Properties

Label 1089.4.a.t.1.2
Level $1089$
Weight $4$
Character 1089.1
Self dual yes
Analytic conductor $64.253$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1089,4,Mod(1,1089)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1089, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1089.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1089.a (trivial)

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: yes
Analytic conductor: \(64.2530799963\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 1089.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.37228 q^{2} +3.37228 q^{4} +3.48913 q^{5} +4.74456 q^{7} -15.6060 q^{8} +O(q^{10})\) \(q+3.37228 q^{2} +3.37228 q^{4} +3.48913 q^{5} +4.74456 q^{7} -15.6060 q^{8} +11.7663 q^{10} +15.0217 q^{13} +16.0000 q^{14} -79.6060 q^{16} +73.1684 q^{17} +78.7011 q^{19} +11.7663 q^{20} -112.000 q^{23} -112.826 q^{25} +50.6576 q^{26} +16.0000 q^{28} +243.125 q^{29} +278.717 q^{31} -143.606 q^{32} +246.745 q^{34} +16.5544 q^{35} +102.380 q^{37} +265.402 q^{38} -54.4512 q^{40} -241.255 q^{41} +280.016 q^{43} -377.696 q^{46} +169.870 q^{47} -320.489 q^{49} -380.481 q^{50} +50.6576 q^{52} +409.652 q^{53} -74.0435 q^{56} +819.886 q^{58} -196.000 q^{59} +701.359 q^{61} +939.913 q^{62} +152.568 q^{64} +52.4128 q^{65} +900.587 q^{67} +246.745 q^{68} +55.8260 q^{70} -756.500 q^{71} +1019.81 q^{73} +345.255 q^{74} +265.402 q^{76} +327.549 q^{79} -277.755 q^{80} -813.581 q^{82} -756.619 q^{83} +255.294 q^{85} +944.293 q^{86} -508.978 q^{89} +71.2716 q^{91} -377.696 q^{92} +572.848 q^{94} +274.598 q^{95} +614.358 q^{97} -1080.78 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + q^{4} - 16 q^{5} - 2 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + q^{4} - 16 q^{5} - 2 q^{7} + 9 q^{8} + 58 q^{10} + 76 q^{13} + 32 q^{14} - 119 q^{16} - 26 q^{17} + 54 q^{19} + 58 q^{20} - 224 q^{23} + 142 q^{25} - 94 q^{26} + 32 q^{28} + 222 q^{29} - 40 q^{31} - 247 q^{32} + 482 q^{34} + 148 q^{35} - 48 q^{37} + 324 q^{38} - 534 q^{40} - 494 q^{41} + 66 q^{43} - 112 q^{46} + 64 q^{47} - 618 q^{49} - 985 q^{50} - 94 q^{52} + 84 q^{53} - 240 q^{56} + 870 q^{58} - 392 q^{59} + 1104 q^{61} + 1696 q^{62} + 713 q^{64} - 1136 q^{65} + 928 q^{67} + 482 q^{68} - 256 q^{70} - 456 q^{71} + 592 q^{73} + 702 q^{74} + 324 q^{76} + 230 q^{79} + 490 q^{80} - 214 q^{82} + 348 q^{83} + 2188 q^{85} + 1452 q^{86} - 972 q^{89} - 340 q^{91} - 112 q^{92} + 824 q^{94} + 756 q^{95} - 1184 q^{97} - 375 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.37228 1.19228 0.596141 0.802880i \(-0.296700\pi\)
0.596141 + 0.802880i \(0.296700\pi\)
\(3\) 0 0
\(4\) 3.37228 0.421535
\(5\) 3.48913 0.312077 0.156038 0.987751i \(-0.450128\pi\)
0.156038 + 0.987751i \(0.450128\pi\)
\(6\) 0 0
\(7\) 4.74456 0.256182 0.128091 0.991762i \(-0.459115\pi\)
0.128091 + 0.991762i \(0.459115\pi\)
\(8\) −15.6060 −0.689693
\(9\) 0 0
\(10\) 11.7663 0.372083
\(11\) 0 0
\(12\) 0 0
\(13\) 15.0217 0.320483 0.160242 0.987078i \(-0.448773\pi\)
0.160242 + 0.987078i \(0.448773\pi\)
\(14\) 16.0000 0.305441
\(15\) 0 0
\(16\) −79.6060 −1.24384
\(17\) 73.1684 1.04388 0.521940 0.852982i \(-0.325209\pi\)
0.521940 + 0.852982i \(0.325209\pi\)
\(18\) 0 0
\(19\) 78.7011 0.950277 0.475138 0.879911i \(-0.342398\pi\)
0.475138 + 0.879911i \(0.342398\pi\)
\(20\) 11.7663 0.131551
\(21\) 0 0
\(22\) 0 0
\(23\) −112.000 −1.01537 −0.507687 0.861541i \(-0.669499\pi\)
−0.507687 + 0.861541i \(0.669499\pi\)
\(24\) 0 0
\(25\) −112.826 −0.902608
\(26\) 50.6576 0.382106
\(27\) 0 0
\(28\) 16.0000 0.107990
\(29\) 243.125 1.55680 0.778399 0.627769i \(-0.216032\pi\)
0.778399 + 0.627769i \(0.216032\pi\)
\(30\) 0 0
\(31\) 278.717 1.61481 0.807405 0.589998i \(-0.200871\pi\)
0.807405 + 0.589998i \(0.200871\pi\)
\(32\) −143.606 −0.793318
\(33\) 0 0
\(34\) 246.745 1.24460
\(35\) 16.5544 0.0799486
\(36\) 0 0
\(37\) 102.380 0.454898 0.227449 0.973790i \(-0.426961\pi\)
0.227449 + 0.973790i \(0.426961\pi\)
\(38\) 265.402 1.13300
\(39\) 0 0
\(40\) −54.4512 −0.215237
\(41\) −241.255 −0.918970 −0.459485 0.888186i \(-0.651966\pi\)
−0.459485 + 0.888186i \(0.651966\pi\)
\(42\) 0 0
\(43\) 280.016 0.993071 0.496536 0.868016i \(-0.334605\pi\)
0.496536 + 0.868016i \(0.334605\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −377.696 −1.21061
\(47\) 169.870 0.527192 0.263596 0.964633i \(-0.415091\pi\)
0.263596 + 0.964633i \(0.415091\pi\)
\(48\) 0 0
\(49\) −320.489 −0.934371
\(50\) −380.481 −1.07616
\(51\) 0 0
\(52\) 50.6576 0.135095
\(53\) 409.652 1.06170 0.530849 0.847466i \(-0.321873\pi\)
