Properties

Label 560.4.a.r.1.2
Level $560$
Weight $4$
Character 560.1
Self dual yes
Analytic conductor $33.041$
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] = [560,4,Mod(1,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("560.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 560.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: \(33.0410696032\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 560.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.65685 q^{3} -5.00000 q^{5} +7.00000 q^{7} -5.31371 q^{9} +O(q^{10})\) \(q+4.65685 q^{3} -5.00000 q^{5} +7.00000 q^{7} -5.31371 q^{9} +52.2548 q^{11} +30.6569 q^{13} -23.2843 q^{15} +37.2254 q^{17} -80.2254 q^{19} +32.5980 q^{21} -25.8335 q^{23} +25.0000 q^{25} -150.480 q^{27} +20.9411 q^{29} +314.558 q^{31} +243.343 q^{33} -35.0000 q^{35} +197.147 q^{37} +142.765 q^{39} +11.3625 q^{41} +33.8335 q^{43} +26.5685 q^{45} +361.676 q^{47} +49.0000 q^{49} +173.353 q^{51} +153.019 q^{53} -261.274 q^{55} -373.598 q^{57} +616.000 q^{59} +15.2649 q^{61} -37.1960 q^{63} -153.284 q^{65} +166.510 q^{67} -120.303 q^{69} +952.000 q^{71} -148.489 q^{73} +116.421 q^{75} +365.784 q^{77} -857.725 q^{79} -557.294 q^{81} -660.528 q^{83} -186.127 q^{85} +97.5198 q^{87} -45.7746 q^{89} +214.598 q^{91} +1464.85 q^{93} +401.127 q^{95} +1682.13 q^{97} -277.667 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 10 q^{5} + 14 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 10 q^{5} + 14 q^{7} + 12 q^{9} + 14 q^{11} + 50 q^{13} + 10 q^{15} - 50 q^{17} - 36 q^{19} - 14 q^{21} - 244 q^{23} + 50 q^{25} - 86 q^{27} - 26 q^{29} + 120 q^{31} + 498 q^{33} - 70 q^{35} + 564 q^{37} + 14 q^{39} - 328 q^{41} + 260 q^{43} - 60 q^{45} + 350 q^{47} + 98 q^{49} + 754 q^{51} - 56 q^{53} - 70 q^{55} - 668 q^{57} + 1232 q^{59} + 336 q^{61} + 84 q^{63} - 250 q^{65} + 152 q^{67} + 1332 q^{69} + 1904 q^{71} + 676 q^{73} - 50 q^{75} + 98 q^{77} - 1014 q^{79} - 1454 q^{81} + 376 q^{83} + 250 q^{85} + 410 q^{87} - 216 q^{89} + 350 q^{91} + 2760 q^{93} + 180 q^{95} + 2742 q^{97} - 940 q^{99}+O(q^{100}) \) 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 0
\(3\) 4.65685 0.896212 0.448106 0.893980i \(-0.352099\pi\)
0.448106 + 0.893980i \(0.352099\pi\)
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 0 0
\(9\) −5.31371 −0.196804
\(10\) 0 0
\(11\) 52.2548 1.43231 0.716156 0.697941i \(-0.245900\pi\)
0.716156 + 0.697941i \(0.245900\pi\)
\(12\) 0 0
\(13\) 30.6569 0.654052 0.327026 0.945015i \(-0.393953\pi\)
0.327026 + 0.945015i \(0.393953\pi\)
\(14\) 0 0
\(15\) −23.2843 −0.400798
\(16\) 0 0
\(17\) 37.2254 0.531087 0.265544 0.964099i \(-0.414449\pi\)
0.265544 + 0.964099i \(0.414449\pi\)
\(18\) 0 0
\(19\) −80.2254 −0.968683 −0.484341 0.874879i \(-0.660941\pi\)
−0.484341 + 0.874879i \(0.660941\pi\)
\(20\) 0 0
\(21\) 32.5980 0.338736
\(22\) 0 0
\(23\) −25.8335 −0.234202 −0.117101 0.993120i \(-0.537360\pi\)
−0.117101 + 0.993120i \(0.537360\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −150.480 −1.07259
\(28\) 0 0
\(29\) 20.9411 0.134092 0.0670460 0.997750i \(-0.478643\pi\)
0.0670460 + 0.997750i \(0.478643\pi\)
\(30\) 0 0
\(31\) 314.558 1.82246 0.911232 0.411894i \(-0.135133\pi\)
0.911232 + 0.411894i \(0.135133\pi\)
\(32\) 0 0
\(33\) 243.343 1.28365
\(34\) 0 0
\(35\) −35.0000 −0.169031
\(36\) 0 0
\(37\) 197.147 0.875968 0.437984 0.898983i \(-0.355693\pi\)
0.437984 + 0.898983i \(0.355693\pi\)
\(38\) 0 0
\(39\) 142.765 0.586170
\(40\) 0 0
\(41\) 11.3625 0.0432810 0.0216405 0.999766i \(-0.493111\pi\)
0.0216405 + 0.999766i \(0.493111\pi\)
\(42\) 0 0
\(43\) 33.8335 0.119990 0.0599948 0.998199i \(-0.480892\pi\)
0.0599948 + 0.998199i \(0.480892\pi\)
\(44\) 0 0
\(45\) 26.5685 0.0880134
\(46\) 0 0
\(47\) 361.676 1.12247 0.561233 0.827658i \(-0.310327\pi\)
0.561233 + 0.827658i \(0.310327\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 173.353 0.475967
\(52\) 0 0
\(53\) 153.019 0.396582 0.198291 0.980143i \(-0.436461\pi\)
0.198291 + 0.980143i \(0.436461\pi\)
\(54\) 0 0
\(55\) −261.274 −0.640549
\(56\) 0 0
\(57\) −373.598 −0.868145
\(58\) 0 0
\(59\) 616.000 1.35926 0.679630 0.733555i \(-0.262140\pi\)
0.679630 + 0.733555i \(0.262140\pi\)
\(60\) 0 0
\(61\) 15.2649 0.0320406 0.0160203 0.999872i \(-0.494900\pi\)
0.0160203 + 0.999872i \(0.494900\pi\)
\(62\) 0 0
