\(285\) 0 0
\(286\) 1320.45i 0.273007i
\(287\) 0 0
\(288\) 0 0
\(289\) 2297.20 + 3978.87i 0.467576 + 0.809866i
\(290\) 337.419 584.426i 0.0683238 0.118340i
\(291\) 0 0
\(292\) 2940.77 1697.85i 0.589368 0.340272i
\(293\) 1557.31 0.310509 0.155254 0.987875i \(-0.450380\pi\)
0.155254 + 0.987875i \(0.450380\pi\)
\(294\) 0 0
\(295\) −3622.20 −0.714890
\(296\) −4042.96 + 2334.20i −0.793892 + 0.458354i
\(297\) 0 0
\(298\) −968.144 + 1676.87i −0.188198 + 0.325969i
\(299\) −1114.54 1930.43i −0.215570 0.373378i
\(300\) 0 0
\(301\) 0 0
\(302\) 1737.62i 0.331088i
\(303\) 0 0
\(304\) 843.205 + 486.825i 0.159083 + 0.0918464i
\(305\) 1478.05 + 853.353i 0.277485 + 0.160206i
\(306\) 0 0
\(307\) 1802.67i 0.335127i 0.985861 + 0.167563i \(0.0535898\pi\)
−0.985861 + 0.167563i \(0.946410\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −973.340 1685.87i −0.178329 0.308875i
\(311\) 4769.98 8261.85i 0.869714 1.50639i 0.00742422 0.999972i \(-0.497637\pi\)
0.862289 0.506416i \(-0.169030\pi\)
\(312\) 0 0
\(313\) −1095.23 + 632.332i −0.197783 + 0.114190i −0.595621 0.803266i \(-0.703094\pi\)
0.397838 + 0.917456i \(0.369761\pi\)
\(314\) 3784.27 0.680124
\(315\) 0 0
\(316\) −3210.10 −0.571463
\(317\) 8265.68 4772.19i 1.46450 0.845530i 0.465287 0.885160i \(-0.345951\pi\)
0.999214 + 0.0396301i \(0.0126180\pi\)
\(318\) 0 0
\(319\) 1323.26 2291.95i 0.232252 0.402271i
\(320\) 563.539 + 976.079i 0.0984463 + 0.170514i
\(321\) 0 0
\(322\) 0 0
\(323\) 2211.44i 0.380954i
\(324\) 0 0
\(325\) 2416.04 + 1394.90i 0.412363 + 0.238078i
\(326\) 5062.47 + 2922.82i 0.860074 + 0.496564i
\(327\) 0 0
\(328\) 9271.39i 1.56075i
\(329\) 0 0
\(330\) 0 0
\(331\) 1353.22 + 2343.85i 0.224712 + 0.389213i 0.956233 0.292606i \(-0.0945225\pi\)
−0.731521 + 0.681819i \(0.761189\pi\)
\(332\) 1589.89 2753.76i 0.262820 0.455218i
\(333\) 0 0
\(334\) 3157.27 1822.85i 0.517239 0.298628i
\(335\) −3057.97 −0.498731
\(336\) 0 0
\(337\) 10550.8 1.70545 0.852727 0.522357i \(-0.174947\pi\)
0.852727 + 0.522357i \(0.174947\pi\)
\(338\) −2048.40 + 1182.64i −0.329640 + 0.190318i
\(339\) 0 0
\(340\) −226.340 + 392.032i −0.0361029 + 0.0625321i
\(341\) −3817.16 6611.51i −0.606189 1.04995i
\(342\) 0 0
\(343\) 0 0
\(344\) 1353.78i 0.212183i
\(345\) 0 0
\(346\) −5677.19 3277.73i −0.882103 0.509283i
\(347\) 8636.09 + 4986.05i 1.33605 + 0.771369i 0.986219 0.165444i \(-0.0529057\pi\)
0.349831 + 0.936813i \(0.386239\pi\)
\(348\) 0 0
\(349\) 1653.11i 0.253549i −0.991932 0.126775i \(-0.959537\pi\)
0.991932 0.126775i \(-0.0404625\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2791.89 + 4835.69i 0.422751 + 0.732226i
\(353\) 1677.25 2905.09i 0.252893 0.438024i −0.711428 0.702759i \(-0.751951\pi\)
0.964321 + 0.264735i \(0.0852846\pi\)
\(354\) 0 0
\(355\) −2193.62 + 1266.49i −0.327959 + 0.189347i
\(356\) 8269.52 1.23113
\(357\) 0 0
\(358\) 3449.47 0.509246
\(359\) −937.988 + 541.548i −0.137897 + 0.0796150i −0.567361 0.823469i \(-0.692036\pi\)
0.429464 + 0.903084i \(0.358702\pi\)
\(360\) 0 0
\(361\) 4245.53 7353.48i 0.618973 1.07209i
\(362\) −3461.92 5996.23i −0.502637 0.870593i
\(363\) 0 0
\(364\) 0 0
\(365\) 2982.76i 0.427739i
\(366\) 0 0
\(367\) −1823.53 1052.81i −0.259366 0.149745i 0.364679 0.931133i \(-0.381179\pi\)
−0.624045 + 0.781388i \(0.714512\pi\)
\(368\) 558.594 + 322.504i 0.0791269 + 0.0456839i
\(369\) 0 0
\(370\) 1647.63i 0.231503i
\(371\) 0 0
\(372\) 0 0
\(373\) −1867.35 3234.34i −0.259216 0.448975i 0.706816 0.707397i \(-0.250131\pi\)
−0.966032 + 0.258422i \(0.916797\pi\)
\(374\) 433.917 751.567i 0.0599928 0.103911i
\(375\) 0 0
\(376\) −7950.66 + 4590.31i −1.09049 + 0.629594i
\(377\) 2395.91 0.327310
\(378\) 0 0
\(379\) −13141.6 −1.78111 −0.890555 0.454876i \(-0.849684\pi\)
