Properties

Label 252.4.e.a.71.18
Level $252$
Weight $4$
Character 252.71
Analytic conductor $14.868$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,4,Mod(71,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.71");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 252.e (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.8684813214\)
Analytic rank: \(0\)
Dimension: \(36\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 71.18
Character \(\chi\) \(=\) 252.71
Dual form 252.4.e.a.71.17

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.614971 + 2.76076i) q^{2} +(-7.24362 - 3.39558i) q^{4} -2.97985i q^{5} +7.00000i q^{7} +(13.8290 - 17.9097i) q^{8} +O(q^{10})\) \(q+(-0.614971 + 2.76076i) q^{2} +(-7.24362 - 3.39558i) q^{4} -2.97985i q^{5} +7.00000i q^{7} +(13.8290 - 17.9097i) q^{8} +(8.22665 + 1.83252i) q^{10} +14.5861 q^{11} +27.9716 q^{13} +(-19.3253 - 4.30480i) q^{14} +(40.9401 + 49.1926i) q^{16} -47.0690i q^{17} +148.649i q^{19} +(-10.1183 + 21.5849i) q^{20} +(-8.97001 + 40.2687i) q^{22} -85.1221 q^{23} +116.121 q^{25} +(-17.2017 + 77.2228i) q^{26} +(23.7690 - 50.7053i) q^{28} +197.212i q^{29} +213.574i q^{31} +(-160.986 + 82.7739i) q^{32} +(129.946 + 28.9461i) q^{34} +20.8589 q^{35} -3.00048 q^{37} +(-410.385 - 91.4149i) q^{38} +(-53.3683 - 41.2083i) q^{40} +4.26768i q^{41} +261.825i q^{43} +(-105.656 - 49.5281i) q^{44} +(52.3476 - 235.002i) q^{46} +96.0780 q^{47} -49.0000 q^{49} +(-71.4108 + 320.581i) q^{50} +(-202.615 - 94.9796i) q^{52} -131.677i q^{53} -43.4643i q^{55} +(125.368 + 96.8030i) q^{56} +(-544.455 - 121.280i) q^{58} -294.327 q^{59} +134.782 q^{61} +(-589.628 - 131.342i) q^{62} +(-129.517 - 495.348i) q^{64} -83.3510i q^{65} +148.648i q^{67} +(-159.827 + 340.950i) q^{68} +(-12.8276 + 57.5866i) q^{70} +796.550 q^{71} -163.335 q^{73} +(1.84521 - 8.28361i) q^{74} +(504.750 - 1076.76i) q^{76} +102.102i q^{77} +1347.64i q^{79} +(146.586 - 121.995i) q^{80} +(-11.7821 - 2.62450i) q^{82} +534.740 q^{83} -140.259 q^{85} +(-722.836 - 161.015i) q^{86} +(201.711 - 261.233i) q^{88} +398.405i q^{89} +195.801i q^{91} +(616.592 + 289.039i) q^{92} +(-59.0852 + 265.249i) q^{94} +442.952 q^{95} +1711.36 q^{97} +(30.1336 - 135.277i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q - 24 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 36 q - 24 q^{4} + 264 q^{10} - 468 q^{16} + 444 q^{22} - 900 q^{25} - 84 q^{28} - 432 q^{34} - 264 q^{37} + 1416 q^{40} + 180 q^{46} - 1764 q^{49} + 2736 q^{52} + 636 q^{58} - 3960 q^{61} + 1392 q^{64} - 504 q^{70} - 2520 q^{76} + 1032 q^{82} - 3144 q^{85} + 2748 q^{88} + 5496 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.614971 + 2.76076i −0.217425 + 0.976077i
\(3\) 0 0
\(4\) −7.24362 3.39558i −0.905453 0.424447i
\(5\) 2.97985i 0.266526i −0.991081 0.133263i \(-0.957455\pi\)
0.991081 0.133263i \(-0.0425454\pi\)
\(6\) 0 0
\(7\) 7.00000i 0.377964i
\(8\) 13.8290 17.9097i 0.611161 0.791506i
\(9\) 0 0
\(10\) 8.22665 + 1.83252i 0.260150 + 0.0579494i
\(11\) 14.5861 0.399806 0.199903 0.979816i \(-0.435937\pi\)
0.199903 + 0.979816i \(0.435937\pi\)
\(12\) 0 0
\(13\) 27.9716 0.596763 0.298381 0.954447i \(-0.403553\pi\)
0.298381 + 0.954447i \(0.403553\pi\)
\(14\) −19.3253 4.30480i −0.368922 0.0821790i
\(15\) 0 0
\(16\) 40.9401 + 49.1926i 0.639689 + 0.768634i
\(17\) 47.0690i 0.671525i −0.941947 0.335762i \(-0.891006\pi\)
0.941947 0.335762i \(-0.108994\pi\)
\(18\) 0 0
\(19\) 148.649i 1.79487i 0.441151 + 0.897433i \(0.354570\pi\)
−0.441151 + 0.897433i \(0.645430\pi\)
\(20\) −10.1183 + 21.5849i −0.113126 + 0.241326i
\(21\) 0 0
\(22\) −8.97001 + 40.2687i −0.0869279 + 0.390241i
\(23\) −85.1221 −0.771704 −0.385852 0.922561i \(-0.626092\pi\)
−0.385852 + 0.922561i \(0.626092\pi\)
\(24\) 0 0
\(25\) 116.121 0.928964
\(26\) −17.2017 + 77.2228i −0.129751 + 0.582486i
\(27\) 0 0
\(28\) 23.7690 50.7053i 0.160426 0.342229i
\(29\) 197.212i 1.26280i 0.775455 + 0.631402i \(0.217520\pi\)
−0.775455 + 0.631402i \(0.782480\pi\)
\(30\) 0 0
\(31\) 213.574i 1.23739i 0.785631 + 0.618695i \(0.212338\pi\)
−0.785631 + 0.618695i \(0.787662\pi\)
\(32\) −160.986 + 82.7739i −0.889330 + 0.457265i
\(33\) 0 0
\(34\) 129.946 + 28.9461i 0.655460 + 0.146006i
\(35\) 20.8589 0.100737
\(36\) 0 0
\(37\) −3.00048 −0.0133318 −0.00666589 0.999978i \(-0.502122\pi\)
−0.00666589 + 0.999978i \(0.502122\pi\)
\(38\) −410.385 91.4149i −1.75193 0.390249i
\(39\) 0 0
\(40\) −53.3683 41.2083i −0.210957 0.162890i
\(41\) 4.26768i 0.0162561i 0.999967 + 0.00812805i \(0.00258727\pi\)
−0.999967 + 0.00812805i \(0.997413\pi\)
\(42\) 0 0
\(43\) 261.825i 0.928556i 0.885689 + 0.464278i \(0.153686\pi\)
−0.885689 + 0.464278i \(0.846314\pi\)
\(44\) −105.656 49.5281i −0.362005 0.169697i
\(45\) 0 0
\(46\) 52.3476 235.002i 0.167788 0.753242i
\(47\) 96.0780 0.298179 0.149090 0.988824i \(-0.452366\pi\)
0.149090 + 0.988824i \(0.452366\pi\)
\(48\) 0 0
\(49\) −49.0000 −0.142857
\(50\) −71.4108 + 320.581i −0.201980 + 0.906740i
\(51\) 0 0
\(52\) −202.615 94.9796i −0.540340 0.253294i
\(53\) 131.677i 0.341268i −0.985334 0.170634i \(-0.945418\pi\)
0.985334 0.170634i \(-0.0545816\pi\)
\(54\) 0 0
\(55\) 43.4643i 0.106559i
\(56\) 125.368 + 96.8030i 0.299161 + 0.230997i
\(57\) 0 0
\(58\) −544.455 121.280i −1.23259 0.274565i
\(59\) −294.327 −0.649460 −0.324730 0.945807i \(-0.605273\pi\)
−0.324730 + 0.945807i \(0.605273\pi\)
\(60\) 0 0
\(61\) 134.782 0.282904 0.141452 0.989945i \(-0.454823\pi\)
0.141452 + 0.989945i \(0.454823\pi\)
\(62\) −589.628 131.342i −1.20779 0.269040i
\(63\) 0 0
