Properties

Label 1323.4.a.bl.1.4
Level $1323$
Weight $4$
Character 1323.1
Self dual yes
Analytic conductor $78.060$
Analytic rank $1$
Dimension $8$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1323,4,Mod(1,1323)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1323, 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("1323.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1323.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: \(78.0595269376\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 54x^{6} + 887x^{4} - 4176x^{2} + 3136 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.959848\) of defining polynomial
Character \(\chi\) \(=\) 1323.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.959848 q^{2} -7.07869 q^{4} +10.2898 q^{5} +14.4733 q^{8} +O(q^{10})\) \(q-0.959848 q^{2} -7.07869 q^{4} +10.2898 q^{5} +14.4733 q^{8} -9.87663 q^{10} +28.3265 q^{11} +5.96269 q^{13} +42.7374 q^{16} -104.664 q^{17} +34.0521 q^{19} -72.8381 q^{20} -27.1891 q^{22} -103.629 q^{23} -19.1205 q^{25} -5.72328 q^{26} -195.695 q^{29} +9.17704 q^{31} -156.807 q^{32} +100.462 q^{34} +245.968 q^{37} -32.6849 q^{38} +148.927 q^{40} +366.457 q^{41} -366.354 q^{43} -200.514 q^{44} +99.4681 q^{46} +244.288 q^{47} +18.3528 q^{50} -42.2080 q^{52} +281.226 q^{53} +291.473 q^{55} +187.838 q^{58} -181.642 q^{59} -24.1701 q^{61} -8.80857 q^{62} -191.388 q^{64} +61.3547 q^{65} -336.788 q^{67} +740.886 q^{68} +196.781 q^{71} -683.417 q^{73} -236.092 q^{74} -241.044 q^{76} +619.337 q^{79} +439.758 q^{80} -351.743 q^{82} -176.475 q^{83} -1076.97 q^{85} +351.644 q^{86} +409.976 q^{88} -761.046 q^{89} +733.557 q^{92} -234.480 q^{94} +350.389 q^{95} -1277.59 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 44 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 44 q^{4} - 132 q^{10} - 336 q^{13} + 204 q^{16} + 288 q^{19} + 484 q^{22} + 152 q^{25} - 120 q^{31} - 1008 q^{34} + 592 q^{37} - 1620 q^{40} - 1872 q^{43} - 1644 q^{46} - 2400 q^{52} - 1344 q^{55} - 1200 q^{58} - 2400 q^{61} - 1388 q^{64} - 1824 q^{73} - 2844 q^{76} + 2368 q^{79} - 2436 q^{82} + 3512 q^{85} + 3780 q^{88} - 4368 q^{94} - 5712 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.959848 −0.339358 −0.169679 0.985499i \(-0.554273\pi\)
−0.169679 + 0.985499i \(0.554273\pi\)
\(3\) 0 0
\(4\) −7.07869 −0.884836
\(5\) 10.2898 0.920346 0.460173 0.887829i \(-0.347787\pi\)
0.460173 + 0.887829i \(0.347787\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 14.4733 0.639634
\(9\) 0 0
\(10\) −9.87663 −0.312326
\(11\) 28.3265 0.776432 0.388216 0.921568i \(-0.373091\pi\)
0.388216 + 0.921568i \(0.373091\pi\)
\(12\) 0 0
\(13\) 5.96269 0.127212 0.0636058 0.997975i \(-0.479740\pi\)
0.0636058 + 0.997975i \(0.479740\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 42.7374 0.667772
\(17\) −104.664 −1.49323 −0.746613 0.665259i \(-0.768321\pi\)
−0.746613 + 0.665259i \(0.768321\pi\)
\(18\) 0 0
\(19\) 34.0521 0.411163 0.205581 0.978640i \(-0.434091\pi\)
0.205581 + 0.978640i \(0.434091\pi\)
\(20\) −72.8381 −0.814355
\(21\) 0 0
\(22\) −27.1891 −0.263488
\(23\) −103.629 −0.939484 −0.469742 0.882804i \(-0.655653\pi\)
−0.469742 + 0.882804i \(0.655653\pi\)
\(24\) 0 0
\(25\) −19.1205 −0.152964
\(26\) −5.72328 −0.0431703
\(27\) 0 0
\(28\) 0 0
\(29\) −195.695 −1.25309 −0.626547 0.779384i \(-0.715532\pi\)
−0.626547 + 0.779384i \(0.715532\pi\)
\(30\) 0 0
\(31\) 9.17704 0.0531692 0.0265846 0.999647i \(-0.491537\pi\)
0.0265846 + 0.999647i \(0.491537\pi\)
\(32\) −156.807 −0.866247
\(33\) 0 0
\(34\) 100.462 0.506737
\(35\) 0 0
\(36\) 0 0
\(37\) 245.968 1.09289 0.546444 0.837496i \(-0.315981\pi\)
0.546444 + 0.837496i \(0.315981\pi\)
\(38\) −32.6849 −0.139531
\(39\) 0 0
\(40\) 148.927 0.588684
\(41\) 366.457 1.39588 0.697939 0.716157i \(-0.254101\pi\)
0.697939 + 0.716157i \(0.254101\pi\)
\(42\) 0 0
\(43\) −366.354 −1.29927 −0.649633 0.760248i \(-0.725078\pi\)
−0.649633 + 0.760248i \(0.725078\pi\)
\(44\) −200.514 −0.687015
\(45\) 0 0
\(46\) 99.4681 0.318821
\(47\) 244.288 0.758152 0.379076 0.925366i \(-0.376242\pi\)
0.379076 + 0.925366i \(0.376242\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 18.3528 0.0519095
\(51\) 0 0
\(52\) −42.2080 −0.112562
\(53\) 281.226 0.728857 0.364428 0.931231i \(-0.381264\pi\)
0.364428 + 0.931231i \(0.381264\pi\)
\(54\) 0 0
\(55\) 291.473 0.714586
\(56\) 0 0
\(57\) 0 0
\(58\) 187.838 0.425247
\(59\) −181.642 −0.400810 −0.200405 0.979713i \(-0.564226\pi\)