−0.890555 + 0.454876i \(0.849684\pi\)
\(380\) −2721.17 + 1571.07i −0.367350 + 0.212090i
\(381\) 0 0
\(382\) −627.645 + 1087.11i −0.0840657 + 0.145606i
\(383\) −2671.88 4627.84i −0.356467 0.617419i 0.630901 0.775863i \(-0.282685\pi\)
−0.987368 + 0.158444i \(0.949352\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 7962.91i 1.05000i
\(387\) 0 0
\(388\) 3043.80 + 1757.34i 0.398262 + 0.229937i
\(389\) 1624.29 + 937.783i 0.211709 + 0.122230i 0.602105 0.798417i \(-0.294329\pi\)
−0.390397 + 0.920647i \(0.627662\pi\)
\(390\) 0 0
\(391\) 1465.00i 0.189484i
\(392\) 0 0
\(393\) 0 0
\(394\) −813.951 1409.81i −0.104077 0.180266i
\(395\) 1409.87 2441.96i 0.179590 0.311059i
\(396\) 0 0
\(397\) −8327.97 + 4808.16i −1.05282 + 0.607845i −0.923437 0.383750i \(-0.874633\pi\)
−0.129381 + 0.991595i \(0.541299\pi\)
\(398\) −3246.07 −0.408821
\(399\) 0 0
\(400\) −807.262 −0.100908
\(401\) 562.762 324.911i 0.0700823 0.0404620i −0.464549 0.885547i \(-0.653784\pi\)
0.534632 + 0.845085i \(0.320450\pi\)
\(402\) 0 0
\(403\) 3455.70 5985.45i 0.427148 0.739842i
\(404\) 797.339 + 1381.03i 0.0981908 + 0.170072i
\(405\) 0 0
\(406\) 0 0
\(407\) 6461.52i 0.786943i
\(408\) 0 0
\(409\) −13491.6 7789.38i −1.63109 0.941711i −0.983757 0.179505i \(-0.942550\pi\)
−0.647334 0.762206i \(-0.724116\pi\)
\(410\) 2833.78 + 1636.08i 0.341343 + 0.197074i
\(411\) 0 0
\(412\) 2402.74i 0.287317i
\(413\) 0 0
\(414\) 0 0
\(415\) 1396.54 + 2418.89i 0.165190 + 0.286117i
\(416\) −2527.52 + 4377.79i −0.297889 + 0.515959i
\(417\) 0 0
\(418\) 5216.77 3011.91i 0.610432 0.352433i
\(419\) 5391.59 0.628631 0.314315 0.949319i \(-0.398225\pi\)
0.314315 + 0.949319i \(0.398225\pi\)
\(420\) 0 0
\(421\) −12506.0 −1.44776 −0.723879 0.689927i \(-0.757642\pi\)
−0.723879 + 0.689927i \(0.757642\pi\)
\(422\) −3692.67 + 2131.97i −0.425963 + 0.245930i
\(423\) 0 0
\(424\) 7348.43 12727.8i 0.841677 1.45783i
\(425\) 916.765 + 1587.88i 0.104634 + 0.181232i
\(426\) 0 0
\(427\) 0 0
\(428\) 4066.57i 0.459264i
\(429\) 0 0
\(430\) 413.780 + 238.896i 0.0464052 + 0.0267920i
\(431\) 7648.94 + 4416.12i 0.854841 + 0.493543i 0.862281 0.506430i \(-0.169035\pi\)
−0.00744041 + 0.999972i \(0.502368\pi\)
\(432\) 0 0
\(433\) 5550.41i 0.616017i 0.951384 + 0.308009i \(0.0996626\pi\)
−0.951384 + 0.308009i \(0.900337\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −318.369 551.432i −0.0349705 0.0605707i
\(437\) 5084.44 8806.51i 0.556571 0.964010i
\(438\) 0 0
\(439\) −9848.19 + 5685.86i −1.07068 + 0.618157i −0.928367 0.371664i \(-0.878787\pi\)
−0.142313 + 0.989822i \(0.545454\pi\)
\(440\) −3068.91 −0.332510
\(441\) 0 0
\(442\) 785.657 0.0845473
\(443\) −6944.93 + 4009.66i −0.744839 + 0.430033i −0.823826 0.566843i \(-0.808165\pi\)
0.0789873 + 0.996876i \(0.474831\pi\)
\(444\) 0 0
\(445\) −3631.94 + 6290.71i −0.386900 + 0.670131i
\(446\) −4029.25 6978.87i −0.427782 0.740940i
\(447\) 0 0
\(448\) 0 0
\(449\) 1039.67i 0.109276i −0.998506 0.0546379i \(-0.982600\pi\)
0.998506 0.0546379i \(-0.0174005\pi\)
\(450\) 0 0
\(451\) 11113.3 + 6416.25i 1.16032 + 0.669910i
\(452\) 421.022 + 243.077i 0.0438124 + 0.0252951i
\(453\) 0 0
\(454\) 5543.69i 0.573079i
\(455\) 0 0
\(456\) 0 0
\(457\) −7151.36 12386.5i −0.732005 1.26787i −0.956025 0.293286i \(-0.905251\pi\)
0.224019 0.974585i \(-0.428082\pi\)
\(458\) −2592.70 + 4490.69i −0.264517 + 0.458158i
\(459\) 0 0
\(460\) −1802.68 + 1040.78i −0.182718 + 0.105492i
\(461\) 12732.1 1.28632 0.643159 0.765733i \(-0.277623\pi\)
0.643159 + 0.765733i \(0.277623\pi\)
\(462\) 0 0
\(463\) 12358.7 1.24051 0.620256 0.784400i \(-0.287029\pi\)
0.620256 + 0.784400i \(0.287029\pi\)
\(464\) −600.402 + 346.642i −0.0600711 + 0.0346820i
\(465\) 0 0
\(466\) −1740.41 + 3014.47i −0.173010 + 0.299662i
