Properties

Label 528.4.a.r.1.2
Level $528$
Weight $4$
Character 528.1
Self dual yes
Analytic conductor $31.153$
Analytic rank $1$
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] = [528,4,Mod(1,528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(528, 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("528.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 528.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: \(31.1530084830\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 264)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 528.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +1.12311 q^{5} -0.876894 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +1.12311 q^{5} -0.876894 q^{7} +9.00000 q^{9} -11.0000 q^{11} -71.0928 q^{13} +3.36932 q^{15} +22.2159 q^{17} -125.693 q^{19} -2.63068 q^{21} +104.600 q^{23} -123.739 q^{25} +27.0000 q^{27} -303.201 q^{29} +196.924 q^{31} -33.0000 q^{33} -0.984845 q^{35} +247.848 q^{37} -213.278 q^{39} -448.462 q^{41} +196.371 q^{43} +10.1080 q^{45} -28.5398 q^{47} -342.231 q^{49} +66.6477 q^{51} -38.4451 q^{53} -12.3542 q^{55} -377.080 q^{57} -14.8599 q^{59} -625.589 q^{61} -7.89205 q^{63} -79.8447 q^{65} -668.742 q^{67} +313.801 q^{69} +179.244 q^{71} +1053.58 q^{73} -371.216 q^{75} +9.64584 q^{77} -458.941 q^{79} +81.0000 q^{81} +626.098 q^{83} +24.9508 q^{85} -909.602 q^{87} -1570.68 q^{89} +62.3409 q^{91} +590.773 q^{93} -141.167 q^{95} +827.322 q^{97} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 6 q^{5} - 10 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} - 6 q^{5} - 10 q^{7} + 18 q^{9} - 22 q^{11} - 2 q^{13} - 18 q^{15} - 104 q^{17} - 4 q^{19} - 30 q^{21} + 102 q^{23} - 198 q^{25} + 54 q^{27} - 392 q^{29} + 64 q^{31} - 66 q^{33} + 64 q^{35} - 164 q^{37} - 6 q^{39} - 732 q^{41} - 168 q^{43} - 54 q^{45} + 314 q^{47} - 602 q^{49} - 312 q^{51} - 382 q^{53} + 66 q^{55} - 12 q^{57} - 508 q^{59} - 6 q^{61} - 90 q^{63} - 572 q^{65} - 216 q^{67} + 306 q^{69} + 878 q^{71} + 260 q^{73} - 594 q^{75} + 110 q^{77} - 118 q^{79} + 162 q^{81} - 496 q^{83} + 924 q^{85} - 1176 q^{87} - 1756 q^{89} - 568 q^{91} + 192 q^{93} - 1008 q^{95} + 1968 q^{97} - 198 q^{99}+O(q^{100}) \) 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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 1.12311 0.100454 0.0502268 0.998738i \(-0.484006\pi\)
0.0502268 + 0.998738i \(0.484006\pi\)
\(6\) 0 0
\(7\) −0.876894 −0.0473478 −0.0236739 0.999720i \(-0.507536\pi\)