−0.200405 + 0.979713i \(0.564226\pi\)
\(60\) 0 0
\(61\) −24.1701 −0.0507323 −0.0253661 0.999678i \(-0.508075\pi\)
−0.0253661 + 0.999678i \(0.508075\pi\)
\(62\) −8.80857 −0.0180434
\(63\) 0 0
\(64\) −191.388 −0.373804
\(65\) 61.3547 0.117079
\(66\) 0 0
\(67\) −336.788 −0.614108 −0.307054 0.951692i \(-0.599343\pi\)
−0.307054 + 0.951692i \(0.599343\pi\)
\(68\) 740.886 1.32126
\(69\) 0 0
\(70\) 0 0
\(71\) 196.781 0.328924 0.164462 0.986383i \(-0.447411\pi\)
0.164462 + 0.986383i \(0.447411\pi\)
\(72\) 0 0
\(73\) −683.417 −1.09573 −0.547863 0.836568i \(-0.684558\pi\)
−0.547863 + 0.836568i \(0.684558\pi\)
\(74\) −236.092 −0.370880
\(75\) 0 0
\(76\) −241.044 −0.363812
\(77\) 0 0
\(78\) 0 0
\(79\) 619.337 0.882036 0.441018 0.897498i \(-0.354617\pi\)
0.441018 + 0.897498i \(0.354617\pi\)
\(80\) 439.758 0.614581
\(81\) 0 0
\(82\) −351.743 −0.473702
\(83\) −176.475 −0.233381 −0.116690 0.993168i \(-0.537229\pi\)
−0.116690 + 0.993168i \(0.537229\pi\)
\(84\) 0 0
\(85\) −1076.97 −1.37428
\(86\) 351.644 0.440916
\(87\) 0 0
\(88\) 409.976 0.496632
\(89\) −761.046 −0.906413 −0.453207 0.891406i \(-0.649720\pi\)
−0.453207 + 0.891406i \(0.649720\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 733.557 0.831290
\(93\) 0 0
\(94\) −234.480 −0.257285
\(95\) 350.389 0.378412
\(96\) 0 0
\(97\) −1277.59 −1.33732 −0.668660 0.743568i \(-0.733132\pi\)
−0.668660 + 0.743568i \(0.733132\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 135.348 0.135348
\(101\) 1555.82 1.53277 0.766387 0.642379i \(-0.222052\pi\)
0.766387 + 0.642379i \(0.222052\pi\)
\(102\) 0 0
\(103\) 246.314 0.235632 0.117816 0.993035i \(-0.462411\pi\)
0.117816 + 0.993035i \(0.462411\pi\)
\(104\) 86.2995 0.0813689
\(105\) 0 0
\(106\) −269.935 −0.247343
\(107\) −1621.50 −1.46501 −0.732504 0.680762i \(-0.761649\pi\)
−0.732504 + 0.680762i \(0.761649\pi\)
\(108\) 0 0
\(109\) 1906.69 1.67548 0.837742 0.546066i \(-0.183875\pi\)
0.837742 + 0.546066i \(0.183875\pi\)
\(110\) −279.770 −0.242500
\(111\) 0 0
\(112\) 0 0
\(113\) −1886.64 −1.57062 −0.785312 0.619100i \(-0.787498\pi\)
−0.785312 + 0.619100i \(0.787498\pi\)
\(114\) 0 0
\(115\) −1066.32 −0.864650
\(116\) 1385.27 1.10878
\(117\) 0 0
\(118\) 174.349 0.136018
\(119\) 0 0
\(120\) 0 0
\(121\) −528.611 −0.397153
\(122\) 23.1997 0.0172164
\(123\) 0 0
\(124\) −64.9614 −0.0470460
\(125\) −1482.97 −1.06113
\(126\) 0 0
\(127\) −594.201 −0.415172 −0.207586 0.978217i \(-0.566561\pi\)
−0.207586 + 0.978217i \(0.566561\pi\)
\(128\) 1438.16 0.993100
\(129\) 0 0
\(130\) −58.8912 −0.0397316
\(131\) −494.116 −0.329551 −0.164775 0.986331i \(-0.552690\pi\)
−0.164775 + 0.986331i \(0.552690\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 323.266 0.208402
\(135\) 0 0
\(136\) −1514.83 −0.955117
\(137\) 1807.01 1.12689 0.563443 0.826155i \(-0.309477\pi\)
0.563443 + 0.826155i \(0.309477\pi\)
\(138\) 0 0
\(139\) −3149.34 −1.92175 −0.960876 0.276979i \(-0.910667\pi\)
−0.960876 + 0.276979i \(0.910667\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −188.880 −0.111623
\(143\) 168.902 0.0987712
\(144\) 0 0
\(145\) −2013.66 −1.15328
\(146\) 655.977 0.371843
\(147\) 0 0
\(148\) −1741.13 −0.967027
\(149\) 1899.54 1.04441 0.522203 0.852821i \(-0.325110\pi\)
0.522203 + 0.852821i \(0.325110\pi\)
\(150\) 0 0
\(151\) 1521.53 0.820004 0.410002 0.912085i \(-0.365528\pi\)
0.410002 + 0.912085i \(0.365528\pi\)
\(152\) 492.845 0.262994
\(153\) 0 0
\(154\) 0 0
\(155\) 94.4297 0.0489340
\(156\) 0 0
\(157\) −3606.96 −1.83355 −0.916773 0.399408i \(-0.869216\pi\)
−0.916773 + 0.399408i \(0.869216\pi\)
\(158\) −594.470 −0.299326
\(159\) 0 0
\(160\) −1613.51 −0.797247
\(161\) 0 0
\(162\) 0 0
\(163\) 887.302 0.426373 0.213187 0.977011i \(-0.431616\pi\)
0.213187 + 0.977011i \(0.431616\pi\)
\(164\) −2594.04 −1.23512
\(165\) 0 0
\(166\) 169.389 0.0791996
\(167\) 1248.96 0.578729 0.289364 0.957219i \(-0.406556\pi\)
0.289364 + 0.957219i \(0.406556\pi\)
\(168\) 0 0
\(169\) −2161.45 −0.983817
\(170\) 1033.73 0.466374
\(171\) 0 0
\(172\) 2593.31 1.14964
\(173\) 2130.55 0.936318 0.468159 0.883644i \(-0.344917\pi\)
0.468159 + 0.883644i \(0.344917\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1210.60 0.518479
\(177\) 0 0
\(178\) 730.489 0.307598
\(179\) −2756.11 −1.15084 −0.575422 0.817857i \(-0.695162\pi\)
−0.575422 + 0.817857i \(0.695162\pi\)
\(180\) 0 0
\(181\) −734.801 −0.301753 −0.150877 0.988553i \(-0.548210\pi\)
−0.150877 + 0.988553i \(0.548210\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −1499.85 −0.600926
