Properties

Label 2760.3.g.a.2161.20
Level $2760$
Weight $3$
Character 2760.2161
Analytic conductor $75.205$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2760,3,Mod(2161,2760)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2760, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2760.2161");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2760.g (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: \(75.2045529634\)
Analytic rank: \(0\)
Dimension: \(96\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2161.20
Character \(\chi\) \(=\) 2760.2161
Dual form 2760.3.g.a.2161.29

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{3} -2.23607i q^{5} +7.31842i q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{3} -2.23607i q^{5} +7.31842i q^{7} +3.00000 q^{9} +16.7777i q^{11} +15.7635 q^{13} +3.87298i q^{15} -0.232991i q^{17} -11.5221i q^{19} -12.6759i q^{21} +(21.9368 + 6.91199i) q^{23} -5.00000 q^{25} -5.19615 q^{27} +50.2947 q^{29} -16.6312 q^{31} -29.0599i q^{33} +16.3645 q^{35} -20.9752i q^{37} -27.3032 q^{39} +40.0718 q^{41} +28.0841i q^{43} -6.70820i q^{45} -16.8642 q^{47} -4.55932 q^{49} +0.403553i q^{51} +2.31406i q^{53} +37.5161 q^{55} +19.9568i q^{57} -82.7783 q^{59} +15.5672i q^{61} +21.9553i q^{63} -35.2483i q^{65} -49.4235i q^{67} +(-37.9957 - 11.9719i) q^{69} -40.2770 q^{71} +33.2936 q^{73} +8.66025 q^{75} -122.786 q^{77} +128.178i q^{79} +9.00000 q^{81} -37.3598i q^{83} -0.520984 q^{85} -87.1130 q^{87} -157.125i q^{89} +115.364i q^{91} +28.8060 q^{93} -25.7641 q^{95} +49.2190i q^{97} +50.3332i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 288 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 96 q + 288 q^{9} - 16 q^{23} - 480 q^{25} - 80 q^{31} + 80 q^{35} + 48 q^{39} + 112 q^{41} + 32 q^{47} - 688 q^{49} - 80 q^{55} - 496 q^{59} - 96 q^{69} - 416 q^{71} - 320 q^{73} + 864 q^{81} + 192 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2760\mathbb{Z}\right)^\times\).

\(n\) \(1201\) \(1381\) \(1657\) \(1841\) \(2071\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(1\)

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 0
\(3\) −1.73205 −0.577350
\(4\) 0 0
\(5\) 2.23607i 0.447214i
\(6\) 0 0
\(7\) 7.31842i 1.04549i 0.852489 + 0.522745i \(0.175092\pi\)
−0.852489 + 0.522745i \(0.824908\pi\)
\(8\) 0 0
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) 16.7777i 1.52525i 0.646842 + 0.762624i \(0.276089\pi\)
−0.646842 + 0.762624i \(0.723911\pi\)
\(12\) 0 0
\(13\) 15.7635 1.21258 0.606289 0.795244i \(-0.292657\pi\)
0.606289 + 0.795244i \(0.292657\pi\)
\(14\) 0 0
\(15\) 3.87298i 0.258199i
\(16\) 0 0
\(17\) 0.232991i 0.0137054i −0.999977 0.00685268i \(-0.997819\pi\)
0.999977 0.00685268i \(-0.00218129\pi\)
\(18\) 0 0
\(19\) 11.5221i 0.606424i −0.952923 0.303212i \(-0.901941\pi\)
0.952923 0.303212i \(-0.0980591\pi\)
\(20\) 0 0
\(21\) 12.6759i 0.603613i
\(22\) 0 0
\(23\) 21.9368 + 6.91199i 0.953775 + 0.300521i
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) 0 0
\(27\) −5.19615 −0.192450
\(28\) 0 0
\(29\) 50.2947 1.73430 0.867150 0.498047i \(-0.165949\pi\)
0.867150 + 0.498047i \(0.165949\pi\)
\(30\) 0 0
\(31\) −16.6312 −0.536489 −0.268245 0.963351i \(-0.586444\pi\)
−0.268245 + 0.963351i \(0.586444\pi\)
\(32\) 0 0
\(33\) 29.0599i 0.880602i
\(34\) 0 0
\(35\) 16.3645 0.467557
\(36\) 0 0
\(37\) 20.9752i 0.566896i −0.958988 0.283448i \(-0.908522\pi\)
0.958988 0.283448i \(-0.0914784\pi\)
\(38\) 0 0
\(39\) −27.3032 −0.700083
\(40\) 0 0