\(467\) −6288.06 10891.2i −0.623077 1.07920i −0.988909 0.148520i \(-0.952549\pi\)
0.365833 0.930681i \(-0.380784\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 3240.14i 0.317992i
\(471\) 0 0
\(472\) −14405.2 8316.82i −1.40477 0.811044i
\(473\) 1622.72 + 936.880i 0.157744 + 0.0910735i
\(474\) 0 0
\(475\) 12726.9i 1.22937i
\(476\) 0 0
\(477\) 0 0
\(478\) 2312.18 + 4004.82i 0.221249 + 0.383214i
\(479\) −6137.19 + 10629.9i −0.585418 + 1.01397i 0.409405 + 0.912353i \(0.365736\pi\)
−0.994823 + 0.101621i \(0.967597\pi\)
\(480\) 0 0
\(481\) 5065.96 2924.83i 0.480224 0.277257i
\(482\) −638.137 −0.0603036
\(483\) 0 0
\(484\) 2316.13 0.217518
\(485\) −2673.65 + 1543.63i −0.250318 + 0.144521i
\(486\) 0 0
\(487\) −3769.96 + 6529.76i −0.350787 + 0.607581i −0.986388 0.164437i \(-0.947419\pi\)
0.635601 + 0.772018i \(0.280753\pi\)
\(488\) 3918.72 + 6787.42i 0.363508 + 0.629614i
\(489\) 0 0
\(490\) 0 0
\(491\) 406.618i 0.0373735i 0.999825 + 0.0186868i \(0.00594853\pi\)
−0.999825 + 0.0186868i \(0.994051\pi\)
\(492\) 0 0
\(493\) 1363.69 + 787.326i 0.124579 + 0.0719258i
\(494\) 4722.78 + 2726.70i 0.430138 + 0.248340i
\(495\) 0 0
\(496\) 1999.89i 0.181044i
\(497\) 0 0
\(498\) 0 0
\(499\) −2752.48 4767.44i −0.246930 0.427695i 0.715742 0.698364i \(-0.246088\pi\)
−0.962672 + 0.270669i \(0.912755\pi\)
\(500\) 2887.66 5001.58i 0.258280 0.447355i
\(501\) 0 0
\(502\) 3537.49 2042.37i 0.314514 0.181585i
\(503\) −6535.76 −0.579354 −0.289677 0.957124i \(-0.593548\pi\)
−0.289677 + 0.957124i \(0.593548\pi\)
\(504\) 0 0
\(505\) −1400.75 −0.123431
\(506\) 3455.93 1995.28i 0.303626 0.175299i
\(507\) 0 0
\(508\) −873.135 + 1512.31i −0.0762581 + 0.132083i
\(509\) 8953.46 + 15507.9i 0.779677 + 1.35044i 0.932128 + 0.362128i \(0.117950\pi\)
−0.152452 + 0.988311i \(0.548717\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 2825.37i 0.243877i
\(513\) 0 0
\(514\) 5967.89 + 3445.56i 0.512125 + 0.295675i
\(515\) −1827.79 1055.28i −0.156392 0.0902932i
\(516\) 0 0
\(517\) 12706.9i 1.08094i
\(518\) 0 0
\(519\) 0 0
\(520\) −1389.15 2406.08i −0.117151 0.202911i
\(521\) 5445.98 9432.71i 0.457951 0.793195i −0.540901 0.841086i \(-0.681917\pi\)
0.998853 + 0.0478914i \(0.0152501\pi\)
\(522\) 0 0
\(523\) 2311.20 1334.37i 0.193235 0.111564i −0.400261 0.916401i \(-0.631081\pi\)
0.593496 + 0.804837i \(0.297747\pi\)
\(524\) 3642.28 0.303652
\(525\) 0 0
\(526\) 11501.6 0.953411
\(527\) 3933.79 2271.17i 0.325158 0.187730i
\(528\) 0 0
\(529\) −2715.24 + 4702.93i −0.223164 + 0.386532i
\(530\) 2593.49 + 4492.06i 0.212555 + 0.368156i
\(531\) 0 0
\(532\) 0 0
\(533\) 11617.4i 0.944097i
\(534\) 0 0
\(535\) −3093.48 1786.02i −0.249987 0.144330i
\(536\) −12161.3 7021.32i −0.980013 0.565811i
\(537\) 0 0
\(538\) 7067.68i 0.566374i
\(539\) 0 0
\(540\) 0 0
\(541\) 5388.29 + 9332.80i 0.428209 + 0.741679i 0.996714 0.0810005i \(-0.0258115\pi\)
−0.568506 + 0.822679i \(0.692478\pi\)
\(542\) 3497.52 6057.89i 0.277180 0.480089i
\(543\) 0 0
\(544\) −2877.19 + 1661.15i −0.226762 + 0.130921i
\(545\) 559.307 0.0439598
\(546\) 0 0
\(547\) −6796.09 −0.531225 −0.265612 0.964080i \(-0.585574\pi\)
−0.265612 + 0.964080i \(0.585574\pi\)
\(548\) 10027.8 5789.54i 0.781689 0.451308i
\(549\) 0 0
\(550\) −2497.20 + 4325.28i −0.193602 + 0.335328i
\(551\) 5464.99 + 9465.64i 0.422534 + 0.731851i
\(552\) 0 0
\(553\) 0 0
\(554\) 4276.94i 0.327996i
\(555\) 0 0
\(556\) 3348.21 + 1933.09i 0.255388 + 0.147448i
\(557\) −15641.9 9030.85i −1.18989 0.686983i −0.231608 0.972809i \(-0.574399\pi\)
−0.958282 + 0.285826i \(0.907732\pi\)
\(558\) 0 0
\(559\) 1696.33i 0.128349i
\(560\) 0 0
\(561\) 0 0
\(562\) −4801.82 8316.99i −0.360413 0.624254i
\(563\) −10523.3 + 18227.0i −0.787755 + 1.36443i 0.139585 + 0.990210i \(0.455423\pi\)