\(185\) 2530.95 1.00583
\(186\) 0 0
\(187\) −2964.77 −1.15939
\(188\) −1729.24 −0.670840
\(189\) 0 0
\(190\) −336.320 −0.128417
\(191\) −2034.49 −0.770736 −0.385368 0.922763i \(-0.625925\pi\)
−0.385368 + 0.922763i \(0.625925\pi\)
\(192\) 0 0
\(193\) −664.168 −0.247709 −0.123855 0.992300i \(-0.539526\pi\)
−0.123855 + 0.992300i \(0.539526\pi\)
\(194\) 1226.30 0.453830
\(195\) 0 0
\(196\) 0 0
\(197\) −1682.40 −0.608455 −0.304228 0.952599i \(-0.598398\pi\)
−0.304228 + 0.952599i \(0.598398\pi\)
\(198\) 0 0
\(199\) −3778.06 −1.34583 −0.672913 0.739721i \(-0.734957\pi\)
−0.672913 + 0.739721i \(0.734957\pi\)
\(200\) −276.736 −0.0978410
\(201\) 0 0
\(202\) −1493.35 −0.520159
\(203\) 0 0
\(204\) 0 0
\(205\) 3770.76 1.28469
\(206\) −236.425 −0.0799635
\(207\) 0 0
\(208\) 254.830 0.0849484
\(209\) 964.576 0.319240
\(210\) 0 0
\(211\) −5122.91 −1.67145 −0.835725 0.549148i \(-0.814952\pi\)
−0.835725 + 0.549148i \(0.814952\pi\)
\(212\) −1990.71 −0.644919
\(213\) 0 0
\(214\) 1556.39 0.497162
\(215\) −3769.70 −1.19577
\(216\) 0 0
\(217\) 0 0
\(218\) −1830.13 −0.568589
\(219\) 0 0
\(220\) −2063.25 −0.632291
\(221\) −624.081 −0.189956
\(222\) 0 0
\(223\) −2118.71 −0.636229 −0.318115 0.948052i \(-0.603050\pi\)
−0.318115 + 0.948052i \(0.603050\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 1810.89 0.533004
\(227\) −4239.57 −1.23960 −0.619802 0.784758i \(-0.712787\pi\)
−0.619802 + 0.784758i \(0.712787\pi\)
\(228\) 0 0
\(229\) 1518.18 0.438098 0.219049 0.975714i \(-0.429705\pi\)
0.219049 + 0.975714i \(0.429705\pi\)
\(230\) 1023.50 0.293426
\(231\) 0 0
\(232\) −2832.35 −0.801521
\(233\) −4596.37 −1.29235 −0.646177 0.763188i \(-0.723633\pi\)
−0.646177 + 0.763188i \(0.723633\pi\)
\(234\) 0 0
\(235\) 2513.67 0.697761
\(236\) 1285.79 0.354651
\(237\) 0 0
\(238\) 0 0
\(239\) 2956.83 0.800256 0.400128 0.916459i \(-0.368966\pi\)
0.400128 + 0.916459i \(0.368966\pi\)
\(240\) 0 0
\(241\) −113.007 −0.0302050 −0.0151025 0.999886i \(-0.504807\pi\)
−0.0151025 + 0.999886i \(0.504807\pi\)
\(242\) 507.387 0.134777
\(243\) 0 0
\(244\) 171.093 0.0448897
\(245\) 0 0
\(246\) 0 0
\(247\) 203.042 0.0523047
\(248\) 132.822 0.0340088
\(249\) 0 0
\(250\) 1423.42 0.360101
\(251\) 6988.70 1.75746 0.878731 0.477318i \(-0.158391\pi\)
0.878731 + 0.477318i \(0.158391\pi\)
\(252\) 0 0
\(253\) −2935.44 −0.729445
\(254\) 570.343 0.140892
\(255\) 0 0
\(256\) 150.683 0.0367880
\(257\) −2969.04 −0.720637 −0.360318 0.932829i \(-0.617332\pi\)
−0.360318 + 0.932829i \(0.617332\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −434.311 −0.103595
\(261\) 0 0
\(262\) 474.277 0.111836
\(263\) −4545.72 −1.06578 −0.532892 0.846183i \(-0.678895\pi\)
−0.532892 + 0.846183i \(0.678895\pi\)
\(264\) 0 0
\(265\) 2893.76 0.670800
\(266\) 0 0
\(267\) 0 0
\(268\) 2384.02 0.543385
\(269\) −1438.64 −0.326079 −0.163040 0.986620i \(-0.552130\pi\)
−0.163040 + 0.986620i \(0.552130\pi\)
\(270\) 0 0
\(271\) 583.420 0.130776 0.0653879 0.997860i \(-0.479172\pi\)
0.0653879 + 0.997860i \(0.479172\pi\)
\(272\) −4473.08 −0.997134
\(273\) 0 0
\(274\) −1734.46 −0.382417
\(275\) −541.616 −0.118766
\(276\) 0 0
\(277\) 1559.79 0.338335 0.169168 0.985587i \(-0.445892\pi\)
0.169168 + 0.985587i \(0.445892\pi\)
\(278\) 3022.89 0.652161
\(279\) 0 0
\(280\) 0 0
\(281\) 8809.34 1.87018 0.935091 0.354407i \(-0.115317\pi\)
0.935091 + 0.354407i \(0.115317\pi\)
\(282\) 0 0
\(283\) −8501.27 −1.78568 −0.892841 0.450371i \(-0.851292\pi\)
−0.892841 + 0.450371i \(0.851292\pi\)
\(284\) −1392.95 −0.291044
\(285\) 0 0
\(286\) −162.120 −0.0335188
\(287\) 0 0
\(288\) 0 0
\(289\) 6041.62 1.22972
\(290\) 1932.81 0.391374
\(291\) 0 0
\(292\) 4837.70 0.969537
\(293\) −8571.41 −1.70903 −0.854517 0.519423i \(-0.826147\pi\)
−0.854517 + 0.519423i \(0.826147\pi\)
\(294\) 0 0
\(295\) −1869.06 −0.368883
\(296\) 3559.95 0.699048
\(297\) 0 0
\(298\) −1823.27 −0.354427
\(299\) −617.907 −0.119513
\(300\) 0 0
\(301\) 0 0
\(302\) −1460.44 −0.278275
\(303\) 0 0
\(304\) 1455.30 0.274563
\(305\) −248.705 −0.0466912
\(306\) 0 0
\(307\) −1751.31 −0.325578 −0.162789 0.986661i \(-0.552049\pi\)
−0.162789 + 0.986661i \(0.552049\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −90.6382 −0.0166061
\(311\) −6681.74 −1.21829 −0.609143 0.793061i \(-0.708486\pi\)
−0.609143 + 0.793061i \(0.708486\pi\)
\(312\) 0 0
\(313\) −6049.69 −1.09249 −0.546244 0.837626i \(-0.683943\pi\)
−0.546244 + 0.837626i \(0.683943\pi\)