\(41\) 40.0718 0.977362 0.488681 0.872463i \(-0.337478\pi\)
0.488681 + 0.872463i \(0.337478\pi\)
\(42\) 0 0
\(43\) 28.0841i 0.653119i 0.945177 + 0.326560i \(0.105889\pi\)
−0.945177 + 0.326560i \(0.894111\pi\)
\(44\) 0 0
\(45\) 6.70820i 0.149071i
\(46\) 0 0
\(47\) −16.8642 −0.358813 −0.179407 0.983775i \(-0.557418\pi\)
−0.179407 + 0.983775i \(0.557418\pi\)
\(48\) 0 0
\(49\) −4.55932 −0.0930473
\(50\) 0 0
\(51\) 0.403553i 0.00791280i
\(52\) 0 0
\(53\) 2.31406i 0.0436615i 0.999762 + 0.0218308i \(0.00694950\pi\)
−0.999762 + 0.0218308i \(0.993050\pi\)
\(54\) 0 0
\(55\) 37.5161 0.682111
\(56\) 0 0
\(57\) 19.9568i 0.350119i
\(58\) 0 0
\(59\) −82.7783 −1.40302 −0.701511 0.712659i \(-0.747491\pi\)
−0.701511 + 0.712659i \(0.747491\pi\)
\(60\) 0 0
\(61\) 15.5672i 0.255200i 0.991826 + 0.127600i \(0.0407273\pi\)
−0.991826 + 0.127600i \(0.959273\pi\)
\(62\) 0 0
\(63\) 21.9553i 0.348496i
\(64\) 0 0
\(65\) 35.2483i 0.542282i
\(66\) 0 0
\(67\) 49.4235i 0.737664i −0.929496 0.368832i \(-0.879758\pi\)
0.929496 0.368832i \(-0.120242\pi\)
\(68\) 0 0
\(69\) −37.9957 11.9719i −0.550662 0.173506i
\(70\) 0 0
\(71\) −40.2770 −0.567282 −0.283641 0.958931i \(-0.591542\pi\)
−0.283641 + 0.958931i \(0.591542\pi\)
\(72\) 0 0
\(73\) 33.2936 0.456077 0.228038 0.973652i \(-0.426769\pi\)
0.228038 + 0.973652i \(0.426769\pi\)
\(74\) 0 0
\(75\) 8.66025 0.115470
\(76\) 0 0
\(77\) −122.786 −1.59463
\(78\) 0 0
\(79\) 128.178i 1.62251i 0.584693 + 0.811255i \(0.301215\pi\)
−0.584693 + 0.811255i \(0.698785\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 37.3598i 0.450119i −0.974345 0.225059i \(-0.927742\pi\)
0.974345 0.225059i \(-0.0722576\pi\)
\(84\) 0 0
\(85\) −0.520984 −0.00612923
\(86\) 0 0
\(87\) −87.1130 −1.00130
\(88\) 0 0
\(89\) 157.125i 1.76544i −0.469896 0.882722i \(-0.655708\pi\)
0.469896 0.882722i \(-0.344292\pi\)
\(90\) 0 0
\(91\) 115.364i 1.26774i
\(92\) 0 0
\(93\) 28.8060 0.309742
\(94\) 0 0
\(95\) −25.7641 −0.271201
\(96\) 0 0
\(97\) 49.2190i 0.507412i 0.967281 + 0.253706i \(0.0816496\pi\)
−0.967281 + 0.253706i \(0.918350\pi\)
\(98\) 0 0
\(99\) 50.3332i 0.508416i
\(100\) 0 0
\(101\) 87.2797 0.864156 0.432078 0.901836i \(-0.357781\pi\)
0.432078 + 0.901836i \(0.357781\pi\)
\(102\) 0 0
\(103\) 128.159i 1.24426i 0.782913 + 0.622132i \(0.213733\pi\)
−0.782913 + 0.622132i \(0.786267\pi\)
\(104\) 0 0
\(105\) −28.3441 −0.269944
\(106\) 0 0
\(107\) 113.516i 1.06090i −0.847716 0.530450i \(-0.822023\pi\)
0.847716 0.530450i \(-0.177977\pi\)
\(108\) 0 0
\(109\) 47.7242i 0.437837i −0.975743 0.218918i \(-0.929747\pi\)
0.975743 0.218918i \(-0.0702528\pi\)
\(110\) 0 0
\(111\) 36.3301i 0.327298i
\(112\) 0 0
\(113\) 48.8684i 0.432464i 0.976342 + 0.216232i \(0.0693768\pi\)
−0.976342 + 0.216232i \(0.930623\pi\)
\(114\) 0 0
\(115\) 15.4557 49.0522i 0.134397 0.426541i
\(116\) 0 0
\(117\) 47.2906 0.404193
\(118\) 0 0
\(119\) 1.70513 0.0143288
\(120\) 0 0
\(121\) −160.492 −1.32638
\(122\) 0 0
\(123\) −69.4064 −0.564280
\(124\) 0 0
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) 7.02563 0.0553199 0.0276600 0.999617i \(-0.491194\pi\)
0.0276600 + 0.999617i \(0.491194\pi\)
\(128\) 0 0
\(129\) 48.6431i 0.377079i
\(130\) 0 0
\(131\) −145.037 −1.10715 −0.553577 0.832798i \(-0.686737\pi\)
−0.553577 + 0.832798i \(0.686737\pi\)
\(132\) 0 0
\(133\) 84.3233 0.634010
\(134\) 0 0
\(135\) 11.6190i 0.0860663i
\(136\) 0 0
\(137\) 129.604i 0.946017i 0.881058 + 0.473008i \(0.156832\pi\)
−0.881058 + 0.473008i \(0.843168\pi\)
\(138\) 0 0
\(139\) 112.583 0.809953 0.404976 0.914327i \(-0.367280\pi\)