−0.0236739 + 0.999720i \(0.507536\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 0 0
\(13\) −71.0928 −1.51674 −0.758369 0.651825i \(-0.774003\pi\)
−0.758369 + 0.651825i \(0.774003\pi\)
\(14\) 0 0
\(15\) 3.36932 0.0579969
\(16\) 0 0
\(17\) 22.2159 0.316950 0.158475 0.987363i \(-0.449342\pi\)
0.158475 + 0.987363i \(0.449342\pi\)
\(18\) 0 0
\(19\) −125.693 −1.51768 −0.758842 0.651275i \(-0.774234\pi\)
−0.758842 + 0.651275i \(0.774234\pi\)
\(20\) 0 0
\(21\) −2.63068 −0.0273363
\(22\) 0 0
\(23\) 104.600 0.948291 0.474145 0.880447i \(-0.342757\pi\)
0.474145 + 0.880447i \(0.342757\pi\)
\(24\) 0 0
\(25\) −123.739 −0.989909
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −303.201 −1.94148 −0.970741 0.240130i \(-0.922810\pi\)
−0.970741 + 0.240130i \(0.922810\pi\)
\(30\) 0 0
\(31\) 196.924 1.14092 0.570462 0.821324i \(-0.306764\pi\)
0.570462 + 0.821324i \(0.306764\pi\)
\(32\) 0 0
\(33\) −33.0000 −0.174078
\(34\) 0 0
\(35\) −0.984845 −0.00475626
\(36\) 0 0
\(37\) 247.848 1.10124 0.550622 0.834755i \(-0.314391\pi\)
0.550622 + 0.834755i \(0.314391\pi\)
\(38\) 0 0
\(39\) −213.278 −0.875689
\(40\) 0 0
\(41\) −448.462 −1.70824 −0.854122 0.520072i \(-0.825905\pi\)
−0.854122 + 0.520072i \(0.825905\pi\)
\(42\) 0 0
\(43\) 196.371 0.696426 0.348213 0.937415i \(-0.386789\pi\)
0.348213 + 0.937415i \(0.386789\pi\)
\(44\) 0 0
\(45\) 10.1080 0.0334845
\(46\) 0 0
\(47\) −28.5398 −0.0885734 −0.0442867 0.999019i \(-0.514102\pi\)
−0.0442867 + 0.999019i \(0.514102\pi\)
\(48\) 0 0
\(49\) −342.231 −0.997758
\(50\) 0 0
\(51\) 66.6477 0.182991
\(52\) 0 0
\(53\) −38.4451 −0.0996385 −0.0498192 0.998758i \(-0.515865\pi\)
−0.0498192 + 0.998758i \(0.515865\pi\)
\(54\) 0 0
\(55\) −12.3542 −0.0302879
\(56\) 0 0
\(57\) −377.080 −0.876235
\(58\) 0 0
\(59\) −14.8599 −0.0327897 −0.0163948 0.999866i \(-0.505219\pi\)
−0.0163948 + 0.999866i \(0.505219\pi\)
\(60\) 0 0
\(61\) −625.589 −1.31309 −0.656545 0.754287i \(-0.727983\pi\)
−0.656545 + 0.754287i \(0.727983\pi\)
\(62\) 0 0
\(63\) −7.89205 −0.0157826
\(64\) 0 0
\(65\) −79.8447 −0.152362
\(66\) 0 0
\(67\) −668.742 −1.21940 −0.609701 0.792632i \(-0.708710\pi\)
−0.609701 + 0.792632i \(0.708710\pi\)
\(68\) 0 0
\(69\) 313.801 0.547496
\(70\) 0 0
\(71\) 179.244 0.299611 0.149806 0.988715i \(-0.452135\pi\)
0.149806 + 0.988715i \(0.452135\pi\)
\(72\) 0 0
\(73\) 1053.58 1.68920 0.844601 0.535397i \(-0.179838\pi\)
0.844601 + 0.535397i \(0.179838\pi\)
\(74\) 0 0
\(75\) −371.216 −0.571524
\(76\) 0 0
\(77\) 9.64584 0.0142759
\(78\) 0 0
\(79\) −458.941 −0.653607 −0.326803 0.945092i \(-0.605971\pi\)