−0.927339 + 0.374221i \(0.877910\pi\)
\(564\) 0 0
\(565\) −369.823 + 213.517i −0.0275373 + 0.0158986i
\(566\) 4984.19 0.370143
\(567\) 0 0
\(568\) −11631.8 −0.859257
\(569\) −10497.9 + 6060.98i −0.773455 + 0.446554i −0.834106 0.551605i \(-0.814016\pi\)
0.0606509 + 0.998159i \(0.480682\pi\)
\(570\) 0 0
\(571\) 1472.07 2549.70i 0.107888 0.186868i −0.807026 0.590516i \(-0.798924\pi\)
0.914915 + 0.403647i \(0.132258\pi\)
\(572\) −2188.91 3791.30i −0.160005 0.277137i
\(573\) 0 0
\(574\) 0 0
\(575\) 8431.12i 0.611482i
\(576\) 0 0
\(577\) 4495.86 + 2595.69i 0.324376 + 0.187279i 0.653342 0.757063i \(-0.273367\pi\)
−0.328965 + 0.944342i \(0.606700\pi\)
\(578\) −6448.60 3723.10i −0.464060 0.267925i
\(579\) 0 0
\(580\) 2237.35i 0.160174i
\(581\) 0 0
\(582\) 0 0
\(583\) 10170.9 + 17616.6i 0.722533 + 1.25146i
\(584\) −6848.63 + 11862.2i −0.485271 + 0.840514i
\(585\) 0 0
\(586\) −2185.81 + 1261.98i −0.154087 + 0.0889620i
\(587\) −21918.4 −1.54118 −0.770588 0.637334i \(-0.780037\pi\)
−0.770588 + 0.637334i \(0.780037\pi\)
\(588\) 0 0
\(589\) 31529.3 2.20568
\(590\) 5084.04 2935.27i 0.354757 0.204819i
\(591\) 0 0
\(592\) −846.334 + 1465.89i −0.0587569 + 0.101770i
\(593\) 10119.7 + 17527.9i 0.700788 + 1.21380i 0.968190 + 0.250216i \(0.0805015\pi\)
−0.267402 + 0.963585i \(0.586165\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6419.55i 0.441200i
\(597\) 0 0
\(598\) 3128.68 + 1806.34i 0.213948 + 0.123523i
\(599\) −7376.06 4258.57i −0.503135 0.290485i 0.226872 0.973925i \(-0.427150\pi\)
−0.730007 + 0.683439i \(0.760483\pi\)
\(600\) 0 0
\(601\) 5040.33i 0.342096i −0.985263 0.171048i \(-0.945285\pi\)
0.985263 0.171048i \(-0.0547152\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −2880.44 4989.07i −0.194046 0.336097i
\(605\) −1017.24 + 1761.90i −0.0683579 + 0.118399i
\(606\) 0 0
\(607\) 23489.5 13561.6i 1.57069 0.906837i 0.574604 0.818432i \(-0.305156\pi\)
0.996085 0.0884057i \(-0.0281772\pi\)
\(608\) −23060.7 −1.53822
\(609\) 0 0
\(610\) −2766.08 −0.183599
\(611\) 9962.44 5751.81i 0.659635 0.380840i
\(612\) 0 0
\(613\) −1467.22 + 2541.30i −0.0966730 + 0.167443i −0.910306 0.413937i \(-0.864153\pi\)
0.813633 + 0.581379i \(0.197487\pi\)
\(614\) −1460.80 2530.19i −0.0960151 0.166303i
\(615\) 0 0
\(616\) 0 0
\(617\) 2908.86i 0.189799i −0.995487 0.0948996i \(-0.969747\pi\)
0.995487 0.0948996i \(-0.0302530\pi\)
\(618\) 0 0
\(619\) 13814.4 + 7975.77i 0.897010 + 0.517889i 0.876229 0.481895i \(-0.160051\pi\)
0.0207811 + 0.999784i \(0.493385\pi\)
\(620\) −5589.33 3227.00i −0.362053 0.209032i
\(621\) 0 0
\(622\) 15461.5i 0.996706i
\(623\) 0 0
\(624\) 0 0
\(625\) −3883.68 6726.73i −0.248556 0.430511i
\(626\) 1024.83 1775.05i 0.0654319 0.113331i
\(627\) 0 0
\(628\) 10865.5 6273.18i 0.690412 0.398610i
\(629\) 3844.54 0.243707
\(630\) 0 0
\(631\) −31218.8 −1.96957 −0.984786 0.173771i \(-0.944405\pi\)
−0.984786 + 0.173771i \(0.944405\pi\)
\(632\) 11213.8 6474.30i 0.705793 0.407490i
\(633\) 0 0
\(634\) −7734.35 + 13396.3i −0.484496 + 0.839171i
\(635\) −766.955 1328.41i −0.0479302 0.0830176i
\(636\) 0 0
\(637\) 0 0
\(638\) 4289.24i 0.266164i
\(639\) 0 0
\(640\) 4504.56 + 2600.71i 0.278216 + 0.160628i
\(641\) −4720.80 2725.55i −0.290890 0.167945i 0.347453 0.937697i \(-0.387047\pi\)
−0.638343 + 0.769752i \(0.720380\pi\)
\(642\) 0 0
\(643\) 15700.8i 0.962956i −0.876458 0.481478i \(-0.840100\pi\)
0.876458 0.481478i \(-0.159900\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 1792.06 + 3103.93i 0.109145 + 0.189044i
\(647\) −11372.0 + 19696.9i −0.691006 + 1.19686i 0.280502 + 0.959853i \(0.409499\pi\)
−0.971509 + 0.237004i \(0.923834\pi\)
\(648\) 0 0
\(649\) 19938.1 11511.3i 1.20592 0.696236i