0.404976 + 0.914327i \(0.367280\pi\)
\(140\) 0 0
\(141\) 29.2097 0.207161
\(142\) 0 0
\(143\) 264.476i 1.84948i
\(144\) 0 0
\(145\) 112.462i 0.775602i
\(146\) 0 0
\(147\) 7.89697 0.0537209
\(148\) 0 0
\(149\) 198.797i 1.33421i 0.744966 + 0.667103i \(0.232466\pi\)
−0.744966 + 0.667103i \(0.767534\pi\)
\(150\) 0 0
\(151\) 227.549 1.50695 0.753474 0.657478i \(-0.228377\pi\)
0.753474 + 0.657478i \(0.228377\pi\)
\(152\) 0 0
\(153\) 0.698974i 0.00456845i
\(154\) 0 0
\(155\) 37.1884i 0.239925i
\(156\) 0 0
\(157\) 84.3025i 0.536958i −0.963285 0.268479i \(-0.913479\pi\)
0.963285 0.268479i \(-0.0865210\pi\)
\(158\) 0 0
\(159\) 4.00807i 0.0252080i
\(160\) 0 0
\(161\) −50.5849 + 160.543i −0.314192 + 0.997161i
\(162\) 0 0
\(163\) 273.495 1.67788 0.838941 0.544223i \(-0.183175\pi\)
0.838941 + 0.544223i \(0.183175\pi\)
\(164\) 0 0
\(165\) −64.9798 −0.393817
\(166\) 0 0
\(167\) 108.536 0.649917 0.324958 0.945728i \(-0.394650\pi\)
0.324958 + 0.945728i \(0.394650\pi\)
\(168\) 0 0
\(169\) 79.4887 0.470347
\(170\) 0 0
\(171\) 34.5662i 0.202141i
\(172\) 0 0
\(173\) −229.540 −1.32682 −0.663410 0.748256i \(-0.730891\pi\)
−0.663410 + 0.748256i \(0.730891\pi\)
\(174\) 0 0
\(175\) 36.5921i 0.209098i
\(176\) 0 0
\(177\) 143.376 0.810035
\(178\) 0 0
\(179\) 200.266 1.11880 0.559402 0.828896i \(-0.311031\pi\)
0.559402 + 0.828896i \(0.311031\pi\)
\(180\) 0 0
\(181\) 327.809i 1.81110i 0.424240 + 0.905550i \(0.360541\pi\)
−0.424240 + 0.905550i \(0.639459\pi\)
\(182\) 0 0
\(183\) 26.9631i 0.147340i
\(184\) 0 0
\(185\) −46.9019 −0.253524
\(186\) 0 0
\(187\) 3.90906 0.0209041
\(188\) 0 0
\(189\) 38.0276i 0.201204i
\(190\) 0 0
\(191\) 254.294i 1.33138i −0.746228 0.665691i \(-0.768137\pi\)
0.746228 0.665691i \(-0.231863\pi\)
\(192\) 0 0
\(193\) −37.4247 −0.193910 −0.0969552 0.995289i \(-0.530910\pi\)
−0.0969552 + 0.995289i \(0.530910\pi\)
\(194\) 0 0
\(195\) 61.0519i 0.313086i
\(196\) 0 0
\(197\) 305.386 1.55018 0.775092 0.631849i \(-0.217704\pi\)
0.775092 + 0.631849i \(0.217704\pi\)
\(198\) 0 0
\(199\) 212.985i 1.07027i 0.844765 + 0.535137i \(0.179740\pi\)
−0.844765 + 0.535137i \(0.820260\pi\)
\(200\) 0 0
\(201\) 85.6040i 0.425890i
\(202\) 0 0
\(203\) 368.078i 1.81319i
\(204\) 0 0
\(205\) 89.6033i 0.437089i
\(206\) 0 0
\(207\) 65.8105 + 20.7360i 0.317925 + 0.100174i
\(208\) 0 0
\(209\) 193.314 0.924947
\(210\) 0 0
\(211\) 241.657 1.14529 0.572647 0.819802i \(-0.305917\pi\)
0.572647 + 0.819802i \(0.305917\pi\)
\(212\) 0 0
\(213\) 69.7618 0.327520
\(214\) 0 0
\(215\) 62.7980 0.292084
\(216\) 0 0
\(217\) 121.714i 0.560894i
\(218\) 0 0
\(219\) −57.6662 −0.263316
\(220\) 0 0
\(221\) 3.67276i 0.0166188i
\(222\) 0 0
\(223\) −268.177 −1.20259 −0.601293 0.799029i \(-0.705348\pi\)
−0.601293 + 0.799029i \(0.705348\pi\)
\(224\) 0 0
\(225\) −15.0000 −0.0666667
\(226\) 0 0
\(227\) 334.274i 1.47257i 0.676669 + 0.736287i \(0.263423\pi\)
−0.676669 + 0.736287i \(0.736577\pi\)
\(228\) 0 0
\(229\) 23.3209i 0.101838i −0.998703 0.0509189i \(-0.983785\pi\)
0.998703 0.0509189i \(-0.0162150\pi\)
\(230\) 0 0
\(231\) 212.672 0.920660
\(232\) 0 0
\(233\) −315.831 −1.35550 −0.677749 0.735294i \(-0.737044\pi\)
−0.677749 + 0.735294i \(0.737044\pi\)
\(234\) 0 0
\(235\) 37.7095i 0.160466i
\(236\) 0 0
\(237\) 222.011i 0.936756i
\(238\) 0 0
\(239\) 57.1360 0.239063 0.119531 0.992830i \(-0.461861\pi\)
0.119531 + 0.992830i \(0.461861\pi\)
\(240\) 0 0
\(241\) 99.0975i 0.411193i 0.978637 + 0.205596i \(0.0659134\pi\)
−0.978637 + 0.205596i \(0.934087\pi\)
\(242\) 0 0
\(243\) −15.5885 −0.0641500
