Properties

Label 441.4.c.b.440.17
Level $441$
Weight $4$
Character 441.440
Analytic conductor $26.020$
Analytic rank $0$
Dimension $24$
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] = [441,4,Mod(440,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.440");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.c (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(26.0198423125\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 440.17
Character \(\chi\) \(=\) 441.440
Dual form 441.4.c.b.440.18

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.70881i q^{2} -5.75528 q^{4} +4.54675 q^{5} -8.32523i q^{8} +O(q^{10})\) \(q-3.70881i q^{2} -5.75528 q^{4} +4.54675 q^{5} -8.32523i q^{8} -16.8631i q^{10} +45.5164i q^{11} -11.6749i q^{13} -76.9190 q^{16} -91.5157 q^{17} -140.085i q^{19} -26.1679 q^{20} +168.812 q^{22} -45.0189i q^{23} -104.327 q^{25} -43.3000 q^{26} -47.7861i q^{29} -238.848i q^{31} +218.676i q^{32} +339.415i q^{34} -148.927 q^{37} -519.549 q^{38} -37.8528i q^{40} -393.755 q^{41} +412.265 q^{43} -261.960i q^{44} -166.967 q^{46} -408.757 q^{47} +386.929i q^{50} +67.1924i q^{52} -167.070i q^{53} +206.952i q^{55} -177.230 q^{58} -137.537 q^{59} -323.364i q^{61} -885.842 q^{62} +195.677 q^{64} -53.0829i q^{65} +424.152 q^{67} +526.699 q^{68} +727.190i q^{71} +1096.11i q^{73} +552.342i q^{74} +806.229i q^{76} -669.260 q^{79} -349.732 q^{80} +1460.36i q^{82} -199.014 q^{83} -416.100 q^{85} -1529.01i q^{86} +378.934 q^{88} +807.348 q^{89} +259.096i q^{92} +1516.00i q^{94} -636.932i q^{95} -701.053i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 96 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 96 q^{4} + 144 q^{16} + 624 q^{22} + 312 q^{25} - 864 q^{37} + 1248 q^{43} - 3888 q^{46} - 7440 q^{58} - 3360 q^{64} - 2688 q^{67} + 480 q^{79} + 13248 q^{85} - 7248 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.70881i − 1.31126i −0.755081 0.655631i \(-0.772403\pi\)
0.755081 0.655631i \(-0.227597\pi\)
\(3\) 0 0
\(4\) −5.75528 −0.719410
\(5\) 4.54675 0.406674 0.203337 0.979109i \(-0.434821\pi\)
0.203337 + 0.979109i \(0.434821\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) − 8.32523i − 0.367927i
\(9\) 0 0
\(10\) − 16.8631i − 0.533257i
\(11\) 45.5164i 1.24761i 0.781580 + 0.623805i \(0.214414\pi\)
−0.781580 + 0.623805i \(0.785586\pi\)
\(12\) 0 0
\(13\) − 11.6749i − 0.249080i −0.992215 0.124540i \(-0.960255\pi\)
0.992215 0.124540i \(-0.0397455\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −76.9190 −1.20186
\(17\) −91.5157 −1.30564 −0.652818 0.757514i \(-0.726414\pi\)
−0.652818 + 0.757514i \(0.726414\pi\)
\(18\) 0 0
\(19\) − 140.085i − 1.69146i −0.533612 0.845730i \(-0.679166\pi\)
0.533612 0.845730i \(-0.320834\pi\)
\(20\) −26.1679 −0.292566
\(21\) 0 0
\(22\) 168.812 1.63594
\(23\) − 45.0189i − 0.408134i −0.978957 0.204067i \(-0.934584\pi\)
0.978957 0.204067i \(-0.0654161\pi\)
\(24\) 0 0
\(25\) −104.327 −0.834616
\(26\) −43.3000 −0.326609
\(27\) 0 0
\(28\) 0 0
\(29\) − 47.7861i − 0.305988i −0.988227 0.152994i \(-0.951108\pi\)
0.988227 0.152994i \(-0.0488916\pi\)
\(30\) 0 0
\(31\) − 238.848i − 1.38382i −0.721985 0.691909i \(-0.756770\pi\)
0.721985 0.691909i \(-0.243230\pi\)
\(32\) 218.676i 1.20803i
\(33\) 0 0
\(34\) 339.415i 1.71203i
\(35\) 0 0
\(36\) 0 0
\(37\) −148.927 −0.661715 −0.330858 0.943681i \(-0.607338\pi\)
−0.330858 + 0.943681i \(0.607338\pi\)
\(38\) −519.549 −2.21795
\(39\) 0 0
\(40\) − 37.8528i − 0.149626i
\(41\) −393.755 −1.49986 −0.749929 0.661518i \(-0.769912\pi\)
−0.749929 + 0.661518i \(0.769912\pi\)
\(42\) 0 0
\(43\) 412.265 1.46209 0.731045 0.682329i \(-0.239033\pi\)
0.731045 + 0.682329i \(0.239033\pi\)
\(44\) − 261.960i − 0.897543i
\(45\) 0 0
\(46\) −166.967 −0.535171
\(47\) −408.757 −1.26858 −0.634292 0.773094i \(-0.718708\pi\)
−0.634292 + 0.773094i \(0.718708\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 386.929i 1.09440i
\(51\) 0 0
\(52\) 67.1924i 0.179191i
\(53\) − 167.070i − 0.432997i −0.976283 0.216498i \(-0.930536\pi\)
0.976283 0.216498i \(-0.0694636\pi\)
\(54\) 0 0
\(55\) 206.952i 0.507370i
\(56\) 0 0
\(57\) 0 0
\(58\) −177.230 −0.401231
\(59\) −137.537 −0.303489 −0.151744 0.988420i \(-0.548489\pi\)
−0.151744 + 0.988420i \(0.548489\pi\)
\(60\) 0 0
\(61\) − 323.364i − 0.678730i −0.940655 0.339365i \(-0.889788\pi\)
0.940655 0.339365i \(-0.110212\pi\)
\(62\) −885.842 −1.81455
\(63\) 0 0
\(64\) 195.677 0.382181
\(65\) − 53.0829i − 0.101294i
\(66\) 0 0
\(67\) 424.152 0.773409 0.386705 0.922204i \(-0.373613\pi\)
0.386705 + 0.922204i \(0.373613\pi\)
\(68\) 526.699 0.939289