\(64\) −129.517 495.348i −0.252963 0.967476i
\(65\) 83.3510i 0.159053i
\(66\) 0 0
\(67\) 148.648i 0.271049i 0.990774 + 0.135525i \(0.0432720\pi\)
−0.990774 + 0.135525i \(0.956728\pi\)
\(68\) −159.827 + 340.950i −0.285027 + 0.608034i
\(69\) 0 0
\(70\) −12.8276 + 57.5866i −0.0219028 + 0.0983273i
\(71\) 796.550 1.33145 0.665726 0.746197i \(-0.268122\pi\)
0.665726 + 0.746197i \(0.268122\pi\)
\(72\) 0 0
\(73\) −163.335 −0.261875 −0.130938 0.991391i \(-0.541799\pi\)
−0.130938 + 0.991391i \(0.541799\pi\)
\(74\) 1.84521 8.28361i 0.00289866 0.0130128i
\(75\) 0 0
\(76\) 504.750 1076.76i 0.761826 1.62517i
\(77\) 102.102i 0.151112i
\(78\) 0 0
\(79\) 1347.64i 1.91925i 0.281275 + 0.959627i \(0.409243\pi\)
−0.281275 + 0.959627i \(0.590757\pi\)
\(80\) 146.586 121.995i 0.204861 0.170494i
\(81\) 0 0
\(82\) −11.7821 2.62450i −0.0158672 0.00353448i
\(83\) 534.740 0.707173 0.353586 0.935402i \(-0.384962\pi\)
0.353586 + 0.935402i \(0.384962\pi\)
\(84\) 0 0
\(85\) −140.259 −0.178979
\(86\) −722.836 161.015i −0.906342 0.201891i
\(87\) 0 0
\(88\) 201.711 261.233i 0.244346 0.316449i
\(89\) 398.405i 0.474504i 0.971448 + 0.237252i \(0.0762466\pi\)
−0.971448 + 0.237252i \(0.923753\pi\)
\(90\) 0 0
\(91\) 195.801i 0.225555i
\(92\) 616.592 + 289.039i 0.698741 + 0.327548i
\(93\) 0 0
\(94\) −59.0852 + 265.249i −0.0648316 + 0.291046i
\(95\) 442.952 0.478378
\(96\) 0 0
\(97\) 1711.36 1.79137 0.895684 0.444691i \(-0.146686\pi\)
0.895684 + 0.444691i \(0.146686\pi\)
\(98\) 30.1336 135.277i 0.0310607 0.139440i
\(99\) 0 0
\(100\) −841.133 394.296i −0.841133 0.394296i
\(101\) 1871.61i 1.84388i −0.387333 0.921940i \(-0.626604\pi\)
0.387333 0.921940i \(-0.373396\pi\)
\(102\) 0 0
\(103\) 11.8187i 0.0113061i −0.999984 0.00565306i \(-0.998201\pi\)
0.999984 0.00565306i \(-0.00179944\pi\)
\(104\) 386.819 500.963i 0.364718 0.472341i
\(105\) 0 0
\(106\) 363.529 + 80.9775i 0.333104 + 0.0742003i
\(107\) 1350.29 1.21998 0.609990 0.792409i \(-0.291173\pi\)
0.609990 + 0.792409i \(0.291173\pi\)
\(108\) 0 0
\(109\) −571.162 −0.501903 −0.250951 0.968000i \(-0.580743\pi\)
−0.250951 + 0.968000i \(0.580743\pi\)
\(110\) 119.994 + 26.7293i 0.104009 + 0.0231685i
\(111\) 0 0
\(112\) −344.348 + 286.581i −0.290516 + 0.241780i
\(113\) 967.624i 0.805544i 0.915300 + 0.402772i \(0.131953\pi\)
−0.915300 + 0.402772i \(0.868047\pi\)
\(114\) 0 0
\(115\) 253.651i 0.205679i
\(116\) 669.649 1428.53i 0.535994 1.14341i
\(117\) 0 0
\(118\) 181.003 812.567i 0.141209 0.633923i
\(119\) 329.483 0.253812
\(120\) 0 0
\(121\) −1118.25 −0.840155
\(122\) −82.8873 + 372.102i −0.0615104 + 0.276136i
\(123\) 0 0
\(124\) 725.209 1547.05i 0.525207 1.12040i
\(125\) 718.502i 0.514118i
\(126\) 0 0
\(127\) 208.578i 0.145734i 0.997342 + 0.0728672i \(0.0232149\pi\)
−0.997342 + 0.0728672i \(0.976785\pi\)
\(128\) 1447.19 52.9421i 0.999332 0.0365583i
\(129\) 0 0
\(130\) 230.112 + 51.2584i 0.155248 + 0.0345820i
\(131\) 1680.47 1.12079 0.560393 0.828227i \(-0.310650\pi\)
0.560393 + 0.828227i \(0.310650\pi\)
\(132\) 0 0
\(133\) −1040.54 −0.678396
\(134\) −410.383 91.4145i −0.264565 0.0589329i
\(135\) 0 0
\(136\) −842.994 650.918i −0.531516 0.410410i
\(137\) 1504.86i 0.938460i −0.883076 0.469230i \(-0.844531\pi\)
0.883076 0.469230i \(-0.155469\pi\)
\(138\) 0 0
\(139\) 1613.01i 0.984273i −0.870518 0.492136i \(-0.836216\pi\)
0.870518 0.492136i \(-0.163784\pi\)
\(140\) −151.094 70.8281i −0.0912128 0.0427576i
\(141\) 0 0
\(142\) −489.855 + 2199.08i −0.289491 + 1.29960i
\(143\) 407.995 0.238589
\(144\) 0 0
\(145\) 587.661 0.336570
\(146\) 100.446 450.929i 0.0569382 0.255610i
\(147\) 0 0
\(148\) 21.7343 + 10.1884i 0.0120713 + 0.00565864i
\(149\) 50.6741i 0.0278616i 0.999903 + 0.0139308i \(0.00443446\pi\)
−0.999903 + 0.0139308i \(0.995566\pi\)
\(150\) 0 0
\(151\) 99.3618i 0.0535493i 0.999641 + 0.0267747i \(0.00852366\pi\)
−0.999641 + 0.0267747i \(0.991476\pi\)
\(152\) 2662.27 + 2055.67i 1.42065 + 1.09695i
\(153\) 0 0
\(154\) −281.881 62.7901i −0.147497 0.0328556i
\(155\) 636.419 0.329796
\(156\) 0 0
\(157\) 2638.39 1.34119 0.670593 0.741825i \(-0.266040\pi\)
0.670593 + 0.741825i \(0.266040\pi\)
\(158\) −3720.51 828.758i −1.87334 0.417294i
\(159\) 0 0
\(160\) 246.654 + 479.714i 0.121873 + 0.237029i
\(161\) 595.855i 0.291677i
\(162\) 0 0
\(163\) 2611.58i 1.25494i −0.778642 0.627468i \(-0.784091\pi\)
0.778642 0.627468i \(-0.215909\pi\)
\(164\) 14.4912 30.9135i 0.00689986 0.0147191i
\(165\) 0 0
\(166\) −328.850 + 1476.29i −0.153757 + 0.690255i
\(167\) −3438.19 −1.59314 −0.796572 0.604544i \(-0.793355\pi\)
−0.796572 + 0.604544i \(0.793355\pi\)
\(168\) 0 0
\(169\) −1414.59 −0.643874
\(170\) 86.2549 387.221i 0.0389144 0.174697i
\(171\) 0 0
\(172\) 889.047 1896.56i 0.394123 0.840764i
\(173\) 3976.76i 1.74767i 0.486219 + 0.873837i \(0.338376\pi\)
−0.486219 + 0.873837i \(0.661624\pi\)
\(174\) 0 0
\(175\) 812.844i 0.351115i
\(176\) 597.155 + 717.526i 0.255751 + 0.307304i
\(177\) 0 0
\(178\) −1099.90 245.007i −0.463152 0.103169i
\(179\) −3605.92 −1.50569 −0.752846 0.658197i \(-0.771319\pi\)
−0.752846 + 0.658197i \(0.771319\pi\)
\(180\) 0 0
\(181\) −1160.80 −0.476693 −0.238346 0.971180i \(-0.576605\pi\)
−0.238346 + 0.971180i \(0.576605\pi\)
\(182\) −540.560 120.412i −0.220159 0.0490413i
\(183\) 0 0
\(184\) −1177.15 + 1524.51i −0.471635 + 0.610808i
\(185\) 8.94097i 0.00355326i
\(186\) 0 0
\(187\) 686.552i 0.268480i
\(188\) −695.953 326.241i −0.269987 0.126561i
\(189\) 0 0
\(190\) −272.403 + 1222.88i −0.104011 + 0.466934i
\(191\) −4106.76 −1.55578 −0.777891 0.628399i \(-0.783710\pi\)
−0.777891 + 0.628399i \(0.783710\pi\)