\(244\) 0 0
\(245\) 10.1949i 0.0416120i
\(246\) 0 0
\(247\) 181.628i 0.735337i
\(248\) 0 0
\(249\) 64.7091i 0.259876i
\(250\) 0 0
\(251\) 109.400i 0.435858i 0.975965 + 0.217929i \(0.0699302\pi\)
−0.975965 + 0.217929i \(0.930070\pi\)
\(252\) 0 0
\(253\) −115.967 + 368.050i −0.458370 + 1.45474i
\(254\) 0 0
\(255\) 0.902371 0.00353871
\(256\) 0 0
\(257\) −360.681 −1.40343 −0.701714 0.712459i \(-0.747582\pi\)
−0.701714 + 0.712459i \(0.747582\pi\)
\(258\) 0 0
\(259\) 153.505 0.592684
\(260\) 0 0
\(261\) 150.884 0.578100
\(262\) 0 0
\(263\) 134.188i 0.510220i −0.966912 0.255110i \(-0.917888\pi\)
0.966912 0.255110i \(-0.0821116\pi\)
\(264\) 0 0
\(265\) 5.17440 0.0195260
\(266\) 0 0
\(267\) 272.148i 1.01928i
\(268\) 0 0
\(269\) −270.290 −1.00480 −0.502398 0.864637i \(-0.667549\pi\)
−0.502398 + 0.864637i \(0.667549\pi\)
\(270\) 0 0
\(271\) −55.5629 −0.205029 −0.102515 0.994731i \(-0.532689\pi\)
−0.102515 + 0.994731i \(0.532689\pi\)
\(272\) 0 0
\(273\) 199.817i 0.731929i
\(274\) 0 0
\(275\) 83.8886i 0.305050i
\(276\) 0 0
\(277\) 68.5001 0.247293 0.123646 0.992326i \(-0.460541\pi\)
0.123646 + 0.992326i \(0.460541\pi\)
\(278\) 0 0
\(279\) −49.8935 −0.178830
\(280\) 0 0
\(281\) 418.773i 1.49030i −0.666899 0.745148i \(-0.732379\pi\)
0.666899 0.745148i \(-0.267621\pi\)
\(282\) 0 0
\(283\) 527.237i 1.86303i 0.363705 + 0.931514i \(0.381512\pi\)
−0.363705 + 0.931514i \(0.618488\pi\)
\(284\) 0 0
\(285\) 44.6248 0.156578
\(286\) 0 0
\(287\) 293.263i 1.02182i
\(288\) 0 0
\(289\) 288.946 0.999812
\(290\) 0 0
\(291\) 85.2498i 0.292955i
\(292\) 0 0
\(293\) 351.549i 1.19983i 0.800065 + 0.599914i \(0.204798\pi\)
−0.800065 + 0.599914i \(0.795202\pi\)
\(294\) 0 0
\(295\) 185.098i 0.627450i
\(296\) 0 0
\(297\) 87.1796i 0.293534i
\(298\) 0 0
\(299\) 345.802 + 108.957i 1.15653 + 0.364406i
\(300\) 0 0
\(301\) −205.532 −0.682829
\(302\) 0 0
\(303\) −151.173 −0.498921
\(304\) 0 0
\(305\) 34.8093 0.114129
\(306\) 0 0
\(307\) −358.319 −1.16716 −0.583582 0.812055i \(-0.698349\pi\)
−0.583582 + 0.812055i \(0.698349\pi\)
\(308\) 0 0
\(309\) 221.978i 0.718376i
\(310\) 0 0
\(311\) 257.124 0.826765 0.413382 0.910558i \(-0.364347\pi\)
0.413382 + 0.910558i \(0.364347\pi\)
\(312\) 0 0
\(313\) 322.388i 1.02999i −0.857192 0.514997i \(-0.827793\pi\)
0.857192 0.514997i \(-0.172207\pi\)
\(314\) 0 0
\(315\) 49.0935 0.155852
\(316\) 0 0
\(317\) −7.69462 −0.0242733 −0.0121366 0.999926i \(-0.503863\pi\)
−0.0121366 + 0.999926i \(0.503863\pi\)
\(318\) 0 0
\(319\) 843.831i 2.64524i
\(320\) 0 0
\(321\) 196.616i 0.612511i
\(322\) 0 0
\(323\) −2.68454 −0.00831127
\(324\) 0 0
\(325\) −78.8176 −0.242516
\(326\) 0 0
\(327\) 82.6608i 0.252785i
\(328\) 0 0
\(329\) 123.419i 0.375135i
\(330\) 0 0
\(331\) 492.799 1.48882 0.744410 0.667723i \(-0.232731\pi\)
0.744410 + 0.667723i \(0.232731\pi\)
\(332\) 0 0
\(333\) 62.9255i 0.188965i
\(334\) 0 0
\(335\) −110.514 −0.329893
\(336\) 0 0
\(337\) 269.923i 0.800957i 0.916306 + 0.400479i \(0.131156\pi\)
−0.916306 + 0.400479i \(0.868844\pi\)
\(338\) 0 0
\(339\) 84.6426i 0.249683i
\(340\) 0 0
\(341\) 279.033i 0.818279i
\(342\) 0 0
\(343\) 325.236i 0.948209i
\(344\) 0 0
\(345\) −26.7700 + 84.9610i −0.0775943 + 0.246264i
\(346\) 0 0
\(347\) −160.179 −0.461611 −0.230805 0.973000i \(-0.574136\pi\)
−0.230805 + 0.973000i \(0.574136\pi\)
\(348\) 0 0
\(349\) 142.660 0.408769 0.204385 0.978891i \(-0.434481\pi\)
0.204385 + 0.978891i \(0.434481\pi\)
\(350\) 0 0
\(351\) −81.9097 −0.233361
\(352\) 0 0
\(353\) −569.504 −1.61333 −0.806663 0.591011i \(-0.798729\pi\)