\(69\) 0 0
\(70\) 0 0
\(71\) 727.190i 1.21551i 0.794123 + 0.607757i \(0.207931\pi\)
−0.794123 + 0.607757i \(0.792069\pi\)
\(72\) 0 0
\(73\) 1096.11i 1.75740i 0.477377 + 0.878699i \(0.341588\pi\)
−0.477377 + 0.878699i \(0.658412\pi\)
\(74\) 552.342i 0.867682i
\(75\) 0 0
\(76\) 806.229i 1.21685i
\(77\) 0 0
\(78\) 0 0
\(79\) −669.260 −0.953135 −0.476568 0.879138i \(-0.658119\pi\)
−0.476568 + 0.879138i \(0.658119\pi\)
\(80\) −349.732 −0.488765
\(81\) 0 0
\(82\) 1460.36i 1.96671i
\(83\) −199.014 −0.263189 −0.131594 0.991304i \(-0.542010\pi\)
−0.131594 + 0.991304i \(0.542010\pi\)
\(84\) 0 0
\(85\) −416.100 −0.530969
\(86\) − 1529.01i − 1.91718i
\(87\) 0 0
\(88\) 378.934 0.459029
\(89\) 807.348 0.961559 0.480779 0.876842i \(-0.340354\pi\)
0.480779 + 0.876842i \(0.340354\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 259.096i 0.293616i
\(93\) 0 0
\(94\) 1516.00i 1.66345i
\(95\) − 636.932i − 0.687872i
\(96\) 0 0
\(97\) − 701.053i − 0.733827i −0.930255 0.366913i \(-0.880415\pi\)
0.930255 0.366913i \(-0.119585\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 600.432 0.600432
\(101\) 1702.58 1.67736 0.838680 0.544625i \(-0.183328\pi\)
0.838680 + 0.544625i \(0.183328\pi\)
\(102\) 0 0
\(103\) − 1134.23i − 1.08504i −0.840042 0.542521i \(-0.817470\pi\)
0.840042 0.542521i \(-0.182530\pi\)
\(104\) −97.1963 −0.0916431
\(105\) 0 0
\(106\) −619.631 −0.567773
\(107\) − 338.726i − 0.306036i −0.988223 0.153018i \(-0.951101\pi\)
0.988223 0.153018i \(-0.0488993\pi\)
\(108\) 0 0
\(109\) 1975.54 1.73598 0.867991 0.496580i \(-0.165411\pi\)
0.867991 + 0.496580i \(0.165411\pi\)
\(110\) 767.545 0.665296
\(111\) 0 0
\(112\) 0 0
\(113\) 1685.74i 1.40337i 0.712488 + 0.701685i \(0.247568\pi\)
−0.712488 + 0.701685i \(0.752432\pi\)
\(114\) 0 0
\(115\) − 204.690i − 0.165978i
\(116\) 275.023i 0.220131i
\(117\) 0 0
\(118\) 510.100i 0.397954i
\(119\) 0 0
\(120\) 0 0
\(121\) −740.739 −0.556529
\(122\) −1199.30 −0.889993
\(123\) 0 0
\(124\) 1374.64i 0.995533i
\(125\) −1042.69 −0.746091
\(126\) 0 0
\(127\) −1155.98 −0.807691 −0.403846 0.914827i \(-0.632327\pi\)
−0.403846 + 0.914827i \(0.632327\pi\)
\(128\) 1023.68i 0.706886i
\(129\) 0 0
\(130\) −196.875 −0.132823
\(131\) −635.067 −0.423558 −0.211779 0.977318i \(-0.567926\pi\)
−0.211779 + 0.977318i \(0.567926\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) − 1573.10i − 1.01414i
\(135\) 0 0
\(136\) 761.890i 0.480379i
\(137\) − 2458.12i − 1.53293i −0.642285 0.766466i \(-0.722013\pi\)
0.642285 0.766466i \(-0.277987\pi\)
\(138\) 0 0
\(139\) − 1094.81i − 0.668062i −0.942562 0.334031i \(-0.891591\pi\)
0.942562 0.334031i \(-0.108409\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2697.01 1.59386
\(143\) 531.399 0.310754
\(144\) 0 0
\(145\) − 217.272i − 0.124437i
\(146\) 4065.27 2.30441
\(147\) 0 0
\(148\) 857.117 0.476045
\(149\) 966.514i 0.531409i 0.964055 + 0.265704i \(0.0856045\pi\)
−0.964055 + 0.265704i \(0.914395\pi\)
\(150\) 0 0
\(151\) −2329.88 −1.25565 −0.627825 0.778354i \(-0.716055\pi\)
−0.627825 + 0.778354i \(0.716055\pi\)
\(152\) −1166.24 −0.622333
\(153\) 0 0
\(154\) 0 0
\(155\) − 1085.98i − 0.562763i
\(156\) 0 0
\(157\) − 3414.89i − 1.73591i −0.496641 0.867956i \(-0.665433\pi\)
0.496641 0.867956i \(-0.334567\pi\)
\(158\) 2482.16i 1.24981i
\(159\) 0 0
\(160\) 994.267i 0.491273i
\(161\) 0 0
\(162\) 0 0
\(163\) −361.669 −0.173792 −0.0868960 0.996217i \(-0.527695\pi\)
−0.0868960 + 0.996217i \(0.527695\pi\)
\(164\) 2266.17 1.07901
\(165\) 0 0
\(166\) 738.107i 0.345110i
\(167\) −1605.57 −0.743969 −0.371984 0.928239i \(-0.621322\pi\)
−0.371984 + 0.928239i \(0.621322\pi\)
\(168\) 0 0
\(169\) 2060.70 0.937959
\(170\) 1543.23i 0.696239i
\(171\) 0 0
\(172\) −2372.70 −1.05184
\(173\) −1466.87 −0.644647 −0.322324 0.946630i \(-0.604464\pi\)
−0.322324 + 0.946630i \(0.604464\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) − 3501.07i − 1.49945i
\(177\) 0 0
\(178\) − 2994.30i − 1.26086i
\(179\) − 981.846i − 0.409981i −0.978764 0.204990i \(-0.934284\pi\)
0.978764 0.204990i \(-0.0657163\pi\)
\(180\) 0 0
\(181\) − 1547.32i − 0.635422i −0.948188 0.317711i \(-0.897086\pi\)
0.948188 0.317711i \(-0.102914\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −374.793 −0.150163
\(185\) −677.135 −0.269102
\(186\) 0 0
\(187\) − 4165.46i − 1.62892i
\(188\) 2352.52 0.912632
\(189\) 0 0
\(190\) −2362.26 −0.901982
\(191\) 4784.63i 1.81259i 0.422650 + 0.906293i \(0.361100\pi\)
−0.422650 + 0.906293i \(0.638900\pi\)
\(192\) 0 0
\(193\) −1276.37 −0.476037 −0.238019 0.971261i \(-0.576498\pi\)
−0.238019 + 0.971261i \(0.576498\pi\)