\(192\) 0 0
\(193\) −2388.95 −0.890987 −0.445494 0.895285i \(-0.646972\pi\)
−0.445494 + 0.895285i \(0.646972\pi\)
\(194\) −1052.44 + 4724.67i −0.389488 + 1.74851i
\(195\) 0 0
\(196\) 354.937 + 166.383i 0.129350 + 0.0606353i
\(197\) 3236.10i 1.17037i 0.810901 + 0.585184i \(0.198978\pi\)
−0.810901 + 0.585184i \(0.801022\pi\)
\(198\) 0 0
\(199\) 3509.90i 1.25030i −0.780503 0.625152i \(-0.785037\pi\)
0.780503 0.625152i \(-0.214963\pi\)
\(200\) 1605.83 2079.69i 0.567747 0.735281i
\(201\) 0 0
\(202\) 5167.06 + 1150.98i 1.79977 + 0.400906i
\(203\) −1380.48 −0.477295
\(204\) 0 0
\(205\) 12.7170 0.00433267
\(206\) 32.6286 + 7.26816i 0.0110357 + 0.00245824i
\(207\) 0 0
\(208\) 1145.16 + 1375.99i 0.381743 + 0.458692i
\(209\) 2168.21i 0.717598i
\(210\) 0 0
\(211\) 485.243i 0.158320i −0.996862 0.0791599i \(-0.974776\pi\)
0.996862 0.0791599i \(-0.0252238\pi\)
\(212\) −447.119 + 953.818i −0.144850 + 0.309002i
\(213\) 0 0
\(214\) −830.392 + 3727.84i −0.265254 + 1.19079i
\(215\) 780.198 0.247484
\(216\) 0 0
\(217\) −1495.02 −0.467690
\(218\) 351.248 1576.84i 0.109126 0.489896i
\(219\) 0 0
\(220\) −147.586 + 314.839i −0.0452285 + 0.0964837i
\(221\) 1316.59i 0.400741i
\(222\) 0 0
\(223\) 96.8412i 0.0290806i 0.999894 + 0.0145403i \(0.00462848\pi\)
−0.999894 + 0.0145403i \(0.995372\pi\)
\(224\) −579.417 1126.90i −0.172830 0.336135i
\(225\) 0 0
\(226\) −2671.38 595.061i −0.786273 0.175145i
\(227\) −3635.58 −1.06300 −0.531502 0.847057i \(-0.678372\pi\)
−0.531502 + 0.847057i \(0.678372\pi\)
\(228\) 0 0
\(229\) 4587.11 1.32369 0.661845 0.749641i \(-0.269774\pi\)
0.661845 + 0.749641i \(0.269774\pi\)
\(230\) −700.270 155.988i −0.200758 0.0447197i
\(231\) 0 0
\(232\) 3532.01 + 2727.24i 0.999518 + 0.771778i
\(233\) 4490.81i 1.26267i −0.775509 0.631336i \(-0.782507\pi\)
0.775509 0.631336i \(-0.217493\pi\)
\(234\) 0 0
\(235\) 286.298i 0.0794724i
\(236\) 2131.99 + 999.411i 0.588055 + 0.275661i
\(237\) 0 0
\(238\) −202.623 + 909.625i −0.0551852 + 0.247741i
\(239\) −3218.42 −0.871057 −0.435528 0.900175i \(-0.643438\pi\)
−0.435528 + 0.900175i \(0.643438\pi\)
\(240\) 0 0
\(241\) 2254.22 0.602520 0.301260 0.953542i \(-0.402593\pi\)
0.301260 + 0.953542i \(0.402593\pi\)
\(242\) 687.689 3087.21i 0.182671 0.820056i
\(243\) 0 0
\(244\) −976.313 457.665i −0.256156 0.120078i
\(245\) 146.013i 0.0380751i
\(246\) 0 0
\(247\) 4157.95i 1.07111i
\(248\) 3825.06 + 2953.52i 0.979402 + 0.756245i
\(249\) 0 0
\(250\) 1983.61 + 441.858i 0.501819 + 0.111782i
\(251\) 3998.17 1.00543 0.502714 0.864453i \(-0.332335\pi\)
0.502714 + 0.864453i \(0.332335\pi\)
\(252\) 0 0
\(253\) −1241.60 −0.308532
\(254\) −575.834 128.269i −0.142248 0.0316863i
\(255\) 0 0
\(256\) −743.817 + 4027.90i −0.181596 + 0.983373i
\(257\) 3387.39i 0.822177i 0.911595 + 0.411088i \(0.134851\pi\)
−0.911595 + 0.411088i \(0.865149\pi\)
\(258\) 0 0
\(259\) 21.0033i 0.00503894i
\(260\) −283.025 + 603.763i −0.0675094 + 0.144015i
\(261\) 0 0
\(262\) −1033.44 + 4639.37i −0.243687 + 1.09397i
\(263\) −3331.72 −0.781151 −0.390575 0.920571i \(-0.627724\pi\)
−0.390575 + 0.920571i \(0.627724\pi\)
\(264\) 0 0
\(265\) −392.377 −0.0909567
\(266\) 639.905 2872.70i 0.147500 0.662166i
\(267\) 0 0
\(268\) 504.748 1076.75i 0.115046 0.245422i
\(269\) 5844.63i 1.32473i −0.749180 0.662367i \(-0.769552\pi\)
0.749180 0.662367i \(-0.230448\pi\)
\(270\) 0 0
\(271\) 6710.76i 1.50424i −0.659025 0.752121i \(-0.729031\pi\)
0.659025 0.752121i \(-0.270969\pi\)
\(272\) 2315.45 1927.01i 0.516157 0.429567i
\(273\) 0 0
\(274\) 4154.57 + 925.447i 0.916009 + 0.204045i
\(275\) 1693.74 0.371405
\(276\) 0 0
\(277\) −4400.29 −0.954469 −0.477235 0.878776i \(-0.658361\pi\)
−0.477235 + 0.878776i \(0.658361\pi\)
\(278\) 4453.14 + 991.956i 0.960726 + 0.214006i
\(279\) 0 0
\(280\) 288.458 373.578i 0.0615667 0.0797341i
\(281\) 4749.75i 1.00835i 0.863601 + 0.504175i \(0.168203\pi\)
−0.863601 + 0.504175i \(0.831797\pi\)
\(282\) 0 0
\(283\) 1497.32i 0.314510i −0.987558 0.157255i \(-0.949735\pi\)
0.987558 0.157255i \(-0.0502645\pi\)
\(284\) −5769.90 2704.75i −1.20557 0.565131i
\(285\) 0 0
\(286\) −250.905 + 1126.38i −0.0518753 + 0.232881i
\(287\) −29.8738 −0.00614423
\(288\) 0 0
\(289\) 2697.51 0.549055
\(290\) −361.395 + 1622.39i −0.0731787 + 0.328518i
\(291\) 0 0
\(292\) 1183.14 + 554.616i 0.237116 + 0.111152i
\(293\) 3651.08i 0.727982i 0.931403 + 0.363991i \(0.118586\pi\)
−0.931403 + 0.363991i \(0.881414\pi\)
\(294\) 0 0
\(295\) 877.050i 0.173098i
\(296\) −41.4936 + 53.7378i −0.00814786 + 0.0105522i
\(297\) 0 0
\(298\) −139.899 31.1631i −0.0271951 0.00605781i
\(299\) −2381.00 −0.460524
\(300\) 0 0
\(301\) −1832.77 −0.350961
\(302\) −274.314 61.1046i −0.0522683 0.0116430i
\(303\) 0 0
\(304\) −7312.43 + 6085.71i −1.37959 + 1.14816i
\(305\) 401.631i 0.0754011i
\(306\) 0 0
\(307\) 210.827i 0.0391940i −0.999808 0.0195970i \(-0.993762\pi\)
0.999808 0.0195970i \(-0.00623832\pi\)
\(308\) 346.697 739.592i 0.0641393 0.136825i
\(309\) 0 0
\(310\) −391.379 + 1757.00i −0.0717060 + 0.321907i
\(311\) −724.581 −0.132113 −0.0660566 0.997816i \(-0.521042\pi\)
−0.0660566 + 0.997816i \(0.521042\pi\)
\(312\) 0 0
\(313\) 8805.01 1.59006 0.795030 0.606571i \(-0.207455\pi\)
0.795030 + 0.606571i \(0.207455\pi\)
\(314\) −1622.53 + 7283.96i −0.291608 + 1.30910i
\(315\) 0 0
\(316\) 4576.01 9761.78i 0.814622 1.73779i
\(317\) 6348.48i 1.12481i −0.826860 0.562407i \(-0.809875\pi\)
0.826860 0.562407i \(-0.190125\pi\)
\(318\) 0 0
\(319\) 2876.55i 0.504877i
\(320\) −1476.06 + 385.942i −0.257857 + 0.0674213i
\(321\) 0 0