−0.806663 + 0.591011i \(0.798729\pi\)
\(354\) 0 0
\(355\) 90.0621i 0.253696i
\(356\) 0 0
\(357\) −2.95337 −0.00827274
\(358\) 0 0
\(359\) 222.167i 0.618850i 0.950924 + 0.309425i \(0.100137\pi\)
−0.950924 + 0.309425i \(0.899863\pi\)
\(360\) 0 0
\(361\) 228.242 0.632249
\(362\) 0 0
\(363\) 277.980 0.765786
\(364\) 0 0
\(365\) 74.4468i 0.203964i
\(366\) 0 0
\(367\) 110.892i 0.302157i −0.988522 0.151078i \(-0.951725\pi\)
0.988522 0.151078i \(-0.0482746\pi\)
\(368\) 0 0
\(369\) 120.215 0.325787
\(370\) 0 0
\(371\) −16.9353 −0.0456477
\(372\) 0 0
\(373\) 40.5503i 0.108714i 0.998522 + 0.0543570i \(0.0173109\pi\)
−0.998522 + 0.0543570i \(0.982689\pi\)
\(374\) 0 0
\(375\) 19.3649i 0.0516398i
\(376\) 0 0
\(377\) 792.822 2.10298
\(378\) 0 0
\(379\) 587.013i 1.54885i 0.632667 + 0.774424i \(0.281960\pi\)
−0.632667 + 0.774424i \(0.718040\pi\)
\(380\) 0 0
\(381\) −12.1687 −0.0319390
\(382\) 0 0
\(383\) 530.874i 1.38609i −0.720892 0.693047i \(-0.756268\pi\)
0.720892 0.693047i \(-0.243732\pi\)
\(384\) 0 0
\(385\) 274.559i 0.713140i
\(386\) 0 0
\(387\) 84.2524i 0.217706i
\(388\) 0 0
\(389\) 543.694i 1.39767i 0.715282 + 0.698836i \(0.246298\pi\)
−0.715282 + 0.698836i \(0.753702\pi\)
\(390\) 0 0
\(391\) 1.61043 5.11109i 0.00411875 0.0130718i
\(392\) 0 0
\(393\) 251.212 0.639215
\(394\) 0 0
\(395\) 286.615 0.725608
\(396\) 0 0
\(397\) 47.3733 0.119328 0.0596641 0.998219i \(-0.480997\pi\)
0.0596641 + 0.998219i \(0.480997\pi\)
\(398\) 0 0
\(399\) −146.052 −0.366046
\(400\) 0 0
\(401\) 100.149i 0.249749i 0.992173 + 0.124874i \(0.0398528\pi\)
−0.992173 + 0.124874i \(0.960147\pi\)
\(402\) 0 0
\(403\) −262.166 −0.650535
\(404\) 0 0
\(405\) 20.1246i 0.0496904i
\(406\) 0 0
\(407\) 351.916 0.864657
\(408\) 0 0
\(409\) −518.693 −1.26820 −0.634099 0.773252i \(-0.718629\pi\)
−0.634099 + 0.773252i \(0.718629\pi\)
\(410\) 0 0
\(411\) 224.481i 0.546183i
\(412\) 0 0
\(413\) 605.806i 1.46684i
\(414\) 0 0
\(415\) −83.5392 −0.201299
\(416\) 0 0
\(417\) −195.000 −0.467627
\(418\) 0 0
\(419\) 89.3169i 0.213167i 0.994304 + 0.106583i \(0.0339911\pi\)
−0.994304 + 0.106583i \(0.966009\pi\)
\(420\) 0 0
\(421\) 389.056i 0.924125i 0.886847 + 0.462062i \(0.152890\pi\)
−0.886847 + 0.462062i \(0.847110\pi\)
\(422\) 0 0
\(423\) −50.5927 −0.119604
\(424\) 0 0
\(425\) 1.16496i 0.00274107i
\(426\) 0 0
\(427\) −113.927 −0.266808
\(428\) 0 0
\(429\) 458.086i 1.06780i
\(430\) 0 0
\(431\) 45.3031i 0.105112i 0.998618 + 0.0525558i \(0.0167367\pi\)
−0.998618 + 0.0525558i \(0.983263\pi\)
\(432\) 0 0
\(433\) 582.634i 1.34557i 0.739836 + 0.672787i \(0.234903\pi\)
−0.739836 + 0.672787i \(0.765097\pi\)
\(434\) 0 0
\(435\) 194.791i 0.447794i
\(436\) 0 0
\(437\) 79.6404 252.758i 0.182243 0.578392i
\(438\) 0 0
\(439\) 93.6105 0.213236 0.106618 0.994300i \(-0.465998\pi\)
0.106618 + 0.994300i \(0.465998\pi\)
\(440\) 0 0
\(441\) −13.6779 −0.0310158
\(442\) 0 0
\(443\) −492.804 −1.11242 −0.556212 0.831040i \(-0.687746\pi\)
−0.556212 + 0.831040i \(0.687746\pi\)
\(444\) 0 0
\(445\) −351.341 −0.789531
\(446\) 0 0
\(447\) 344.326i 0.770304i
\(448\) 0 0
\(449\) 672.198 1.49710 0.748550 0.663078i \(-0.230750\pi\)
0.748550 + 0.663078i \(0.230750\pi\)
\(450\) 0 0
\(451\) 672.314i 1.49072i
\(452\) 0 0
\(453\) −394.127 −0.870037
\(454\) 0 0
\(455\) 257.962 0.566950
\(456\) 0 0
\(457\) 259.087i 0.566930i 0.958983 + 0.283465i \(0.0914839\pi\)
−0.958983 + 0.283465i \(0.908516\pi\)
\(458\) 0 0
\(459\) 1.21066i 0.00263760i
\(460\) 0 0
\(461\) −285.784 −0.619922 −0.309961 0.950749i \(-0.600316\pi\)