\(194\) −2600.08 −0.962240
\(195\) 0 0
\(196\) 0 0
\(197\) − 491.111i − 0.177615i −0.996049 0.0888076i \(-0.971694\pi\)
0.996049 0.0888076i \(-0.0283056\pi\)
\(198\) 0 0
\(199\) − 2988.44i − 1.06455i −0.846573 0.532273i \(-0.821338\pi\)
0.846573 0.532273i \(-0.178662\pi\)
\(200\) 868.547i 0.307078i
\(201\) 0 0
\(202\) − 6314.56i − 2.19946i
\(203\) 0 0
\(204\) 0 0
\(205\) −1790.31 −0.609953
\(206\) −4206.66 −1.42278
\(207\) 0 0
\(208\) 898.022i 0.299359i
\(209\) 6376.16 2.11028
\(210\) 0 0
\(211\) 978.467 0.319244 0.159622 0.987178i \(-0.448972\pi\)
0.159622 + 0.987178i \(0.448972\pi\)
\(212\) 961.535i 0.311502i
\(213\) 0 0
\(214\) −1256.27 −0.401294
\(215\) 1874.47 0.594594
\(216\) 0 0
\(217\) 0 0
\(218\) − 7326.89i − 2.27633i
\(219\) 0 0
\(220\) − 1191.07i − 0.365007i
\(221\) 1068.44i 0.325208i
\(222\) 0 0
\(223\) 727.150i 0.218357i 0.994022 + 0.109178i \(0.0348220\pi\)
−0.994022 + 0.109178i \(0.965178\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 6252.08 1.84019
\(227\) 6230.08 1.82161 0.910803 0.412841i \(-0.135463\pi\)
0.910803 + 0.412841i \(0.135463\pi\)
\(228\) 0 0
\(229\) − 5670.06i − 1.63619i −0.575080 0.818097i \(-0.695029\pi\)
0.575080 0.818097i \(-0.304971\pi\)
\(230\) −759.156 −0.217640
\(231\) 0 0
\(232\) −397.830 −0.112581
\(233\) − 3953.76i − 1.11167i −0.831292 0.555836i \(-0.812398\pi\)
0.831292 0.555836i \(-0.187602\pi\)
\(234\) 0 0
\(235\) −1858.52 −0.515900
\(236\) 791.567 0.218333
\(237\) 0 0
\(238\) 0 0
\(239\) 3916.55i 1.06000i 0.847997 + 0.530001i \(0.177808\pi\)
−0.847997 + 0.530001i \(0.822192\pi\)
\(240\) 0 0
\(241\) − 3271.69i − 0.874474i −0.899346 0.437237i \(-0.855957\pi\)
0.899346 0.437237i \(-0.144043\pi\)
\(242\) 2747.26i 0.729755i
\(243\) 0 0
\(244\) 1861.05i 0.488285i
\(245\) 0 0
\(246\) 0 0
\(247\) −1635.48 −0.421308
\(248\) −1988.46 −0.509144
\(249\) 0 0
\(250\) 3867.15i 0.978321i
\(251\) 1401.75 0.352502 0.176251 0.984345i \(-0.443603\pi\)
0.176251 + 0.984345i \(0.443603\pi\)
\(252\) 0 0
\(253\) 2049.10 0.509192
\(254\) 4287.32i 1.05910i
\(255\) 0 0
\(256\) 5362.05 1.30910
\(257\) 4088.34 0.992310 0.496155 0.868234i \(-0.334745\pi\)
0.496155 + 0.868234i \(0.334745\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 305.507i 0.0728721i
\(261\) 0 0
\(262\) 2355.34i 0.555396i
\(263\) − 4612.24i − 1.08138i −0.841222 0.540690i \(-0.818163\pi\)
0.841222 0.540690i \(-0.181837\pi\)
\(264\) 0 0
\(265\) − 759.626i − 0.176089i
\(266\) 0 0
\(267\) 0 0
\(268\) −2441.12 −0.556399
\(269\) 3678.31 0.833720 0.416860 0.908971i \(-0.363131\pi\)
0.416860 + 0.908971i \(0.363131\pi\)
\(270\) 0 0
\(271\) 7159.95i 1.60493i 0.596699 + 0.802465i \(0.296479\pi\)
−0.596699 + 0.802465i \(0.703521\pi\)
\(272\) 7039.30 1.56919
\(273\) 0 0
\(274\) −9116.72 −2.01008
\(275\) − 4748.59i − 1.04127i
\(276\) 0 0
\(277\) 1130.70 0.245260 0.122630 0.992452i \(-0.460867\pi\)
0.122630 + 0.992452i \(0.460867\pi\)
\(278\) −4060.45 −0.876005
\(279\) 0 0
\(280\) 0 0
\(281\) − 3919.67i − 0.832128i −0.909335 0.416064i \(-0.863409\pi\)
0.909335 0.416064i \(-0.136591\pi\)
\(282\) 0 0
\(283\) 4506.76i 0.946640i 0.880890 + 0.473320i \(0.156945\pi\)
−0.880890 + 0.473320i \(0.843055\pi\)
\(284\) − 4185.18i − 0.874454i
\(285\) 0 0
\(286\) − 1970.86i − 0.407480i
\(287\) 0 0
\(288\) 0 0
\(289\) 3462.13 0.704688
\(290\) −805.820 −0.163170
\(291\) 0 0
\(292\) − 6308.42i − 1.26429i
\(293\) −8956.63 −1.78584 −0.892922 0.450212i \(-0.851348\pi\)
−0.892922 + 0.450212i \(0.851348\pi\)
\(294\) 0 0
\(295\) −625.349 −0.123421
\(296\) 1239.85i 0.243463i
\(297\) 0 0
\(298\) 3584.62 0.696817
\(299\) −525.591 −0.101658
\(300\) 0 0
\(301\) 0 0
\(302\) 8641.10i 1.64649i
\(303\) 0 0
\(304\) 10775.2i 2.03290i
\(305\) − 1470.26i − 0.276022i
\(306\) 0 0
\(307\) 3522.69i 0.654888i 0.944871 + 0.327444i \(0.106187\pi\)
−0.944871 + 0.327444i \(0.893813\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −4027.71 −0.737930
\(311\) 3555.82 0.648335 0.324168 0.946000i \(-0.394916\pi\)
0.324168 + 0.946000i \(0.394916\pi\)
\(312\) 0 0
\(313\) 6339.77i 1.14487i 0.819949 + 0.572436i \(0.194001\pi\)
−0.819949 + 0.572436i \(0.805999\pi\)
\(314\) −12665.2 −2.27624
\(315\) 0 0
\(316\) 3851.78 0.685695
\(317\) 2544.86i 0.450895i 0.974255 + 0.225447i \(0.0723844\pi\)
−0.974255 + 0.225447i \(0.927616\pi\)
\(318\) 0 0
\(319\) 2175.05 0.381754
\(320\) 889.695 0.155423
\(321\) 0 0
\(322\) 0 0
\(323\) 12820.0i 2.20843i
\(324\) 0 0
\(325\) 1218.01i 0.207886i
\(326\) 1341.36i 0.227887i
\(327\) 0 0
\(328\) 3278.10i 0.551838i
\(329\) 0 0
\(330\) 0 0
\(331\) 11398.4 1.89279 0.946396 0.323008i \(-0.104694\pi\)