0.530849 + 0.847466i \(0.321873\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −74.0435 −0.176687
\(57\) 0 0
\(58\) 819.886 1.85614
\(59\) −196.000 −0.432492 −0.216246 0.976339i \(-0.569381\pi\)
−0.216246 + 0.976339i \(0.569381\pi\)
\(60\) 0 0
\(61\) 701.359 1.47213 0.736064 0.676912i \(-0.236682\pi\)
0.736064 + 0.676912i \(0.236682\pi\)
\(62\) 939.913 1.92531
\(63\) 0 0
\(64\) 152.568 0.297984
\(65\) 52.4128 0.100015
\(66\) 0 0
\(67\) 900.587 1.64215 0.821076 0.570819i \(-0.193374\pi\)
0.821076 + 0.570819i \(0.193374\pi\)
\(68\) 246.745 0.440032
\(69\) 0 0
\(70\) 55.8260 0.0953212
\(71\) −756.500 −1.26451 −0.632254 0.774762i \(-0.717870\pi\)
−0.632254 + 0.774762i \(0.717870\pi\)
\(72\) 0 0
\(73\) 1019.81 1.63507 0.817536 0.575877i \(-0.195339\pi\)
0.817536 + 0.575877i \(0.195339\pi\)
\(74\) 345.255 0.542367
\(75\) 0 0
\(76\) 265.402 0.400575
\(77\) 0 0
\(78\) 0 0
\(79\) 327.549 0.466483 0.233241 0.972419i \(-0.425067\pi\)
0.233241 + 0.972419i \(0.425067\pi\)
\(80\) −277.755 −0.388175
\(81\) 0 0
\(82\) −813.581 −1.09567
\(83\) −756.619 −1.00060 −0.500300 0.865852i \(-0.666777\pi\)
−0.500300 + 0.865852i \(0.666777\pi\)
\(84\) 0 0
\(85\) 255.294 0.325771
\(86\) 944.293 1.18402
\(87\) 0 0
\(88\) 0 0
\(89\) −508.978 −0.606198 −0.303099 0.952959i \(-0.598021\pi\)
−0.303099 + 0.952959i \(0.598021\pi\)
\(90\) 0 0
\(91\) 71.2716 0.0821022
\(92\) −377.696 −0.428016
\(93\) 0 0
\(94\) 572.848 0.628561
\(95\) 274.598 0.296559
\(96\) 0 0
\(97\) 614.358 0.643079 0.321539 0.946896i \(-0.395800\pi\)
0.321539 + 0.946896i \(0.395800\pi\)
\(98\) −1080.78 −1.11403
\(99\) 0 0
\(100\) −380.481 −0.380481
\(101\) −1015.92 −1.00087 −0.500434 0.865775i \(-0.666826\pi\)
−0.500434 + 0.865775i \(0.666826\pi\)
\(102\) 0 0
\(103\) 1102.16 1.05436 0.527181 0.849753i \(-0.323249\pi\)
0.527181 + 0.849753i \(0.323249\pi\)
\(104\) −234.429 −0.221035
\(105\) 0 0
\(106\) 1381.46 1.26584
\(107\) 1377.58 1.24463 0.622315 0.782767i \(-0.286192\pi\)
0.622315 + 0.782767i \(0.286192\pi\)
\(108\) 0 0
\(109\) −320.217 −0.281388 −0.140694 0.990053i \(-0.544933\pi\)
−0.140694 + 0.990053i \(0.544933\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −377.696 −0.318651
\(113\) 1629.45 1.35651 0.678254 0.734828i \(-0.262737\pi\)
0.678254 + 0.734828i \(0.262737\pi\)
\(114\) 0 0
\(115\) −390.782 −0.316875
\(116\) 819.886 0.656245
\(117\) 0 0
\(118\) −660.967 −0.515652
\(119\) 347.152 0.267423
\(120\) 0 0
\(121\) 0 0
\(122\) 2365.18 1.75519
\(123\) 0 0
\(124\) 939.913 0.680699
\(125\) −829.805 −0.593760
\(126\) 0 0
\(127\) −2291.26 −1.60091 −0.800457 0.599390i \(-0.795410\pi\)
−0.800457 + 0.599390i \(0.795410\pi\)
\(128\) 1663.35 1.14860
\(129\) 0 0
\(130\) 176.751 0.119247
\(131\) −1147.41 −0.765267 −0.382633 0.923900i \(-0.624983\pi\)
−0.382633 + 0.923900i \(0.624983\pi\)
\(132\) 0 0
\(133\) 373.402 0.243444
\(134\) 3037.03 1.95791
\(135\) 0 0
\(136\) −1141.86 −0.719956
\(137\) −1268.60 −0.791121 −0.395561 0.918440i \(-0.629450\pi\)
−0.395561 + 0.918440i \(0.629450\pi\)
\(138\) 0 0
\(139\) 486.288 0.296737 0.148368 0.988932i \(-0.452598\pi\)
0.148368 + 0.988932i \(0.452598\pi\)
\(140\) 55.8260 0.0337011
\(141\) 0 0
\(142\) −2551.13 −1.50765
\(143\) 0 0
\(144\) 0 0
\(145\) 848.293 0.485841
\(146\) 3439.10 1.94947
\(147\) 0 0
\(148\) 345.255 0.191756
\(149\) 2354.11 1.29434 0.647169 0.762346i \(-0.275953\pi\)
0.647169 + 0.762346i \(0.275953\pi\)
\(150\) 0 0
\(151\) 570.070 0.307229 0.153615 0.988131i \(-0.450909\pi\)
0.153615 + 0.988131i \(0.450909\pi\)
\(152\) −1228.21 −0.655399
\(153\) 0 0
\(154\) 0 0
\(155\) 972.479 0.503945
\(156\) 0 0
\(157\) −2072.67 −1.05361 −0.526807 0.849985i \(-0.676611\pi\)
−0.526807 + 0.849985i \(0.676611\pi\)
\(158\) 1104.59 0.556179
\(159\) 0 0
\(160\) −501.059 −0.247576
\(161\) −531.391 −0.260121
\(162\) 0 0
\(163\) 2676.51 1.28614 0.643069 0.765808i \(-0.277661\pi\)
0.643069 + 0.765808i \(0.277661\pi\)
\(164\) −813.581 −0.387378
\(165\) 0 0
\(166\) −2551.53 −1.19300
\(167\) −1188.12 −0.550536 −0.275268 0.961368i \(-0.588767\pi\)
−0.275268 + 0.961368i \(0.588767\pi\)
\(168\) 0 0
\(169\) −1971.35 −0.897290
\(170\) 860.923 0.388410
\(171\) 0 0
\(172\) 944.293 0.418615
\(173\) 807.147 0.354718 0.177359 0.984146i \(-0.443245\pi\)
0.177359 + 0.984146i \(0.443245\pi\)
\(174\) 0 0
\(175\) −535.310 −0.231232
\(176\) 0 0
\(177\) 0 0
\(178\) −1716.42 −0.722758
\(179\) 1950.39 0.814408 0.407204 0.913337i \(-0.366504\pi\)
0.407204 + 0.913337i \(0.366504\pi\)
\(180\) 0 0
\(181\) 1061.61 0.435959 0.217980 0.975953i \(-0.430053\pi\)
0.217980 + 0.975953i \(0.430053\pi\)