−0.309961 + 0.950749i \(0.600316\pi\)
\(462\) 0 0
\(463\) −826.544 −1.78519 −0.892596 0.450857i \(-0.851119\pi\)
−0.892596 + 0.450857i \(0.851119\pi\)
\(464\) 0 0
\(465\) 64.4122i 0.138521i
\(466\) 0 0
\(467\) 524.184i 1.12245i 0.827663 + 0.561225i \(0.189670\pi\)
−0.827663 + 0.561225i \(0.810330\pi\)
\(468\) 0 0
\(469\) 361.702 0.771220
\(470\) 0 0
\(471\) 146.016i 0.310013i
\(472\) 0 0
\(473\) −471.188 −0.996169
\(474\) 0 0
\(475\) 57.6103i 0.121285i
\(476\) 0 0
\(477\) 6.94218i 0.0145538i
\(478\) 0 0
\(479\) 825.685i 1.72377i −0.507105 0.861884i \(-0.669284\pi\)
0.507105 0.861884i \(-0.330716\pi\)
\(480\) 0 0
\(481\) 330.643i 0.687406i
\(482\) 0 0
\(483\) 87.6156 278.069i 0.181399 0.575711i
\(484\) 0 0
\(485\) 110.057 0.226922
\(486\) 0 0
\(487\) −170.367 −0.349829 −0.174915 0.984584i \(-0.555965\pi\)
−0.174915 + 0.984584i \(0.555965\pi\)
\(488\) 0 0
\(489\) −473.707 −0.968725
\(490\) 0 0
\(491\) −817.579 −1.66513 −0.832565 0.553928i \(-0.813128\pi\)
−0.832565 + 0.553928i \(0.813128\pi\)
\(492\) 0 0
\(493\) 11.7182i 0.0237692i
\(494\) 0 0
\(495\) 112.548 0.227370
\(496\) 0 0
\(497\) 294.764i 0.593087i
\(498\) 0 0
\(499\) 633.750 1.27004 0.635020 0.772496i \(-0.280992\pi\)
0.635020 + 0.772496i \(0.280992\pi\)
\(500\) 0 0
\(501\) −187.990 −0.375229
\(502\) 0 0
\(503\) 94.8682i 0.188605i 0.995544 + 0.0943023i \(0.0300620\pi\)
−0.995544 + 0.0943023i \(0.969938\pi\)
\(504\) 0 0
\(505\) 195.163i 0.386462i
\(506\) 0 0
\(507\) −137.678 −0.271555
\(508\) 0 0
\(509\) 553.528 1.08748 0.543741 0.839253i \(-0.317007\pi\)
0.543741 + 0.839253i \(0.317007\pi\)
\(510\) 0 0
\(511\) 243.657i 0.476823i
\(512\) 0 0
\(513\) 59.8704i 0.116706i
\(514\) 0 0
\(515\) 286.573 0.556452
\(516\) 0 0
\(517\) 282.943i 0.547279i
\(518\) 0 0
\(519\) 397.575 0.766040
\(520\) 0 0
\(521\) 62.9334i 0.120793i −0.998174 0.0603967i \(-0.980763\pi\)
0.998174 0.0603967i \(-0.0192366\pi\)
\(522\) 0 0
\(523\) 729.483i 1.39480i −0.716680 0.697402i \(-0.754339\pi\)
0.716680 0.697402i \(-0.245661\pi\)
\(524\) 0 0
\(525\) 63.3794i 0.120723i
\(526\) 0 0
\(527\) 3.87492i 0.00735278i
\(528\) 0 0
\(529\) 433.449 + 303.254i 0.819374 + 0.573260i
\(530\) 0 0
\(531\) −248.335 −0.467674
\(532\) 0 0
\(533\) 631.673 1.18513
\(534\) 0 0
\(535\) −253.830 −0.474449
\(536\) 0 0
\(537\) −346.871 −0.645942
\(538\) 0 0
\(539\) 76.4949i 0.141920i
\(540\) 0 0
\(541\) 407.066 0.752433 0.376216 0.926532i \(-0.377225\pi\)
0.376216 + 0.926532i \(0.377225\pi\)
\(542\) 0 0
\(543\) 567.782i 1.04564i
\(544\) 0 0
\(545\) −106.715 −0.195807
\(546\) 0 0
\(547\) 694.565 1.26977 0.634886 0.772606i \(-0.281047\pi\)
0.634886 + 0.772606i \(0.281047\pi\)
\(548\) 0 0
\(549\) 46.7015i 0.0850665i
\(550\) 0 0
\(551\) 579.499i 1.05172i
\(552\) 0 0
\(553\) −938.063 −1.69632
\(554\) 0 0
\(555\) 81.2365 0.146372
\(556\) 0 0
\(557\) 104.688i 0.187950i −0.995575 0.0939749i \(-0.970043\pi\)
0.995575 0.0939749i \(-0.0299574\pi\)
\(558\) 0 0
\(559\) 442.705i 0.791959i
\(560\) 0 0
\(561\) −6.77069 −0.0120690
\(562\) 0 0
\(563\) 191.118i 0.339464i −0.985490 0.169732i \(-0.945710\pi\)
0.985490 0.169732i \(-0.0542902\pi\)
\(564\) 0 0
\(565\) 109.273 0.193404
\(566\) 0 0
\(567\) 65.8658i 0.116165i
\(568\) 0 0
\(569\) 960.237i 1.68759i −0.536668 0.843794i \(-0.680317\pi\)
0.536668 0.843794i \(-0.319683\pi\)
\(570\) 0 0
\(571\) 374.770i 0.656339i 0.944619 + 0.328170i \(0.106432\pi\)
−0.944619 + 0.328170i \(0.893568\pi\)
\(572\) 0 0
\(573\) 440.450i 0.768674i
\(574\) 0 0