\(63\) −37.1960 −0.0743849
\(64\) 0 0
\(65\) −153.284 −0.292501
\(66\) 0 0
\(67\) 166.510 0.303618 0.151809 0.988410i \(-0.451490\pi\)
0.151809 + 0.988410i \(0.451490\pi\)
\(68\) 0 0
\(69\) −120.303 −0.209895
\(70\) 0 0
\(71\) 952.000 1.59129 0.795645 0.605763i \(-0.207132\pi\)
0.795645 + 0.605763i \(0.207132\pi\)
\(72\) 0 0
\(73\) −148.489 −0.238074 −0.119037 0.992890i \(-0.537981\pi\)
−0.119037 + 0.992890i \(0.537981\pi\)
\(74\) 0 0
\(75\) 116.421 0.179242
\(76\) 0 0
\(77\) 365.784 0.541363
\(78\) 0 0
\(79\) −857.725 −1.22154 −0.610770 0.791808i \(-0.709140\pi\)
−0.610770 + 0.791808i \(0.709140\pi\)
\(80\) 0 0
\(81\) −557.294 −0.764464
\(82\) 0 0
\(83\) −660.528 −0.873523 −0.436761 0.899577i \(-0.643875\pi\)
−0.436761 + 0.899577i \(0.643875\pi\)
\(84\) 0 0
\(85\) −186.127 −0.237509
\(86\) 0 0
\(87\) 97.5198 0.120175
\(88\) 0 0
\(89\) −45.7746 −0.0545180 −0.0272590 0.999628i \(-0.508678\pi\)
−0.0272590 + 0.999628i \(0.508678\pi\)
\(90\) 0 0
\(91\) 214.598 0.247209
\(92\) 0 0
\(93\) 1464.85 1.63331
\(94\) 0 0
\(95\) 401.127 0.433208
\(96\) 0 0
\(97\) 1682.13 1.76076 0.880382 0.474265i \(-0.157286\pi\)
0.880382 + 0.474265i \(0.157286\pi\)
\(98\) 0 0
\(99\) −277.667 −0.281885
\(100\) 0 0
\(101\) −434.167 −0.427734 −0.213867 0.976863i \(-0.568606\pi\)
−0.213867 + 0.976863i \(0.568606\pi\)
\(102\) 0 0
\(103\) −345.577 −0.330589 −0.165295 0.986244i \(-0.552858\pi\)
−0.165295 + 0.986244i \(0.552858\pi\)
\(104\) 0 0
\(105\) −162.990 −0.151487
\(106\) 0 0
\(107\) −217.119 −0.196165 −0.0980825 0.995178i \(-0.531271\pi\)
−0.0980825 + 0.995178i \(0.531271\pi\)
\(108\) 0 0
\(109\) 1734.41 1.52409 0.762047 0.647521i \(-0.224194\pi\)
0.762047 + 0.647521i \(0.224194\pi\)
\(110\) 0 0
\(111\) 918.086 0.785053
\(112\) 0 0
\(113\) −1854.20 −1.54362 −0.771809 0.635855i \(-0.780648\pi\)
−0.771809 + 0.635855i \(0.780648\pi\)
\(114\) 0 0
\(115\) 129.167 0.104738
\(116\) 0 0
\(117\) −162.902 −0.128720
\(118\) 0 0
\(119\) 260.578 0.200732
\(120\) 0 0
\(121\) 1399.57 1.05152
\(122\) 0 0
\(123\) 52.9134 0.0387890
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −1394.51 −0.974352 −0.487176 0.873304i \(-0.661973\pi\)
−0.487176 + 0.873304i \(0.661973\pi\)
\(128\) 0 0
\(129\) 157.558 0.107536
\(130\) 0 0
\(131\) −1762.42 −1.17544 −0.587722 0.809063i \(-0.699975\pi\)
−0.587722 + 0.809063i \(0.699975\pi\)
\(132\) 0 0
\(133\) −561.578 −0.366128
\(134\) 0 0
\(135\) 752.401 0.479677
\(136\) 0 0
\(137\) −922.949 −0.575568 −0.287784 0.957695i \(-0.592919\pi\)
−0.287784 + 0.957695i \(0.592919\pi\)
\(138\) 0 0
\(139\) 196.039 0.119624 0.0598122 0.998210i \(-0.480950\pi\)
0.0598122 + 0.998210i \(0.480950\pi\)
\(140\) 0 0
\(141\) 1684.27 1.00597
\(142\) 0 0
\(143\) 1601.97 0.936807
\(144\) 0 0
\(145\) −104.706 −0.0599678
\(146\) 0 0
\(147\) 228.186 0.128030
\(148\) 0 0
\(149\) 780.372 0.429064 0.214532 0.976717i \(-0.431177\pi\)
0.214532 + 0.976717i \(0.431177\pi\)
\(150\) 0 0
\(151\) 2319.43 1.25002 0.625008 0.780618i \(-0.285096\pi\)
0.625008 + 0.780618i \(0.285096\pi\)
\(152\) 0 0
\(153\) −197.805 −0.104520
\(154\) 0 0
\(155\) −1572.79 −0.815030
\(156\) 0 0
\(157\) 1022.90 0.519977 0.259989 0.965612i \(-0.416281\pi\)
0.259989 + 0.965612i \(0.416281\pi\)
\(158\) 0 0
\(159\) 712.589 0.355421
\(160\) 0 0
\(161\) −180.834 −0.0885201
\(162\) 0 0
\(163\) 1350.63 0.649013 0.324507 0.945883i \(-0.394802\pi\)
0.324507 + 0.945883i \(0.394802\pi\)
\(164\) 0 0
\(165\) −1216.72 −0.574068
\(166\) 0 0
\(167\) 1230.58 0.570209 0.285105 0.958496i \(-0.407972\pi\)
0.285105 + 0.958496i \(0.407972\pi\)
\(168\) 0 0
\(169\) −1257.16 −0.572215
\(170\) 0 0
\(171\) 426.294 0.190641
\(172\) 0 0
\(173\) −2487.65 −1.09325 −0.546626 0.837377i \(-0.684088\pi\)
−0.546626 + 0.837377i \(0.684088\pi\)
\(174\) 0 0
\(175\) 175.000 0.0755929
\(176\) 0 0
\(177\) 2868.62 1.21819
\(178\) 0 0
\(179\) −1621.18 −0.676941 −0.338471 0.940977i \(-0.609910\pi\)
−0.338471 + 0.940977i \(0.609910\pi\)
\(180\) 0 0
\(181\) 2593.69 1.06512 0.532561 0.846392i \(-0.321230\pi\)
0.532561 + 0.846392i \(0.321230\pi\)
\(182\) 0 0
\(183\) 71.0866 0.0287151
\(184\) 0 0
\(185\) −985.736 −0.391745
\(186\) 0 0
\(187\) 1945.21 0.760682
\(188\) 0 0
\(189\) −1053.36 −0.405401
\(190\) 0 0
\(191\) 1823.08 0.690645 0.345323 0.938484i \(-0.387769\pi\)
0.345323 + 0.938484i \(0.387769\pi\)
\(192\) 0 0