−0.326803 + 0.945092i \(0.605971\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 626.098 0.827991 0.413995 0.910279i \(-0.364133\pi\)
0.413995 + 0.910279i \(0.364133\pi\)
\(84\) 0 0
\(85\) 24.9508 0.0318388
\(86\) 0 0
\(87\) −909.602 −1.12091
\(88\) 0 0
\(89\) −1570.68 −1.87070 −0.935348 0.353729i \(-0.884913\pi\)
−0.935348 + 0.353729i \(0.884913\pi\)
\(90\) 0 0
\(91\) 62.3409 0.0718143
\(92\) 0 0
\(93\) 590.773 0.658713
\(94\) 0 0
\(95\) −141.167 −0.152457
\(96\) 0 0
\(97\) 827.322 0.865998 0.432999 0.901394i \(-0.357455\pi\)
0.432999 + 0.901394i \(0.357455\pi\)
\(98\) 0 0
\(99\) −99.0000 −0.100504
\(100\) 0 0
\(101\) 137.852 0.135810 0.0679050 0.997692i \(-0.478369\pi\)
0.0679050 + 0.997692i \(0.478369\pi\)
\(102\) 0 0
\(103\) −623.352 −0.596318 −0.298159 0.954516i \(-0.596373\pi\)
−0.298159 + 0.954516i \(0.596373\pi\)
\(104\) 0 0
\(105\) −2.95454 −0.00274603
\(106\) 0 0
\(107\) −996.068 −0.899940 −0.449970 0.893044i \(-0.648565\pi\)
−0.449970 + 0.893044i \(0.648565\pi\)
\(108\) 0 0
\(109\) 567.180 0.498404 0.249202 0.968452i \(-0.419832\pi\)
0.249202 + 0.968452i \(0.419832\pi\)
\(110\) 0 0
\(111\) 743.545 0.635804
\(112\) 0 0
\(113\) −440.644 −0.366834 −0.183417 0.983035i \(-0.558716\pi\)
−0.183417 + 0.983035i \(0.558716\pi\)
\(114\) 0 0
\(115\) 117.477 0.0952592
\(116\) 0 0
\(117\) −639.835 −0.505579
\(118\) 0 0
\(119\) −19.4810 −0.0150069
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 0 0
\(123\) −1345.39 −0.986255
\(124\) 0 0
\(125\) −279.360 −0.199894
\(126\) 0 0
\(127\) −533.032 −0.372433 −0.186216 0.982509i \(-0.559623\pi\)
−0.186216 + 0.982509i \(0.559623\pi\)
\(128\) 0 0
\(129\) 589.114 0.402082
\(130\) 0 0
\(131\) −1983.45 −1.32286 −0.661432 0.750005i \(-0.730051\pi\)
−0.661432 + 0.750005i \(0.730051\pi\)
\(132\) 0 0
\(133\) 110.220 0.0718591
\(134\) 0 0
\(135\) 30.3239 0.0193323
\(136\) 0 0
\(137\) −1756.75 −1.09554 −0.547770 0.836629i \(-0.684523\pi\)
−0.547770 + 0.836629i \(0.684523\pi\)
\(138\) 0 0
\(139\) −1453.38 −0.886865 −0.443433 0.896308i \(-0.646239\pi\)
−0.443433 + 0.896308i \(0.646239\pi\)
\(140\) 0 0
\(141\) −85.6193 −0.0511379
\(142\) 0 0
\(143\) 782.021 0.457314
\(144\) 0 0
\(145\) −340.526 −0.195029
\(146\) 0 0
\(147\) −1026.69 −0.576056
\(148\) 0 0
\(149\) 745.151 0.409699 0.204850 0.978793i \(-0.434329\pi\)
0.204850 + 0.978793i \(0.434329\pi\)
\(150\) 0 0
\(151\) 2647.53 1.42684 0.713420 0.700737i \(-0.247145\pi\)
0.713420 + 0.700737i \(0.247145\pi\)
\(152\) 0 0
\(153\) 199.943 0.105650
\(154\) 0 0