\(182\) 240.348 0.0978889
\(183\) 0 0
\(184\) 1747.87 0.700297
\(185\) 357.218 0.141963
\(186\) 0 0
\(187\) 0 0
\(188\) 572.848 0.222230
\(189\) 0 0
\(190\) 926.021 0.353582
\(191\) −2136.41 −0.809348 −0.404674 0.914461i \(-0.632615\pi\)
−0.404674 + 0.914461i \(0.632615\pi\)
\(192\) 0 0
\(193\) −3947.76 −1.47236 −0.736181 0.676784i \(-0.763373\pi\)
−0.736181 + 0.676784i \(0.763373\pi\)
\(194\) 2071.79 0.766731
\(195\) 0 0
\(196\) −1080.78 −0.393870
\(197\) 923.886 0.334133 0.167066 0.985946i \(-0.446571\pi\)
0.167066 + 0.985946i \(0.446571\pi\)
\(198\) 0 0
\(199\) −476.152 −0.169616 −0.0848078 0.996397i \(-0.527028\pi\)
−0.0848078 + 0.996397i \(0.527028\pi\)
\(200\) 1760.76 0.622522
\(201\) 0 0
\(202\) −3425.96 −1.19332
\(203\) 1153.52 0.398824
\(204\) 0 0
\(205\) −841.770 −0.286789
\(206\) 3716.80 1.25710
\(207\) 0 0
\(208\) −1195.82 −0.398631
\(209\) 0 0
\(210\) 0 0
\(211\) 4918.24 1.60467 0.802336 0.596872i \(-0.203590\pi\)
0.802336 + 0.596872i \(0.203590\pi\)
\(212\) 1381.46 0.447543
\(213\) 0 0
\(214\) 4645.57 1.48395
\(215\) 977.012 0.309915
\(216\) 0 0
\(217\) 1322.39 0.413686
\(218\) −1079.86 −0.335494
\(219\) 0 0
\(220\) 0 0
\(221\) 1099.12 0.334546
\(222\) 0 0
\(223\) 2100.29 0.630700 0.315350 0.948975i \(-0.397878\pi\)
0.315350 + 0.948975i \(0.397878\pi\)
\(224\) −681.348 −0.203234
\(225\) 0 0
\(226\) 5494.95 1.61734
\(227\) −2257.16 −0.659970 −0.329985 0.943986i \(-0.607044\pi\)
−0.329985 + 0.943986i \(0.607044\pi\)
\(228\) 0 0
\(229\) −5311.07 −1.53260 −0.766301 0.642482i \(-0.777905\pi\)
−0.766301 + 0.642482i \(0.777905\pi\)
\(230\) −1317.83 −0.377804
\(231\) 0 0
\(232\) −3794.20 −1.07371
\(233\) 2466.27 0.693435 0.346718 0.937970i \(-0.387296\pi\)
0.346718 + 0.937970i \(0.387296\pi\)
\(234\) 0 0
\(235\) 592.696 0.164524
\(236\) −660.967 −0.182311
\(237\) 0 0
\(238\) 1170.70 0.318844
\(239\) 1429.40 0.386863 0.193432 0.981114i \(-0.438038\pi\)
0.193432 + 0.981114i \(0.438038\pi\)
\(240\) 0 0
\(241\) 978.989 0.261669 0.130835 0.991404i \(-0.458234\pi\)
0.130835 + 0.991404i \(0.458234\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 2365.18 0.620553
\(245\) −1118.23 −0.291595
\(246\) 0 0
\(247\) 1182.23 0.304548
\(248\) −4349.65 −1.11372
\(249\) 0 0
\(250\) −2798.33 −0.707929
\(251\) 6530.63 1.64227 0.821135 0.570734i \(-0.193341\pi\)
0.821135 + 0.570734i \(0.193341\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −7726.76 −1.90874
\(255\) 0 0
\(256\) 4388.74 1.07147
\(257\) −8130.26 −1.97335 −0.986676 0.162696i \(-0.947981\pi\)
−0.986676 + 0.162696i \(0.947981\pi\)
\(258\) 0 0
\(259\) 485.750 0.116537
\(260\) 176.751 0.0421600
\(261\) 0 0
\(262\) −3869.40 −0.912414
\(263\) −4549.42 −1.06665 −0.533326 0.845910i \(-0.679058\pi\)
−0.533326 + 0.845910i \(0.679058\pi\)
\(264\) 0 0
\(265\) 1429.33 0.331332
\(266\) 1259.22 0.290254
\(267\) 0 0
\(268\) 3037.03 0.692225
\(269\) 29.1522 0.00660760 0.00330380 0.999995i \(-0.498948\pi\)
0.00330380 + 0.999995i \(0.498948\pi\)
\(270\) 0 0
\(271\) −7711.22 −1.72850 −0.864250 0.503063i \(-0.832206\pi\)
−0.864250 + 0.503063i \(0.832206\pi\)
\(272\) −5824.64 −1.29842
\(273\) 0 0
\(274\) −4278.07 −0.943239
\(275\) 0 0
\(276\) 0 0
\(277\) −1127.52 −0.244571 −0.122286 0.992495i \(-0.539022\pi\)
−0.122286 + 0.992495i \(0.539022\pi\)
\(278\) 1639.90 0.353794
\(279\) 0 0
\(280\) −258.347 −0.0551400
\(281\) −1872.47 −0.397517 −0.198758 0.980049i \(-0.563691\pi\)
−0.198758 + 0.980049i \(0.563691\pi\)
\(282\) 0 0
\(283\) −2124.48 −0.446245 −0.223123 0.974790i \(-0.571625\pi\)
−0.223123 + 0.974790i \(0.571625\pi\)
\(284\) −2551.13 −0.533034
\(285\) 0 0
\(286\) 0 0
\(287\) −1144.65 −0.235424
\(288\) 0 0
\(289\) 440.621 0.0896846
\(290\) 2860.68 0.579259
\(291\) 0 0
\(292\) 3439.10 0.689241
\(293\) −3324.19 −0.662802 −0.331401 0.943490i \(-0.607521\pi\)
−0.331401 + 0.943490i \(0.607521\pi\)
\(294\) 0 0
\(295\) −683.869 −0.134971
\(296\) −1597.75 −0.313740
\(297\) 0 0
\(298\) 7938.73 1.54322
\(299\) −1682.44 −0.325411
\(300\) 0 0
\(301\) 1328.55 0.254407
\(302\) 1922.44 0.366304
\(303\) 0 0
\(304\) −6265.07 −1.18200
\(305\) 2447.13 0.459417
\(306\) 0 0
\(307\) 1698.94 0.315843 0.157921 0.987452i \(-0.449521\pi\)
0.157921 + 0.987452i \(0.449521\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 3279.47 0.600844
\(311\) −6928.83 −1.26334 −0.631668 0.775239i \(-0.717630\pi\)
−0.631668 + 0.775239i \(0.717630\pi\)
\(312\) 0 0
\(313\) −3560.75 −0.643020 −0.321510 0.946906i \(-0.604190\pi\)
−0.321510 + 0.946906i \(0.604190\pi\)
\(314\) −6989.64 −1.25620
\(315\) 0 0
\(316\) 1104.59 0.196639