\(314\) 3462.14 0.622228
\(315\) 0 0
\(316\) −4384.10 −0.780458
\(317\) 3678.59 0.651767 0.325884 0.945410i \(-0.394338\pi\)
0.325884 + 0.945410i \(0.394338\pi\)
\(318\) 0 0
\(319\) −5543.36 −0.972942
\(320\) −1969.34 −0.344029
\(321\) 0 0
\(322\) 0 0
\(323\) −3564.04 −0.613959
\(324\) 0 0
\(325\) −114.010 −0.0194588
\(326\) −851.675 −0.144693
\(327\) 0 0
\(328\) 5303.83 0.892850
\(329\) 0 0
\(330\) 0 0
\(331\) 10462.0 1.73729 0.868647 0.495432i \(-0.164990\pi\)
0.868647 + 0.495432i \(0.164990\pi\)
\(332\) 1249.21 0.206504
\(333\) 0 0
\(334\) −1198.82 −0.196396
\(335\) −3465.48 −0.565192
\(336\) 0 0
\(337\) 1072.17 0.173308 0.0866539 0.996238i \(-0.472383\pi\)
0.0866539 + 0.996238i \(0.472383\pi\)
\(338\) 2074.66 0.333866
\(339\) 0 0
\(340\) 7623.56 1.21602
\(341\) 259.953 0.0412823
\(342\) 0 0
\(343\) 0 0
\(344\) −5302.34 −0.831055
\(345\) 0 0
\(346\) −2045.01 −0.317747
\(347\) 9562.41 1.47936 0.739679 0.672960i \(-0.234977\pi\)
0.739679 + 0.672960i \(0.234977\pi\)
\(348\) 0 0
\(349\) −4159.34 −0.637950 −0.318975 0.947763i \(-0.603339\pi\)
−0.318975 + 0.947763i \(0.603339\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −4441.80 −0.672582
\(353\) 6003.78 0.905238 0.452619 0.891704i \(-0.350490\pi\)
0.452619 + 0.891704i \(0.350490\pi\)
\(354\) 0 0
\(355\) 2024.83 0.302723
\(356\) 5387.21 0.802027
\(357\) 0 0
\(358\) 2645.44 0.390548
\(359\) 11515.2 1.69290 0.846448 0.532472i \(-0.178737\pi\)
0.846448 + 0.532472i \(0.178737\pi\)
\(360\) 0 0
\(361\) −5699.45 −0.830945
\(362\) 705.298 0.102402
\(363\) 0 0
\(364\) 0 0
\(365\) −7032.21 −1.00845
\(366\) 0 0
\(367\) 8631.92 1.22774 0.613872 0.789405i \(-0.289611\pi\)
0.613872 + 0.789405i \(0.289611\pi\)
\(368\) −4428.83 −0.627361
\(369\) 0 0
\(370\) −2429.33 −0.341338
\(371\) 0 0
\(372\) 0 0
\(373\) −11586.5 −1.60838 −0.804189 0.594373i \(-0.797400\pi\)
−0.804189 + 0.594373i \(0.797400\pi\)
\(374\) 2845.73 0.393447
\(375\) 0 0
\(376\) 3535.65 0.484939
\(377\) −1166.87 −0.159408
\(378\) 0 0
\(379\) 11.3954 0.00154443 0.000772216 1.00000i \(-0.499754\pi\)
0.000772216 1.00000i \(0.499754\pi\)
\(380\) −2480.29 −0.334833
\(381\) 0 0
\(382\) 1952.80 0.261555
\(383\) −7649.91 −1.02061 −0.510303 0.859995i \(-0.670467\pi\)
−0.510303 + 0.859995i \(0.670467\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 637.501 0.0840620
\(387\) 0 0
\(388\) 9043.69 1.18331
\(389\) 1018.92 0.132805 0.0664027 0.997793i \(-0.478848\pi\)
0.0664027 + 0.997793i \(0.478848\pi\)
\(390\) 0 0
\(391\) 10846.3 1.40286
\(392\) 0 0
\(393\) 0 0
\(394\) 1614.84 0.206484
\(395\) 6372.84 0.811778
\(396\) 0 0
\(397\) −1213.68 −0.153433 −0.0767164 0.997053i \(-0.524444\pi\)
−0.0767164 + 0.997053i \(0.524444\pi\)
\(398\) 3626.36 0.456717
\(399\) 0 0
\(400\) −817.161 −0.102145
\(401\) −7176.55 −0.893715 −0.446858 0.894605i \(-0.647457\pi\)
−0.446858 + 0.894605i \(0.647457\pi\)
\(402\) 0 0
\(403\) 54.7198 0.00676374
\(404\) −11013.2 −1.35625
\(405\) 0 0
\(406\) 0 0
\(407\) 6967.40 0.848553
\(408\) 0 0
\(409\) −14888.8 −1.80001 −0.900006 0.435878i \(-0.856438\pi\)
−0.900006 + 0.435878i \(0.856438\pi\)
\(410\) −3619.36 −0.435969
\(411\) 0 0
\(412\) −1743.58 −0.208496
\(413\) 0 0
\(414\) 0 0
\(415\) −1815.89 −0.214791
\(416\) −934.994 −0.110197
\(417\) 0 0
\(418\) −925.847 −0.108337
\(419\) −7504.96 −0.875039 −0.437519 0.899209i \(-0.644143\pi\)
−0.437519 + 0.899209i \(0.644143\pi\)
\(420\) 0 0
\(421\) 8568.94 0.991983 0.495991 0.868327i \(-0.334805\pi\)
0.495991 + 0.868327i \(0.334805\pi\)
\(422\) 4917.22 0.567219
\(423\) 0 0
\(424\) 4070.26 0.466201
\(425\) 2001.23 0.228410
\(426\) 0 0
\(427\) 0 0
\(428\) 11478.1 1.29629
\(429\) 0 0
\(430\) 3618.34 0.405795
\(431\) 1869.85 0.208974 0.104487 0.994526i \(-0.466680\pi\)
0.104487 + 0.994526i \(0.466680\pi\)
\(432\) 0 0
\(433\) −11202.1 −1.24327 −0.621636 0.783306i \(-0.713532\pi\)
−0.621636 + 0.783306i \(0.713532\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −13496.9 −1.48253
\(437\) −3528.79 −0.386281
\(438\) 0 0
\(439\) −9815.68 −1.06715 −0.533573 0.845754i \(-0.679151\pi\)
−0.533573 + 0.845754i \(0.679151\pi\)
\(440\) 4218.56 0.457073
\(441\) 0 0
\(442\) 599.023 0.0644629
\(443\) 5807.00 0.622797 0.311399 0.950279i \(-0.399203\pi\)
0.311399 + 0.950279i \(0.399203\pi\)
\(444\) 0 0
\(445\) −7831.00 −0.834213
\(446\) 2033.64 0.215909
\(447\) 0 0
\(448\) 0 0
\(449\) −257.367 −0.0270510 −0.0135255 0.999909i \(-0.504305\pi\)
−0.0135255 + 0.999909i \(0.504305\pi\)
\(450\) 0 0