\(155\) 221.167 0.114610
\(156\) 0 0
\(157\) 2537.55 1.28993 0.644963 0.764214i \(-0.276873\pi\)
0.644963 + 0.764214i \(0.276873\pi\)
\(158\) 0 0
\(159\) −115.335 −0.0575263
\(160\) 0 0
\(161\) −91.7235 −0.0448995
\(162\) 0 0
\(163\) 84.8636 0.0407793 0.0203897 0.999792i \(-0.493509\pi\)
0.0203897 + 0.999792i \(0.493509\pi\)
\(164\) 0 0
\(165\) −37.0625 −0.0174867
\(166\) 0 0
\(167\) 4133.03 1.91511 0.957556 0.288247i \(-0.0930723\pi\)
0.957556 + 0.288247i \(0.0930723\pi\)
\(168\) 0 0
\(169\) 2857.19 1.30049
\(170\) 0 0
\(171\) −1131.24 −0.505895
\(172\) 0 0
\(173\) −1177.29 −0.517386 −0.258693 0.965960i \(-0.583292\pi\)
−0.258693 + 0.965960i \(0.583292\pi\)
\(174\) 0 0
\(175\) 108.506 0.0468701
\(176\) 0 0
\(177\) −44.5796 −0.0189311
\(178\) 0 0
\(179\) 241.087 0.100669 0.0503343 0.998732i \(-0.483971\pi\)
0.0503343 + 0.998732i \(0.483971\pi\)
\(180\) 0 0
\(181\) 2125.85 0.873002 0.436501 0.899704i \(-0.356218\pi\)
0.436501 + 0.899704i \(0.356218\pi\)
\(182\) 0 0
\(183\) −1876.77 −0.758113
\(184\) 0 0
\(185\) 278.360 0.110624
\(186\) 0 0
\(187\) −244.375 −0.0955640
\(188\) 0 0
\(189\) −23.6761 −0.00911210
\(190\) 0 0
\(191\) −3348.26 −1.26844 −0.634219 0.773154i \(-0.718678\pi\)
−0.634219 + 0.773154i \(0.718678\pi\)
\(192\) 0 0
\(193\) −1437.61 −0.536173 −0.268087 0.963395i \(-0.586391\pi\)
−0.268087 + 0.963395i \(0.586391\pi\)
\(194\) 0 0
\(195\) −239.534 −0.0879661
\(196\) 0 0
\(197\) −3276.09 −1.18483 −0.592415 0.805633i \(-0.701825\pi\)
−0.592415 + 0.805633i \(0.701825\pi\)
\(198\) 0 0
\(199\) 4479.61 1.59573 0.797867 0.602833i \(-0.205962\pi\)
0.797867 + 0.602833i \(0.205962\pi\)
\(200\) 0 0
\(201\) −2006.23 −0.704022
\(202\) 0 0
\(203\) 265.875 0.0919250
\(204\) 0 0
\(205\) −503.670 −0.171599
\(206\) 0 0
\(207\) 941.403 0.316097
\(208\) 0 0
\(209\) 1382.62 0.457599
\(210\) 0 0
\(211\) 4326.58 1.41163 0.705816 0.708395i \(-0.250581\pi\)
0.705816 + 0.708395i \(0.250581\pi\)
\(212\) 0 0
\(213\) 537.733 0.172981
\(214\) 0 0
\(215\) 220.546 0.0699585
\(216\) 0 0
\(217\) −172.682 −0.0540203
\(218\) 0 0
\(219\) 3160.73 0.975261
\(220\) 0 0
\(221\) −1579.39 −0.480730
\(222\) 0 0
\(223\) −898.969 −0.269953 −0.134976 0.990849i \(-0.543096\pi\)
−0.134976 + 0.990849i \(0.543096\pi\)
\(224\) 0 0
\(225\) −1113.65 −0.329970
\(226\) 0 0
\(227\) −601.530 −0.175881 −0.0879405 0.996126i \(-0.528029\pi\)
−0.0879405 + 0.996126i \(0.528029\pi\)
\(228\) 0 0
\(229\) 3143.94 0.907238 0.453619 0.891196i \(-0.350133\pi\)
0.453619 + 0.891196i \(0.350133\pi\)
\(230\) 0 0
\(231\) 28.9375 0.00824220