\(317\) −332.750 −0.0589561 −0.0294780 0.999565i \(-0.509385\pi\)
−0.0294780 + 0.999565i \(0.509385\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 532.329 0.0929940
\(321\) 0 0
\(322\) −1792.00 −0.310137
\(323\) 5758.43 0.991975
\(324\) 0 0
\(325\) −1694.84 −0.289271
\(326\) 9025.94 1.53344
\(327\) 0 0
\(328\) 3765.02 0.633807
\(329\) 805.957 0.135057
\(330\) 0 0
\(331\) −541.445 −0.0899108 −0.0449554 0.998989i \(-0.514315\pi\)
−0.0449554 + 0.998989i \(0.514315\pi\)
\(332\) −2551.53 −0.421788
\(333\) 0 0
\(334\) −4006.67 −0.656393
\(335\) 3142.26 0.512478
\(336\) 0 0
\(337\) −816.531 −0.131986 −0.0659930 0.997820i \(-0.521022\pi\)
−0.0659930 + 0.997820i \(0.521022\pi\)
\(338\) −6647.94 −1.06982
\(339\) 0 0
\(340\) 860.923 0.137324
\(341\) 0 0
\(342\) 0 0
\(343\) −3147.97 −0.495552
\(344\) −4369.92 −0.684914
\(345\) 0 0
\(346\) 2721.93 0.422924
\(347\) 6260.53 0.968539 0.484269 0.874919i \(-0.339086\pi\)
0.484269 + 0.874919i \(0.339086\pi\)
\(348\) 0 0
\(349\) 12768.5 1.95840 0.979198 0.202906i \(-0.0650386\pi\)
0.979198 + 0.202906i \(0.0650386\pi\)
\(350\) −1805.22 −0.275694
\(351\) 0 0
\(352\) 0 0
\(353\) 2649.28 0.399453 0.199727 0.979852i \(-0.435995\pi\)
0.199727 + 0.979852i \(0.435995\pi\)
\(354\) 0 0
\(355\) −2639.52 −0.394623
\(356\) −1716.42 −0.255534
\(357\) 0 0
\(358\) 6577.27 0.971004
\(359\) −3203.91 −0.471020 −0.235510 0.971872i \(-0.575676\pi\)
−0.235510 + 0.971872i \(0.575676\pi\)
\(360\) 0 0
\(361\) −665.143 −0.0969737
\(362\) 3580.04 0.519786
\(363\) 0 0
\(364\) 240.348 0.0346089
\(365\) 3558.26 0.510268
\(366\) 0 0
\(367\) −8429.40 −1.19894 −0.599470 0.800397i \(-0.704622\pi\)
−0.599470 + 0.800397i \(0.704622\pi\)
\(368\) 8915.87 1.26297
\(369\) 0 0
\(370\) 1204.64 0.169260
\(371\) 1943.62 0.271988
\(372\) 0 0
\(373\) 9388.53 1.30327 0.651635 0.758533i \(-0.274083\pi\)
0.651635 + 0.758533i \(0.274083\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −2650.98 −0.363600
\(377\) 3652.16 0.498928
\(378\) 0 0
\(379\) −14264.5 −1.93329 −0.966647 0.256112i \(-0.917558\pi\)
−0.966647 + 0.256112i \(0.917558\pi\)
\(380\) 926.021 0.125010
\(381\) 0 0
\(382\) −7204.58 −0.964970
\(383\) −13462.2 −1.79605 −0.898026 0.439942i \(-0.854999\pi\)
−0.898026 + 0.439942i \(0.854999\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −13313.0 −1.75547
\(387\) 0 0
\(388\) 2071.79 0.271080
\(389\) 941.881 0.122764 0.0613821 0.998114i \(-0.480449\pi\)
0.0613821 + 0.998114i \(0.480449\pi\)
\(390\) 0 0
\(391\) −8194.87 −1.05993
\(392\) 5001.54 0.644429
\(393\) 0 0
\(394\) 3115.60 0.398380
\(395\) 1142.86 0.145578
\(396\) 0 0
\(397\) −847.839 −0.107183 −0.0535917 0.998563i \(-0.517067\pi\)
−0.0535917 + 0.998563i \(0.517067\pi\)
\(398\) −1605.72 −0.202230
\(399\) 0 0
\(400\) 8981.62 1.12270
\(401\) −12203.6 −1.51975 −0.759875 0.650069i \(-0.774740\pi\)
−0.759875 + 0.650069i \(0.774740\pi\)
\(402\) 0 0
\(403\) 4186.82 0.517520
\(404\) −3425.96 −0.421901
\(405\) 0 0
\(406\) 3890.00 0.475511
\(407\) 0 0
\(408\) 0 0
\(409\) −8759.53 −1.05900 −0.529500 0.848310i \(-0.677620\pi\)
−0.529500 + 0.848310i \(0.677620\pi\)
\(410\) −2838.69 −0.341934
\(411\) 0 0
\(412\) 3716.80 0.444451
\(413\) −929.934 −0.110797
\(414\) 0 0
\(415\) −2639.94 −0.312264
\(416\) −2157.21 −0.254245
\(417\) 0 0
\(418\) 0 0
\(419\) 11188.4 1.30451 0.652256 0.757999i \(-0.273823\pi\)
0.652256 + 0.757999i \(0.273823\pi\)
\(420\) 0 0
\(421\) −14082.3 −1.63023 −0.815116 0.579298i \(-0.803327\pi\)
−0.815116 + 0.579298i \(0.803327\pi\)
\(422\) 16585.7 1.91322
\(423\) 0 0
\(424\) −6393.02 −0.732246
\(425\) −8255.30 −0.942214
\(426\) 0 0
\(427\) 3327.64 0.377133
\(428\) 4645.57 0.524655
\(429\) 0 0
\(430\) 3294.76 0.369505
\(431\) −5616.05 −0.627647 −0.313823 0.949481i \(-0.601610\pi\)
−0.313823 + 0.949481i \(0.601610\pi\)
\(432\) 0 0
\(433\) 7195.75 0.798627 0.399314 0.916814i \(-0.369248\pi\)
0.399314 + 0.916814i \(0.369248\pi\)
\(434\) 4459.48 0.493230
\(435\) 0 0
\(436\) −1079.86 −0.118615
\(437\) −8814.52 −0.964887
\(438\) 0 0
\(439\) −101.959 −0.0110848 −0.00554240 0.999985i \(-0.501764\pi\)
−0.00554240 + 0.999985i \(0.501764\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 3706.53 0.398873
\(443\) −4953.74 −0.531285 −0.265642 0.964072i \(-0.585584\pi\)
−0.265642 + 0.964072i \(0.585584\pi\)
\(444\) 0 0
\(445\) −1775.89 −0.189180
\(446\) 7082.78 0.751972
\(447\) 0 0
\(448\) 723.869 0.0763383
\(449\) 11602.0 1.21945 0.609723 0.792615i \(-0.291281\pi\)
0.609723 + 0.792615i \(0.291281\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 5494.95 0.571816