0.946396 + 0.323008i \(0.104694\pi\)
\(332\) 1145.38 0.189341
\(333\) 0 0
\(334\) 5954.76i 0.975539i
\(335\) 1928.52 0.314525
\(336\) 0 0
\(337\) −551.119 −0.0890842 −0.0445421 0.999008i \(-0.514183\pi\)
−0.0445421 + 0.999008i \(0.514183\pi\)
\(338\) − 7642.74i − 1.22991i
\(339\) 0 0
\(340\) 2394.77 0.381984
\(341\) 10871.5 1.72646
\(342\) 0 0
\(343\) 0 0
\(344\) − 3432.20i − 0.537942i
\(345\) 0 0
\(346\) 5440.34i 0.845302i
\(347\) − 2887.43i − 0.446702i −0.974738 0.223351i \(-0.928300\pi\)
0.974738 0.223351i \(-0.0716996\pi\)
\(348\) 0 0
\(349\) 2290.34i 0.351286i 0.984454 + 0.175643i \(0.0562005\pi\)
−0.984454 + 0.175643i \(0.943800\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −9953.34 −1.50714
\(353\) 1409.52 0.212525 0.106262 0.994338i \(-0.466112\pi\)
0.106262 + 0.994338i \(0.466112\pi\)
\(354\) 0 0
\(355\) 3306.35i 0.494318i
\(356\) −4646.52 −0.691755
\(357\) 0 0
\(358\) −3641.48 −0.537593
\(359\) − 6952.66i − 1.02214i −0.859540 0.511068i \(-0.829250\pi\)
0.859540 0.511068i \(-0.170750\pi\)
\(360\) 0 0
\(361\) −12764.8 −1.86103
\(362\) −5738.72 −0.833205
\(363\) 0 0
\(364\) 0 0
\(365\) 4983.74i 0.714688i
\(366\) 0 0
\(367\) 1502.10i 0.213648i 0.994278 + 0.106824i \(0.0340682\pi\)
−0.994278 + 0.106824i \(0.965932\pi\)
\(368\) 3462.81i 0.490520i
\(369\) 0 0
\(370\) 2511.37i 0.352864i
\(371\) 0 0
\(372\) 0 0
\(373\) −7055.03 −0.979345 −0.489673 0.871906i \(-0.662884\pi\)
−0.489673 + 0.871906i \(0.662884\pi\)
\(374\) −15448.9 −2.13595
\(375\) 0 0
\(376\) 3403.00i 0.466746i
\(377\) −557.898 −0.0762155
\(378\) 0 0
\(379\) 5163.51 0.699820 0.349910 0.936783i \(-0.386212\pi\)
0.349910 + 0.936783i \(0.386212\pi\)
\(380\) 3665.73i 0.494863i
\(381\) 0 0
\(382\) 17745.3 2.37678
\(383\) 2517.92 0.335926 0.167963 0.985793i \(-0.446281\pi\)
0.167963 + 0.985793i \(0.446281\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 4733.82i 0.624210i
\(387\) 0 0
\(388\) 4034.76i 0.527923i
\(389\) 4786.74i 0.623900i 0.950099 + 0.311950i \(0.100982\pi\)
−0.950099 + 0.311950i \(0.899018\pi\)
\(390\) 0 0
\(391\) 4119.94i 0.532875i
\(392\) 0 0
\(393\) 0 0
\(394\) −1821.44 −0.232900
\(395\) −3042.96 −0.387615
\(396\) 0 0
\(397\) − 9594.49i − 1.21293i −0.795110 0.606466i \(-0.792587\pi\)
0.795110 0.606466i \(-0.207413\pi\)
\(398\) −11083.6 −1.39590
\(399\) 0 0
\(400\) 8024.73 1.00309
\(401\) − 13547.4i − 1.68709i −0.537059 0.843544i \(-0.680465\pi\)
0.537059 0.843544i \(-0.319535\pi\)
\(402\) 0 0
\(403\) −2788.53 −0.344681
\(404\) −9798.85 −1.20671
\(405\) 0 0
\(406\) 0 0
\(407\) − 6778.62i − 0.825562i
\(408\) 0 0
\(409\) − 2162.24i − 0.261408i −0.991421 0.130704i \(-0.958276\pi\)
0.991421 0.130704i \(-0.0417237\pi\)
\(410\) 6639.91i 0.799809i
\(411\) 0 0
\(412\) 6527.83i 0.780590i
\(413\) 0 0
\(414\) 0 0
\(415\) −904.870 −0.107032
\(416\) 2553.02 0.300895
\(417\) 0 0
\(418\) − 23648.0i − 2.76713i
\(419\) −4907.59 −0.572199 −0.286099 0.958200i \(-0.592359\pi\)
−0.286099 + 0.958200i \(0.592359\pi\)
\(420\) 0 0
\(421\) 5719.49 0.662116 0.331058 0.943610i \(-0.392594\pi\)
0.331058 + 0.943610i \(0.392594\pi\)
\(422\) − 3628.95i − 0.418613i
\(423\) 0 0
\(424\) −1390.90 −0.159311
\(425\) 9547.56 1.08971
\(426\) 0 0
\(427\) 0 0
\(428\) 1949.46i 0.220166i
\(429\) 0 0
\(430\) − 6952.05i − 0.779669i
\(431\) 3826.81i 0.427682i 0.976868 + 0.213841i \(0.0685975\pi\)
−0.976868 + 0.213841i \(0.931403\pi\)
\(432\) 0 0
\(433\) 9120.21i 1.01222i 0.862470 + 0.506108i \(0.168916\pi\)
−0.862470 + 0.506108i \(0.831084\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −11369.8 −1.24888
\(437\) −6306.47 −0.690342
\(438\) 0 0
\(439\) 2944.49i 0.320120i 0.987107 + 0.160060i \(0.0511688\pi\)
−0.987107 + 0.160060i \(0.948831\pi\)
\(440\) 1722.92 0.186675
\(441\) 0 0
\(442\) 3962.63 0.426433
\(443\) 4263.66i 0.457275i 0.973512 + 0.228637i \(0.0734270\pi\)
−0.973512 + 0.228637i \(0.926573\pi\)
\(444\) 0 0
\(445\) 3670.81 0.391041
\(446\) 2696.86 0.286323
\(447\) 0 0
\(448\) 0 0
\(449\) − 1608.94i − 0.169110i −0.996419 0.0845550i \(-0.973053\pi\)
0.996419 0.0845550i \(-0.0269469\pi\)
\(450\) 0 0
\(451\) − 17922.3i − 1.87124i
\(452\) − 9701.89i − 1.00960i
\(453\) 0 0
\(454\) − 23106.2i − 2.38861i
\(455\) 0 0
\(456\) 0 0
\(457\) −5253.81 −0.537775 −0.268887 0.963172i \(-0.586656\pi\)
−0.268887 + 0.963172i \(0.586656\pi\)
\(458\) −21029.2 −2.14548
\(459\) 0 0
\(460\) 1178.05i 0.119406i
\(461\) 19229.1 1.94271 0.971356 0.237627i \(-0.0763696\pi\)
0.971356 + 0.237627i \(0.0763696\pi\)
\(462\) 0 0
\(463\) 6539.63 0.656420 0.328210 0.944605i \(-0.393555\pi\)