\(322\) 1645.01 + 366.433i 0.284699 + 0.0634178i
\(323\) 6996.77 1.20530
\(324\) 0 0
\(325\) 3248.07 0.554371
\(326\) 7209.95 + 1606.05i 1.22491 + 0.272855i
\(327\) 0 0
\(328\) 76.4331 + 59.0178i 0.0128668 + 0.00993510i
\(329\) 672.546i 0.112701i
\(330\) 0 0
\(331\) 5556.10i 0.922631i −0.887236 0.461315i \(-0.847378\pi\)
0.887236 0.461315i \(-0.152622\pi\)
\(332\) −3873.45 1815.75i −0.640312 0.300158i
\(333\) 0 0
\(334\) 2114.39 9492.02i 0.346390 1.55503i
\(335\) 442.950 0.0722416
\(336\) 0 0
\(337\) −3816.67 −0.616936 −0.308468 0.951235i \(-0.599816\pi\)
−0.308468 + 0.951235i \(0.599816\pi\)
\(338\) 869.933 3905.35i 0.139994 0.628471i
\(339\) 0 0
\(340\) 1015.98 + 476.259i 0.162057 + 0.0759670i
\(341\) 3115.21i 0.494716i
\(342\) 0 0
\(343\) 343.000i 0.0539949i
\(344\) 4689.21 + 3620.78i 0.734958 + 0.567498i
\(345\) 0 0
\(346\) −10978.9 2445.59i −1.70586 0.379988i
\(347\) 12314.6 1.90514 0.952571 0.304317i \(-0.0984281\pi\)
0.952571 + 0.304317i \(0.0984281\pi\)
\(348\) 0 0
\(349\) −8216.49 −1.26023 −0.630113 0.776503i \(-0.716991\pi\)
−0.630113 + 0.776503i \(0.716991\pi\)
\(350\) −2244.07 499.875i −0.342716 0.0763413i
\(351\) 0 0
\(352\) −2348.15 + 1207.35i −0.355560 + 0.182817i
\(353\) 9603.40i 1.44798i −0.689810 0.723990i \(-0.742306\pi\)
0.689810 0.723990i \(-0.257694\pi\)
\(354\) 0 0
\(355\) 2373.60i 0.354866i
\(356\) 1352.81 2885.89i 0.201402 0.429640i
\(357\) 0 0
\(358\) 2217.53 9955.08i 0.327375 1.46967i
\(359\) −7618.71 −1.12006 −0.560028 0.828474i \(-0.689210\pi\)
−0.560028 + 0.828474i \(0.689210\pi\)
\(360\) 0 0
\(361\) −15237.6 −2.22154
\(362\) 713.856 3204.68i 0.103645 0.465289i
\(363\) 0 0
\(364\) 664.857 1418.31i 0.0957363 0.204229i
\(365\) 486.713i 0.0697965i
\(366\) 0 0
\(367\) 7619.79i 1.08379i 0.840447 + 0.541894i \(0.182292\pi\)
−0.840447 + 0.541894i \(0.817708\pi\)
\(368\) −3484.91 4187.37i −0.493650 0.593158i
\(369\) 0 0
\(370\) −24.6839 5.49844i −0.00346825 0.000772568i
\(371\) 921.739 0.128987
\(372\) 0 0
\(373\) −102.877 −0.0142809 −0.00714045 0.999975i \(-0.502273\pi\)
−0.00714045 + 0.999975i \(0.502273\pi\)
\(374\) 1895.41 + 422.210i 0.262057 + 0.0583742i
\(375\) 0 0
\(376\) 1328.66 1720.73i 0.182236 0.236011i
\(377\) 5516.33i 0.753595i
\(378\) 0 0
\(379\) 10008.1i 1.35641i −0.734871 0.678207i \(-0.762757\pi\)
0.734871 0.678207i \(-0.237243\pi\)
\(380\) −3208.57 1504.08i −0.433148 0.203046i
\(381\) 0 0
\(382\) 2525.54 11337.8i 0.338266 1.51856i
\(383\) 6622.81 0.883576 0.441788 0.897119i \(-0.354344\pi\)
0.441788 + 0.897119i \(0.354344\pi\)
\(384\) 0 0
\(385\) 304.250 0.0402753
\(386\) 1469.14 6595.33i 0.193723 0.869672i
\(387\) 0 0
\(388\) −12396.5 5811.07i −1.62200 0.760341i
\(389\) 10793.4i 1.40681i 0.710791 + 0.703403i \(0.248337\pi\)
−0.710791 + 0.703403i \(0.751663\pi\)
\(390\) 0 0
\(391\) 4006.62i 0.518218i
\(392\) −677.621 + 877.577i −0.0873088 + 0.113072i
\(393\) 0 0
\(394\) −8934.10 1990.11i −1.14237 0.254467i
\(395\) 4015.75 0.511530
\(396\) 0 0
\(397\) 4491.35 0.567795 0.283897 0.958855i \(-0.408373\pi\)
0.283897 + 0.958855i \(0.408373\pi\)
\(398\) 9690.01 + 2158.49i 1.22039 + 0.271847i
\(399\) 0 0
\(400\) 4753.98 + 5712.27i 0.594248 + 0.714033i
\(401\) 14004.7i 1.74404i −0.489467 0.872022i \(-0.662809\pi\)
0.489467 0.872022i \(-0.337191\pi\)
\(402\) 0 0
\(403\) 5974.01i 0.738428i
\(404\) −6355.19 + 13557.2i −0.782630 + 1.66955i
\(405\) 0 0
\(406\) 848.957 3811.19i 0.103776 0.465877i
\(407\) −43.7652 −0.00533012
\(408\) 0 0
\(409\) 14406.8 1.74173 0.870867 0.491519i \(-0.163558\pi\)
0.870867 + 0.491519i \(0.163558\pi\)
\(410\) −7.82061 + 35.1087i −0.000942030 + 0.00422902i
\(411\) 0 0
\(412\) −40.1313 + 85.6102i −0.00479886 + 0.0102372i
\(413\) 2060.29i 0.245473i
\(414\) 0 0
\(415\) 1593.44i 0.188480i
\(416\) −4503.03 + 2315.31i −0.530719 + 0.272879i
\(417\) 0 0
\(418\) −5985.90 1333.38i −0.700431 0.156024i
\(419\) 13240.1 1.54372 0.771862 0.635790i \(-0.219326\pi\)
0.771862 + 0.635790i \(0.219326\pi\)
\(420\) 0 0
\(421\) 4321.30 0.500255 0.250128 0.968213i \(-0.419527\pi\)
0.250128 + 0.968213i \(0.419527\pi\)
\(422\) 1339.64 + 298.410i 0.154532 + 0.0344227i
\(423\) 0 0
\(424\) −2358.30 1820.96i −0.270116 0.208570i
\(425\) 5465.68i 0.623822i
\(426\) 0 0
\(427\) 943.477i 0.106928i
\(428\) −9781.02 4585.03i −1.10463 0.517817i
\(429\) 0 0
\(430\) −479.799 + 2153.94i −0.0538092 + 0.241564i
\(431\) −1296.30 −0.144874 −0.0724371 0.997373i \(-0.523078\pi\)
−0.0724371 + 0.997373i \(0.523078\pi\)
\(432\) 0 0
\(433\) −10543.7 −1.17020 −0.585101 0.810961i \(-0.698945\pi\)
−0.585101 + 0.810961i \(0.698945\pi\)
\(434\) 919.395 4127.40i 0.101687 0.456501i
\(435\) 0 0
\(436\) 4137.28 + 1939.43i 0.454449 + 0.213031i
\(437\) 12653.3i 1.38510i
\(438\) 0 0
\(439\) 13198.6i 1.43493i 0.696595 + 0.717465i \(0.254698\pi\)
−0.696595 + 0.717465i \(0.745302\pi\)
\(440\) −778.433 601.067i −0.0843417 0.0651245i
\(441\) 0 0
\(442\) 3634.80 + 809.668i 0.391154 + 0.0871311i
\(443\) −2362.08 −0.253331 −0.126666 0.991945i \(-0.540427\pi\)
−0.126666 + 0.991945i \(0.540427\pi\)
\(444\) 0 0
\(445\) 1187.19 0.126467
\(446\) −267.355 59.5545i −0.0283849 0.00632284i
\(447\) 0 0
\(448\) 3467.43 906.621i 0.365671 0.0956112i
\(449\) 12464.5i 1.31010i 0.755583 + 0.655052i \(0.227353\pi\)
−0.755583 + 0.655052i \(0.772647\pi\)
\(450\) 0 0
\(451\) 62.2487i 0.00649928i
\(452\) 3285.64 7009.11i 0.341911 0.729382i
\(453\) 0 0
\(454\) 2235.78 10037.0i 0.231124 1.03757i
\(455\) 583.457 0.0601162
\(456\) 0 0
\(457\) 14842.4 1.51925 0.759627 0.650359i \(-0.225381\pi\)
0.759627 + 0.650359i \(0.225381\pi\)