\(451\) 10380.4 1.08380
\(452\) 13355.0 1.38975
\(453\) 0 0
\(454\) 4069.34 0.420669
\(455\) 0 0
\(456\) 0 0
\(457\) 1448.74 0.148292 0.0741458 0.997247i \(-0.476377\pi\)
0.0741458 + 0.997247i \(0.476377\pi\)
\(458\) −1457.23 −0.148672
\(459\) 0 0
\(460\) 7548.14 0.765074
\(461\) −8713.18 −0.880289 −0.440144 0.897927i \(-0.645073\pi\)
−0.440144 + 0.897927i \(0.645073\pi\)
\(462\) 0 0
\(463\) 17766.2 1.78329 0.891646 0.452734i \(-0.149551\pi\)
0.891646 + 0.452734i \(0.149551\pi\)
\(464\) −8363.51 −0.836780
\(465\) 0 0
\(466\) 4411.82 0.438570
\(467\) 13735.0 1.36098 0.680492 0.732755i \(-0.261766\pi\)
0.680492 + 0.732755i \(0.261766\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −2412.74 −0.236791
\(471\) 0 0
\(472\) −2628.95 −0.256371
\(473\) −10377.5 −1.00879
\(474\) 0 0
\(475\) −651.094 −0.0628931
\(476\) 0 0
\(477\) 0 0
\(478\) −2838.11 −0.271573
\(479\) −4821.63 −0.459929 −0.229964 0.973199i \(-0.573861\pi\)
−0.229964 + 0.973199i \(0.573861\pi\)
\(480\) 0 0
\(481\) 1466.63 0.139028
\(482\) 108.469 0.0102503
\(483\) 0 0
\(484\) 3741.88 0.351416
\(485\) −13146.2 −1.23080
\(486\) 0 0
\(487\) −13603.0 −1.26573 −0.632864 0.774263i \(-0.718121\pi\)
−0.632864 + 0.774263i \(0.718121\pi\)
\(488\) −349.821 −0.0324501
\(489\) 0 0
\(490\) 0 0
\(491\) 6137.42 0.564110 0.282055 0.959398i \(-0.408984\pi\)
0.282055 + 0.959398i \(0.408984\pi\)
\(492\) 0 0
\(493\) 20482.3 1.87115
\(494\) −194.890 −0.0177500
\(495\) 0 0
\(496\) 392.203 0.0355049
\(497\) 0 0
\(498\) 0 0
\(499\) −13674.1 −1.22673 −0.613363 0.789801i \(-0.710184\pi\)
−0.613363 + 0.789801i \(0.710184\pi\)
\(500\) 10497.5 0.938922
\(501\) 0 0
\(502\) −6708.09 −0.596408
\(503\) −4346.13 −0.385257 −0.192629 0.981272i \(-0.561701\pi\)
−0.192629 + 0.981272i \(0.561701\pi\)
\(504\) 0 0
\(505\) 16009.1 1.41068
\(506\) 2817.58 0.247543
\(507\) 0 0
\(508\) 4206.17 0.367359
\(509\) 19414.9 1.69067 0.845336 0.534235i \(-0.179400\pi\)
0.845336 + 0.534235i \(0.179400\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −11649.9 −1.00558
\(513\) 0 0
\(514\) 2849.83 0.244554
\(515\) 2534.52 0.216863
\(516\) 0 0
\(517\) 6919.83 0.588653
\(518\) 0 0
\(519\) 0 0
\(520\) 888.003 0.0748875
\(521\) 2749.25 0.231184 0.115592 0.993297i \(-0.463124\pi\)
0.115592 + 0.993297i \(0.463124\pi\)
\(522\) 0 0
\(523\) −3313.72 −0.277054 −0.138527 0.990359i \(-0.544237\pi\)
−0.138527 + 0.990359i \(0.544237\pi\)
\(524\) 3497.70 0.291598
\(525\) 0 0
\(526\) 4363.21 0.361682
\(527\) −960.509 −0.0793936
\(528\) 0 0
\(529\) −1428.04 −0.117370
\(530\) −2777.57 −0.227641
\(531\) 0 0
\(532\) 0 0
\(533\) 2185.07 0.177572
\(534\) 0 0
\(535\) −16684.8 −1.34831
\(536\) −4874.42 −0.392804
\(537\) 0 0
\(538\) 1380.87 0.110657
\(539\) 0 0
\(540\) 0 0
\(541\) −1490.04 −0.118414 −0.0592069 0.998246i \(-0.518857\pi\)
−0.0592069 + 0.998246i \(0.518857\pi\)
\(542\) −559.995 −0.0443798
\(543\) 0 0
\(544\) 16412.1 1.29350
\(545\) 19619.4 1.54202
\(546\) 0 0
\(547\) 18962.2 1.48220 0.741102 0.671392i \(-0.234303\pi\)
0.741102 + 0.671392i \(0.234303\pi\)
\(548\) −12791.3 −0.997109
\(549\) 0 0
\(550\) 519.870 0.0403042
\(551\) −6663.84 −0.515225
\(552\) 0 0
\(553\) 0 0
\(554\) −1497.16 −0.114817
\(555\) 0 0
\(556\) 22293.2 1.70044
\(557\) −4846.56 −0.368681 −0.184341 0.982862i \(-0.559015\pi\)
−0.184341 + 0.982862i \(0.559015\pi\)
\(558\) 0 0
\(559\) −2184.45 −0.165282
\(560\) 0 0
\(561\) 0 0
\(562\) −8455.63 −0.634661
\(563\) −12390.7 −0.927540 −0.463770 0.885956i \(-0.653504\pi\)
−0.463770 + 0.885956i \(0.653504\pi\)
\(564\) 0 0
\(565\) −19413.2 −1.44552
\(566\) 8159.93 0.605985
\(567\) 0 0
\(568\) 2848.06 0.210391
\(569\) −8706.19 −0.641445 −0.320723 0.947173i \(-0.603926\pi\)
−0.320723 + 0.947173i \(0.603926\pi\)
\(570\) 0 0
\(571\) −22592.6 −1.65582 −0.827909 0.560862i \(-0.810470\pi\)
−0.827909 + 0.560862i \(0.810470\pi\)
\(572\) −1195.60 −0.0873964
\(573\) 0 0
\(574\) 0 0
\(575\) 1981.44 0.143707
\(576\) 0 0
\(577\) 11721.9 0.845734 0.422867 0.906192i \(-0.361024\pi\)
0.422867 + 0.906192i \(0.361024\pi\)
\(578\) −5799.04 −0.417315
\(579\) 0 0
\(580\) 14254.1 1.02046
\(581\) 0 0
\(582\) 0 0
\(583\) 7966.15 0.565908
\(584\) −9891.27 −0.700863
\(585\) 0 0
\(586\) 8227.25 0.579974
\(587\) 18102.6 1.27287 0.636433 0.771332i \(-0.280409\pi\)
0.636433 + 0.771332i \(0.280409\pi\)
\(588\) 0 0
\(589\) 312.498 0.0218612
\(590\) 1794.01 0.125183
\(591\) 0 0
\(592\) 10512.0 0.729800
\(593\) 538.675 0.0373031 0.0186516 0.999826i \(-0.494063\pi\)