0.328210 + 0.944605i \(0.393555\pi\)
\(464\) 3675.66i 0.367755i
\(465\) 0 0
\(466\) −14663.8 −1.45769
\(467\) −9017.76 −0.893559 −0.446780 0.894644i \(-0.647429\pi\)
−0.446780 + 0.894644i \(0.647429\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 6892.90i 0.676480i
\(471\) 0 0
\(472\) 1145.03i 0.111662i
\(473\) 18764.8i 1.82412i
\(474\) 0 0
\(475\) 14614.7i 1.41172i
\(476\) 0 0
\(477\) 0 0
\(478\) 14525.7 1.38994
\(479\) −8225.75 −0.784643 −0.392322 0.919828i \(-0.628328\pi\)
−0.392322 + 0.919828i \(0.628328\pi\)
\(480\) 0 0
\(481\) 1738.71i 0.164820i
\(482\) −12134.1 −1.14666
\(483\) 0 0
\(484\) 4263.17 0.400372
\(485\) − 3187.52i − 0.298428i
\(486\) 0 0
\(487\) −18034.8 −1.67810 −0.839048 0.544058i \(-0.816887\pi\)
−0.839048 + 0.544058i \(0.816887\pi\)
\(488\) −2692.08 −0.249723
\(489\) 0 0
\(490\) 0 0
\(491\) − 12148.2i − 1.11658i −0.829646 0.558290i \(-0.811458\pi\)
0.829646 0.558290i \(-0.188542\pi\)
\(492\) 0 0
\(493\) 4373.18i 0.399510i
\(494\) 6065.69i 0.552446i
\(495\) 0 0
\(496\) 18371.9i 1.66315i
\(497\) 0 0
\(498\) 0 0
\(499\) −2946.55 −0.264340 −0.132170 0.991227i \(-0.542195\pi\)
−0.132170 + 0.991227i \(0.542195\pi\)
\(500\) 6001.00 0.536746
\(501\) 0 0
\(502\) − 5198.84i − 0.462222i
\(503\) −11290.2 −1.00080 −0.500402 0.865793i \(-0.666814\pi\)
−0.500402 + 0.865793i \(0.666814\pi\)
\(504\) 0 0
\(505\) 7741.23 0.682139
\(506\) − 7599.71i − 0.667684i
\(507\) 0 0
\(508\) 6653.00 0.581062
\(509\) −6428.81 −0.559827 −0.279914 0.960025i \(-0.590306\pi\)
−0.279914 + 0.960025i \(0.590306\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 11697.4i − 1.00968i
\(513\) 0 0
\(514\) − 15162.9i − 1.30118i
\(515\) − 5157.08i − 0.441258i
\(516\) 0 0
\(517\) − 18605.2i − 1.58270i
\(518\) 0 0
\(519\) 0 0
\(520\) −441.928 −0.0372689
\(521\) 6300.76 0.529830 0.264915 0.964272i \(-0.414656\pi\)
0.264915 + 0.964272i \(0.414656\pi\)
\(522\) 0 0
\(523\) 13190.3i 1.10281i 0.834238 + 0.551405i \(0.185908\pi\)
−0.834238 + 0.551405i \(0.814092\pi\)
\(524\) 3654.99 0.304712
\(525\) 0 0
\(526\) −17105.9 −1.41797
\(527\) 21858.3i 1.80676i
\(528\) 0 0
\(529\) 10140.3 0.833427
\(530\) −2817.31 −0.230898
\(531\) 0 0
\(532\) 0 0
\(533\) 4597.05i 0.373584i
\(534\) 0 0
\(535\) − 1540.10i − 0.124457i
\(536\) − 3531.16i − 0.284558i
\(537\) 0 0
\(538\) − 13642.2i − 1.09323i
\(539\) 0 0
\(540\) 0 0
\(541\) 11893.0 0.945137 0.472568 0.881294i \(-0.343327\pi\)
0.472568 + 0.881294i \(0.343327\pi\)
\(542\) 26554.9 2.10449
\(543\) 0 0
\(544\) − 20012.3i − 1.57724i
\(545\) 8982.27 0.705979
\(546\) 0 0
\(547\) 1348.19 0.105383 0.0526915 0.998611i \(-0.483220\pi\)
0.0526915 + 0.998611i \(0.483220\pi\)
\(548\) 14147.2i 1.10281i
\(549\) 0 0
\(550\) −17611.6 −1.36539
\(551\) −6694.12 −0.517567
\(552\) 0 0
\(553\) 0 0
\(554\) − 4193.54i − 0.321600i
\(555\) 0 0
\(556\) 6300.95i 0.480611i
\(557\) − 19008.3i − 1.44598i −0.690861 0.722988i \(-0.742768\pi\)
0.690861 0.722988i \(-0.257232\pi\)
\(558\) 0 0
\(559\) − 4813.16i − 0.364177i
\(560\) 0 0
\(561\) 0 0
\(562\) −14537.3 −1.09114
\(563\) −2730.52 −0.204401 −0.102201 0.994764i \(-0.532588\pi\)
−0.102201 + 0.994764i \(0.532588\pi\)
\(564\) 0 0
\(565\) 7664.63i 0.570714i
\(566\) 16714.7 1.24129
\(567\) 0 0
\(568\) 6054.02 0.447220
\(569\) 1541.26i 0.113556i 0.998387 + 0.0567778i \(0.0180827\pi\)
−0.998387 + 0.0567778i \(0.981917\pi\)
\(570\) 0 0
\(571\) −3470.00 −0.254317 −0.127158 0.991882i \(-0.540586\pi\)
−0.127158 + 0.991882i \(0.540586\pi\)
\(572\) −3058.35 −0.223560
\(573\) 0 0
\(574\) 0 0
\(575\) 4696.69i 0.340635i
\(576\) 0 0
\(577\) 4653.81i 0.335773i 0.985806 + 0.167886i \(0.0536941\pi\)
−0.985806 + 0.167886i \(0.946306\pi\)
\(578\) − 12840.4i − 0.924031i
\(579\) 0 0
\(580\) 1250.46i 0.0895216i
\(581\) 0 0
\(582\) 0 0
\(583\) 7604.42 0.540211
\(584\) 9125.37 0.646593
\(585\) 0 0
\(586\) 33218.5i 2.34171i
\(587\) −12056.1 −0.847717 −0.423858 0.905728i \(-0.639325\pi\)
−0.423858 + 0.905728i \(0.639325\pi\)
\(588\) 0 0
\(589\) −33459.0 −2.34067
\(590\) 2319.30i 0.161837i
\(591\) 0 0
\(592\) 11455.3 0.795288
\(593\) 12900.0 0.893324 0.446662 0.894703i \(-0.352613\pi\)
0.446662 + 0.894703i \(0.352613\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) − 5562.56i − 0.382301i
\(597\) 0 0
\(598\) 1949.32i 0.133300i
\(599\) − 3507.81i − 0.239274i −0.992818 0.119637i \(-0.961827\pi\)
0.992818 0.119637i \(-0.0381731\pi\)
\(600\) 0 0
\(601\) 1767.94i 0.119993i 0.998199 + 0.0599966i \(0.0191090\pi\)
−0.998199 + 0.0599966i \(0.980891\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 13409.1 0.903328
\(605\) −3367.96 −0.226326