\(193\) −1541.03 −0.574744 −0.287372 0.957819i \(-0.592782\pi\)
−0.287372 + 0.957819i \(0.592782\pi\)
\(194\) 0 0
\(195\) −713.823 −0.262143
\(196\) 0 0
\(197\) 701.243 0.253612 0.126806 0.991928i \(-0.459527\pi\)
0.126806 + 0.991928i \(0.459527\pi\)
\(198\) 0 0
\(199\) −3294.96 −1.17374 −0.586868 0.809682i \(-0.699639\pi\)
−0.586868 + 0.809682i \(0.699639\pi\)
\(200\) 0 0
\(201\) 775.411 0.272106
\(202\) 0 0
\(203\) 146.588 0.0506820
\(204\) 0 0
\(205\) −56.8124 −0.0193559
\(206\) 0 0
\(207\) 137.272 0.0460920
\(208\) 0 0
\(209\) −4192.16 −1.38746
\(210\) 0 0
\(211\) −4082.35 −1.33195 −0.665974 0.745975i \(-0.731984\pi\)
−0.665974 + 0.745975i \(0.731984\pi\)
\(212\) 0 0
\(213\) 4433.33 1.42613
\(214\) 0 0
\(215\) −169.167 −0.0536610
\(216\) 0 0
\(217\) 2201.91 0.688826
\(218\) 0 0
\(219\) −691.494 −0.213364
\(220\) 0 0
\(221\) 1141.21 0.347359
\(222\) 0 0
\(223\) −747.161 −0.224366 −0.112183 0.993688i \(-0.535784\pi\)
−0.112183 + 0.993688i \(0.535784\pi\)
\(224\) 0 0
\(225\) −132.843 −0.0393608
\(226\) 0 0
\(227\) −1665.67 −0.487025 −0.243513 0.969898i \(-0.578300\pi\)
−0.243513 + 0.969898i \(0.578300\pi\)
\(228\) 0 0
\(229\) −6628.35 −1.91272 −0.956362 0.292183i \(-0.905618\pi\)
−0.956362 + 0.292183i \(0.905618\pi\)
\(230\) 0 0
\(231\) 1703.40 0.485176
\(232\) 0 0
\(233\) −432.431 −0.121586 −0.0607929 0.998150i \(-0.519363\pi\)
−0.0607929 + 0.998150i \(0.519363\pi\)
\(234\) 0 0
\(235\) −1808.38 −0.501982
\(236\) 0 0
\(237\) −3994.30 −1.09476
\(238\) 0 0
\(239\) −5580.44 −1.51033 −0.755165 0.655535i \(-0.772443\pi\)
−0.755165 + 0.655535i \(0.772443\pi\)
\(240\) 0 0
\(241\) −6296.87 −1.68306 −0.841529 0.540212i \(-0.818344\pi\)
−0.841529 + 0.540212i \(0.818344\pi\)
\(242\) 0 0
\(243\) 1467.73 0.387468
\(244\) 0 0
\(245\) −245.000 −0.0638877
\(246\) 0 0
\(247\) −2459.46 −0.633569
\(248\) 0 0
\(249\) −3075.98 −0.782862
\(250\) 0 0
\(251\) −311.921 −0.0784393 −0.0392197 0.999231i \(-0.512487\pi\)
−0.0392197 + 0.999231i \(0.512487\pi\)
\(252\) 0 0
\(253\) −1349.92 −0.335451
\(254\) 0 0
\(255\) −866.766 −0.212859
\(256\) 0 0
\(257\) −7861.39 −1.90809 −0.954046 0.299659i \(-0.903127\pi\)
−0.954046 + 0.299659i \(0.903127\pi\)
\(258\) 0 0
\(259\) 1380.03 0.331085
\(260\) 0 0
\(261\) −111.275 −0.0263899
\(262\) 0 0
\(263\) −5227.09 −1.22554 −0.612769 0.790262i \(-0.709944\pi\)
−0.612769 + 0.790262i \(0.709944\pi\)
\(264\) 0 0
\(265\) −765.097 −0.177357
\(266\) 0 0
\(267\) −213.166 −0.0488596
\(268\) 0 0
\(269\) 1281.71 0.290510 0.145255 0.989394i \(-0.453600\pi\)
0.145255 + 0.989394i \(0.453600\pi\)
\(270\) 0 0
\(271\) −4704.14 −1.05445 −0.527226 0.849725i \(-0.676768\pi\)
−0.527226 + 0.849725i \(0.676768\pi\)
\(272\) 0 0
\(273\) 999.352 0.221551
\(274\) 0 0
\(275\) 1306.37 0.286462
\(276\) 0 0
\(277\) 8958.56 1.94321 0.971603 0.236619i \(-0.0760393\pi\)
0.971603 + 0.236619i \(0.0760393\pi\)
\(278\) 0 0
\(279\) −1671.47 −0.358668
\(280\) 0 0
\(281\) −370.904 −0.0787412 −0.0393706 0.999225i \(-0.512535\pi\)
−0.0393706 + 0.999225i \(0.512535\pi\)
\(282\) 0 0
\(283\) 5822.26 1.22296 0.611479 0.791261i \(-0.290575\pi\)
0.611479 + 0.791261i \(0.290575\pi\)
\(284\) 0 0
\(285\) 1867.99 0.388246
\(286\) 0 0
\(287\) 79.5374 0.0163587
\(288\) 0 0
\(289\) −3527.27 −0.717946
\(290\) 0 0
\(291\) 7833.42 1.57802
\(292\) 0 0
\(293\) 7443.79 1.48420 0.742100 0.670289i \(-0.233830\pi\)
0.742100 + 0.670289i \(0.233830\pi\)
\(294\) 0 0
\(295\) −3080.00 −0.607880
\(296\) 0 0
\(297\) −7863.32 −1.53628
\(298\) 0 0
\(299\) −791.973 −0.153181
\(300\) 0 0
\(301\) 236.834 0.0453518
\(302\) 0 0
\(303\) −2021.85 −0.383341
\(304\) 0 0
\(305\) −76.3247 −0.0143290
\(306\) 0 0
\(307\) 761.674 0.141600 0.0707998 0.997491i \(-0.477445\pi\)
0.0707998 + 0.997491i \(0.477445\pi\)
\(308\) 0 0
\(309\) −1609.30 −0.296278
\(310\) 0 0
\(311\) −7718.69 −1.40735 −0.703677 0.710520i \(-0.748460\pi\)
−0.703677 + 0.710520i \(0.748460\pi\)
\(312\) 0 0
\(313\) 8556.00 1.54509 0.772546 0.634959i \(-0.218983\pi\)
0.772546 + 0.634959i \(0.218983\pi\)
\(314\) 0 0
\(315\) 185.980 0.0332660
\(316\) 0 0
\(317\) −7780.95 −1.37862 −0.689309 0.724468i \(-0.742086\pi\)
−0.689309 + 0.724468i \(0.742086\pi\)
\(318\) 0 0
\(319\) 1094.28 0.192062
\(320\) 0 0
\(321\) −1011.09 −0.175805
\(322\) 0 0
\(323\) −2986.42 −0.514455
\(324\) 0 0
\(325\) 766.421 0.130810
\(326\) 0 0