\(453\) 0 0
\(454\) −7611.79 −0.786870
\(455\) 248.676 0.0256222
\(456\) 0 0
\(457\) 3530.68 0.361397 0.180698 0.983539i \(-0.442164\pi\)
0.180698 + 0.983539i \(0.442164\pi\)
\(458\) −17910.4 −1.82729
\(459\) 0 0
\(460\) −1317.83 −0.133574
\(461\) 11566.3 1.16854 0.584271 0.811559i \(-0.301381\pi\)
0.584271 + 0.811559i \(0.301381\pi\)
\(462\) 0 0
\(463\) 10888.5 1.09294 0.546470 0.837479i \(-0.315971\pi\)
0.546470 + 0.837479i \(0.315971\pi\)
\(464\) −19354.2 −1.93641
\(465\) 0 0
\(466\) 8316.94 0.826770
\(467\) −10688.0 −1.05906 −0.529529 0.848292i \(-0.677631\pi\)
−0.529529 + 0.848292i \(0.677631\pi\)
\(468\) 0 0
\(469\) 4272.89 0.420690
\(470\) 1998.74 0.196159
\(471\) 0 0
\(472\) 3058.77 0.298287
\(473\) 0 0
\(474\) 0 0
\(475\) −8879.53 −0.857728
\(476\) 1170.70 0.112728
\(477\) 0 0
\(478\) 4820.35 0.461250
\(479\) 2341.90 0.223391 0.111696 0.993742i \(-0.464372\pi\)
0.111696 + 0.993742i \(0.464372\pi\)
\(480\) 0 0
\(481\) 1537.93 0.145787
\(482\) 3301.43 0.311983
\(483\) 0 0
\(484\) 0 0
\(485\) 2143.57 0.200690
\(486\) 0 0
\(487\) 6748.91 0.627972 0.313986 0.949428i \(-0.398335\pi\)
0.313986 + 0.949428i \(0.398335\pi\)
\(488\) −10945.4 −1.01532
\(489\) 0 0
\(490\) −3770.98 −0.347664
\(491\) 7361.40 0.676609 0.338305 0.941037i \(-0.390147\pi\)
0.338305 + 0.941037i \(0.390147\pi\)
\(492\) 0 0
\(493\) 17789.1 1.62511
\(494\) 3986.80 0.363107
\(495\) 0 0
\(496\) −22187.6 −2.00857
\(497\) −3589.26 −0.323944
\(498\) 0 0
\(499\) 10381.7 0.931359 0.465680 0.884953i \(-0.345810\pi\)
0.465680 + 0.884953i \(0.345810\pi\)
\(500\) −2798.33 −0.250291
\(501\) 0 0
\(502\) 22023.1 1.95805
\(503\) 19149.0 1.69744 0.848721 0.528840i \(-0.177373\pi\)
0.848721 + 0.528840i \(0.177373\pi\)
\(504\) 0 0
\(505\) −3544.67 −0.312348
\(506\) 0 0
\(507\) 0 0
\(508\) −7726.76 −0.674841
\(509\) −16073.2 −1.39967 −0.699836 0.714303i \(-0.746744\pi\)
−0.699836 + 0.714303i \(0.746744\pi\)
\(510\) 0 0
\(511\) 4838.58 0.418877
\(512\) 1493.27 0.128894
\(513\) 0 0
\(514\) −27417.5 −2.35279
\(515\) 3845.58 0.329042
\(516\) 0 0
\(517\) 0 0
\(518\) 1638.09 0.138945
\(519\) 0 0
\(520\) −817.952 −0.0689799
\(521\) 18955.3 1.59395 0.796975 0.604012i \(-0.206432\pi\)
0.796975 + 0.604012i \(0.206432\pi\)
\(522\) 0 0
\(523\) 4442.19 0.371402 0.185701 0.982606i \(-0.440544\pi\)
0.185701 + 0.982606i \(0.440544\pi\)
\(524\) −3869.40 −0.322587
\(525\) 0 0
\(526\) −15341.9 −1.27175
\(527\) 20393.3 1.68567
\(528\) 0 0
\(529\) 377.000 0.0309855
\(530\) 4820.09 0.395041
\(531\) 0 0
\(532\) 1259.22 0.102620
\(533\) −3624.08 −0.294515
\(534\) 0 0
\(535\) 4806.54 0.388420
\(536\) −14054.5 −1.13258
\(537\) 0 0
\(538\) 98.3096 0.00787812
\(539\) 0 0
\(540\) 0 0
\(541\) −2180.90 −0.173316 −0.0866580 0.996238i \(-0.527619\pi\)
−0.0866580 + 0.996238i \(0.527619\pi\)
\(542\) −26004.4 −2.06086
\(543\) 0 0
\(544\) −10507.4 −0.828129
\(545\) −1117.28 −0.0878146
\(546\) 0 0
\(547\) −8225.04 −0.642920 −0.321460 0.946923i \(-0.604174\pi\)
−0.321460 + 0.946923i \(0.604174\pi\)
\(548\) −4278.07 −0.333485
\(549\) 0 0
\(550\) 0 0
\(551\) 19134.2 1.47939
\(552\) 0 0
\(553\) 1554.08 0.119505
\(554\) −3802.32 −0.291598
\(555\) 0 0
\(556\) 1639.90 0.125085
\(557\) −25181.9 −1.91561 −0.957804 0.287423i \(-0.907201\pi\)
−0.957804 + 0.287423i \(0.907201\pi\)
\(558\) 0 0
\(559\) 4206.33 0.318263
\(560\) −1317.83 −0.0994435
\(561\) 0 0
\(562\) −6314.50 −0.473952
\(563\) −4504.50 −0.337197 −0.168599 0.985685i \(-0.553924\pi\)
−0.168599 + 0.985685i \(0.553924\pi\)
\(564\) 0 0
\(565\) 5685.34 0.423335
\(566\) −7164.36 −0.532050
\(567\) 0 0
\(568\) 11805.9 0.872122
\(569\) −13447.0 −0.990732 −0.495366 0.868684i \(-0.664966\pi\)
−0.495366 + 0.868684i \(0.664966\pi\)
\(570\) 0 0
\(571\) 2605.52 0.190959 0.0954795 0.995431i \(-0.469562\pi\)
0.0954795 + 0.995431i \(0.469562\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −3860.09 −0.280691
\(575\) 12636.5 0.916485
\(576\) 0 0
\(577\) 6339.65 0.457406 0.228703 0.973496i \(-0.426552\pi\)
0.228703 + 0.973496i \(0.426552\pi\)
\(578\) 1485.90 0.106929
\(579\) 0 0
\(580\) 2860.68 0.204799
\(581\) −3589.83 −0.256336
\(582\) 0 0
\(583\) 0 0
\(584\) −15915.2 −1.12770
\(585\) 0 0
\(586\) −11210.1 −0.790247
\(587\) 13370.6 0.940140 0.470070 0.882629i \(-0.344229\pi\)
0.470070 + 0.882629i \(0.344229\pi\)
\(588\) 0 0
\(589\) 21935.3 1.53452
\(590\) −2306.20 −0.160923
\(591\) 0 0
\(592\) −8150.09 −0.565822
\(593\) 14319.3 0.991608 0.495804 0.868434i \(-0.334873\pi\)
0.495804 + 0.868434i \(0.334873\pi\)
\(594\) 0 0
\(595\) 1211.26 0.0834567
\(596\) 7938.73 0.545609
\(597\) 0 0