\(606\) 0 0
\(607\) − 13887.7i − 0.928639i −0.885668 0.464320i \(-0.846299\pi\)
0.885668 0.464320i \(-0.153701\pi\)
\(608\) 30633.3 2.04333
\(609\) 0 0
\(610\) −5452.90 −0.361937
\(611\) 4772.20i 0.315978i
\(612\) 0 0
\(613\) 22509.8 1.48314 0.741569 0.670877i \(-0.234082\pi\)
0.741569 + 0.670877i \(0.234082\pi\)
\(614\) 13065.0 0.858730
\(615\) 0 0
\(616\) 0 0
\(617\) 1591.00i 0.103811i 0.998652 + 0.0519053i \(0.0165294\pi\)
−0.998652 + 0.0519053i \(0.983471\pi\)
\(618\) 0 0
\(619\) 13273.5i 0.861886i 0.902379 + 0.430943i \(0.141819\pi\)
−0.902379 + 0.430943i \(0.858181\pi\)
\(620\) 6250.14i 0.404857i
\(621\) 0 0
\(622\) − 13187.9i − 0.850138i
\(623\) 0 0
\(624\) 0 0
\(625\) 8300.01 0.531200
\(626\) 23513.0 1.50123
\(627\) 0 0
\(628\) 19653.7i 1.24883i
\(629\) 13629.2 0.863960
\(630\) 0 0
\(631\) −11933.2 −0.752860 −0.376430 0.926445i \(-0.622848\pi\)
−0.376430 + 0.926445i \(0.622848\pi\)
\(632\) 5571.75i 0.350684i
\(633\) 0 0
\(634\) 9438.41 0.591242
\(635\) −5255.96 −0.328467
\(636\) 0 0
\(637\) 0 0
\(638\) − 8066.85i − 0.500580i
\(639\) 0 0
\(640\) 4654.42i 0.287472i
\(641\) 27542.5i 1.69713i 0.529089 + 0.848566i \(0.322534\pi\)
−0.529089 + 0.848566i \(0.677466\pi\)
\(642\) 0 0
\(643\) − 19003.1i − 1.16549i −0.812655 0.582746i \(-0.801978\pi\)
0.812655 0.582746i \(-0.198022\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 47546.9 2.89583
\(647\) 25048.4 1.52203 0.761016 0.648733i \(-0.224701\pi\)
0.761016 + 0.648733i \(0.224701\pi\)
\(648\) 0 0
\(649\) − 6260.20i − 0.378636i
\(650\) 4517.36 0.272593
\(651\) 0 0
\(652\) 2081.51 0.125028
\(653\) − 13745.1i − 0.823718i −0.911248 0.411859i \(-0.864880\pi\)
0.911248 0.411859i \(-0.135120\pi\)
\(654\) 0 0
\(655\) −2887.49 −0.172250
\(656\) 30287.2 1.80262
\(657\) 0 0
\(658\) 0 0
\(659\) − 6965.25i − 0.411726i −0.978581 0.205863i \(-0.934000\pi\)
0.978581 0.205863i \(-0.0660002\pi\)
\(660\) 0 0
\(661\) 11853.6i 0.697504i 0.937215 + 0.348752i \(0.113394\pi\)
−0.937215 + 0.348752i \(0.886606\pi\)
\(662\) − 42274.6i − 2.48195i
\(663\) 0 0
\(664\) 1656.84i 0.0968342i
\(665\) 0 0
\(666\) 0 0
\(667\) −2151.28 −0.124884
\(668\) 9240.51 0.535219
\(669\) 0 0
\(670\) − 7152.50i − 0.412426i
\(671\) 14718.3 0.846789
\(672\) 0 0
\(673\) −30769.5 −1.76238 −0.881188 0.472766i \(-0.843255\pi\)
−0.881188 + 0.472766i \(0.843255\pi\)
\(674\) 2044.00i 0.116813i
\(675\) 0 0
\(676\) −11859.9 −0.674778
\(677\) −1309.49 −0.0743393 −0.0371696 0.999309i \(-0.511834\pi\)
−0.0371696 + 0.999309i \(0.511834\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 3464.12i 0.195358i
\(681\) 0 0
\(682\) − 40320.3i − 2.26385i
\(683\) − 23134.7i − 1.29608i −0.761606 0.648041i \(-0.775589\pi\)
0.761606 0.648041i \(-0.224411\pi\)
\(684\) 0 0
\(685\) − 11176.5i − 0.623403i
\(686\) 0 0
\(687\) 0 0
\(688\) −31711.0 −1.75723
\(689\) −1950.53 −0.107851
\(690\) 0 0
\(691\) 24284.5i 1.33694i 0.743739 + 0.668471i \(0.233051\pi\)
−0.743739 + 0.668471i \(0.766949\pi\)
\(692\) 8442.24 0.463766
\(693\) 0 0
\(694\) −10708.9 −0.585744
\(695\) − 4977.84i − 0.271684i
\(696\) 0 0
\(697\) 36034.8 1.95827
\(698\) 8494.43 0.460629
\(699\) 0 0
\(700\) 0 0
\(701\) − 20398.8i − 1.09907i −0.835469 0.549537i \(-0.814804\pi\)
0.835469 0.549537i \(-0.185196\pi\)
\(702\) 0 0
\(703\) 20862.5i 1.11926i
\(704\) 8906.50i 0.476813i
\(705\) 0 0
\(706\) − 5227.65i − 0.278676i
\(707\) 0 0
\(708\) 0 0
\(709\) −30857.0 −1.63450 −0.817250 0.576283i \(-0.804502\pi\)
−0.817250 + 0.576283i \(0.804502\pi\)
\(710\) 12262.6 0.648181
\(711\) 0 0
\(712\) − 6721.36i − 0.353783i
\(713\) −10752.7 −0.564783
\(714\) 0 0
\(715\) 2416.14 0.126376
\(716\) 5650.80i 0.294945i
\(717\) 0 0
\(718\) −25786.1 −1.34029
\(719\) −34407.2 −1.78466 −0.892332 0.451380i \(-0.850932\pi\)
−0.892332 + 0.451380i \(0.850932\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 47342.4i 2.44030i
\(723\) 0 0
\(724\) 8905.26i 0.457129i
\(725\) 4985.38i 0.255383i
\(726\) 0 0
\(727\) − 11996.1i − 0.611983i −0.952034 0.305992i \(-0.901012\pi\)
0.952034 0.305992i \(-0.0989880\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 18483.8 0.937144
\(731\) −37728.8 −1.90896
\(732\) 0 0
\(733\) − 32123.2i − 1.61868i −0.587337 0.809342i \(-0.699824\pi\)
0.587337 0.809342i \(-0.300176\pi\)
\(734\) 5571.00 0.280149
\(735\) 0 0
\(736\) 9844.55 0.493037
\(737\) 19305.9i 0.964913i
\(738\) 0 0
\(739\) 22871.9 1.13851 0.569254 0.822162i \(-0.307232\pi\)
0.569254 + 0.822162i \(0.307232\pi\)
\(740\) 3897.10 0.193595
\(741\) 0 0
\(742\) 0 0
\(743\) − 20982.3i − 1.03602i −0.855373 0.518012i \(-0.826672\pi\)