\(327\) 8076.89 1.36591
\(328\) 0 0
\(329\) 2531.73 0.424252
\(330\) 0 0
\(331\) 4932.12 0.819015 0.409507 0.912307i \(-0.365701\pi\)
0.409507 + 0.912307i \(0.365701\pi\)
\(332\) 0 0
\(333\) −1047.58 −0.172394
\(334\) 0 0
\(335\) −832.548 −0.135782
\(336\) 0 0
\(337\) −7121.13 −1.15108 −0.575538 0.817775i \(-0.695207\pi\)
−0.575538 + 0.817775i \(0.695207\pi\)
\(338\) 0 0
\(339\) −8634.76 −1.38341
\(340\) 0 0
\(341\) 16437.2 2.61034
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 0 0
\(345\) 601.514 0.0938679
\(346\) 0 0
\(347\) −9540.58 −1.47598 −0.737991 0.674811i \(-0.764225\pi\)
−0.737991 + 0.674811i \(0.764225\pi\)
\(348\) 0 0
\(349\) 1281.65 0.196576 0.0982880 0.995158i \(-0.468663\pi\)
0.0982880 + 0.995158i \(0.468663\pi\)
\(350\) 0 0
\(351\) −4613.25 −0.701530
\(352\) 0 0
\(353\) 5798.07 0.874221 0.437110 0.899408i \(-0.356002\pi\)
0.437110 + 0.899408i \(0.356002\pi\)
\(354\) 0 0
\(355\) −4760.00 −0.711647
\(356\) 0 0
\(357\) 1213.47 0.179899
\(358\) 0 0
\(359\) −2267.29 −0.333323 −0.166662 0.986014i \(-0.553299\pi\)
−0.166662 + 0.986014i \(0.553299\pi\)
\(360\) 0 0
\(361\) −422.886 −0.0616541
\(362\) 0 0
\(363\) 6517.58 0.942381
\(364\) 0 0
\(365\) 742.447 0.106470
\(366\) 0 0
\(367\) 7372.85 1.04866 0.524332 0.851514i \(-0.324315\pi\)
0.524332 + 0.851514i \(0.324315\pi\)
\(368\) 0 0
\(369\) −60.3769 −0.00851788
\(370\) 0 0
\(371\) 1071.14 0.149894
\(372\) 0 0
\(373\) 6447.14 0.894961 0.447480 0.894294i \(-0.352321\pi\)
0.447480 + 0.894294i \(0.352321\pi\)
\(374\) 0 0
\(375\) −582.107 −0.0801596
\(376\) 0 0
\(377\) 641.989 0.0877032
\(378\) 0 0
\(379\) 4247.57 0.575680 0.287840 0.957678i \(-0.407063\pi\)
0.287840 + 0.957678i \(0.407063\pi\)
\(380\) 0 0
\(381\) −6494.03 −0.873226
\(382\) 0 0
\(383\) 6681.86 0.891454 0.445727 0.895169i \(-0.352945\pi\)
0.445727 + 0.895169i \(0.352945\pi\)
\(384\) 0 0
\(385\) −1828.92 −0.242105
\(386\) 0 0
\(387\) −179.781 −0.0236145
\(388\) 0 0
\(389\) −6371.78 −0.830494 −0.415247 0.909709i \(-0.636305\pi\)
−0.415247 + 0.909709i \(0.636305\pi\)
\(390\) 0 0
\(391\) −961.661 −0.124382
\(392\) 0 0
\(393\) −8207.33 −1.05345
\(394\) 0 0
\(395\) 4288.62 0.546289
\(396\) 0 0
\(397\) 4247.93 0.537021 0.268510 0.963277i \(-0.413469\pi\)
0.268510 + 0.963277i \(0.413469\pi\)
\(398\) 0 0
\(399\) −2615.19 −0.328128
\(400\) 0 0
\(401\) −8833.62 −1.10008 −0.550038 0.835140i \(-0.685387\pi\)
−0.550038 + 0.835140i \(0.685387\pi\)
\(402\) 0 0
\(403\) 9643.37 1.19199
\(404\) 0 0
\(405\) 2786.47 0.341879
\(406\) 0 0
\(407\) 10301.9 1.25466
\(408\) 0 0
\(409\) −319.205 −0.0385908 −0.0192954 0.999814i \(-0.506142\pi\)
−0.0192954 + 0.999814i \(0.506142\pi\)
\(410\) 0 0
\(411\) −4298.04 −0.515831
\(412\) 0 0
\(413\) 4312.00 0.513752
\(414\) 0 0
\(415\) 3302.64 0.390651
\(416\) 0 0
\(417\) 912.924 0.107209
\(418\) 0 0
\(419\) 12789.2 1.49115 0.745577 0.666420i \(-0.232174\pi\)
0.745577 + 0.666420i \(0.232174\pi\)
\(420\) 0 0
\(421\) −6747.40 −0.781112 −0.390556 0.920579i \(-0.627717\pi\)
−0.390556 + 0.920579i \(0.627717\pi\)
\(422\) 0 0
\(423\) −1921.84 −0.220906
\(424\) 0 0
\(425\) 930.635 0.106217
\(426\) 0 0
\(427\) 106.855 0.0121102
\(428\) 0 0
\(429\) 7460.14 0.839577
\(430\) 0 0
\(431\) 5184.75 0.579444 0.289722 0.957111i \(-0.406437\pi\)
0.289722 + 0.957111i \(0.406437\pi\)
\(432\) 0 0
\(433\) −4242.03 −0.470806 −0.235403 0.971898i \(-0.575641\pi\)
−0.235403 + 0.971898i \(0.575641\pi\)
\(434\) 0 0
\(435\) −487.599 −0.0537439
\(436\) 0 0
\(437\) 2072.50 0.226868
\(438\) 0 0
\(439\) 5434.12 0.590789 0.295394 0.955375i \(-0.404549\pi\)
0.295394 + 0.955375i \(0.404549\pi\)
\(440\) 0 0
\(441\) −260.372 −0.0281149
\(442\) 0 0
\(443\) 11493.8 1.23270 0.616350 0.787472i \(-0.288611\pi\)
0.616350 + 0.787472i \(0.288611\pi\)
\(444\) 0 0
\(445\) 228.873 0.0243812
\(446\) 0 0
\(447\) 3634.08 0.384532
\(448\) 0 0
\(449\) −16849.3 −1.77098 −0.885489 0.464661i \(-0.846176\pi\)
−0.885489 + 0.464661i \(0.846176\pi\)
\(450\) 0 0
\(451\) 593.745 0.0619919
\(452\) 0 0
\(453\) 10801.2 1.12028
\(454\) 0 0
\(455\) −1072.99 −0.110555
\(456\) 0 0
\(457\) 15348.5 1.57106 0.785528 0.618826i \(-0.212391\pi\)
0.785528 + 0.618826i \(0.212391\pi\)
\(458\) 0 0
\(459\) −5601.69 −0.569639
\(460\) 0 0
\(461\) 14038.4 1.41830 0.709148 0.705059i \(-0.249080\pi\)
0.709148 + 0.705059i \(0.249080\pi\)
\(462\) 0 0