\(575\) −109.684 34.5600i −0.190755 0.0601043i
\(576\) 0 0
\(577\) 18.0628 0.0313047 0.0156523 0.999877i \(-0.495018\pi\)
0.0156523 + 0.999877i \(0.495018\pi\)
\(578\) 0 0
\(579\) 64.8215 0.111954
\(580\) 0 0
\(581\) 273.415 0.470594
\(582\) 0 0
\(583\) −38.8247 −0.0665946
\(584\) 0 0
\(585\) 105.745i 0.180761i
\(586\) 0 0
\(587\) 40.2519 0.0685723 0.0342861 0.999412i \(-0.489084\pi\)
0.0342861 + 0.999412i \(0.489084\pi\)
\(588\) 0 0
\(589\) 191.625i 0.325340i
\(590\) 0 0
\(591\) −528.944 −0.894999
\(592\) 0 0
\(593\) −903.069 −1.52288 −0.761441 0.648235i \(-0.775508\pi\)
−0.761441 + 0.648235i \(0.775508\pi\)
\(594\) 0 0
\(595\) 3.81278i 0.00640804i
\(596\) 0 0
\(597\) 368.900i 0.617923i
\(598\) 0 0
\(599\) 9.82481 0.0164020 0.00820101 0.999966i \(-0.497390\pi\)
0.00820101 + 0.999966i \(0.497390\pi\)
\(600\) 0 0
\(601\) −380.948 −0.633857 −0.316929 0.948449i \(-0.602652\pi\)
−0.316929 + 0.948449i \(0.602652\pi\)
\(602\) 0 0
\(603\) 148.270i 0.245888i
\(604\) 0 0
\(605\) 358.871i 0.593175i
\(606\) 0 0
\(607\) −604.962 −0.996642 −0.498321 0.866993i \(-0.666050\pi\)
−0.498321 + 0.866993i \(0.666050\pi\)
\(608\) 0 0
\(609\) 637.530i 1.04685i
\(610\) 0 0
\(611\) −265.839 −0.435089
\(612\) 0 0
\(613\) 223.785i 0.365065i −0.983200 0.182533i \(-0.941570\pi\)
0.983200 0.182533i \(-0.0584295\pi\)
\(614\) 0 0
\(615\) 155.198i 0.252354i
\(616\) 0 0
\(617\) 693.990i 1.12478i 0.826872 + 0.562391i \(0.190118\pi\)
−0.826872 + 0.562391i \(0.809882\pi\)
\(618\) 0 0
\(619\) 428.128i 0.691644i −0.938300 0.345822i \(-0.887600\pi\)
0.938300 0.345822i \(-0.112400\pi\)
\(620\) 0 0
\(621\) −113.987 35.9158i −0.183554 0.0578354i
\(622\) 0 0
\(623\) 1149.90 1.84575
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) −334.830 −0.534019
\(628\) 0 0
\(629\) −4.88703 −0.00776952
\(630\) 0 0
\(631\) 439.125i 0.695919i −0.937510 0.347959i \(-0.886875\pi\)
0.937510 0.347959i \(-0.113125\pi\)
\(632\) 0 0
\(633\) −418.562 −0.661235
\(634\) 0 0
\(635\) 15.7098i 0.0247398i
\(636\) 0 0
\(637\) −71.8709 −0.112827
\(638\) 0 0
\(639\) −120.831 −0.189094
\(640\) 0 0
\(641\) 1024.86i 1.59884i −0.600770 0.799422i \(-0.705139\pi\)
0.600770 0.799422i \(-0.294861\pi\)
\(642\) 0 0
\(643\) 785.445i 1.22153i 0.791811 + 0.610766i \(0.209139\pi\)
−0.791811 + 0.610766i \(0.790861\pi\)
\(644\) 0 0
\(645\) −108.769 −0.168635
\(646\) 0 0
\(647\) 904.821 1.39849 0.699243 0.714884i \(-0.253520\pi\)
0.699243 + 0.714884i \(0.253520\pi\)
\(648\) 0 0
\(649\) 1388.83i 2.13996i
\(650\) 0 0
\(651\) 210.815i 0.323832i
\(652\) 0 0
\(653\) 480.213 0.735396 0.367698 0.929945i \(-0.380146\pi\)
0.367698 + 0.929945i \(0.380146\pi\)
\(654\) 0 0
\(655\) 324.313i 0.495134i
\(656\) 0 0
\(657\) 99.8809 0.152026
\(658\) 0 0
\(659\) 847.972i 1.28676i −0.765549 0.643378i \(-0.777532\pi\)
0.765549 0.643378i \(-0.222468\pi\)
\(660\) 0 0
\(661\) 22.9727i 0.0347544i −0.999849 0.0173772i \(-0.994468\pi\)
0.999849 0.0173772i \(-0.00553161\pi\)
\(662\) 0 0
\(663\) 6.36141i 0.00959489i
\(664\) 0 0
\(665\) 188.553i 0.283538i
\(666\) 0 0
\(667\) 1103.31 + 347.637i 1.65413 + 0.521194i
\(668\) 0 0
\(669\) 464.496 0.694313
\(670\) 0 0
\(671\) −261.182 −0.389242
\(672\) 0 0
\(673\) 548.716 0.815329 0.407664 0.913132i \(-0.366343\pi\)
0.407664 + 0.913132i \(0.366343\pi\)
\(674\) 0 0
\(675\) 25.9808 0.0384900
\(676\) 0 0
\(677\) 151.484i 0.223758i −0.993722 0.111879i \(-0.964313\pi\)
0.993722 0.111879i \(-0.0356870\pi\)
\(678\) 0 0
\(679\) −360.205 −0.530494
\(680\) 0 0
\(681\) 578.980i 0.850191i
\(682\) 0 0