\(598\) −5673.65 −0.387981
\(599\) 5788.63 0.394853 0.197427 0.980318i \(-0.436742\pi\)
0.197427 + 0.980318i \(0.436742\pi\)
\(600\) 0 0
\(601\) −23968.1 −1.62675 −0.813375 0.581739i \(-0.802372\pi\)
−0.813375 + 0.581739i \(0.802372\pi\)
\(602\) 4480.26 0.303325
\(603\) 0 0
\(604\) 1922.44 0.129508
\(605\) 0 0
\(606\) 0 0
\(607\) 23526.6 1.57317 0.786585 0.617482i \(-0.211847\pi\)
0.786585 + 0.617482i \(0.211847\pi\)
\(608\) −11301.9 −0.753872
\(609\) 0 0
\(610\) 8252.40 0.547754
\(611\) 2551.74 0.168956
\(612\) 0 0
\(613\) −1228.07 −0.0809159 −0.0404579 0.999181i \(-0.512882\pi\)
−0.0404579 + 0.999181i \(0.512882\pi\)
\(614\) 5729.31 0.376573
\(615\) 0 0
\(616\) 0 0
\(617\) 9844.90 0.642368 0.321184 0.947017i \(-0.395919\pi\)
0.321184 + 0.947017i \(0.395919\pi\)
\(618\) 0 0
\(619\) −6551.68 −0.425419 −0.212709 0.977115i \(-0.568229\pi\)
−0.212709 + 0.977115i \(0.568229\pi\)
\(620\) 3279.47 0.212430
\(621\) 0 0
\(622\) −23365.9 −1.50625
\(623\) −2414.88 −0.155297
\(624\) 0 0
\(625\) 11208.0 0.717309
\(626\) −12007.8 −0.766661
\(627\) 0 0
\(628\) −6989.64 −0.444135
\(629\) 7491.01 0.474859
\(630\) 0 0
\(631\) −26440.5 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(632\) −5111.72 −0.321730
\(633\) 0 0
\(634\) −1122.13 −0.0702923
\(635\) −7994.48 −0.499608
\(636\) 0 0
\(637\) −4814.31 −0.299450
\(638\) 0 0
\(639\) 0 0
\(640\) 5803.64 0.358451
\(641\) 27927.2 1.72084 0.860421 0.509584i \(-0.170201\pi\)
0.860421 + 0.509584i \(0.170201\pi\)
\(642\) 0 0
\(643\) −16737.7 −1.02655 −0.513274 0.858225i \(-0.671568\pi\)
−0.513274 + 0.858225i \(0.671568\pi\)
\(644\) −1792.00 −0.109650
\(645\) 0 0
\(646\) 19419.1 1.18271
\(647\) −7818.70 −0.475092 −0.237546 0.971376i \(-0.576343\pi\)
−0.237546 + 0.971376i \(0.576343\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −5715.49 −0.344892
\(651\) 0 0
\(652\) 9025.94 0.542152
\(653\) −19747.6 −1.18344 −0.591719 0.806144i \(-0.701550\pi\)
−0.591719 + 0.806144i \(0.701550\pi\)
\(654\) 0 0
\(655\) −4003.47 −0.238822
\(656\) 19205.4 1.14305
\(657\) 0 0
\(658\) 2717.91 0.161026
\(659\) 7867.72 0.465072 0.232536 0.972588i \(-0.425298\pi\)
0.232536 + 0.972588i \(0.425298\pi\)
\(660\) 0 0
\(661\) 4227.41 0.248755 0.124378 0.992235i \(-0.460307\pi\)
0.124378 + 0.992235i \(0.460307\pi\)
\(662\) −1825.90 −0.107199
\(663\) 0 0
\(664\) 11807.8 0.690106
\(665\) 1302.85 0.0759733
\(666\) 0 0
\(667\) −27230.0 −1.58073
\(668\) −4006.67 −0.232070
\(669\) 0 0
\(670\) 10596.6 0.611018
\(671\) 0 0
\(672\) 0 0
\(673\) −29397.6 −1.68379 −0.841897 0.539638i \(-0.818561\pi\)
−0.841897 + 0.539638i \(0.818561\pi\)
\(674\) −2753.57 −0.157364
\(675\) 0 0
\(676\) −6647.94 −0.378239
\(677\) 5737.14 0.325696 0.162848 0.986651i \(-0.447932\pi\)
0.162848 + 0.986651i \(0.447932\pi\)
\(678\) 0 0
\(679\) 2914.86 0.164745
\(680\) −3984.11 −0.224682
\(681\) 0 0
\(682\) 0 0
\(683\) −32097.6 −1.79821 −0.899107 0.437729i \(-0.855783\pi\)
−0.899107 + 0.437729i \(0.855783\pi\)
\(684\) 0 0
\(685\) −4426.30 −0.246891
\(686\) −10615.8 −0.590837
\(687\) 0 0
\(688\) −22291.0 −1.23523
\(689\) 6153.69 0.340257
\(690\) 0 0
\(691\) −16456.2 −0.905965 −0.452983 0.891519i \(-0.649640\pi\)
−0.452983 + 0.891519i \(0.649640\pi\)
\(692\) 2721.93 0.149526
\(693\) 0 0
\(694\) 21112.3 1.15477
\(695\) 1696.72 0.0926047
\(696\) 0 0
\(697\) −17652.3 −0.959294
\(698\) 43058.9 2.33496
\(699\) 0 0
\(700\) −1805.22 −0.0974725
\(701\) 27238.1 1.46758 0.733788 0.679379i \(-0.237751\pi\)
0.733788 + 0.679379i \(0.237751\pi\)
\(702\) 0 0
\(703\) 8057.44 0.432279
\(704\) 0 0
\(705\) 0 0
\(706\) 8934.12 0.476261
\(707\) −4820.09 −0.256405
\(708\) 0 0
\(709\) 28761.4 1.52349 0.761747 0.647875i \(-0.224342\pi\)
0.761747 + 0.647875i \(0.224342\pi\)
\(710\) −8901.21 −0.470502
\(711\) 0 0
\(712\) 7943.10 0.418090
\(713\) −31216.3 −1.63964
\(714\) 0 0
\(715\) 0 0
\(716\) 6577.27 0.343302
\(717\) 0 0
\(718\) −10804.5 −0.561588
\(719\) 27272.0 1.41456 0.707282 0.706931i \(-0.249921\pi\)
0.707282 + 0.706931i \(0.249921\pi\)
\(720\) 0 0
\(721\) 5229.28 0.270109
\(722\) −2243.05 −0.115620
\(723\) 0 0
\(724\) 3580.04 0.183772
\(725\) −27430.8 −1.40518
\(726\) 0 0
\(727\) 3979.75 0.203027 0.101514 0.994834i \(-0.467631\pi\)
0.101514 + 0.994834i \(0.467631\pi\)
\(728\) −1112.26 −0.0566253
\(729\) 0 0
\(730\) 11999.5 0.608383
\(731\) 20488.3 1.03665
\(732\) 0 0
\(733\) −9342.48 −0.470767 −0.235384 0.971903i \(-0.575635\pi\)
−0.235384 + 0.971903i \(0.575635\pi\)
\(734\) −28426.3 −1.42947
\(735\) 0 0
\(736\) 16083.9 0.805515
\(737\) 0 0
\(738\) 0 0