\(463\) 8661.23 0.869377 0.434689 0.900581i \(-0.356858\pi\)
0.434689 + 0.900581i \(0.356858\pi\)
\(464\) 0 0
\(465\) −7324.26 −0.730440
\(466\) 0 0
\(467\) −7014.71 −0.695079 −0.347539 0.937665i \(-0.612983\pi\)
−0.347539 + 0.937665i \(0.612983\pi\)
\(468\) 0 0
\(469\) 1165.57 0.114757
\(470\) 0 0
\(471\) 4763.50 0.466010
\(472\) 0 0
\(473\) 1767.96 0.171863
\(474\) 0 0
\(475\) −2005.63 −0.193737
\(476\) 0 0
\(477\) −813.100 −0.0780488
\(478\) 0 0
\(479\) −18134.7 −1.72984 −0.864922 0.501907i \(-0.832632\pi\)
−0.864922 + 0.501907i \(0.832632\pi\)
\(480\) 0 0
\(481\) 6043.91 0.572929
\(482\) 0 0
\(483\) −842.119 −0.0793328
\(484\) 0 0
\(485\) −8410.63 −0.787438
\(486\) 0 0
\(487\) −16537.8 −1.53881 −0.769405 0.638761i \(-0.779447\pi\)
−0.769405 + 0.638761i \(0.779447\pi\)
\(488\) 0 0
\(489\) 6289.67 0.581654
\(490\) 0 0
\(491\) −220.608 −0.0202768 −0.0101384 0.999949i \(-0.503227\pi\)
−0.0101384 + 0.999949i \(0.503227\pi\)
\(492\) 0 0
\(493\) 779.542 0.0712146
\(494\) 0 0
\(495\) 1388.33 0.126063
\(496\) 0 0
\(497\) 6664.00 0.601451
\(498\) 0 0
\(499\) −5939.04 −0.532801 −0.266401 0.963862i \(-0.585834\pi\)
−0.266401 + 0.963862i \(0.585834\pi\)
\(500\) 0 0
\(501\) 5730.62 0.511029
\(502\) 0 0
\(503\) 11604.8 1.02869 0.514345 0.857584i \(-0.328035\pi\)
0.514345 + 0.857584i \(0.328035\pi\)
\(504\) 0 0
\(505\) 2170.83 0.191289
\(506\) 0 0
\(507\) −5854.40 −0.512826
\(508\) 0 0
\(509\) −1867.67 −0.162639 −0.0813193 0.996688i \(-0.525913\pi\)
−0.0813193 + 0.996688i \(0.525913\pi\)
\(510\) 0 0
\(511\) −1039.43 −0.0899834
\(512\) 0 0
\(513\) 12072.3 1.03900
\(514\) 0 0
\(515\) 1727.88 0.147844
\(516\) 0 0
\(517\) 18899.3 1.60772
\(518\) 0 0
\(519\) −11584.6 −0.979786
\(520\) 0 0
\(521\) 6117.21 0.514395 0.257197 0.966359i \(-0.417201\pi\)
0.257197 + 0.966359i \(0.417201\pi\)
\(522\) 0 0
\(523\) 16685.6 1.39505 0.697524 0.716561i \(-0.254285\pi\)
0.697524 + 0.716561i \(0.254285\pi\)
\(524\) 0 0
\(525\) 814.949 0.0677473
\(526\) 0 0
\(527\) 11709.6 0.967887
\(528\) 0 0
\(529\) −11499.6 −0.945149
\(530\) 0 0
\(531\) −3273.24 −0.267508
\(532\) 0 0
\(533\) 348.338 0.0283081
\(534\) 0 0
\(535\) 1085.59 0.0877276
\(536\) 0 0
\(537\) −7549.59 −0.606683
\(538\) 0 0
\(539\) 2560.49 0.204616
\(540\) 0 0
\(541\) 9309.03 0.739790 0.369895 0.929074i \(-0.379394\pi\)
0.369895 + 0.929074i \(0.379394\pi\)
\(542\) 0 0
\(543\) 12078.4 0.954575
\(544\) 0 0
\(545\) −8672.05 −0.681596
\(546\) 0 0
\(547\) −10894.7 −0.851598 −0.425799 0.904818i \(-0.640007\pi\)
−0.425799 + 0.904818i \(0.640007\pi\)
\(548\) 0 0
\(549\) −81.1134 −0.00630571
\(550\) 0 0
\(551\) −1680.01 −0.129893
\(552\) 0 0
\(553\) −6004.07 −0.461698
\(554\) 0 0
\(555\) −4590.43 −0.351086
\(556\) 0 0
\(557\) −7873.90 −0.598973 −0.299486 0.954101i \(-0.596815\pi\)
−0.299486 + 0.954101i \(0.596815\pi\)
\(558\) 0 0
\(559\) 1037.23 0.0784796
\(560\) 0 0
\(561\) 9058.55 0.681733
\(562\) 0 0
\(563\) −21770.7 −1.62971 −0.814854 0.579666i \(-0.803183\pi\)
−0.814854 + 0.579666i \(0.803183\pi\)
\(564\) 0 0
\(565\) 9271.02 0.690327
\(566\) 0 0
\(567\) −3901.06 −0.288940
\(568\) 0 0
\(569\) −12381.3 −0.912213 −0.456106 0.889925i \(-0.650756\pi\)
−0.456106 + 0.889925i \(0.650756\pi\)
\(570\) 0 0
\(571\) 5768.38 0.422765 0.211383 0.977403i \(-0.432203\pi\)
0.211383 + 0.977403i \(0.432203\pi\)
\(572\) 0 0
\(573\) 8489.81 0.618965
\(574\) 0 0
\(575\) −645.837 −0.0468405
\(576\) 0 0
\(577\) 4733.38 0.341513 0.170757 0.985313i \(-0.445379\pi\)
0.170757 + 0.985313i \(0.445379\pi\)
\(578\) 0 0
\(579\) −7176.34 −0.515093
\(580\) 0 0
\(581\) −4623.70 −0.330161
\(582\) 0 0
\(583\) 7996.00 0.568028
\(584\) 0 0
\(585\) 814.508 0.0575654
\(586\) 0 0
\(587\) 8441.67 0.593569 0.296785 0.954944i \(-0.404086\pi\)
0.296785 + 0.954944i \(0.404086\pi\)
\(588\) 0 0
\(589\) −25235.6 −1.76539
\(590\) 0 0
\(591\) 3265.59 0.227290
\(592\) 0 0
\(593\) 18939.9 1.31158 0.655791 0.754943i \(-0.272335\pi\)
0.655791 + 0.754943i \(0.272335\pi\)
\(594\) 0 0
\(595\) −1302.89 −0.0897701
\(596\) 0 0
\(597\) −15344.2 −1.05192
\(598\) 0 0
\(599\) −22655.3 −1.54536 −0.772681 0.634794i \(-0.781085\pi\)
−0.772681 + 0.634794i \(0.781085\pi\)
\(600\) 0 0
\(601\) −15947.4 −1.08237 −0.541187 0.840902i \(-0.682025\pi\)
−0.541187 + 0.840902i \(0.682025\pi\)
\(602\) 0 0
\(603\) −884.784 −0.0597532
\(604\) 0 0
\(605\) −6997.84 −0.470252