0.855373 0.518012i \(-0.173328\pi\)
\(744\) 0 0
\(745\) 4394.50i 0.216110i
\(746\) 26165.8i 1.28418i
\(747\) 0 0
\(748\) 23973.4i 1.17187i
\(749\) 0 0
\(750\) 0 0
\(751\) 18642.5 0.905827 0.452914 0.891554i \(-0.350385\pi\)
0.452914 + 0.891554i \(0.350385\pi\)
\(752\) 31441.2 1.52466
\(753\) 0 0
\(754\) 2069.14i 0.0999385i
\(755\) −10593.4 −0.510641
\(756\) 0 0
\(757\) 4135.46 0.198555 0.0992774 0.995060i \(-0.468347\pi\)
0.0992774 + 0.995060i \(0.468347\pi\)
\(758\) − 19150.5i − 0.917648i
\(759\) 0 0
\(760\) −5302.61 −0.253087
\(761\) 3581.69 0.170612 0.0853062 0.996355i \(-0.472813\pi\)
0.0853062 + 0.996355i \(0.472813\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) − 27536.9i − 1.30399i
\(765\) 0 0
\(766\) − 9338.50i − 0.440488i
\(767\) 1605.74i 0.0755929i
\(768\) 0 0
\(769\) − 11427.1i − 0.535852i −0.963439 0.267926i \(-0.913662\pi\)
0.963439 0.267926i \(-0.0863383\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 7345.88 0.342466
\(773\) −1483.61 −0.0690318 −0.0345159 0.999404i \(-0.510989\pi\)
−0.0345159 + 0.999404i \(0.510989\pi\)
\(774\) 0 0
\(775\) 24918.3i 1.15496i
\(776\) −5836.43 −0.269994
\(777\) 0 0
\(778\) 17753.1 0.818097
\(779\) 55159.2i 2.53695i
\(780\) 0 0
\(781\) −33099.0 −1.51649
\(782\) 15280.1 0.698739
\(783\) 0 0
\(784\) 0 0
\(785\) − 15526.7i − 0.705951i
\(786\) 0 0
\(787\) − 17520.2i − 0.793557i −0.917914 0.396778i \(-0.870128\pi\)
0.917914 0.396778i \(-0.129872\pi\)
\(788\) 2826.48i 0.127778i
\(789\) 0 0
\(790\) 11285.8i 0.508266i
\(791\) 0 0
\(792\) 0 0
\(793\) −3775.24 −0.169058
\(794\) −35584.2 −1.59047
\(795\) 0 0
\(796\) 17199.3i 0.765846i
\(797\) 2687.85 0.119459 0.0597293 0.998215i \(-0.480976\pi\)
0.0597293 + 0.998215i \(0.480976\pi\)
\(798\) 0 0
\(799\) 37407.7 1.65631
\(800\) − 22813.8i − 1.00824i
\(801\) 0 0
\(802\) −50244.6 −2.21222
\(803\) −49890.9 −2.19254
\(804\) 0 0
\(805\) 0 0
\(806\) 10342.1i 0.451967i
\(807\) 0 0
\(808\) − 14174.4i − 0.617145i
\(809\) − 841.639i − 0.0365766i −0.999833 0.0182883i \(-0.994178\pi\)
0.999833 0.0182883i \(-0.00582167\pi\)
\(810\) 0 0
\(811\) 37115.6i 1.60704i 0.595281 + 0.803518i \(0.297041\pi\)
−0.595281 + 0.803518i \(0.702959\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −25140.6 −1.08253
\(815\) −1644.42 −0.0706767
\(816\) 0 0
\(817\) − 57752.2i − 2.47306i
\(818\) −8019.33 −0.342774
\(819\) 0 0
\(820\) 10303.7 0.438807
\(821\) − 27601.6i − 1.17333i −0.809831 0.586664i \(-0.800441\pi\)
0.809831 0.586664i \(-0.199559\pi\)
\(822\) 0 0
\(823\) 7365.20 0.311950 0.155975 0.987761i \(-0.450148\pi\)
0.155975 + 0.987761i \(0.450148\pi\)
\(824\) −9442.75 −0.399216
\(825\) 0 0
\(826\) 0 0
\(827\) 46801.3i 1.96789i 0.178485 + 0.983943i \(0.442880\pi\)
−0.178485 + 0.983943i \(0.557120\pi\)
\(828\) 0 0
\(829\) 10169.1i 0.426039i 0.977048 + 0.213020i \(0.0683298\pi\)
−0.977048 + 0.213020i \(0.931670\pi\)
\(830\) 3355.99i 0.140347i
\(831\) 0 0
\(832\) − 2284.51i − 0.0951936i
\(833\) 0 0
\(834\) 0 0
\(835\) −7300.13 −0.302553
\(836\) −36696.6 −1.51816
\(837\) 0 0
\(838\) 18201.3i 0.750303i
\(839\) −45477.6 −1.87135 −0.935674 0.352864i \(-0.885208\pi\)
−0.935674 + 0.352864i \(0.885208\pi\)
\(840\) 0 0
\(841\) 22105.5 0.906371
\(842\) − 21212.5i − 0.868209i
\(843\) 0 0
\(844\) −5631.36 −0.229667
\(845\) 9369.48 0.381444
\(846\) 0 0
\(847\) 0 0
\(848\) 12850.9i 0.520401i
\(849\) 0 0
\(850\) − 35410.1i − 1.42889i
\(851\) 6704.53i 0.270068i
\(852\) 0 0
\(853\) 3005.89i 0.120656i 0.998179 + 0.0603281i \(0.0192147\pi\)
−0.998179 + 0.0603281i \(0.980785\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −2819.97 −0.112599
\(857\) −44853.2 −1.78781 −0.893907 0.448252i \(-0.852047\pi\)
−0.893907 + 0.448252i \(0.852047\pi\)
\(858\) 0 0
\(859\) − 13090.8i − 0.519968i −0.965613 0.259984i \(-0.916283\pi\)
0.965613 0.259984i \(-0.0837172\pi\)
\(860\) −10788.1 −0.427757
\(861\) 0 0
\(862\) 14192.9 0.560804
\(863\) 21219.2i 0.836977i 0.908222 + 0.418488i \(0.137440\pi\)
−0.908222 + 0.418488i \(0.862560\pi\)
\(864\) 0 0
\(865\) −6669.49 −0.262161
\(866\) 33825.2 1.32728
\(867\) 0 0
\(868\) 0 0
\(869\) − 30462.3i − 1.18914i
\(870\) 0 0
\(871\) − 4951.93i − 0.192641i
\(872\) − 16446.8i − 0.638714i
\(873\) 0 0
\(874\) 23389.5i 0.905220i
\(875\) 0 0
\(876\) 0 0
\(877\) −41876.4 −1.61239 −0.806195 0.591649i \(-0.798477\pi\)
−0.806195 + 0.591649i \(0.798477\pi\)
\(878\) 10920.5 0.419762
\(879\) 0 0
\(880\) − 15918.5i − 0.609788i
\(881\) 16277.8 0.622490 0.311245 0.950330i \(-0.399254\pi\)
0.311245 + 0.950330i \(0.399254\pi\)
\(882\) 0 0
\(883\) −32041.9 −1.22117 −0.610586 0.791950i \(-0.709066\pi\)