\(650\) −4521.47 −0.272841
\(651\) 0 0
\(652\) 19380.6 1.16411
\(653\) −4984.17 + 2877.61i −0.298691 + 0.172450i −0.641855 0.766826i \(-0.721835\pi\)
0.343163 + 0.939276i \(0.388502\pi\)
\(654\) 0 0
\(655\) −1599.68 + 2770.72i −0.0954268 + 0.165284i
\(656\) −1680.81 2911.24i −0.100037 0.173270i
\(657\) 0 0
\(658\) 0 0
\(659\) 24579.2i 1.45292i −0.687211 0.726458i \(-0.741165\pi\)
0.687211 0.726458i \(-0.258835\pi\)
\(660\) 0 0
\(661\) 4251.57 + 2454.64i 0.250177 + 0.144440i 0.619845 0.784724i \(-0.287195\pi\)
−0.369669 + 0.929164i \(0.620529\pi\)
\(662\) −3798.70 2193.18i −0.223022 0.128762i
\(663\) 0 0
\(664\) 12826.2i 0.749631i
\(665\) 0 0
\(666\) 0 0
\(667\) 3620.36 + 6270.65i 0.210166 + 0.364019i
\(668\) 6043.46 10467.6i 0.350043 0.606291i
\(669\) 0 0
\(670\) 4292.10 2478.05i 0.247490 0.142888i
\(671\) −10847.8 −0.624103
\(672\) 0 0
\(673\) 2635.70 0.150964 0.0754819 0.997147i \(-0.475950\pi\)
0.0754819 + 0.997147i \(0.475950\pi\)
\(674\) −14808.8 + 8549.89i −0.846314 + 0.488619i
\(675\) 0 0
\(676\) −3920.93 + 6791.25i −0.223084 + 0.386393i
\(677\) −3191.45 5527.76i −0.181178 0.313809i 0.761104 0.648630i \(-0.224658\pi\)
−0.942282 + 0.334821i \(0.891324\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 1825.97i 0.102975i
\(681\) 0 0
\(682\) 10715.3 + 6186.51i 0.601630 + 0.347351i
\(683\) −11952.8 6900.95i −0.669636 0.386614i 0.126303 0.991992i \(-0.459689\pi\)
−0.795939 + 0.605377i \(0.793022\pi\)
\(684\) 0 0
\(685\) 10171.0i 0.567318i
\(686\) 0 0
\(687\) 0 0
\(688\) −245.426 425.091i −0.0136000 0.0235559i
\(689\) −9207.82 + 15948.4i −0.509129 + 0.881838i
\(690\) 0 0
\(691\) −1516.12 + 875.333i −0.0834674 + 0.0481899i −0.541153 0.840924i \(-0.682012\pi\)
0.457685 + 0.889114i \(0.348679\pi\)
\(692\) −21733.9 −1.19393
\(693\) 0 0
\(694\) −16161.9 −0.884001
\(695\) −2941.04 + 1698.01i −0.160518 + 0.0926752i
\(696\) 0 0
\(697\) −3817.61 + 6612.29i −0.207464 + 0.359338i
\(698\) 1339.60 + 2320.26i 0.0726429 + 0.125821i
\(699\) 0 0
\(700\) 0 0
\(701\) 26927.8i 1.45085i 0.688300 + 0.725426i \(0.258357\pi\)
−0.688300 + 0.725426i \(0.741643\pi\)
\(702\) 0 0
\(703\) 23110.5 + 13342.9i 1.23987 + 0.715841i
\(704\) −6203.92 3581.83i −0.332129 0.191755i
\(705\) 0 0
\(706\) 5436.69i 0.289819i
\(707\) 0 0
\(708\) 0 0
\(709\) 5801.80 + 10049.0i 0.307322 + 0.532297i 0.977776 0.209654i \(-0.0672338\pi\)
−0.670454 + 0.741952i \(0.733900\pi\)
\(710\) 2052.61 3555.23i 0.108497 0.187923i
\(711\) 0 0
\(712\) −28887.8 + 16678.4i −1.52053 + 0.877877i
\(713\) 20887.1 1.09709
\(714\) 0 0
\(715\) 3845.44 0.201135
\(716\) 9904.17 5718.17i 0.516950 0.298461i
\(717\) 0 0
\(718\) 877.693 1520.21i 0.0456201 0.0790163i
\(719\) 6750.67 + 11692.5i 0.350150 + 0.606477i 0.986275 0.165108i \(-0.0527974\pi\)
−0.636126 + 0.771585i \(0.719464\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 13761.6i 0.709353i
\(723\) 0 0
\(724\) −19879.8 11477.6i −1.02048 0.589175i
\(725\) −7848.06 4531.08i −0.402027 0.232110i
\(726\) 0 0
\(727\) 1116.75i 0.0569708i 0.999594 + 0.0284854i \(0.00906842\pi\)
−0.999594 + 0.0284854i \(0.990932\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −2417.10 4186.54i −0.122549 0.212261i
\(731\) −557.435 + 965.505i −0.0282045 + 0.0488516i
\(732\) 0 0
\(733\) 25019.4 14444.9i 1.26072 0.727879i 0.287510 0.957778i \(-0.407173\pi\)
0.973215 + 0.229898i \(0.0738394\pi\)
\(734\) 3412.62 0.171610
\(735\) 0 0
\(736\) −15276.9 −0.765101
\(737\) 16832.4 9718.18i 0.841287 0.485717i
\(738\) 0 0
\(739\) 996.244 1725.55i 0.0495906 0.0858934i −0.840165 0.542332i \(-0.817542\pi\)
0.889755 + 0.456438i \(0.150875\pi\)
\(740\) −2731.27 4730.69i −0.135680 0.235005i
\(741\) 0 0
\(742\) 0 0
\(743\) 32395.7i 1.59957i −0.600284 0.799787i \(-0.704946\pi\)