\(232\) 0 0
\(233\) 1699.18 0.477756 0.238878 0.971050i \(-0.423220\pi\)
0.238878 + 0.971050i \(0.423220\pi\)
\(234\) 0 0
\(235\) −32.0532 −0.00889752
\(236\) 0 0
\(237\) −1376.82 −0.377360
\(238\) 0 0
\(239\) 7204.05 1.94975 0.974877 0.222744i \(-0.0715014\pi\)
0.974877 + 0.222744i \(0.0715014\pi\)
\(240\) 0 0
\(241\) 3347.54 0.894747 0.447374 0.894347i \(-0.352359\pi\)
0.447374 + 0.894347i \(0.352359\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) −384.362 −0.100228
\(246\) 0 0
\(247\) 8935.88 2.30193
\(248\) 0 0
\(249\) 1878.30 0.478041
\(250\) 0 0
\(251\) −50.8673 −0.0127917 −0.00639585 0.999980i \(-0.502036\pi\)
−0.00639585 + 0.999980i \(0.502036\pi\)
\(252\) 0 0
\(253\) −1150.60 −0.285920
\(254\) 0 0
\(255\) 74.8524 0.0183821
\(256\) 0 0
\(257\) 4216.53 1.02342 0.511712 0.859157i \(-0.329012\pi\)
0.511712 + 0.859157i \(0.329012\pi\)
\(258\) 0 0
\(259\) −217.337 −0.0521415
\(260\) 0 0
\(261\) −2728.81 −0.647161
\(262\) 0 0
\(263\) 2000.38 0.469006 0.234503 0.972115i \(-0.424654\pi\)
0.234503 + 0.972115i \(0.424654\pi\)
\(264\) 0 0
\(265\) −43.1779 −0.0100090
\(266\) 0 0
\(267\) −4712.05 −1.08005
\(268\) 0 0
\(269\) 2305.18 0.522487 0.261244 0.965273i \(-0.415867\pi\)
0.261244 + 0.965273i \(0.415867\pi\)
\(270\) 0 0
\(271\) 4668.18 1.04639 0.523195 0.852213i \(-0.324740\pi\)
0.523195 + 0.852213i \(0.324740\pi\)
\(272\) 0 0
\(273\) 187.023 0.0414620
\(274\) 0 0
\(275\) 1361.12 0.298469
\(276\) 0 0
\(277\) −2933.96 −0.636406 −0.318203 0.948023i \(-0.603079\pi\)
−0.318203 + 0.948023i \(0.603079\pi\)
\(278\) 0 0
\(279\) 1772.32 0.380308
\(280\) 0 0
\(281\) −4395.40 −0.933124 −0.466562 0.884488i \(-0.654508\pi\)
−0.466562 + 0.884488i \(0.654508\pi\)
\(282\) 0 0
\(283\) 1527.07 0.320759 0.160379 0.987055i \(-0.448728\pi\)
0.160379 + 0.987055i \(0.448728\pi\)
\(284\) 0 0
\(285\) −423.500 −0.0880210
\(286\) 0 0
\(287\) 393.254 0.0808817
\(288\) 0 0
\(289\) −4419.45 −0.899543
\(290\) 0 0
\(291\) 2481.97 0.499984
\(292\) 0 0
\(293\) 4641.14 0.925386 0.462693 0.886519i \(-0.346883\pi\)
0.462693 + 0.886519i \(0.346883\pi\)
\(294\) 0 0
\(295\) −16.6892 −0.00329384
\(296\) 0 0
\(297\) −297.000 −0.0580259
\(298\) 0 0
\(299\) −7436.33 −1.43831
\(300\) 0 0
\(301\) −172.197 −0.0329743
\(302\) 0 0
\(303\) 413.557 0.0784099
\(304\) 0 0
\(305\) −702.602 −0.131905
\(306\) 0 0
\(307\) −5918.64 −1.10031 −0.550154 0.835063i \(-0.685431\pi\)
−0.550154 + 0.835063i \(0.685431\pi\)
\(308\) 0 0
\(309\) −1870.06 −0.344284
\(310\) 0 0