\(739\) 28928.0 1.43997 0.719983 0.693992i \(-0.244150\pi\)
0.719983 + 0.693992i \(0.244150\pi\)
\(740\) 1204.64 0.0598425
\(741\) 0 0
\(742\) 6554.43 0.324287
\(743\) 4857.04 0.239822 0.119911 0.992785i \(-0.461739\pi\)
0.119911 + 0.992785i \(0.461739\pi\)
\(744\) 0 0
\(745\) 8213.80 0.403933
\(746\) 31660.8 1.55386
\(747\) 0 0
\(748\) 0 0
\(749\) 6536.00 0.318852
\(750\) 0 0
\(751\) 14355.4 0.697517 0.348759 0.937213i \(-0.386603\pi\)
0.348759 + 0.937213i \(0.386603\pi\)
\(752\) −13522.6 −0.655744
\(753\) 0 0
\(754\) 12316.1 0.594863
\(755\) 1989.05 0.0958792
\(756\) 0 0
\(757\) −17714.9 −0.850538 −0.425269 0.905067i \(-0.639821\pi\)
−0.425269 + 0.905067i \(0.639821\pi\)
\(758\) −48103.9 −2.30503
\(759\) 0 0
\(760\) −4285.37 −0.204535
\(761\) −7945.82 −0.378497 −0.189248 0.981929i \(-0.560605\pi\)
−0.189248 + 0.981929i \(0.560605\pi\)
\(762\) 0 0
\(763\) −1519.29 −0.0720866
\(764\) −7204.58 −0.341168
\(765\) 0 0
\(766\) −45398.4 −2.14140
\(767\) −2944.26 −0.138606
\(768\) 0 0
\(769\) −27308.1 −1.28057 −0.640284 0.768139i \(-0.721183\pi\)
−0.640284 + 0.768139i \(0.721183\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −13313.0 −0.620653
\(773\) 18872.6 0.878136 0.439068 0.898454i \(-0.355309\pi\)
0.439068 + 0.898454i \(0.355309\pi\)
\(774\) 0 0
\(775\) −31446.6 −1.45754
\(776\) −9587.65 −0.443527
\(777\) 0 0
\(778\) 3176.29 0.146369
\(779\) −18987.1 −0.873276
\(780\) 0 0
\(781\) 0 0
\(782\) −27635.4 −1.26373
\(783\) 0 0
\(784\) 25512.8 1.16221
\(785\) −7231.82 −0.328808
\(786\) 0 0
\(787\) 14512.1 0.657307 0.328654 0.944450i \(-0.393405\pi\)
0.328654 + 0.944450i \(0.393405\pi\)
\(788\) 3115.60 0.140849
\(789\) 0 0
\(790\) 3854.04 0.173570
\(791\) 7731.01 0.347513
\(792\) 0 0
\(793\) 10535.6 0.471792
\(794\) −2859.15 −0.127793
\(795\) 0 0
\(796\) −1605.72 −0.0714989
\(797\) −29108.9 −1.29371 −0.646856 0.762612i \(-0.723917\pi\)
−0.646856 + 0.762612i \(0.723917\pi\)
\(798\) 0 0
\(799\) 12429.1 0.550325
\(800\) 16202.5 0.716056
\(801\) 0 0
\(802\) −41154.0 −1.81197
\(803\) 0 0
\(804\) 0 0
\(805\) −1854.09 −0.0811777
\(806\) 14119.1 0.617029
\(807\) 0 0
\(808\) 15854.4 0.690291
\(809\) −3000.83 −0.130413 −0.0652063 0.997872i \(-0.520771\pi\)
−0.0652063 + 0.997872i \(0.520771\pi\)
\(810\) 0 0
\(811\) −6239.39 −0.270154 −0.135077 0.990835i \(-0.543128\pi\)
−0.135077 + 0.990835i \(0.543128\pi\)
\(812\) 3890.00 0.168118
\(813\) 0 0
\(814\) 0 0
\(815\) 9338.68 0.401374
\(816\) 0 0
\(817\) 22037.6 0.943693
\(818\) −29539.6 −1.26263
\(819\) 0 0
\(820\) −2838.69 −0.120892
\(821\) 14922.4 0.634342 0.317171 0.948368i \(-0.397267\pi\)
0.317171 + 0.948368i \(0.397267\pi\)
\(822\) 0 0
\(823\) −25737.8 −1.09011 −0.545057 0.838399i \(-0.683492\pi\)
−0.545057 + 0.838399i \(0.683492\pi\)
\(824\) −17200.3 −0.727186
\(825\) 0 0
\(826\) −3136.00 −0.132101
\(827\) 27043.4 1.13711 0.568555 0.822645i \(-0.307503\pi\)
0.568555 + 0.822645i \(0.307503\pi\)
\(828\) 0 0
\(829\) −9795.41 −0.410384 −0.205192 0.978722i \(-0.565782\pi\)
−0.205192 + 0.978722i \(0.565782\pi\)
\(830\) −8902.62 −0.372306
\(831\) 0 0
\(832\) 2291.84 0.0954990
\(833\) −23449.7 −0.975370
\(834\) 0 0
\(835\) −4145.50 −0.171809
\(836\) 0 0
\(837\) 0 0
\(838\) 37730.5 1.55535
\(839\) −28875.5 −1.18819 −0.594095 0.804395i \(-0.702490\pi\)
−0.594095 + 0.804395i \(0.702490\pi\)
\(840\) 0 0
\(841\) 34720.7 1.42362
\(842\) −47489.3 −1.94369
\(843\) 0 0
\(844\) 16585.7 0.676426
\(845\) −6878.28 −0.280024
\(846\) 0 0
\(847\) 0 0
\(848\) −32610.7 −1.32059
\(849\) 0 0
\(850\) −27839.2 −1.12338
\(851\) −11466.6 −0.461892
\(852\) 0 0
\(853\) 47157.1 1.89288 0.946441 0.322878i \(-0.104650\pi\)
0.946441 + 0.322878i \(0.104650\pi\)
\(854\) 11221.7 0.449649
\(855\) 0 0
\(856\) −21498.4 −0.858412
\(857\) 5021.31 0.200145 0.100073 0.994980i \(-0.468092\pi\)
0.100073 + 0.994980i \(0.468092\pi\)
\(858\) 0 0
\(859\) −22921.1 −0.910428 −0.455214 0.890382i \(-0.650437\pi\)
−0.455214 + 0.890382i \(0.650437\pi\)
\(860\) 3294.76 0.130640
\(861\) 0 0
\(862\) −18938.9 −0.748332
\(863\) 19488.1 0.768693 0.384347 0.923189i \(-0.374427\pi\)
0.384347 + 0.923189i \(0.374427\pi\)
\(864\) 0 0
\(865\) 2816.24 0.110699
\(866\) 24266.1 0.952188
\(867\) 0 0
\(868\) 4459.48 0.174383
\(869\) 0 0
\(870\) 0 0
\(871\) 13528.4 0.526282
\(872\) 4997.30 0.194071
\(873\) 0 0
\(874\) −29725.0 −1.15042
\(875\) −3937.06 −0.152111
\(876\) 0 0
\(877\) 8455.67 0.325573 0.162787 0.986661i \(-0.447952\pi\)
0.162787 + 0.986661i \(0.447952\pi\)
\(878\) −343.834 −0.0132162
\(879\) 0 0
\(880\) 0 0
\(881\) 11291.2 0.431794 0.215897 0.976416i \(-0.430732\pi\)