0.600284 0.799787i \(-0.295054\pi\)
\(744\) 0 0
\(745\) −4883.42 2819.44i −0.240154 0.138653i
\(746\) 5241.93 + 3026.43i 0.257266 + 0.148533i
\(747\) 0 0
\(748\) 2877.21i 0.140643i
\(749\) 0 0
\(750\) 0 0
\(751\) 3796.22 + 6575.24i 0.184455 + 0.319486i 0.943393 0.331677i \(-0.107615\pi\)
−0.758937 + 0.651163i \(0.774281\pi\)
\(752\) −1664.35 + 2882.75i −0.0807084 + 0.139791i
\(753\) 0 0
\(754\) −3362.85 + 1941.54i −0.162424 + 0.0937755i
\(755\) 5060.32 0.243926
\(756\) 0 0
\(757\) 29398.9 1.41152 0.705760 0.708451i \(-0.250605\pi\)
0.705760 + 0.708451i \(0.250605\pi\)
\(758\) 18445.3 10649.4i 0.883857 0.510295i
\(759\) 0 0
\(760\) 6337.22 10976.4i 0.302467 0.523888i
\(761\) 16277.5 + 28193.4i 0.775373 + 1.34299i 0.934585 + 0.355741i \(0.115771\pi\)
−0.159212 + 0.987244i \(0.550895\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 4161.78i 0.197078i
\(765\) 0 0
\(766\) 7500.39 + 4330.35i 0.353786 + 0.204258i
\(767\) 18050.1 + 10421.2i 0.849742 + 0.490599i
\(768\) 0 0
\(769\) 34767.8i 1.63038i −0.579195 0.815189i \(-0.696633\pi\)
0.579195 0.815189i \(-0.303367\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −13200.1 22863.2i −0.615391 1.06589i
\(773\) 2994.63 5186.85i 0.139339 0.241343i −0.787907 0.615794i \(-0.788835\pi\)
0.927247 + 0.374451i \(0.122169\pi\)
\(774\) 0 0
\(775\) −22639.0 + 13070.6i −1.04931 + 0.605821i
\(776\) −14177.2 −0.655838
\(777\) 0 0
\(778\) −3039.75 −0.140078
\(779\) −45897.2 + 26498.8i −2.11096 + 1.21876i
\(780\) 0 0
\(781\) 8049.74 13942.6i 0.368812 0.638802i
\(782\) 1187.17 + 2056.25i 0.0542880 + 0.0940296i
\(783\) 0 0
\(784\) 0 0
\(785\) 11020.6i 0.501074i
\(786\) 0 0
\(787\) −36037.5 20806.3i −1.63227 0.942393i −0.983390 0.181504i \(-0.941903\pi\)
−0.648882 0.760889i \(-0.724763\pi\)
\(788\) −4674.06 2698.57i −0.211302 0.121996i
\(789\) 0 0
\(790\) 4569.97i 0.205813i
\(791\) 0 0
\(792\) 0 0
\(793\) −4910.28 8504.86i −0.219885 0.380853i
\(794\) 7792.64 13497.2i 0.348300 0.603274i
\(795\) 0 0
\(796\) −9320.15 + 5380.99i −0.415005 + 0.239603i
\(797\) 2077.70 0.0923412 0.0461706 0.998934i \(-0.485298\pi\)
0.0461706 + 0.998934i \(0.485298\pi\)
\(798\) 0 0
\(799\) 7560.47 0.334756
\(800\) 16558.3 9559.94i 0.731780 0.422494i
\(801\) 0 0
\(802\) −526.587 + 912.075i −0.0231851 + 0.0401577i
\(803\) −9479.16 16418.4i −0.416578 0.721535i
\(804\) 0 0
\(805\) 0 0
\(806\) 11201.4i 0.489519i
\(807\) 0 0
\(808\) −5570.67 3216.23i −0.242544 0.140033i
\(809\) 21392.6 + 12351.0i 0.929697 + 0.536761i 0.886716 0.462315i \(-0.152981\pi\)
0.0429815 + 0.999076i \(0.486314\pi\)
\(810\) 0 0
\(811\) 30991.5i 1.34187i 0.741516 + 0.670936i \(0.234107\pi\)
−0.741516 + 0.670936i \(0.765893\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 5236.13 + 9069.24i 0.225462 + 0.390512i
\(815\) −8511.88 + 14743.0i −0.365838 + 0.633650i
\(816\) 0 0
\(817\) −6701.77 + 3869.27i −0.286983 + 0.165690i
\(818\) 25248.7 1.07922
\(819\) 0 0
\(820\) 10848.5 0.462008
\(821\) −24401.5 + 14088.2i −1.03730 + 0.598883i −0.919066 0.394103i \(-0.871055\pi\)
−0.118230 + 0.992986i \(0.537722\pi\)
\(822\) 0 0
\(823\) −8539.10 + 14790.2i −0.361670 + 0.626431i −0.988236 0.152938i \(-0.951127\pi\)
0.626566 + 0.779368i \(0.284460\pi\)
\(824\) −4845.97 8393.47i −0.204876 0.354855i
\(825\) 0 0
\(826\) 0 0
\(827\) 19778.2i 0.831626i −0.909450 0.415813i \(-0.863497\pi\)
0.909450 0.415813i \(-0.136503\pi\)
\(828\) 0 0
\(829\) −10941.2 6316.89i −0.458387 0.264650i 0.252979 0.967472i \(-0.418590\pi\)
−0.711366 + 0.702822i \(0.751923\pi\)
\(830\) −3920.32 2263.40i −0.163947 0.0946549i
\(831\) 0 0
\(832\) 6485.32i 0.270238i
\(833\) 0 0
\(834\) 0 0
\(835\) 5308.53 + 9194.64i 0.220011 + 0.381070i