\(458\) −2820.94 + 12663.9i −0.287803 + 1.29202i
\(459\) 0 0
\(460\) 861.291 1837.35i 0.0872998 0.186232i
\(461\) 9687.52i 0.978727i 0.872080 + 0.489363i \(0.162771\pi\)
−0.872080 + 0.489363i \(0.837229\pi\)
\(462\) 0 0
\(463\) 15024.8i 1.50812i 0.656803 + 0.754062i \(0.271908\pi\)
−0.656803 + 0.754062i \(0.728092\pi\)
\(464\) −9701.36 + 8073.88i −0.970635 + 0.807802i
\(465\) 0 0
\(466\) 12398.1 + 2761.72i 1.23246 + 0.274537i
\(467\) 10567.3 1.04710 0.523549 0.851995i \(-0.324608\pi\)
0.523549 + 0.851995i \(0.324608\pi\)
\(468\) 0 0
\(469\) −1040.54 −0.102447
\(470\) 790.400 + 176.065i 0.0775712 + 0.0172793i
\(471\) 0 0
\(472\) −4070.25 + 5271.32i −0.396925 + 0.514051i
\(473\) 3819.00i 0.371242i
\(474\) 0 0
\(475\) 17261.2i 1.66737i
\(476\) −2386.65 1118.79i −0.229815 0.107730i
\(477\) 0 0
\(478\) 1979.24 8885.31i 0.189390 0.850219i
\(479\) −18844.8 −1.79758 −0.898788 0.438384i \(-0.855551\pi\)
−0.898788 + 0.438384i \(0.855551\pi\)
\(480\) 0 0
\(481\) −83.9281 −0.00795590
\(482\) −1386.28 + 6223.38i −0.131003 + 0.588106i
\(483\) 0 0
\(484\) 8100.15 + 3797.09i 0.760721 + 0.356602i
\(485\) 5099.60i 0.477446i
\(486\) 0 0
\(487\) 9562.33i 0.889755i 0.895591 + 0.444878i \(0.146753\pi\)
−0.895591 + 0.444878i \(0.853247\pi\)
\(488\) 1863.91 2413.92i 0.172900 0.223920i
\(489\) 0 0
\(490\) −403.106 89.7935i −0.0371642 0.00827848i
\(491\) 10643.7 0.978298 0.489149 0.872200i \(-0.337307\pi\)
0.489149 + 0.872200i \(0.337307\pi\)
\(492\) 0 0
\(493\) 9282.58 0.848005
\(494\) −11479.1 2557.02i −1.04548 0.232886i
\(495\) 0 0
\(496\) −10506.3 + 8743.76i −0.951100 + 0.791545i
\(497\) 5575.85i 0.503241i
\(498\) 0 0
\(499\) 7550.79i 0.677394i 0.940895 + 0.338697i \(0.109986\pi\)
−0.940895 + 0.338697i \(0.890014\pi\)
\(500\) −2439.73 + 5204.56i −0.218216 + 0.465510i
\(501\) 0 0
\(502\) −2458.76 + 11038.0i −0.218605 + 0.981374i
\(503\) −1090.04 −0.0966255 −0.0483127 0.998832i \(-0.515384\pi\)
−0.0483127 + 0.998832i \(0.515384\pi\)
\(504\) 0 0
\(505\) −5577.10 −0.491441
\(506\) 763.546 3427.75i 0.0670825 0.301151i
\(507\) 0 0
\(508\) 708.242 1510.86i 0.0618566 0.131956i
\(509\) 7957.32i 0.692931i −0.938063 0.346465i \(-0.887382\pi\)
0.938063 0.346465i \(-0.112618\pi\)
\(510\) 0 0
\(511\) 1143.34i 0.0989795i
\(512\) −10662.6 4530.54i −0.920364 0.391062i
\(513\) 0 0
\(514\) −9351.77 2083.15i −0.802508 0.178762i
\(515\) −35.2179 −0.00301337
\(516\) 0 0
\(517\) 1401.40 0.119214
\(518\) 57.9853 + 12.9165i 0.00491839 + 0.00109559i
\(519\) 0 0
\(520\) −1492.79 1152.66i −0.125891 0.0972068i
\(521\) 6770.45i 0.569326i −0.958628 0.284663i \(-0.908118\pi\)
0.958628 0.284663i \(-0.0918816\pi\)
\(522\) 0 0
\(523\) 9837.70i 0.822510i −0.911520 0.411255i \(-0.865091\pi\)
0.911520 0.411255i \(-0.134909\pi\)
\(524\) −12172.7 5706.15i −1.01482 0.475714i
\(525\) 0 0
\(526\) 2048.91 9198.09i 0.169842 0.762463i
\(527\) 10052.7 0.830938
\(528\) 0 0
\(529\) −4921.23 −0.404474
\(530\) 241.301 1083.26i 0.0197763 0.0887808i
\(531\) 0 0
\(532\) 7537.31 + 3533.25i 0.614255 + 0.287943i
\(533\) 119.374i 0.00970103i
\(534\) 0 0
\(535\) 4023.67i 0.325156i
\(536\) 2662.25 + 2055.66i 0.214537 + 0.165655i
\(537\) 0 0
\(538\) 16135.6 + 3594.28i 1.29304 + 0.288030i
\(539\) −714.717 −0.0571151
\(540\) 0 0
\(541\) 11900.7 0.945749 0.472874 0.881130i \(-0.343216\pi\)
0.472874 + 0.881130i \(0.343216\pi\)
\(542\) 18526.8 + 4126.92i 1.46826 + 0.327060i
\(543\) 0 0
\(544\) 3896.09 + 7577.46i 0.307065 + 0.597207i
\(545\) 1701.98i 0.133770i
\(546\) 0 0
\(547\) 4883.36i 0.381714i 0.981618 + 0.190857i \(0.0611267\pi\)
−0.981618 + 0.190857i \(0.938873\pi\)
\(548\) −5109.88 + 10900.7i −0.398327 + 0.849731i
\(549\) 0 0
\(550\) −1041.60 + 4676.02i −0.0807529 + 0.362520i
\(551\) −29315.4 −2.26657
\(552\) 0 0
\(553\) −9433.46 −0.725410
\(554\) 2706.05 12148.2i 0.207526 0.931636i
\(555\) 0 0
\(556\) −5477.11 + 11684.0i −0.417772 + 0.891212i
\(557\) 1481.08i 0.112667i −0.998412 0.0563334i \(-0.982059\pi\)
0.998412 0.0563334i \(-0.0179410\pi\)
\(558\) 0 0
\(559\) 7323.65i 0.554128i
\(560\) 853.967 + 1026.10i 0.0644405 + 0.0774300i
\(561\) 0 0
\(562\) −13112.9 2920.96i −0.984228 0.219241i
\(563\) 20443.6 1.53037 0.765183 0.643813i \(-0.222649\pi\)
0.765183 + 0.643813i \(0.222649\pi\)
\(564\) 0 0
\(565\) 2883.37 0.214698
\(566\) 4133.74 + 920.808i 0.306986 + 0.0683824i
\(567\) 0 0
\(568\) 11015.5 14266.0i 0.813732 1.05385i
\(569\) 17488.2i 1.28848i 0.764824 + 0.644240i \(0.222826\pi\)
−0.764824 + 0.644240i \(0.777174\pi\)
\(570\) 0 0
\(571\) 6945.88i 0.509065i −0.967064 0.254532i \(-0.918078\pi\)
0.967064 0.254532i \(-0.0819215\pi\)
\(572\) −2955.36 1385.38i −0.216031 0.101269i
\(573\) 0 0
\(574\) 18.3715 82.4744i 0.00133591 0.00599724i
\(575\) −9884.42 −0.716885
\(576\) 0 0
\(577\) 24227.0 1.74797 0.873987 0.485949i \(-0.161526\pi\)
0.873987 + 0.485949i \(0.161526\pi\)
\(578\) −1658.89 + 7447.17i −0.119378 + 0.535920i
\(579\) 0 0
\(580\) −4256.80 1995.45i −0.304748 0.142856i
\(581\) 3743.18i 0.267286i
\(582\) 0 0
\(583\) 1920.65i 0.136441i
\(584\) −2258.76 + 2925.28i −0.160048 + 0.207276i
\(585\) 0 0
\(586\) −10079.8 2245.31i −0.710566 0.158281i
\(587\) −15665.7 −1.10152 −0.550760 0.834663i \(-0.685662\pi\)
−0.550760 + 0.834663i \(0.685662\pi\)
\(588\) 0 0
\(589\) −31747.7 −2.22095
\(590\) −2421.33 539.360i −0.168957 0.0376358i
\(591\) 0 0
\(592\) −122.840 147.601i −0.00852819 0.0102473i
\(593\) 3668.32i 0.254030i 0.991901 + 0.127015i \(0.0405397\pi\)
−0.991901 + 0.127015i \(0.959460\pi\)
\(594\) 0 0
\(595\) 981.810i 0.0676475i