0.215897 + 0.976416i \(0.430732\pi\)
\(882\) 0 0
\(883\) 31818.1 1.21264 0.606322 0.795219i \(-0.292644\pi\)
0.606322 + 0.795219i \(0.292644\pi\)
\(884\) 3706.53 0.141023
\(885\) 0 0
\(886\) −16705.4 −0.633441
\(887\) 17481.1 0.661732 0.330866 0.943678i \(-0.392659\pi\)
0.330866 + 0.943678i \(0.392659\pi\)
\(888\) 0 0
\(889\) −10871.0 −0.410126
\(890\) −5988.80 −0.225556
\(891\) 0 0
\(892\) 7082.78 0.265862
\(893\) 13368.9 0.500978
\(894\) 0 0
\(895\) 6805.16 0.254158
\(896\) 7891.87 0.294251
\(897\) 0 0
\(898\) 39125.1 1.45392
\(899\) 67763.1 2.51393
\(900\) 0 0
\(901\) 29973.6 1.10829
\(902\) 0 0
\(903\) 0 0
\(904\) −25429.1 −0.935574
\(905\) 3704.08 0.136053
\(906\) 0 0
\(907\) 10607.4 0.388326 0.194163 0.980969i \(-0.437801\pi\)
0.194163 + 0.980969i \(0.437801\pi\)
\(908\) −7611.79 −0.278201
\(909\) 0 0
\(910\) 838.604 0.0305489
\(911\) 41249.2 1.50016 0.750080 0.661347i \(-0.230015\pi\)
0.750080 + 0.661347i \(0.230015\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 11906.5 0.430887
\(915\) 0 0
\(916\) −17910.4 −0.646045
\(917\) −5443.97 −0.196048
\(918\) 0 0
\(919\) 13858.1 0.497429 0.248714 0.968577i \(-0.419992\pi\)
0.248714 + 0.968577i \(0.419992\pi\)
\(920\) 6098.53 0.218546
\(921\) 0 0
\(922\) 39004.9 1.39323
\(923\) −11363.9 −0.405253
\(924\) 0 0
\(925\) −11551.2 −0.410595
\(926\) 36719.0 1.30309
\(927\) 0 0
\(928\) −34914.2 −1.23504
\(929\) −20893.7 −0.737890 −0.368945 0.929451i \(-0.620281\pi\)
−0.368945 + 0.929451i \(0.620281\pi\)
\(930\) 0 0
\(931\) −25222.8 −0.887911
\(932\) 8316.94 0.292307
\(933\) 0 0
\(934\) −36042.9 −1.26270
\(935\) 0 0
\(936\) 0 0
\(937\) −3203.52 −0.111691 −0.0558454 0.998439i \(-0.517785\pi\)
−0.0558454 + 0.998439i \(0.517785\pi\)
\(938\) 14409.4 0.501581
\(939\) 0 0
\(940\) 1998.74 0.0693528
\(941\) 19951.6 0.691182 0.345591 0.938385i \(-0.387678\pi\)
0.345591 + 0.938385i \(0.387678\pi\)
\(942\) 0 0
\(943\) 27020.6 0.933099
\(944\) 15602.8 0.537952
\(945\) 0 0
\(946\) 0 0
\(947\) 38216.7 1.31138 0.655689 0.755031i \(-0.272378\pi\)
0.655689 + 0.755031i \(0.272378\pi\)
\(948\) 0 0
\(949\) 15319.4 0.524014
\(950\) −29944.3 −1.02265
\(951\) 0 0
\(952\) −5417.65 −0.184440
\(953\) −47661.4 −1.62004 −0.810022 0.586399i \(-0.800545\pi\)
−0.810022 + 0.586399i \(0.800545\pi\)
\(954\) 0 0
\(955\) −7454.21 −0.252579
\(956\) 4820.35 0.163077
\(957\) 0 0
\(958\) 7897.56 0.266345
\(959\) −6018.94 −0.202671
\(960\) 0 0
\(961\) 47892.3 1.60761
\(962\) 5186.34 0.173819
\(963\) 0 0
\(964\) 3301.43 0.110303
\(965\) −13774.2 −0.459490
\(966\) 0 0
\(967\) −18933.2 −0.629628 −0.314814 0.949153i \(-0.601942\pi\)
−0.314814 + 0.949153i \(0.601942\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 7228.73 0.239279
\(971\) 40660.3 1.34382 0.671911 0.740632i \(-0.265474\pi\)
0.671911 + 0.740632i \(0.265474\pi\)
\(972\) 0 0
\(973\) 2307.23 0.0760188
\(974\) 22759.2 0.748720
\(975\) 0 0
\(976\) −55832.3 −1.83110
\(977\) 22502.8 0.736876 0.368438 0.929652i \(-0.379893\pi\)
0.368438 + 0.929652i \(0.379893\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −3770.98 −0.122918
\(981\) 0 0
\(982\) 24824.7 0.806709
\(983\) 4435.20 0.143907 0.0719536 0.997408i \(-0.477077\pi\)
0.0719536 + 0.997408i \(0.477077\pi\)
\(984\) 0 0
\(985\) 3223.55 0.104275
\(986\) 59989.8 1.93759
\(987\) 0 0
\(988\) 3986.80 0.128378
\(989\) −31361.8 −1.00834
\(990\) 0 0
\(991\) 7362.76 0.236010 0.118005 0.993013i \(-0.462350\pi\)
0.118005 + 0.993013i \(0.462350\pi\)
\(992\) −40025.5 −1.28106
\(993\) 0 0
\(994\) −12104.0 −0.386233
\(995\) −1661.35 −0.0529331
\(996\) 0 0
\(997\) 53480.1 1.69883 0.849413 0.527728i \(-0.176956\pi\)
0.849413 + 0.527728i \(0.176956\pi\)
\(998\) 35010.0 1.11044
\(999\) 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 1089.4.a.t.1.2 2
3.2 odd 2 363.4.a.j.1.1 2
11.10 odd 2 99.4.a.e.1.1 2
33.32 even 2 33.4.a.d.1.2 2
44.43 even 2 1584.4.a.x.1.2 2
55.54 odd 2 2475.4.a.o.1.2 2
132.131 odd 2 528.4.a.o.1.1 2
165.32 odd 4 825.4.c.i.199.4 4
165.98 odd 4 825.4.c.i.199.1 4
165.164 even 2 825.4.a.k.1.1 2
231.230 odd 2 1617.4.a.j.1.2 2
264.131 odd 2 2112.4.a.bh.1.2 2
264.197 even 2 2112.4.a.ba.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.d.1.2 2 33.32 even 2
99.4.a.e.1.1 2 11.10 odd 2
363.4.a.j.1.1 2 3.2 odd 2
528.4.a.o.1.1 2 132.131 odd 2
825.4.a.k.1.1 2 165.164 even 2
825.4.c.i.199.1 4 165.98 odd 4
825.4.c.i.199.4 4 165.32 odd 4
1089.4.a.t.1.2 2 1.1 even 1 trivial
1584.4.a.x.1.2 2 44.43 even 2
1617.4.a.j.1.2 2 231.230 odd 2
2112.4.a.ba.1.2 2 264.197 even 2
2112.4.a.bh.1.2 2 264.131 odd 2
2475.4.a.o.1.2 2 55.54 odd 2