0.0186516 + 0.999826i \(0.494063\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −13446.3 −0.924128
\(597\) 0 0
\(598\) 593.097 0.0405578
\(599\) 23628.2 1.61172 0.805861 0.592105i \(-0.201703\pi\)
0.805861 + 0.592105i \(0.201703\pi\)
\(600\) 0 0
\(601\) 3930.39 0.266762 0.133381 0.991065i \(-0.457417\pi\)
0.133381 + 0.991065i \(0.457417\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −10770.5 −0.725569
\(605\) −5439.29 −0.365518
\(606\) 0 0
\(607\) 9149.06 0.611778 0.305889 0.952067i \(-0.401046\pi\)
0.305889 + 0.952067i \(0.401046\pi\)
\(608\) −5339.63 −0.356169
\(609\) 0 0
\(610\) 238.719 0.0158450
\(611\) 1456.62 0.0964457
\(612\) 0 0
\(613\) −16844.1 −1.10983 −0.554917 0.831905i \(-0.687250\pi\)
−0.554917 + 0.831905i \(0.687250\pi\)
\(614\) 1680.99 0.110487
\(615\) 0 0
\(616\) 0 0
\(617\) −25080.5 −1.63647 −0.818235 0.574884i \(-0.805047\pi\)
−0.818235 + 0.574884i \(0.805047\pi\)
\(618\) 0 0
\(619\) 23686.3 1.53802 0.769009 0.639238i \(-0.220750\pi\)
0.769009 + 0.639238i \(0.220750\pi\)
\(620\) −668.439 −0.0432986
\(621\) 0 0
\(622\) 6413.46 0.413434
\(623\) 0 0
\(624\) 0 0
\(625\) −12869.3 −0.823638
\(626\) 5806.78 0.370744
\(627\) 0 0
\(628\) 25532.6 1.62239
\(629\) −25744.0 −1.63193
\(630\) 0 0
\(631\) 11732.0 0.740162 0.370081 0.929000i \(-0.379330\pi\)
0.370081 + 0.929000i \(0.379330\pi\)
\(632\) 8963.82 0.564180
\(633\) 0 0
\(634\) −3530.89 −0.221182
\(635\) −6114.20 −0.382102
\(636\) 0 0
\(637\) 0 0
\(638\) 5320.78 0.330175
\(639\) 0 0
\(640\) 14798.4 0.913996
\(641\) −9158.83 −0.564356 −0.282178 0.959362i \(-0.591057\pi\)
−0.282178 + 0.959362i \(0.591057\pi\)
\(642\) 0 0
\(643\) 29594.0 1.81504 0.907522 0.420003i \(-0.137971\pi\)
0.907522 + 0.420003i \(0.137971\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 3420.94 0.208352
\(647\) 4569.57 0.277663 0.138832 0.990316i \(-0.455665\pi\)
0.138832 + 0.990316i \(0.455665\pi\)
\(648\) 0 0
\(649\) −5145.28 −0.311201
\(650\) 109.432 0.00660350
\(651\) 0 0
\(652\) −6280.94 −0.377271
\(653\) 21264.7 1.27435 0.637176 0.770718i \(-0.280102\pi\)
0.637176 + 0.770718i \(0.280102\pi\)
\(654\) 0 0
\(655\) −5084.35 −0.303300
\(656\) 15661.4 0.932128
\(657\) 0 0
\(658\) 0 0
\(659\) 9214.64 0.544691 0.272345 0.962200i \(-0.412201\pi\)
0.272345 + 0.962200i \(0.412201\pi\)
\(660\) 0 0
\(661\) −7730.00 −0.454859 −0.227430 0.973794i \(-0.573032\pi\)
−0.227430 + 0.973794i \(0.573032\pi\)
\(662\) −10041.9 −0.589564
\(663\) 0 0
\(664\) −2554.16 −0.149278
\(665\) 0 0
\(666\) 0 0
\(667\) 20279.7 1.17726
\(668\) −8841.03 −0.512080
\(669\) 0 0
\(670\) 3326.33 0.191802
\(671\) −684.654 −0.0393901
\(672\) 0 0
\(673\) 14627.9 0.837835 0.418917 0.908024i \(-0.362410\pi\)
0.418917 + 0.908024i \(0.362410\pi\)
\(674\) −1029.12 −0.0588133
\(675\) 0 0
\(676\) 15300.2 0.870517
\(677\) 27179.8 1.54299 0.771496 0.636234i \(-0.219509\pi\)
0.771496 + 0.636234i \(0.219509\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −15587.3 −0.879038
\(681\) 0 0
\(682\) −249.516 −0.0140095
\(683\) 34448.5 1.92992 0.964959 0.262402i \(-0.0845147\pi\)
0.964959 + 0.262402i \(0.0845147\pi\)
\(684\) 0 0
\(685\) 18593.7 1.03712
\(686\) 0 0
\(687\) 0 0
\(688\) −15657.0 −0.867614
\(689\) 1676.86 0.0927191
\(690\) 0 0
\(691\) −13766.2 −0.757872 −0.378936 0.925423i \(-0.623710\pi\)
−0.378936 + 0.925423i \(0.623710\pi\)
\(692\) −15081.5 −0.828488
\(693\) 0 0
\(694\) −9178.46 −0.502031
\(695\) −32406.0 −1.76868
\(696\) 0 0
\(697\) −38355.0 −2.08436
\(698\) 3992.34 0.216493
\(699\) 0 0
\(700\) 0 0
\(701\) 28107.2 1.51440 0.757200 0.653183i \(-0.226567\pi\)
0.757200 + 0.653183i \(0.226567\pi\)
\(702\) 0 0
\(703\) 8375.72 0.449355
\(704\) −5421.34 −0.290234
\(705\) 0 0
\(706\) −5762.72 −0.307200
\(707\) 0 0
\(708\) 0 0
\(709\) 29866.8 1.58205 0.791023 0.611786i \(-0.209549\pi\)
0.791023 + 0.611786i \(0.209549\pi\)
\(710\) −1943.53 −0.102732
\(711\) 0 0
\(712\) −11014.8 −0.579772
\(713\) −951.007 −0.0499516
\(714\) 0 0
\(715\) 1737.96 0.0909037
\(716\) 19509.6 1.01831
\(717\) 0 0
\(718\) −11052.9 −0.574497
\(719\) −20815.8 −1.07969 −0.539845 0.841764i \(-0.681517\pi\)
−0.539845 + 0.841764i \(0.681517\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 5470.61 0.281988
\(723\) 0 0
\(724\) 5201.43 0.267002
\(725\) 3741.79 0.191678
\(726\) 0 0
\(727\) 2638.00 0.134578 0.0672888 0.997734i \(-0.478565\pi\)
0.0672888 + 0.997734i \(0.478565\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 6749.86 0.342224