\(596\) 172.068 367.064i 0.0118258 0.0252274i
\(597\) 0 0
\(598\) 1464.24 6573.37i 0.100129 0.449507i
\(599\) −11412.5 −0.778469 −0.389235 0.921139i \(-0.627260\pi\)
−0.389235 + 0.921139i \(0.627260\pi\)
\(600\) 0 0
\(601\) −10557.7 −0.716566 −0.358283 0.933613i \(-0.616638\pi\)
−0.358283 + 0.933613i \(0.616638\pi\)
\(602\) 1127.10 5059.85i 0.0763078 0.342565i
\(603\) 0 0
\(604\) 337.391 719.739i 0.0227289 0.0484864i
\(605\) 3332.20i 0.223923i
\(606\) 0 0
\(607\) 6370.55i 0.425985i 0.977054 + 0.212992i \(0.0683209\pi\)
−0.977054 + 0.212992i \(0.931679\pi\)
\(608\) −12304.3 23930.4i −0.820730 1.59623i
\(609\) 0 0
\(610\) 1108.81 + 246.992i 0.0735973 + 0.0163941i
\(611\) 2687.45 0.177942
\(612\) 0 0
\(613\) −11921.8 −0.785506 −0.392753 0.919644i \(-0.628477\pi\)
−0.392753 + 0.919644i \(0.628477\pi\)
\(614\) 582.045 + 129.653i 0.0382564 + 0.00852176i
\(615\) 0 0
\(616\) 1828.63 + 1411.98i 0.119606 + 0.0923541i
\(617\) 10885.0i 0.710233i −0.934822 0.355117i \(-0.884441\pi\)
0.934822 0.355117i \(-0.115559\pi\)
\(618\) 0 0
\(619\) 22866.9i 1.48481i −0.669949 0.742407i \(-0.733684\pi\)
0.669949 0.742407i \(-0.266316\pi\)
\(620\) −4609.98 2161.01i −0.298615 0.139981i
\(621\) 0 0
\(622\) 445.596 2000.40i 0.0287247 0.128953i
\(623\) −2788.83 −0.179345
\(624\) 0 0
\(625\) 12374.0 0.791938
\(626\) −5414.83 + 24308.5i −0.345719 + 1.55202i
\(627\) 0 0
\(628\) −19111.5 8958.85i −1.21438 0.569263i
\(629\) 141.230i 0.00895261i
\(630\) 0 0
\(631\) 627.472i 0.0395868i −0.999804 0.0197934i \(-0.993699\pi\)
0.999804 0.0197934i \(-0.00630085\pi\)
\(632\) 24135.8 + 18636.5i 1.51910 + 1.17297i
\(633\) 0 0
\(634\) 17526.6 + 3904.13i 1.09791 + 0.244563i
\(635\) 621.530 0.0388420
\(636\) 0 0
\(637\) −1370.61 −0.0852518
\(638\) −7941.46 1768.99i −0.492799 0.109773i
\(639\) 0 0
\(640\) −157.759 4312.39i −0.00974373 0.266347i
\(641\) 12854.0i 0.792047i 0.918240 + 0.396024i \(0.129610\pi\)
−0.918240 + 0.396024i \(0.870390\pi\)
\(642\) 0 0
\(643\) 31056.4i 1.90473i −0.304956 0.952366i \(-0.598642\pi\)
0.304956 0.952366i \(-0.401358\pi\)
\(644\) −2023.27 + 4316.15i −0.123801 + 0.264099i
\(645\) 0 0
\(646\) −4302.81 + 19316.4i −0.262062 + 1.17646i
\(647\) −9340.79 −0.567580 −0.283790 0.958886i \(-0.591592\pi\)
−0.283790 + 0.958886i \(0.591592\pi\)
\(648\) 0 0
\(649\) −4293.07 −0.259658
\(650\) −1997.47 + 8967.16i −0.120534 + 0.541109i
\(651\) 0 0
\(652\) −8867.82 + 18917.3i −0.532654 + 1.13629i
\(653\) 4651.62i 0.278763i 0.990239 + 0.139381i \(0.0445114\pi\)
−0.990239 + 0.139381i \(0.955489\pi\)
\(654\) 0 0
\(655\) 5007.53i 0.298718i
\(656\) −209.938 + 174.719i −0.0124950 + 0.0103988i
\(657\) 0 0
\(658\) −1856.74 413.597i −0.110005 0.0245041i
\(659\) −30458.7 −1.80046 −0.900231 0.435413i \(-0.856602\pi\)
−0.900231 + 0.435413i \(0.856602\pi\)
\(660\) 0 0
\(661\) 22311.1 1.31286 0.656431 0.754386i \(-0.272065\pi\)
0.656431 + 0.754386i \(0.272065\pi\)
\(662\) 15339.1 + 3416.84i 0.900559 + 0.200603i
\(663\) 0 0
\(664\) 7394.92 9577.05i 0.432197 0.559732i
\(665\) 3100.66i 0.180810i
\(666\) 0 0
\(667\) 16787.1i 0.974511i
\(668\) 24904.9 + 11674.6i 1.44252 + 0.676206i
\(669\) 0 0
\(670\) −272.401 + 1222.88i −0.0157071 + 0.0705133i
\(671\) 1965.95 0.113107
\(672\) 0 0
\(673\) −12390.1 −0.709661 −0.354831 0.934931i \(-0.615461\pi\)
−0.354831 + 0.934931i \(0.615461\pi\)
\(674\) 2347.14 10536.9i 0.134137 0.602177i
\(675\) 0 0
\(676\) 10246.8 + 4803.36i 0.582998 + 0.273291i
\(677\) 32115.6i 1.82319i −0.411084 0.911597i \(-0.634850\pi\)
0.411084 0.911597i \(-0.365150\pi\)
\(678\) 0 0
\(679\) 11979.5i 0.677074i
\(680\) −1939.64 + 2511.99i −0.109385 + 0.141663i
\(681\) 0 0
\(682\) −8600.36 1915.77i −0.482881 0.107564i
\(683\) 22387.2 1.25421 0.627104 0.778936i \(-0.284240\pi\)
0.627104 + 0.778936i \(0.284240\pi\)
\(684\) 0 0
\(685\) −4484.26 −0.250124
\(686\) 946.942 + 210.935i 0.0527032 + 0.0117399i
\(687\) 0 0
\(688\) −12879.8 + 10719.1i −0.713720 + 0.593987i
\(689\) 3683.21i 0.203656i
\(690\) 0 0
\(691\) 18455.0i 1.01601i −0.861354 0.508005i \(-0.830383\pi\)
0.861354 0.508005i \(-0.169617\pi\)
\(692\) 13503.4 28806.1i 0.741795 1.58244i
\(693\) 0 0
\(694\) −7573.14 + 33997.8i −0.414226 + 1.85957i
\(695\) −4806.53 −0.262334
\(696\) 0 0
\(697\) 200.876 0.0109164
\(698\) 5052.91 22683.8i 0.274005 1.23008i
\(699\) 0 0
\(700\) 2760.07 5887.93i 0.149030 0.317918i
\(701\) 19758.3i 1.06457i 0.846566 + 0.532283i \(0.178666\pi\)
−0.846566 + 0.532283i \(0.821334\pi\)
\(702\) 0 0
\(703\) 446.019i 0.0239287i
\(704\) −1889.15 7225.17i −0.101136 0.386803i
\(705\) 0 0
\(706\) 26512.7 + 5905.81i 1.41334 + 0.314827i
\(707\) 13101.2 0.696921
\(708\) 0 0
\(709\) −5354.63 −0.283635 −0.141818 0.989893i \(-0.545295\pi\)
−0.141818 + 0.989893i \(0.545295\pi\)
\(710\) 6552.94 + 1459.69i 0.346376 + 0.0771568i
\(711\) 0 0
\(712\) 7135.32 + 5509.54i 0.375572 + 0.289998i
\(713\) 18179.9i 0.954899i
\(714\) 0 0
\(715\) 1215.76i 0.0635902i
\(716\) 26119.9 + 12244.2i 1.36333 + 0.639087i
\(717\) 0 0
\(718\) 4685.29 21033.5i 0.243528 1.09326i
\(719\) 6327.40 0.328195 0.164097 0.986444i \(-0.447529\pi\)
0.164097 + 0.986444i \(0.447529\pi\)
\(720\) 0 0
\(721\) 82.7309 0.00427331
\(722\) 9370.67 42067.3i 0.483019 2.16840i
\(723\) 0 0
\(724\) 8408.37 + 3941.58i 0.431622 + 0.202331i
\(725\) 22900.4i 1.17310i
\(726\) 0 0
\(727\) 17148.8i 0.874846i −0.899256 0.437423i \(-0.855891\pi\)
0.899256 0.437423i \(-0.144109\pi\)
\(728\) 3506.74 + 2707.73i 0.178528 + 0.137851i
\(729\) 0 0
\(730\) −1343.70 299.314i −0.0681267 0.0151755i
\(731\) 12323.8 0.623548