−0.610586 + 0.791950i \(0.709066\pi\)
\(884\) − 6149.16i − 0.233958i
\(885\) 0 0
\(886\) 15813.1 0.599607
\(887\) −46153.6 −1.74711 −0.873555 0.486726i \(-0.838191\pi\)
−0.873555 + 0.486726i \(0.838191\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) − 13614.4i − 0.512758i
\(891\) 0 0
\(892\) − 4184.96i − 0.157088i
\(893\) 57260.8i 2.14576i
\(894\) 0 0
\(895\) − 4464.21i − 0.166729i
\(896\) 0 0
\(897\) 0 0
\(898\) −5967.24 −0.221748
\(899\) −11413.6 −0.423432
\(900\) 0 0
\(901\) 15289.5i 0.565337i
\(902\) −66470.4 −2.45368
\(903\) 0 0
\(904\) 14034.1 0.516337
\(905\) − 7035.28i − 0.258410i
\(906\) 0 0
\(907\) 27229.7 0.996854 0.498427 0.866932i \(-0.333911\pi\)
0.498427 + 0.866932i \(0.333911\pi\)
\(908\) −35855.9 −1.31048
\(909\) 0 0
\(910\) 0 0
\(911\) 11012.2i 0.400494i 0.979745 + 0.200247i \(0.0641745\pi\)
−0.979745 + 0.200247i \(0.935826\pi\)
\(912\) 0 0
\(913\) − 9058.41i − 0.328357i
\(914\) 19485.4i 0.705164i
\(915\) 0 0
\(916\) 32632.8i 1.17709i
\(917\) 0 0
\(918\) 0 0
\(919\) 21777.5 0.781692 0.390846 0.920456i \(-0.372182\pi\)
0.390846 + 0.920456i \(0.372182\pi\)
\(920\) −1704.09 −0.0610676
\(921\) 0 0
\(922\) − 71317.3i − 2.54741i
\(923\) 8489.87 0.302760
\(924\) 0 0
\(925\) 15537.1 0.552278
\(926\) − 24254.3i − 0.860739i
\(927\) 0 0
\(928\) 10449.7 0.369642
\(929\) −12372.3 −0.436945 −0.218472 0.975843i \(-0.570107\pi\)
−0.218472 + 0.975843i \(0.570107\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 22755.0i 0.799749i
\(933\) 0 0
\(934\) 33445.2i 1.17169i
\(935\) − 18939.3i − 0.662441i
\(936\) 0 0
\(937\) − 41577.7i − 1.44961i −0.688954 0.724806i \(-0.741930\pi\)
0.688954 0.724806i \(-0.258070\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 10696.3 0.371144
\(941\) −32490.8 −1.12558 −0.562789 0.826601i \(-0.690272\pi\)
−0.562789 + 0.826601i \(0.690272\pi\)
\(942\) 0 0
\(943\) 17726.4i 0.612143i
\(944\) 10579.2 0.364751
\(945\) 0 0
\(946\) 69595.2 2.39190
\(947\) − 50830.7i − 1.74422i −0.489309 0.872110i \(-0.662751\pi\)
0.489309 0.872110i \(-0.337249\pi\)
\(948\) 0 0
\(949\) 12797.0 0.437732
\(950\) 54203.0 1.85114
\(951\) 0 0
\(952\) 0 0
\(953\) − 4892.40i − 0.166296i −0.996537 0.0831482i \(-0.973503\pi\)
0.996537 0.0831482i \(-0.0264975\pi\)
\(954\) 0 0
\(955\) 21754.5i 0.737132i
\(956\) − 22540.9i − 0.762577i
\(957\) 0 0
\(958\) 30507.8i 1.02887i
\(959\) 0 0
\(960\) 0 0
\(961\) −27257.3 −0.914952
\(962\) 6448.54 0.216122
\(963\) 0 0
\(964\) 18829.5i 0.629105i
\(965\) −5803.35 −0.193592
\(966\) 0 0
\(967\) −41170.9 −1.36915 −0.684575 0.728943i \(-0.740012\pi\)
−0.684575 + 0.728943i \(0.740012\pi\)
\(968\) 6166.83i 0.204762i
\(969\) 0 0
\(970\) −11821.9 −0.391318
\(971\) 43287.6 1.43065 0.715327 0.698790i \(-0.246278\pi\)
0.715327 + 0.698790i \(0.246278\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 66887.5i 2.20042i
\(975\) 0 0
\(976\) 24872.8i 0.815737i
\(977\) 18073.1i 0.591821i 0.955216 + 0.295911i \(0.0956231\pi\)
−0.955216 + 0.295911i \(0.904377\pi\)
\(978\) 0 0
\(979\) 36747.5i 1.19965i
\(980\) 0 0
\(981\) 0 0
\(982\) −45055.4 −1.46413
\(983\) −19730.6 −0.640192 −0.320096 0.947385i \(-0.603715\pi\)
−0.320096 + 0.947385i \(0.603715\pi\)
\(984\) 0 0
\(985\) − 2232.96i − 0.0722315i
\(986\) 16219.3 0.523862
\(987\) 0 0
\(988\) 9412.65 0.303093
\(989\) − 18559.7i − 0.596729i
\(990\) 0 0
\(991\) −13692.3 −0.438899 −0.219450 0.975624i \(-0.570426\pi\)
−0.219450 + 0.975624i \(0.570426\pi\)
\(992\) 52230.4 1.67169
\(993\) 0 0
\(994\) 0 0
\(995\) − 13587.7i − 0.432923i
\(996\) 0 0
\(997\) − 3085.56i − 0.0980148i −0.998798 0.0490074i \(-0.984394\pi\)
0.998798 0.0490074i \(-0.0156058\pi\)
\(998\) 10928.2i 0.346620i
\(999\) 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.c.b.440.17 yes 24
3.2 odd 2 inner 441.4.c.b.440.8 yes 24
7.2 even 3 441.4.p.d.80.3 48
7.3 odd 6 441.4.p.d.215.22 48
7.4 even 3 441.4.p.d.215.21 48
7.5 odd 6 441.4.p.d.80.4 48
7.6 odd 2 inner 441.4.c.b.440.7 24
21.2 odd 6 441.4.p.d.80.22 48
21.5 even 6 441.4.p.d.80.21 48
21.11 odd 6 441.4.p.d.215.4 48
21.17 even 6 441.4.p.d.215.3 48
21.20 even 2 inner 441.4.c.b.440.18 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
441.4.c.b.440.7 24 7.6 odd 2 inner
441.4.c.b.440.8 yes 24 3.2 odd 2 inner
441.4.c.b.440.17 yes 24 1.1 even 1 trivial
441.4.c.b.440.18 yes 24 21.20 even 2 inner
441.4.p.d.80.3 48 7.2 even 3
441.4.p.d.80.4 48 7.5 odd 6
441.4.p.d.80.21 48 21.5 even 6
441.4.p.d.80.22 48 21.2 odd 6
441.4.p.d.215.3 48 21.17 even 6
441.4.p.d.215.4 48 21.11 odd 6
441.4.p.d.215.21 48 7.4 even 3
441.4.p.d.215.22 48 7.3 odd 6