\(731\) 38344.2 1.94010
\(732\) 0 0
\(733\) 19483.1 0.981754 0.490877 0.871229i \(-0.336677\pi\)
0.490877 + 0.871229i \(0.336677\pi\)
\(734\) −8285.33 −0.416645
\(735\) 0 0
\(736\) 16249.8 0.813825
\(737\) −9540.02 −0.476813
\(738\) 0 0
\(739\) −2190.08 −0.109017 −0.0545084 0.998513i \(-0.517359\pi\)
−0.0545084 + 0.998513i \(0.517359\pi\)
\(740\) −17915.8 −0.889999
\(741\) 0 0
\(742\) 0 0
\(743\) 21777.5 1.07529 0.537644 0.843172i \(-0.319315\pi\)
0.537644 + 0.843172i \(0.319315\pi\)
\(744\) 0 0
\(745\) 19545.8 0.961214
\(746\) 11121.3 0.545815
\(747\) 0 0
\(748\) 20986.7 1.02587
\(749\) 0 0
\(750\) 0 0
\(751\) −14790.8 −0.718672 −0.359336 0.933208i \(-0.616997\pi\)
−0.359336 + 0.933208i \(0.616997\pi\)
\(752\) 10440.2 0.506272
\(753\) 0 0
\(754\) 1120.02 0.0540964
\(755\) 15656.2 0.754687
\(756\) 0 0
\(757\) 30374.8 1.45838 0.729188 0.684313i \(-0.239898\pi\)
0.729188 + 0.684313i \(0.239898\pi\)
\(758\) −10.9378 −0.000524115 0
\(759\) 0 0
\(760\) 5071.27 0.242045
\(761\) −14854.4 −0.707584 −0.353792 0.935324i \(-0.615108\pi\)
−0.353792 + 0.935324i \(0.615108\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 14401.5 0.681975
\(765\) 0 0
\(766\) 7342.75 0.346350
\(767\) −1083.07 −0.0509877
\(768\) 0 0
\(769\) −3016.48 −0.141453 −0.0707264 0.997496i \(-0.522532\pi\)
−0.0707264 + 0.997496i \(0.522532\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4701.44 0.219182
\(773\) −5047.57 −0.234862 −0.117431 0.993081i \(-0.537466\pi\)
−0.117431 + 0.993081i \(0.537466\pi\)
\(774\) 0 0
\(775\) −175.470 −0.00813298
\(776\) −18490.9 −0.855395
\(777\) 0 0
\(778\) −978.008 −0.0450685
\(779\) 12478.6 0.573933
\(780\) 0 0
\(781\) 5574.11 0.255387
\(782\) −10410.8 −0.476072
\(783\) 0 0
\(784\) 0 0
\(785\) −37114.8 −1.68750
\(786\) 0 0
\(787\) −19962.3 −0.904166 −0.452083 0.891976i \(-0.649319\pi\)
−0.452083 + 0.891976i \(0.649319\pi\)
\(788\) 11909.2 0.538383
\(789\) 0 0
\(790\) −6116.96 −0.275483
\(791\) 0 0
\(792\) 0 0
\(793\) −144.119 −0.00645374
\(794\) 1164.95 0.0520686
\(795\) 0 0
\(796\) 26743.7 1.19084
\(797\) 806.334 0.0358367 0.0179183 0.999839i \(-0.494296\pi\)
0.0179183 + 0.999839i \(0.494296\pi\)
\(798\) 0 0
\(799\) −25568.3 −1.13209
\(800\) 2998.24 0.132505
\(801\) 0 0
\(802\) 6888.40 0.303289
\(803\) −19358.8 −0.850756
\(804\) 0 0
\(805\) 0 0
\(806\) −52.5227 −0.00229533
\(807\) 0 0
\(808\) 22517.8 0.980414
\(809\) −42013.3 −1.82584 −0.912922 0.408134i \(-0.866180\pi\)
−0.912922 + 0.408134i \(0.866180\pi\)
\(810\) 0 0
\(811\) 2938.48 0.127231 0.0636153 0.997974i \(-0.479737\pi\)
0.0636153 + 0.997974i \(0.479737\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −6687.64 −0.287963
\(815\) 9130.14 0.392411
\(816\) 0 0
\(817\) −12475.1 −0.534210
\(818\) 14291.0 0.610848
\(819\) 0 0
\(820\) −26692.1 −1.13674
\(821\) 24232.8 1.03012 0.515060 0.857154i \(-0.327770\pi\)
0.515060 + 0.857154i \(0.327770\pi\)
\(822\) 0 0
\(823\) −27959.8 −1.18422 −0.592112 0.805856i \(-0.701706\pi\)
−0.592112 + 0.805856i \(0.701706\pi\)
\(824\) 3564.97 0.150718
\(825\) 0 0
\(826\) 0 0
\(827\) −11855.3 −0.498487 −0.249244 0.968441i \(-0.580182\pi\)
−0.249244 + 0.968441i \(0.580182\pi\)
\(828\) 0 0
\(829\) −43362.5 −1.81670 −0.908349 0.418213i \(-0.862657\pi\)
−0.908349 + 0.418213i \(0.862657\pi\)
\(830\) 1742.97 0.0728910
\(831\) 0 0
\(832\) −1141.19 −0.0475523
\(833\) 0 0
\(834\) 0 0
\(835\) 12851.6 0.532630
\(836\) −6827.94 −0.282475
\(837\) 0 0
\(838\) 7203.62 0.296951
\(839\) 20467.7 0.842221 0.421111 0.907009i \(-0.361640\pi\)
0.421111 + 0.907009i \(0.361640\pi\)
\(840\) 0 0
\(841\) 13907.7 0.570243
\(842\) −8224.89 −0.336637
\(843\) 0 0
\(844\) 36263.5 1.47896
\(845\) −22240.8 −0.905452
\(846\) 0 0
\(847\) 0 0
\(848\) 12018.9 0.486710
\(849\) 0 0
\(850\) −1920.88 −0.0775126
\(851\) −25489.4 −1.02675
\(852\) 0 0
\(853\) −17910.2 −0.718914 −0.359457 0.933162i \(-0.617038\pi\)
−0.359457 + 0.933162i \(0.617038\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −23468.3 −0.937069
\(857\) 4494.31 0.179140 0.0895698 0.995981i \(-0.471451\pi\)
0.0895698 + 0.995981i \(0.471451\pi\)
\(858\) 0 0
\(859\) −14367.1 −0.570663 −0.285331 0.958429i \(-0.592104\pi\)
−0.285331 + 0.958429i \(0.592104\pi\)
\(860\) 26684.5 1.05806
\(861\) 0 0
\(862\) −1794.78 −0.0709169
\(863\) 12240.1 0.482802 0.241401 0.970425i \(-0.422393\pi\)
0.241401 + 0.970425i \(0.422393\pi\)
\(864\) 0 0
\(865\) 21922.9 0.861736
\(866\) 10752.3 0.421914
\(867\) 0 0
\(868\) 0 0