\(606\) 0 0
\(607\) 25993.2 1.73811 0.869053 0.494719i \(-0.164729\pi\)
0.869053 + 0.494719i \(0.164729\pi\)
\(608\) 0 0
\(609\) 682.638 0.0454218
\(610\) 0 0
\(611\) 11087.9 0.734152
\(612\) 0 0
\(613\) 665.408 0.0438427 0.0219213 0.999760i \(-0.493022\pi\)
0.0219213 + 0.999760i \(0.493022\pi\)
\(614\) 0 0
\(615\) −264.567 −0.0173470
\(616\) 0 0
\(617\) 18401.3 1.20066 0.600330 0.799752i \(-0.295036\pi\)
0.600330 + 0.799752i \(0.295036\pi\)
\(618\) 0 0
\(619\) 11150.6 0.724040 0.362020 0.932170i \(-0.382087\pi\)
0.362020 + 0.932170i \(0.382087\pi\)
\(620\) 0 0
\(621\) 3887.43 0.251203
\(622\) 0 0
\(623\) −320.422 −0.0206059
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) −19522.3 −1.24345
\(628\) 0 0
\(629\) 7338.88 0.465215
\(630\) 0 0
\(631\) −5381.79 −0.339534 −0.169767 0.985484i \(-0.554301\pi\)
−0.169767 + 0.985484i \(0.554301\pi\)
\(632\) 0 0
\(633\) −19010.9 −1.19371
\(634\) 0 0
\(635\) 6972.55 0.435744
\(636\) 0 0
\(637\) 1502.19 0.0934361
\(638\) 0 0
\(639\) −5058.65 −0.313172
\(640\) 0 0
\(641\) −19455.1 −1.19880 −0.599398 0.800451i \(-0.704593\pi\)
−0.599398 + 0.800451i \(0.704593\pi\)
\(642\) 0 0
\(643\) 14695.8 0.901317 0.450658 0.892696i \(-0.351189\pi\)
0.450658 + 0.892696i \(0.351189\pi\)
\(644\) 0 0
\(645\) −787.788 −0.0480917
\(646\) 0 0
\(647\) 12694.8 0.771383 0.385691 0.922628i \(-0.373963\pi\)
0.385691 + 0.922628i \(0.373963\pi\)
\(648\) 0 0
\(649\) 32189.0 1.94688
\(650\) 0 0
\(651\) 10254.0 0.617334
\(652\) 0 0
\(653\) −12385.6 −0.742247 −0.371124 0.928583i \(-0.621027\pi\)
−0.371124 + 0.928583i \(0.621027\pi\)
\(654\) 0 0
\(655\) 8812.09 0.525675
\(656\) 0 0
\(657\) 789.030 0.0468539
\(658\) 0 0
\(659\) 2072.18 0.122489 0.0612447 0.998123i \(-0.480493\pi\)
0.0612447 + 0.998123i \(0.480493\pi\)
\(660\) 0 0
\(661\) 1074.36 0.0632193 0.0316096 0.999500i \(-0.489937\pi\)
0.0316096 + 0.999500i \(0.489937\pi\)
\(662\) 0 0
\(663\) 5314.47 0.311307
\(664\) 0 0
\(665\) 2807.89 0.163737
\(666\) 0 0
\(667\) −540.982 −0.0314047
\(668\) 0 0
\(669\) −3479.42 −0.201080
\(670\) 0 0
\(671\) 797.667 0.0458921
\(672\) 0 0
\(673\) 26195.2 1.50037 0.750186 0.661226i \(-0.229964\pi\)
0.750186 + 0.661226i \(0.229964\pi\)
\(674\) 0 0
\(675\) −3762.01 −0.214518
\(676\) 0 0
\(677\) −4228.44 −0.240047 −0.120024 0.992771i \(-0.538297\pi\)
−0.120024 + 0.992771i \(0.538297\pi\)
\(678\) 0 0
\(679\) 11774.9 0.665506
\(680\) 0 0
\(681\) −7756.80 −0.436478
\(682\) 0 0
\(683\) −27525.5 −1.54207 −0.771036 0.636792i \(-0.780261\pi\)
−0.771036 + 0.636792i \(0.780261\pi\)
\(684\) 0 0
\(685\) 4614.74 0.257402
\(686\) 0 0
\(687\) −30867.3 −1.71421
\(688\) 0 0
\(689\) 4691.09 0.259385
\(690\) 0 0
\(691\) 33324.4 1.83462 0.917309 0.398177i \(-0.130357\pi\)
0.917309 + 0.398177i \(0.130357\pi\)
\(692\) 0 0
\(693\) −1943.67 −0.106542
\(694\) 0 0
\(695\) −980.193 −0.0534976
\(696\) 0 0
\(697\) 422.973 0.0229860
\(698\) 0 0
\(699\) −2013.77 −0.108967
\(700\) 0 0
\(701\) −33262.9 −1.79219 −0.896094 0.443864i \(-0.853607\pi\)
−0.896094 + 0.443864i \(0.853607\pi\)
\(702\) 0 0
\(703\) −15816.2 −0.848534
\(704\) 0 0
\(705\) −8421.37 −0.449882
\(706\) 0 0
\(707\) −3039.17 −0.161668
\(708\) 0 0
\(709\) 13703.0 0.725851 0.362926 0.931818i \(-0.381778\pi\)
0.362926 + 0.931818i \(0.381778\pi\)
\(710\) 0 0
\(711\) 4557.70 0.240404
\(712\) 0 0
\(713\) −8126.14 −0.426825
\(714\) 0 0
\(715\) −8009.84 −0.418953
\(716\) 0 0
\(717\) −25987.3 −1.35358
\(718\) 0 0
\(719\) 8074.93 0.418838 0.209419 0.977826i \(-0.432843\pi\)
0.209419 + 0.977826i \(0.432843\pi\)
\(720\) 0 0
\(721\) −2419.04 −0.124951
\(722\) 0 0
\(723\) −29323.6 −1.50838
\(724\) 0 0
\(725\) 523.528 0.0268184
\(726\) 0 0
\(727\) 3668.70 0.187159 0.0935794 0.995612i \(-0.470169\pi\)
0.0935794 + 0.995612i \(0.470169\pi\)
\(728\) 0 0
\(729\) 21881.9 1.11172
\(730\) 0 0
\(731\) 1259.46 0.0637250
\(732\) 0 0
\(733\) −14980.3 −0.754857 −0.377428 0.926039i \(-0.623192\pi\)
−0.377428 + 0.926039i \(0.623192\pi\)
\(734\) 0 0
\(735\) −1140.93 −0.0572569
\(736\) 0 0
\(737\) 8700.94 0.434875
\(738\) 0 0
\(739\) −6530.59 −0.325077 −0.162538 0.986702i \(-0.551968\pi\)
−0.162538 + 0.986702i \(0.551968\pi\)
\(740\) 0 0
\(741\) −11453.3 −0.567812
\(742\) 0 0
\(743\) −25952.0 −1.28141 −0.640704 0.767788i \(-0.721357\pi\)
−0.640704 + 0.767788i \(0.721357\pi\)
\(744\) 0 0