\(836\) 9985.65 17295.6i 0.413111 0.715529i
\(837\) 0 0
\(838\) −7567.51 + 4369.10i −0.311951 + 0.180105i
\(839\) 39650.4 1.63157 0.815784 0.578357i \(-0.196306\pi\)
0.815784 + 0.578357i \(0.196306\pi\)
\(840\) 0 0
\(841\) 16606.3 0.680894
\(842\) 17553.2 10134.3i 0.718434 0.414788i
\(843\) 0 0
\(844\) −7068.30 + 12242.7i −0.288271 + 0.499301i
\(845\) −3444.11 5965.38i −0.140214 0.242858i
\(846\) 0 0
\(847\) 0 0
\(848\) 5328.78i 0.215791i
\(849\) 0 0
\(850\) −2573.50 1485.81i −0.103847 0.0599564i
\(851\) 15309.9 + 8839.18i 0.616706 + 0.356056i
\(852\) 0 0
\(853\) 27542.7i 1.10556i −0.833327 0.552780i \(-0.813567\pi\)
0.833327 0.552780i \(-0.186433\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −8201.67 14205.7i −0.327485 0.567221i
\(857\) 20097.4 34809.8i 0.801068 1.38749i −0.117846 0.993032i \(-0.537599\pi\)
0.918914 0.394458i \(-0.129068\pi\)
\(858\) 0 0
\(859\) 10338.3 5968.81i 0.410638 0.237082i −0.280426 0.959876i \(-0.590476\pi\)
0.691064 + 0.722794i \(0.257142\pi\)
\(860\) 1584.07 0.0628095
\(861\) 0 0
\(862\) −14314.5 −0.565608
\(863\) −28785.5 + 16619.3i −1.13542 + 0.655536i −0.945293 0.326223i \(-0.894224\pi\)
−0.190129 + 0.981759i \(0.560891\pi\)
\(864\) 0 0
\(865\) 9545.45 16533.2i 0.375208 0.649880i
\(866\) −4497.80 7790.43i −0.176491 0.305692i
\(867\) 0 0
\(868\) 0 0
\(869\) 17922.1i 0.699615i
\(870\) 0 0
\(871\) 15238.5 + 8797.94i 0.592809 + 0.342258i
\(872\) 2224.31 + 1284.21i 0.0863815 + 0.0498724i
\(873\) 0 0
\(874\) 16480.8i 0.637840i
\(875\) 0 0
\(876\) 0 0
\(877\) −10709.6 18549.6i −0.412358 0.714226i 0.582789 0.812624i \(-0.301961\pi\)
−0.995147 + 0.0983981i \(0.968628\pi\)
\(878\) 9215.13 15961.1i 0.354209 0.613508i
\(879\) 0 0
\(880\) −963.646 + 556.362i −0.0369142 + 0.0213124i
\(881\) 31353.3 1.19900 0.599501 0.800374i \(-0.295366\pi\)
0.599501 + 0.800374i \(0.295366\pi\)
\(882\) 0 0
\(883\) −35483.7 −1.35235 −0.676173 0.736743i \(-0.736363\pi\)
−0.676173 + 0.736743i \(0.736363\pi\)
\(884\) 2255.79 1302.38i 0.0858262 0.0495518i
\(885\) 0 0
\(886\) 6498.50 11255.7i 0.246412 0.426799i
\(887\) −10127.6 17541.6i −0.383374 0.664024i 0.608168 0.793808i \(-0.291905\pi\)
−0.991542 + 0.129785i \(0.958571\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 11772.7i 0.443394i
\(891\) 0 0
\(892\) −23137.7 13358.6i −0.868506 0.501432i
\(893\) 45447.9 + 26239.4i 1.70309 + 0.983278i
\(894\) 0 0
\(895\) 10045.6i 0.375182i
\(896\) 0 0
\(897\) 0 0
\(898\) 842.499 + 1459.25i 0.0313080 + 0.0542270i
\(899\) −11225.2 + 19442.6i −0.416442 + 0.721298i
\(900\) 0 0
\(901\) −10481.7 + 6051.61i −0.387565 + 0.223761i
\(902\) −20797.8 −0.767727
\(903\) 0 0
\(904\) −1961.00 −0.0721481
\(905\) 17462.3 10081.9i 0.641400 0.370312i
\(906\) 0 0
\(907\) 12848.0 22253.5i 0.470355 0.814680i −0.529070 0.848578i \(-0.677459\pi\)
0.999425 + 0.0338987i \(0.0107924\pi\)
\(908\) −9189.75 15917.1i −0.335873 0.581749i
\(909\) 0 0
\(910\) 0 0
\(911\) 33200.2i 1.20743i −0.797199 0.603716i \(-0.793686\pi\)
0.797199 0.603716i \(-0.206314\pi\)
\(912\) 0 0
\(913\) −15374.3 8876.38i −0.557302 0.321758i
\(914\) 20075.0 + 11590.3i 0.726500 + 0.419445i
\(915\) 0 0
\(916\) 17191.6i 0.620118i
\(917\) 0 0
\(918\) 0 0
\(919\) −1068.12 1850.03i −0.0383394 0.0664058i 0.846219 0.532835i \(-0.178873\pi\)
−0.884558 + 0.466429i \(0.845540\pi\)
\(920\) 4198.18 7271.46i 0.150446 0.260579i
\(921\) 0 0
\(922\) −17870.5 + 10317.5i −0.638322 + 0.368535i
\(923\) 14575.0 0.519763
\(924\) 0 0
\(925\) −22125.4 −0.786464
\(926\) −17346.4 + 10014.9i −0.615591 + 0.355412i
\(927\) 0 0
\(928\) 8210.17 14220.4i 0.290422 0.503026i
\(929\) −526.275 911.535i −0.0185861 0.0321921i 0.856583 0.516010i \(-0.172583\pi\)