\(869\) 17543.6 0.684841
\(870\) 0 0
\(871\) −2008.16 −0.0781217
\(872\) 27596.0 1.07170
\(873\) 0 0
\(874\) 3387.10 0.131087
\(875\) 0 0
\(876\) 0 0
\(877\) −18127.6 −0.697978 −0.348989 0.937127i \(-0.613475\pi\)
−0.348989 + 0.937127i \(0.613475\pi\)
\(878\) 9421.57 0.362144
\(879\) 0 0
\(880\) 12456.8 0.477180
\(881\) −28737.3 −1.09896 −0.549480 0.835507i \(-0.685174\pi\)
−0.549480 + 0.835507i \(0.685174\pi\)
\(882\) 0 0
\(883\) 29899.7 1.13953 0.569765 0.821807i \(-0.307034\pi\)
0.569765 + 0.821807i \(0.307034\pi\)
\(884\) 4417.67 0.168080
\(885\) 0 0
\(886\) −5573.84 −0.211351
\(887\) 35540.7 1.34537 0.672683 0.739931i \(-0.265142\pi\)
0.672683 + 0.739931i \(0.265142\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 7516.57 0.283097
\(891\) 0 0
\(892\) 14997.7 0.562959
\(893\) 8318.54 0.311724
\(894\) 0 0
\(895\) −28359.7 −1.05917
\(896\) 0 0
\(897\) 0 0
\(898\) 247.034 0.00917998
\(899\) −1795.90 −0.0666260
\(900\) 0 0
\(901\) −29434.4 −1.08835
\(902\) −9963.64 −0.367797
\(903\) 0 0
\(904\) −27305.9 −1.00462
\(905\) −7560.94 −0.277717
\(906\) 0 0
\(907\) −16816.0 −0.615617 −0.307808 0.951448i \(-0.599596\pi\)
−0.307808 + 0.951448i \(0.599596\pi\)
\(908\) 30010.6 1.09685
\(909\) 0 0
\(910\) 0 0
\(911\) −21626.8 −0.786527 −0.393264 0.919426i \(-0.628654\pi\)
−0.393264 + 0.919426i \(0.628654\pi\)
\(912\) 0 0
\(913\) −4998.91 −0.181204
\(914\) −1390.57 −0.0503239
\(915\) 0 0
\(916\) −10746.8 −0.387645
\(917\) 0 0
\(918\) 0 0
\(919\) 31512.2 1.13111 0.565556 0.824710i \(-0.308662\pi\)
0.565556 + 0.824710i \(0.308662\pi\)
\(920\) −15433.1 −0.553059
\(921\) 0 0
\(922\) 8363.33 0.298733
\(923\) 1173.34 0.0418429
\(924\) 0 0
\(925\) −4703.03 −0.167173
\(926\) −17052.8 −0.605174
\(927\) 0 0
\(928\) 30686.5 1.08549
\(929\) −25267.9 −0.892373 −0.446186 0.894940i \(-0.647218\pi\)
−0.446186 + 0.894940i \(0.647218\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 32536.3 1.14352
\(933\) 0 0
\(934\) −13183.5 −0.461861
\(935\) −30506.8 −1.06704
\(936\) 0 0
\(937\) −14694.4 −0.512321 −0.256161 0.966634i \(-0.582458\pi\)
−0.256161 + 0.966634i \(0.582458\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −17793.5 −0.617405
\(941\) −7004.20 −0.242647 −0.121323 0.992613i \(-0.538714\pi\)
−0.121323 + 0.992613i \(0.538714\pi\)
\(942\) 0 0
\(943\) −37975.6 −1.31140
\(944\) −7762.91 −0.267649
\(945\) 0 0
\(946\) 9960.84 0.342341
\(947\) 10065.2 0.345379 0.172690 0.984976i \(-0.444754\pi\)
0.172690 + 0.984976i \(0.444754\pi\)
\(948\) 0 0
\(949\) −4075.00 −0.139389
\(950\) 624.951 0.0213433
\(951\) 0 0
\(952\) 0 0
\(953\) −15265.2 −0.518876 −0.259438 0.965760i \(-0.583537\pi\)
−0.259438 + 0.965760i \(0.583537\pi\)
\(954\) 0 0
\(955\) −20934.5 −0.709343
\(956\) −20930.5 −0.708096
\(957\) 0 0
\(958\) 4628.03 0.156080
\(959\) 0 0
\(960\) 0 0
\(961\) −29706.8 −0.997173
\(962\) −1407.74 −0.0471802
\(963\) 0 0
\(964\) 799.940 0.0267265
\(965\) −6834.14 −0.227978
\(966\) 0 0
\(967\) −32705.8 −1.08764 −0.543820 0.839202i \(-0.683022\pi\)
−0.543820 + 0.839202i \(0.683022\pi\)
\(968\) −7650.73 −0.254033
\(969\) 0 0
\(970\) 12618.3 0.417680
\(971\) 48338.6 1.59759 0.798795 0.601603i \(-0.205471\pi\)
0.798795 + 0.601603i \(0.205471\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 13056.8 0.429535
\(975\) 0 0
\(976\) −1032.97 −0.0338776
\(977\) −21230.1 −0.695200 −0.347600 0.937643i \(-0.613003\pi\)
−0.347600 + 0.937643i \(0.613003\pi\)
\(978\) 0 0
\(979\) −21557.8 −0.703768
\(980\) 0 0
\(981\) 0 0
\(982\) −5890.99 −0.191435
\(983\) −54979.2 −1.78389 −0.891946 0.452142i \(-0.850660\pi\)
−0.891946 + 0.452142i \(0.850660\pi\)
\(984\) 0 0
\(985\) −17311.5 −0.559989
\(986\) −19659.9 −0.634989
\(987\) 0 0
\(988\) −1437.27 −0.0462811
\(989\) 37964.9 1.22064
\(990\) 0 0
\(991\) 12954.3 0.415245 0.207622 0.978209i \(-0.433427\pi\)
0.207622 + 0.978209i \(0.433427\pi\)
\(992\) −1439.03 −0.0460577
\(993\) 0 0
\(994\) 0 0
\(995\) −38875.4 −1.23863
\(996\) 0 0
\(997\) −13382.0 −0.425088 −0.212544 0.977151i \(-0.568175\pi\)
−0.212544 + 0.977151i \(0.568175\pi\)
\(998\) 13125.0 0.416299
\(999\) 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 1323.4.a.bl.1.4 8
3.2 odd 2 inner 1323.4.a.bl.1.5 yes 8
7.6 odd 2 1323.4.a.bm.1.4 yes 8
21.20 even 2 1323.4.a.bm.1.5 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1323.4.a.bl.1.4 8 1.1 even 1 trivial
1323.4.a.bl.1.5 yes 8 3.2 odd 2 inner
1323.4.a.bm.1.4 yes 8 7.6 odd 2
1323.4.a.bm.1.5 yes 8 21.20 even 2