\(745\) −3901.86 −0.191883
\(746\) 0 0
\(747\) 3509.85 0.171913
\(748\) 0 0
\(749\) −1519.83 −0.0741434
\(750\) 0 0
\(751\) 14093.9 0.684813 0.342407 0.939552i \(-0.388758\pi\)
0.342407 + 0.939552i \(0.388758\pi\)
\(752\) 0 0
\(753\) −1452.57 −0.0702983
\(754\) 0 0
\(755\) −11597.1 −0.559024
\(756\) 0 0
\(757\) −2554.41 −0.122644 −0.0613220 0.998118i \(-0.519532\pi\)
−0.0613220 + 0.998118i \(0.519532\pi\)
\(758\) 0 0
\(759\) −6286.40 −0.300635
\(760\) 0 0
\(761\) 2219.08 0.105705 0.0528527 0.998602i \(-0.483169\pi\)
0.0528527 + 0.998602i \(0.483169\pi\)
\(762\) 0 0
\(763\) 12140.9 0.576054
\(764\) 0 0
\(765\) 989.025 0.0467428
\(766\) 0 0
\(767\) 18884.6 0.889028
\(768\) 0 0
\(769\) −22466.2 −1.05352 −0.526758 0.850015i \(-0.676592\pi\)
−0.526758 + 0.850015i \(0.676592\pi\)
\(770\) 0 0
\(771\) −36609.3 −1.71006
\(772\) 0 0
\(773\) 9674.79 0.450165 0.225083 0.974340i \(-0.427735\pi\)
0.225083 + 0.974340i \(0.427735\pi\)
\(774\) 0 0
\(775\) 7863.96 0.364493
\(776\) 0 0
\(777\) 6426.60 0.296722
\(778\) 0 0
\(779\) −911.560 −0.0419256
\(780\) 0 0
\(781\) 49746.6 2.27922
\(782\) 0 0
\(783\) −3151.23 −0.143826
\(784\) 0 0
\(785\) −5114.51 −0.232541
\(786\) 0 0
\(787\) 20942.8 0.948577 0.474288 0.880370i \(-0.342705\pi\)
0.474288 + 0.880370i \(0.342705\pi\)
\(788\) 0 0
\(789\) −24341.8 −1.09834
\(790\) 0 0
\(791\) −12979.4 −0.583433
\(792\) 0 0
\(793\) 467.975 0.0209562
\(794\) 0 0
\(795\) −3562.94 −0.158949
\(796\) 0 0
\(797\) 23526.6 1.04561 0.522807 0.852451i \(-0.324885\pi\)
0.522807 + 0.852451i \(0.324885\pi\)
\(798\) 0 0
\(799\) 13463.5 0.596127
\(800\) 0 0
\(801\) 243.233 0.0107294
\(802\) 0 0
\(803\) −7759.29 −0.340996
\(804\) 0 0
\(805\) 904.172 0.0395874
\(806\) 0 0
\(807\) 5968.72 0.260358
\(808\) 0 0
\(809\) −18202.2 −0.791047 −0.395523 0.918456i \(-0.629437\pi\)
−0.395523 + 0.918456i \(0.629437\pi\)
\(810\) 0 0
\(811\) 2510.24 0.108689 0.0543443 0.998522i \(-0.482693\pi\)
0.0543443 + 0.998522i \(0.482693\pi\)
\(812\) 0 0
\(813\) −21906.5 −0.945012
\(814\) 0 0
\(815\) −6753.13 −0.290248
\(816\) 0 0
\(817\) −2714.30 −0.116232
\(818\) 0 0
\(819\) −1140.31 −0.0486516
\(820\) 0 0
\(821\) 17899.6 0.760903 0.380451 0.924801i \(-0.375769\pi\)
0.380451 + 0.924801i \(0.375769\pi\)
\(822\) 0 0
\(823\) −14039.5 −0.594637 −0.297318 0.954778i \(-0.596092\pi\)
−0.297318 + 0.954778i \(0.596092\pi\)
\(824\) 0 0
\(825\) 6083.58 0.256731
\(826\) 0 0
\(827\) −15127.4 −0.636073 −0.318036 0.948079i \(-0.603023\pi\)
−0.318036 + 0.948079i \(0.603023\pi\)
\(828\) 0 0
\(829\) 21986.5 0.921136 0.460568 0.887624i \(-0.347646\pi\)
0.460568 + 0.887624i \(0.347646\pi\)
\(830\) 0 0
\(831\) 41718.7 1.74152
\(832\) 0 0
\(833\) 1824.04 0.0758696
\(834\) 0 0
\(835\) −6152.89 −0.255005
\(836\) 0 0
\(837\) −47334.8 −1.95476
\(838\) 0 0
\(839\) 2276.89 0.0936914 0.0468457 0.998902i \(-0.485083\pi\)
0.0468457 + 0.998902i \(0.485083\pi\)
\(840\) 0 0
\(841\) −23950.5 −0.982019
\(842\) 0 0
\(843\) −1727.25 −0.0705688
\(844\) 0 0
\(845\) 6285.79 0.255903
\(846\) 0 0
\(847\) 9796.97 0.397436
\(848\) 0 0
\(849\) 27113.4 1.09603
\(850\) 0 0
\(851\) −5093.00 −0.205154
\(852\) 0 0
\(853\) 13342.6 0.535570 0.267785 0.963479i \(-0.413708\pi\)
0.267785 + 0.963479i \(0.413708\pi\)
\(854\) 0 0
\(855\) −2131.47 −0.0852571
\(856\) 0 0
\(857\) 18690.9 0.745003 0.372502 0.928032i \(-0.378500\pi\)
0.372502 + 0.928032i \(0.378500\pi\)
\(858\) 0 0
\(859\) −18318.9 −0.727628 −0.363814 0.931472i \(-0.618526\pi\)
−0.363814 + 0.931472i \(0.618526\pi\)
\(860\) 0 0
\(861\) 370.394 0.0146609
\(862\) 0 0
\(863\) −38133.1 −1.50413 −0.752067 0.659087i \(-0.770943\pi\)
−0.752067 + 0.659087i \(0.770943\pi\)
\(864\) 0 0
\(865\) 12438.3 0.488917
\(866\) 0 0
\(867\) −16426.0 −0.643432
\(868\) 0 0
\(869\) −44820.3 −1.74962
\(870\) 0 0
\(871\) 5104.66 0.198582
\(872\) 0 0
\(873\) −8938.33 −0.346525
\(874\) 0 0
\(875\) −875.000 −0.0338062
\(876\) 0 0
\(877\) −19707.5 −0.758807 −0.379404 0.925231i \(-0.623871\pi\)
−0.379404 + 0.925231i \(0.623871\pi\)
\(878\) 0 0
\(879\) 34664.6 1.33016
\(880\) 0 0
\(881\) −14091.5 −0.538883 −0.269441 0.963017i \(-0.586839\pi\)
−0.269441 + 0.963017i \(0.586839\pi\)
\(882\) 0 0
\(883\) −3115.87 −0.118751 −0.0593757 0.998236i \(-0.518911\pi\)
−0.0593757 + 0.998236i \(0.518911\pi\)
\(884\) 0 0
\(885\) −14343.1 −0.544789
\(886\) 0 0