−0.875169 + 0.483818i \(0.839250\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 11540.3i 0.405594i
\(933\) 0 0
\(934\) 17651.6 + 10191.1i 0.618391 + 0.357028i
\(935\) 2188.72 + 1263.66i 0.0765550 + 0.0441990i
\(936\) 0 0
\(937\) 14333.0i 0.499721i −0.968282 0.249861i \(-0.919615\pi\)
0.968282 0.249861i \(-0.0803848\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −5371.16 9303.12i −0.186370 0.322803i
\(941\) 6563.30 11368.0i 0.227372 0.393821i −0.729656 0.683814i \(-0.760320\pi\)
0.957029 + 0.289994i \(0.0936532\pi\)
\(942\) 0 0
\(943\) −30405.3 + 17554.5i −1.04998 + 0.606207i
\(944\) −6031.02 −0.207937
\(945\) 0 0
\(946\) −3036.82 −0.104372
\(947\) −7602.28 + 4389.18i −0.260867 + 0.150611i −0.624730 0.780841i \(-0.714791\pi\)
0.363863 + 0.931452i \(0.381458\pi\)
\(948\) 0 0
\(949\) 8581.56 14863.7i 0.293540 0.508425i
\(950\) −10313.3 17863.2i −0.352219 0.610061i
\(951\) 0 0
\(952\) 0 0
\(953\) 4130.75i 0.140407i −0.997533 0.0702036i \(-0.977635\pi\)
0.997533 0.0702036i \(-0.0223649\pi\)
\(954\) 0 0
\(955\) −3165.91 1827.84i −0.107274 0.0619344i
\(956\) 13277.5 + 7665.80i 0.449191 + 0.259341i
\(957\) 0 0
\(958\) 19893.2i 0.670899i
\(959\) 0 0
\(960\) 0 0
\(961\) 17485.4 + 30285.6i 0.586936 + 1.01660i
\(962\) −4740.31 + 8210.45i −0.158871 + 0.275172i
\(963\) 0 0
\(964\) −1832.23 + 1057.84i −0.0612158 + 0.0353430i
\(965\) 23189.7 0.773579
\(966\) 0 0
\(967\) 2697.99 0.0897222 0.0448611 0.998993i \(-0.485715\pi\)
0.0448611 + 0.998993i \(0.485715\pi\)
\(968\) −8090.91 + 4671.29i −0.268648 + 0.155104i
\(969\) 0 0
\(970\) 2501.79 4333.22i 0.0828119 0.143434i
\(971\) −8411.94 14569.9i −0.278014 0.481535i 0.692877 0.721056i \(-0.256343\pi\)
−0.970891 + 0.239521i \(0.923010\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 12220.0i 0.402007i
\(975\) 0 0
\(976\) 2460.98 + 1420.85i 0.0807110 + 0.0465985i
\(977\) −33352.8 19256.3i −1.09217 0.630565i −0.158017 0.987436i \(-0.550510\pi\)
−0.934154 + 0.356871i \(0.883843\pi\)
\(978\) 0 0
\(979\) 46169.0i 1.50722i
\(980\) 0 0
\(981\) 0 0
\(982\) −329.505 570.719i −0.0107077 0.0185462i
\(983\) −13465.4 + 23322.7i −0.436905 + 0.756742i −0.997449 0.0713826i \(-0.977259\pi\)
0.560544 + 0.828125i \(0.310592\pi\)
\(984\) 0 0
\(985\) 4105.66 2370.40i 0.132809 0.0766774i
\(986\) −2552.06 −0.0824281
\(987\) 0 0
\(988\) 18080.2 0.582193
\(989\) −4439.68 + 2563.25i −0.142744 + 0.0824132i
\(990\) 0 0
\(991\) 5796.70 10040.2i 0.185810 0.321833i −0.758039 0.652209i \(-0.773842\pi\)
0.943849 + 0.330376i \(0.107176\pi\)
\(992\) −23683.6 41021.2i −0.758018 1.31293i
\(993\) 0 0
\(994\) 0 0
\(995\) 9453.25i 0.301194i
\(996\) 0 0
\(997\) −43168.7 24923.4i −1.37128 0.791708i −0.380190 0.924908i \(-0.624141\pi\)
−0.991089 + 0.133200i \(0.957475\pi\)
\(998\) 7726.65 + 4460.98i 0.245073 + 0.141493i
\(999\) 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.p.d.80.9 48
3.2 odd 2 inner 441.4.p.d.80.16 48
7.2 even 3 inner 441.4.p.d.215.15 48
7.3 odd 6 441.4.c.b.440.15 yes 24
7.4 even 3 441.4.c.b.440.9 24
7.5 odd 6 inner 441.4.p.d.215.16 48
7.6 odd 2 inner 441.4.p.d.80.10 48
21.2 odd 6 inner 441.4.p.d.215.10 48
21.5 even 6 inner 441.4.p.d.215.9 48
21.11 odd 6 441.4.c.b.440.16 yes 24
21.17 even 6 441.4.c.b.440.10 yes 24
21.20 even 2 inner 441.4.p.d.80.15 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
441.4.c.b.440.9 24 7.4 even 3
441.4.c.b.440.10 yes 24 21.17 even 6
441.4.c.b.440.15 yes 24 7.3 odd 6
441.4.c.b.440.16 yes 24 21.11 odd 6
441.4.p.d.80.9 48 1.1 even 1 trivial
441.4.p.d.80.10 48 7.6 odd 2 inner
441.4.p.d.80.15 48 21.20 even 2 inner
441.4.p.d.80.16 48 3.2 odd 2 inner
441.4.p.d.215.9 48 21.5 even 6 inner
441.4.p.d.215.10 48 21.2 odd 6 inner
441.4.p.d.215.15 48 7.2 even 3 inner
441.4.p.d.215.16 48 7.5 odd 6 inner