\(732\) 0 0
\(733\) −14960.0 −0.753832 −0.376916 0.926247i \(-0.623016\pi\)
−0.376916 + 0.926247i \(0.623016\pi\)
\(734\) −21036.4 4685.95i −1.05786 0.235643i
\(735\) 0 0
\(736\) 13703.5 7045.89i 0.686299 0.352873i
\(737\) 2168.20i 0.108367i
\(738\) 0 0
\(739\) 11540.8i 0.574470i −0.957860 0.287235i \(-0.907264\pi\)
0.957860 0.287235i \(-0.0927361\pi\)
\(740\) 30.3598 64.7650i 0.00150817 0.00321731i
\(741\) 0 0
\(742\) −566.843 + 2544.70i −0.0280451 + 0.125902i
\(743\) 22310.2 1.10159 0.550795 0.834640i \(-0.314325\pi\)
0.550795 + 0.834640i \(0.314325\pi\)
\(744\) 0 0
\(745\) 151.001 0.00742583
\(746\) 63.2665 284.019i 0.00310503 0.0139393i
\(747\) 0 0
\(748\) −2331.24 + 4973.12i −0.113955 + 0.243096i
\(749\) 9452.06i 0.461109i
\(750\) 0 0
\(751\) 21276.7i 1.03382i −0.856040 0.516909i \(-0.827083\pi\)
0.856040 0.516909i \(-0.172917\pi\)
\(752\) 3933.44 + 4726.33i 0.190742 + 0.229191i
\(753\) 0 0
\(754\) −15229.3 3392.38i −0.735567 0.163850i
\(755\) 296.083 0.0142723
\(756\) 0 0
\(757\) −18995.8 −0.912042 −0.456021 0.889969i \(-0.650726\pi\)
−0.456021 + 0.889969i \(0.650726\pi\)
\(758\) 27629.9 + 6154.68i 1.32396 + 0.294918i
\(759\) 0 0
\(760\) 6125.58 7933.15i 0.292366 0.378639i
\(761\) 24525.2i 1.16825i −0.811664 0.584124i \(-0.801438\pi\)
0.811664 0.584124i \(-0.198562\pi\)
\(762\) 0 0
\(763\) 3998.13i 0.189701i
\(764\) 29747.8 + 13944.8i 1.40869 + 0.660348i
\(765\) 0 0
\(766\) −4072.84 + 18284.0i −0.192112 + 0.862438i
\(767\) −8232.79 −0.387573
\(768\) 0 0
\(769\) 12318.1 0.577636 0.288818 0.957384i \(-0.406738\pi\)
0.288818 + 0.957384i \(0.406738\pi\)
\(770\) −187.105 + 839.961i −0.00875687 + 0.0393118i
\(771\) 0 0
\(772\) 17304.7 + 8111.88i 0.806747 + 0.378177i
\(773\) 29609.2i 1.37771i −0.724901 0.688853i \(-0.758114\pi\)
0.724901 0.688853i \(-0.241886\pi\)
\(774\) 0 0
\(775\) 24800.4i 1.14949i
\(776\) 23666.5 30650.1i 1.09482 1.41788i
\(777\) 0 0
\(778\) −29798.0 6637.64i −1.37315 0.305875i
\(779\) −634.387 −0.0291775
\(780\) 0 0
\(781\) 11618.5 0.532322
\(782\) −11061.3 2463.95i −0.505821 0.112674i
\(783\) 0 0
\(784\) −2006.06 2410.44i −0.0913841 0.109805i
\(785\) 7861.99i 0.357461i
\(786\) 0 0
\(787\) 23458.7i 1.06253i −0.847206 0.531265i \(-0.821717\pi\)
0.847206 0.531265i \(-0.178283\pi\)
\(788\) 10988.4 23441.1i 0.496759 1.05971i
\(789\) 0 0
\(790\) −2469.57 + 11086.5i −0.111220 + 0.499293i
\(791\) −6773.37 −0.304467
\(792\) 0 0
\(793\) 3770.08 0.168826
\(794\) −2762.05 + 12399.6i −0.123453 + 0.554211i
\(795\) 0 0
\(796\) −11918.2 + 25424.4i −0.530688 + 1.13209i
\(797\) 11962.9i 0.531680i −0.964017 0.265840i \(-0.914351\pi\)
0.964017 0.265840i \(-0.0856492\pi\)
\(798\) 0 0
\(799\) 4522.30i 0.200235i
\(800\) −18693.8 + 9611.74i −0.826156 + 0.424783i
\(801\) 0 0
\(802\) 38663.7 + 8612.49i 1.70232 + 0.379199i
\(803\) −2382.41 −0.104699
\(804\) 0 0
\(805\) −1775.56 −0.0777393
\(806\) −16492.8 3673.84i −0.720763 0.160553i
\(807\) 0 0
\(808\) −33520.0 25882.5i −1.45944 1.12691i
\(809\) 3777.90i 0.164183i −0.996625 0.0820914i \(-0.973840\pi\)
0.996625 0.0820914i \(-0.0261600\pi\)
\(810\) 0 0
\(811\) 133.384i 0.00577527i −0.999996 0.00288763i \(-0.999081\pi\)
0.999996 0.00288763i \(-0.000919164\pi\)
\(812\) 9999.70 + 4687.54i 0.432168 + 0.202587i
\(813\) 0 0
\(814\) 26.9143 120.825i 0.00115890 0.00520261i
\(815\) −7782.10 −0.334473
\(816\) 0 0
\(817\) −38920.0 −1.66663
\(818\) −8859.75 + 39773.7i −0.378697 + 1.70007i
\(819\) 0 0
\(820\) −92.1174 43.1817i −0.00392302 0.00183899i
\(821\) 19162.7i 0.814595i 0.913296 + 0.407298i \(0.133529\pi\)
−0.913296 + 0.407298i \(0.866471\pi\)
\(822\) 0 0
\(823\) 7574.02i 0.320794i −0.987053 0.160397i \(-0.948722\pi\)
0.987053 0.160397i \(-0.0512775\pi\)
\(824\) −211.670 163.441i −0.00894887 0.00690987i
\(825\) 0 0
\(826\) 5687.97 + 1267.02i 0.239600 + 0.0533719i
\(827\) 34129.2 1.43505 0.717527 0.696531i \(-0.245274\pi\)
0.717527 + 0.696531i \(0.245274\pi\)
\(828\) 0 0
\(829\) 34706.3 1.45404 0.727021 0.686615i \(-0.240904\pi\)
0.727021 + 0.686615i \(0.240904\pi\)
\(830\) 4399.12 + 979.922i 0.183971 + 0.0409802i
\(831\) 0 0
\(832\) −3622.80 13855.6i −0.150959 0.577353i
\(833\) 2306.38i 0.0959321i
\(834\) 0 0
\(835\) 10245.3i 0.424614i
\(836\) 7362.32 15705.7i 0.304583 0.649751i
\(837\) 0 0
\(838\) −8142.27 + 36552.7i −0.335644 + 1.50679i
\(839\) −39467.7 −1.62405 −0.812025 0.583623i \(-0.801635\pi\)
−0.812025 + 0.583623i \(0.801635\pi\)
\(840\) 0 0
\(841\) −14503.5 −0.594676
\(842\) −2657.48 + 11930.1i −0.108768 + 0.488287i
\(843\) 0 0
\(844\) −1647.68 + 3514.91i −0.0671984 + 0.143351i
\(845\) 4215.27i 0.171609i
\(846\) 0 0
\(847\) 7827.73i 0.317549i
\(848\) 6477.53 5390.87i 0.262310 0.218306i
\(849\) 0 0
\(850\) 15089.4 + 3361.24i 0.608899 + 0.135635i
\(851\) 255.407 0.0102882
\(852\) 0 0
\(853\) 15904.1 0.638389 0.319194 0.947689i \(-0.396588\pi\)
0.319194 + 0.947689i \(0.396588\pi\)
\(854\) −2604.72 580.211i −0.104370 0.0232487i
\(855\) 0 0
\(856\) 18673.2 24183.4i 0.745605 0.965621i
\(857\) 21592.9i 0.860677i 0.902668 + 0.430338i \(0.141606\pi\)
−0.902668 + 0.430338i \(0.858394\pi\)
\(858\) 0 0
\(859\) 9787.59i 0.388764i 0.980926 + 0.194382i \(0.0622701\pi\)
−0.980926 + 0.194382i \(0.937730\pi\)
\(860\) −5651.46 2649.22i −0.224085 0.105044i
\(861\) 0 0
\(862\) 797.189 3578.79i 0.0314993 0.141408i
\(863\) 15577.8 0.614455 0.307227 0.951636i \(-0.400599\pi\)
0.307227 + 0.951636i \(0.400599\pi\)
\(864\) 0 0
\(865\) 11850.1 0.465800
\(866\) 6484.06 29108.6i 0.254431 1.14221i
\(867\) 0 0
\(868\) 10829.4 + 5076.46i 0.423471 + 0.198510i