\(887\) −38734.6 −1.46627 −0.733134 0.680084i \(-0.761943\pi\)
−0.733134 + 0.680084i \(0.761943\pi\)
\(888\) 0 0
\(889\) −9761.57 −0.368270
\(890\) 0 0
\(891\) −29121.3 −1.09495
\(892\) 0 0
\(893\) −29015.6 −1.08731
\(894\) 0 0
\(895\) 8105.89 0.302737
\(896\) 0 0
\(897\) −3688.10 −0.137282
\(898\) 0 0
\(899\) 6587.21 0.244378
\(900\) 0 0
\(901\) 5696.21 0.210619
\(902\) 0 0
\(903\) 1102.90 0.0406449
\(904\) 0 0
\(905\) −12968.4 −0.476337
\(906\) 0 0
\(907\) −19242.9 −0.704464 −0.352232 0.935913i \(-0.614577\pi\)
−0.352232 + 0.935913i \(0.614577\pi\)
\(908\) 0 0
\(909\) 2307.03 0.0841799
\(910\) 0 0
\(911\) −34613.3 −1.25882 −0.629412 0.777072i \(-0.716704\pi\)
−0.629412 + 0.777072i \(0.716704\pi\)
\(912\) 0 0
\(913\) −34515.8 −1.25116
\(914\) 0 0
\(915\) −355.433 −0.0128418
\(916\) 0 0
\(917\) −12336.9 −0.444276
\(918\) 0 0
\(919\) −25826.4 −0.927022 −0.463511 0.886091i \(-0.653411\pi\)
−0.463511 + 0.886091i \(0.653411\pi\)
\(920\) 0 0
\(921\) 3547.01 0.126903
\(922\) 0 0
\(923\) 29185.3 1.04079
\(924\) 0 0
\(925\) 4928.68 0.175194
\(926\) 0 0
\(927\) 1836.29 0.0650613
\(928\) 0 0
\(929\) 19451.6 0.686960 0.343480 0.939160i \(-0.388394\pi\)
0.343480 + 0.939160i \(0.388394\pi\)
\(930\) 0 0
\(931\) −3931.04 −0.138383
\(932\) 0 0
\(933\) −35944.8 −1.26129
\(934\) 0 0
\(935\) −9726.03 −0.340188
\(936\) 0 0
\(937\) 34469.1 1.20177 0.600884 0.799336i \(-0.294815\pi\)
0.600884 + 0.799336i \(0.294815\pi\)
\(938\) 0 0
\(939\) 39844.1 1.38473
\(940\) 0 0
\(941\) 14156.4 0.490419 0.245209 0.969470i \(-0.421143\pi\)
0.245209 + 0.969470i \(0.421143\pi\)
\(942\) 0 0
\(943\) −293.532 −0.0101365
\(944\) 0 0
\(945\) 5266.81 0.181301
\(946\) 0 0
\(947\) 38092.4 1.30711 0.653557 0.756877i \(-0.273276\pi\)
0.653557 + 0.756877i \(0.273276\pi\)
\(948\) 0 0
\(949\) −4552.22 −0.155713
\(950\) 0 0
\(951\) −36234.8 −1.23553
\(952\) 0 0
\(953\) 5037.40 0.171225 0.0856126 0.996329i \(-0.472715\pi\)
0.0856126 + 0.996329i \(0.472715\pi\)
\(954\) 0 0
\(955\) −9115.39 −0.308866
\(956\) 0 0
\(957\) 5095.88 0.172128
\(958\) 0 0
\(959\) −6460.64 −0.217544
\(960\) 0 0
\(961\) 69156.0 2.32137
\(962\) 0 0
\(963\) 1153.71 0.0386060
\(964\) 0 0
\(965\) 7705.14 0.257033
\(966\) 0 0
\(967\) −11495.3 −0.382278 −0.191139 0.981563i \(-0.561218\pi\)
−0.191139 + 0.981563i \(0.561218\pi\)
\(968\) 0 0
\(969\) −13907.3 −0.461061
\(970\) 0 0
\(971\) −22352.7 −0.738757 −0.369379 0.929279i \(-0.620429\pi\)
−0.369379 + 0.929279i \(0.620429\pi\)
\(972\) 0 0
\(973\) 1372.27 0.0452138
\(974\) 0 0
\(975\) 3569.11 0.117234
\(976\) 0 0
\(977\) 14345.7 0.469765 0.234882 0.972024i \(-0.424530\pi\)
0.234882 + 0.972024i \(0.424530\pi\)
\(978\) 0 0
\(979\) −2391.94 −0.0780867
\(980\) 0 0
\(981\) −9216.15 −0.299948
\(982\) 0 0
\(983\) −34460.9 −1.11814 −0.559070 0.829120i \(-0.688842\pi\)
−0.559070 + 0.829120i \(0.688842\pi\)
\(984\) 0 0
\(985\) −3506.21 −0.113419
\(986\) 0 0
\(987\) 11789.9 0.380220
\(988\) 0 0
\(989\) −874.036 −0.0281019
\(990\) 0 0
\(991\) 35189.6 1.12799 0.563993 0.825780i \(-0.309265\pi\)
0.563993 + 0.825780i \(0.309265\pi\)
\(992\) 0 0
\(993\) 22968.2 0.734011
\(994\) 0 0
\(995\) 16474.8 0.524911
\(996\) 0 0
\(997\) −50730.0 −1.61147 −0.805734 0.592277i \(-0.798229\pi\)
−0.805734 + 0.592277i \(0.798229\pi\)
\(998\) 0 0
\(999\) −29666.8 −0.939554
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 560.4.a.r.1.2 2
4.3 odd 2 35.4.a.b.1.2 2
8.3 odd 2 2240.4.a.bn.1.2 2
8.5 even 2 2240.4.a.bo.1.1 2
12.11 even 2 315.4.a.f.1.1 2
20.3 even 4 175.4.b.c.99.1 4
20.7 even 4 175.4.b.c.99.4 4
20.19 odd 2 175.4.a.c.1.1 2
28.3 even 6 245.4.e.i.226.1 4
28.11 odd 6 245.4.e.h.226.1 4
28.19 even 6 245.4.e.i.116.1 4
28.23 odd 6 245.4.e.h.116.1 4
28.27 even 2 245.4.a.k.1.2 2
60.59 even 2 1575.4.a.z.1.2 2
84.83 odd 2 2205.4.a.u.1.1 2
140.139 even 2 1225.4.a.m.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.b.1.2 2 4.3 odd 2
175.4.a.c.1.1 2 20.19 odd 2
175.4.b.c.99.1 4 20.3 even 4
175.4.b.c.99.4 4 20.7 even 4
245.4.a.k.1.2 2 28.27 even 2
245.4.e.h.116.1 4 28.23 odd 6
245.4.e.h.226.1 4 28.11 odd 6
245.4.e.i.116.1 4 28.19 even 6
245.4.e.i.226.1 4 28.3 even 6
315.4.a.f.1.1 2 12.11 even 2
560.4.a.r.1.2 2 1.1 even 1 trivial
1225.4.a.m.1.1 2 140.139 even 2
1575.4.a.z.1.2 2 60.59 even 2
2205.4.a.u.1.1 2 84.83 odd 2
2240.4.a.bn.1.2 2 8.3 odd 2
2240.4.a.bo.1.1 2 8.5 even 2