\(683\) −1302.11 −1.90645 −0.953227 0.302255i \(-0.902261\pi\)
−0.953227 + 0.302255i \(0.902261\pi\)
\(684\) 0 0
\(685\) 289.804 0.423071
\(686\) 0 0
\(687\) 40.3929i 0.0587961i
\(688\) 0 0
\(689\) 36.4778i 0.0529430i
\(690\) 0 0
\(691\) 539.778 0.781156 0.390578 0.920570i \(-0.372275\pi\)
0.390578 + 0.920570i \(0.372275\pi\)
\(692\) 0 0
\(693\) −368.359 −0.531543
\(694\) 0 0
\(695\) 251.744i 0.362222i
\(696\) 0 0
\(697\) 9.33638i 0.0133951i
\(698\) 0 0
\(699\) 547.035 0.782597
\(700\) 0 0
\(701\) 68.3481i 0.0975009i 0.998811 + 0.0487505i \(0.0155239\pi\)
−0.998811 + 0.0487505i \(0.984476\pi\)
\(702\) 0 0
\(703\) −241.677 −0.343780
\(704\) 0 0
\(705\) 65.3148i 0.0926452i
\(706\) 0 0
\(707\) 638.750i 0.903465i
\(708\) 0 0
\(709\) 397.854i 0.561148i −0.959832 0.280574i \(-0.909475\pi\)
0.959832 0.280574i \(-0.0905248\pi\)
\(710\) 0 0
\(711\) 384.535i 0.540837i
\(712\) 0 0
\(713\) −364.835 114.954i −0.511690 0.161226i
\(714\) 0 0
\(715\) 591.386 0.827114
\(716\) 0 0
\(717\) −98.9624 −0.138023
\(718\) 0 0
\(719\) −387.514 −0.538962 −0.269481 0.963006i \(-0.586852\pi\)
−0.269481 + 0.963006i \(0.586852\pi\)
\(720\) 0 0
\(721\) −937.923 −1.30086
\(722\) 0 0
\(723\) 171.642i 0.237402i
\(724\) 0 0
\(725\) −251.473 −0.346860
\(726\) 0 0
\(727\) 201.611i 0.277320i 0.990340 + 0.138660i \(0.0442795\pi\)
−0.990340 + 0.138660i \(0.955721\pi\)
\(728\) 0 0
\(729\) 27.0000 0.0370370
\(730\) 0 0
\(731\) 6.54335 0.00895124
\(732\) 0 0
\(733\) 42.9999i 0.0586629i 0.999570 + 0.0293314i \(0.00933782\pi\)
−0.999570 + 0.0293314i \(0.990662\pi\)
\(734\) 0 0
\(735\) 17.6582i 0.0240247i
\(736\) 0 0
\(737\) 829.214 1.12512
\(738\) 0 0
\(739\) −32.7333 −0.0442940 −0.0221470 0.999755i \(-0.507050\pi\)
−0.0221470 + 0.999755i \(0.507050\pi\)
\(740\) 0 0
\(741\) 314.589i 0.424547i
\(742\) 0 0
\(743\) 570.140i 0.767349i −0.923468 0.383675i \(-0.874658\pi\)
0.923468 0.383675i \(-0.125342\pi\)
\(744\) 0 0
\(745\) 444.523 0.596675
\(746\) 0 0
\(747\) 112.080i 0.150040i
\(748\) 0 0
\(749\) 830.761 1.10916
\(750\) 0 0
\(751\) 1005.37i 1.33871i 0.742944 + 0.669354i \(0.233429\pi\)
−0.742944 + 0.669354i \(0.766571\pi\)
\(752\) 0 0
\(753\) 189.487i 0.251643i
\(754\) 0 0
\(755\) 508.815i 0.673928i
\(756\) 0 0
\(757\) 282.779i 0.373552i −0.982403 0.186776i \(-0.940196\pi\)
0.982403 0.186776i \(-0.0598039\pi\)
\(758\) 0 0
\(759\) 200.862 637.481i 0.264640 0.839896i
\(760\) 0 0
\(761\) −744.849 −0.978777 −0.489389 0.872066i \(-0.662780\pi\)
−0.489389 + 0.872066i \(0.662780\pi\)
\(762\) 0 0
\(763\) 349.266 0.457754
\(764\) 0 0
\(765\) −1.56295 −0.00204308
\(766\) 0 0
\(767\) −1304.88 −1.70127
\(768\) 0 0
\(769\) 939.496i 1.22171i −0.791742 0.610856i \(-0.790826\pi\)
0.791742 0.610856i \(-0.209174\pi\)
\(770\) 0 0
\(771\) 624.718 0.810270
\(772\) 0 0
\(773\) 859.775i 1.11226i 0.831096 + 0.556129i \(0.187714\pi\)
−0.831096 + 0.556129i \(0.812286\pi\)
\(774\) 0 0
\(775\) 83.1558 0.107298
\(776\) 0 0
\(777\) −265.879 −0.342186
\(778\) 0 0
\(779\) 461.710i 0.592696i
\(780\) 0 0
\(781\) 675.757i 0.865245i
\(782\) 0 0
\(783\) −261.339 −0.333766
\(784\) 0 0
\(785\) −188.506 −0.240135
\(786\) 0 0
\(787\) 606.069i 0.770100i 0.922896 + 0.385050i \(0.125816\pi\)
−0.922896 + 0.385050i \(0.874184\pi\)
\(788\) 0 0
\(789\) 232.420i 0.294575i
\(790\) 0 0
\(791\) −357.640 −0.452136
\(792\) 0 0
\(793\) 245.393i 0.309450i
\(794\) 0 0
\(795\) −8.96232 −0.0112734
\(796\) 0 0
\(797\) 18.7215i 0.0234900i −0.999931 0.0117450i \(-0.996261\pi\)
0.999931 0.0117450i \(-0.00373863\pi\)
\(798\) 0 0
\