\(869\) 19656.7i 0.767329i
\(870\) 0 0
\(871\) 4157.93i 0.161752i
\(872\) −7898.60 + 10229.4i −0.306744 + 0.397259i
\(873\) 0 0
\(874\) 34932.8 + 7781.43i 1.35197 + 0.301157i
\(875\) 5029.52 0.194318
\(876\) 0 0
\(877\) 9974.67 0.384060 0.192030 0.981389i \(-0.438493\pi\)
0.192030 + 0.981389i \(0.438493\pi\)
\(878\) −36438.2 8116.75i −1.40060 0.311990i
\(879\) 0 0
\(880\) 2138.12 1779.43i 0.0819045 0.0681643i
\(881\) 8716.38i 0.333328i −0.986014 0.166664i \(-0.946700\pi\)
0.986014 0.166664i \(-0.0532996\pi\)
\(882\) 0 0
\(883\) 31703.2i 1.20826i 0.796884 + 0.604132i \(0.206480\pi\)
−0.796884 + 0.604132i \(0.793520\pi\)
\(884\) −4470.60 + 9536.91i −0.170093 + 0.362852i
\(885\) 0 0
\(886\) 1452.61 6521.13i 0.0550805 0.247271i
\(887\) −46045.2 −1.74301 −0.871503 0.490390i \(-0.836854\pi\)
−0.871503 + 0.490390i \(0.836854\pi\)
\(888\) 0 0
\(889\) −1460.04 −0.0550825
\(890\) −730.085 + 3277.54i −0.0274972 + 0.123442i
\(891\) 0 0
\(892\) 328.832 701.481i 0.0123432 0.0263311i
\(893\) 14281.9i 0.535192i
\(894\) 0 0
\(895\) 10745.1i 0.401305i
\(896\) 370.595 + 10130.3i 0.0138177 + 0.377712i
\(897\) 0 0
\(898\) −34411.6 7665.32i −1.27876 0.284850i
\(899\) −42119.4 −1.56258
\(900\) 0 0
\(901\) −6197.91 −0.229170
\(902\) −171.854 38.2812i −0.00634380 0.00141311i
\(903\) 0 0
\(904\) 17329.9 + 13381.3i 0.637593 + 0.492317i
\(905\) 3459.00i 0.127051i
\(906\) 0 0
\(907\) 30436.7i 1.11426i −0.830425 0.557131i \(-0.811902\pi\)
0.830425 0.557131i \(-0.188098\pi\)
\(908\) 26334.8 + 12344.9i 0.962500 + 0.451189i
\(909\) 0 0
\(910\) −358.809 + 1610.79i −0.0130708 + 0.0586781i
\(911\) −33110.9 −1.20419 −0.602093 0.798426i \(-0.705666\pi\)
−0.602093 + 0.798426i \(0.705666\pi\)
\(912\) 0 0
\(913\) 7799.76 0.282732
\(914\) −9127.66 + 40976.4i −0.330324 + 1.48291i
\(915\) 0 0
\(916\) −33227.3 15575.9i −1.19854 0.561836i
\(917\) 11763.3i 0.423617i
\(918\) 0 0
\(919\) 36489.9i 1.30978i 0.755723 + 0.654891i \(0.227285\pi\)
−0.755723 + 0.654891i \(0.772715\pi\)
\(920\) 4542.82 + 3507.74i 0.162796 + 0.125703i
\(921\) 0 0
\(922\) −26745.0 5957.55i −0.955312 0.212800i
\(923\) 22280.7 0.794561
\(924\) 0 0
\(925\) −348.417 −0.0123847
\(926\) −41479.9 9239.82i −1.47205 0.327904i
\(927\) 0 0
\(928\) −16324.0 31748.4i −0.577437 1.12305i
\(929\) 1922.94i 0.0679113i −0.999423 0.0339556i \(-0.989190\pi\)
0.999423 0.0339556i \(-0.0108105\pi\)
\(930\) 0 0
\(931\) 7283.81i 0.256409i
\(932\) −15248.9 + 32529.7i −0.535938 + 1.14329i
\(933\) 0 0
\(934\) −6498.57 + 29173.7i −0.227665 + 1.02205i
\(935\) −2045.82 −0.0715567
\(936\) 0 0
\(937\) 820.948 0.0286224 0.0143112 0.999898i \(-0.495444\pi\)
0.0143112 + 0.999898i \(0.495444\pi\)
\(938\) 639.902 2872.68i 0.0222745 0.0999962i
\(939\) 0 0
\(940\) −972.147 + 2073.83i −0.0337318 + 0.0719585i
\(941\) 2949.16i 0.102168i 0.998694 + 0.0510839i \(0.0162676\pi\)
−0.998694 + 0.0510839i \(0.983732\pi\)
\(942\) 0 0
\(943\) 363.274i 0.0125449i
\(944\) −12049.8 14478.7i −0.415452 0.499197i
\(945\) 0 0
\(946\) −10543.3 2348.57i −0.362361 0.0807174i
\(947\) 14834.6 0.509038 0.254519 0.967068i \(-0.418083\pi\)
0.254519 + 0.967068i \(0.418083\pi\)
\(948\) 0 0
\(949\) −4568.73 −0.156277
\(950\) −47654.1 10615.1i −1.62748 0.362527i
\(951\) 0 0
\(952\) 4556.43 5900.96i 0.155120 0.200894i
\(953\) 23219.2i 0.789237i −0.918845 0.394619i \(-0.870877\pi\)
0.918845 0.394619i \(-0.129123\pi\)
\(954\) 0 0
\(955\) 12237.5i 0.414656i
\(956\) 23313.0 + 10928.4i 0.788701 + 0.369718i
\(957\) 0 0
\(958\) 11589.0 52025.9i 0.390838 1.75457i
\(959\) 10534.0 0.354705
\(960\) 0 0
\(961\) −15823.0 −0.531135
\(962\) 51.6133 231.705i 0.00172981 0.00776558i
\(963\) 0 0
\(964\) −16328.7 7654.39i −0.545553 0.255738i
\(965\) 7118.71i 0.237471i
\(966\) 0 0
\(967\) 8504.08i 0.282805i 0.989952 + 0.141403i \(0.0451613\pi\)
−0.989952 + 0.141403i \(0.954839\pi\)
\(968\) −15464.2 + 20027.5i −0.513470 + 0.664988i
\(969\) 0 0
\(970\) 14078.8 + 3136.11i 0.466024 + 0.103809i
\(971\) 23347.8 0.771645 0.385822 0.922573i \(-0.373918\pi\)
0.385822 + 0.922573i \(0.373918\pi\)
\(972\) 0 0
\(973\) 11291.1 0.372020
\(974\) −26399.3 5880.56i −0.868470 0.193455i
\(975\) 0 0
\(976\) 5518.01 + 6630.30i 0.180970 + 0.217449i
\(977\) 38899.9i 1.27382i 0.770940 + 0.636908i \(0.219787\pi\)
−0.770940 + 0.636908i \(0.780213\pi\)
\(978\) 0 0
\(979\) 5811.16i 0.189709i
\(980\) 495.797 1057.66i 0.0161609 0.0344752i
\(981\) 0 0
\(982\) −6545.58 + 29384.8i −0.212707 + 0.954894i
\(983\) 9378.14 0.304289 0.152145 0.988358i \(-0.451382\pi\)
0.152145 + 0.988358i \(0.451382\pi\)
\(984\) 0 0
\(985\) 9643.08 0.311933
\(986\) −5708.52 + 25627.0i −0.184377 + 0.827718i
\(987\) 0 0
\(988\) 14118.6 30118.6i 0.454629 0.969839i
\(989\) 22287.1i 0.716570i
\(990\) 0 0
\(991\) 30417.6i 0.975023i 0.873117 + 0.487511i \(0.162095\pi\)
−0.873117 + 0.487511i \(0.837905\pi\)
\(992\) −17678.4 34382.5i −0.565816 1.10045i
\(993\) 0 0
\(994\) −15393.6 3428.99i −0.491202 0.109417i
\(995\) −10459.0 −0.333238
\(996\) 0 0
\(997\) −42253.4 −1.34221 −0.671103 0.741364i \(-0.734179\pi\)
−0.671103 + 0.741364i \(0.734179\pi\)
\(998\) −20845.9 4643.52i −0.661189 0.147282i
\(999\) 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 252.4.e.a.71.18 yes 36
3.2 odd 2 inner 252.4.e.a.71.19 yes 36
4.3 odd 2 inner 252.4.e.a.71.20 yes 36
12.11 even 2 inner 252.4.e.a.71.17 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.4.e.a.71.17 36 12.11 even 2 inner
252.4.e.a.71.18 yes 36 1.1 even 1 trivial
252.4.e.a.71.19 yes 36 3.2 odd 2 inner
252.4.e.a.71.20 yes 36 4.3 odd 2 inner