\(311\) −4622.47 −0.842817 −0.421409 0.906871i \(-0.638464\pi\)
−0.421409 + 0.906871i \(0.638464\pi\)
\(312\) 0 0
\(313\) 9862.20 1.78097 0.890486 0.455010i \(-0.150364\pi\)
0.890486 + 0.455010i \(0.150364\pi\)
\(314\) 0 0
\(315\) −8.86361 −0.00158542
\(316\) 0 0
\(317\) 6797.67 1.20440 0.602201 0.798345i \(-0.294291\pi\)
0.602201 + 0.798345i \(0.294291\pi\)
\(318\) 0 0
\(319\) 3335.21 0.585379
\(320\) 0 0
\(321\) −2988.20 −0.519580
\(322\) 0 0
\(323\) −2792.39 −0.481030
\(324\) 0 0
\(325\) 8796.93 1.50143
\(326\) 0 0
\(327\) 1701.54 0.287753
\(328\) 0 0
\(329\) 25.0263 0.00419376
\(330\) 0 0
\(331\) 5847.55 0.971029 0.485514 0.874229i \(-0.338632\pi\)
0.485514 + 0.874229i \(0.338632\pi\)
\(332\) 0 0
\(333\) 2230.64 0.367081
\(334\) 0 0
\(335\) −751.068 −0.122493
\(336\) 0 0
\(337\) 3797.52 0.613840 0.306920 0.951735i \(-0.400702\pi\)
0.306920 + 0.951735i \(0.400702\pi\)
\(338\) 0 0
\(339\) −1321.93 −0.211792
\(340\) 0 0
\(341\) −2166.17 −0.344001
\(342\) 0 0
\(343\) 600.875 0.0945895
\(344\) 0 0
\(345\) 352.432 0.0549979
\(346\) 0 0
\(347\) −11221.2 −1.73599 −0.867993 0.496577i \(-0.834590\pi\)
−0.867993 + 0.496577i \(0.834590\pi\)
\(348\) 0 0
\(349\) −8130.46 −1.24703 −0.623515 0.781811i \(-0.714296\pi\)
−0.623515 + 0.781811i \(0.714296\pi\)
\(350\) 0 0
\(351\) −1919.51 −0.291896
\(352\) 0 0
\(353\) −11060.0 −1.66761 −0.833804 0.552061i \(-0.813842\pi\)
−0.833804 + 0.552061i \(0.813842\pi\)
\(354\) 0 0
\(355\) 201.310 0.0300970
\(356\) 0 0
\(357\) −58.4430 −0.00866423
\(358\) 0 0
\(359\) −9040.68 −1.32911 −0.664553 0.747242i \(-0.731378\pi\)
−0.664553 + 0.747242i \(0.731378\pi\)
\(360\) 0 0
\(361\) 8939.77 1.30336
\(362\) 0 0
\(363\) 363.000 0.0524864
\(364\) 0 0
\(365\) 1183.28 0.169686
\(366\) 0 0
\(367\) 2896.46 0.411973 0.205986 0.978555i \(-0.433960\pi\)
0.205986 + 0.978555i \(0.433960\pi\)
\(368\) 0 0
\(369\) −4036.16 −0.569415
\(370\) 0 0
\(371\) 33.7123 0.00471767
\(372\) 0 0
\(373\) −12226.8 −1.69726 −0.848631 0.528985i \(-0.822573\pi\)
−0.848631 + 0.528985i \(0.822573\pi\)
\(374\) 0 0
\(375\) −838.079 −0.115409
\(376\) 0 0
\(377\) 21555.4 2.94472
\(378\) 0 0
\(379\) −10363.3 −1.40455 −0.702277 0.711903i \(-0.747833\pi\)
−0.702277 + 0.711903i \(0.747833\pi\)
\(380\) 0 0
\(381\) −1599.10 −0.215024
\(382\) 0 0
\(383\) −4571.03 −0.609840 −0.304920 0.952378i \(-0.598630\pi\)
−0.304920 + 0.952378i \(0.598630\pi\)
\(384\) 0 0
\(385\) 10.8333 0.00143407
\(386\) 0 0
\(387\) 1767.34 0.232142
\(388\) 0 0
\(389\) 7931.49 1.03378 0.516892 0.856050i \(-0.327089\pi\)
0.516892 + 0.856050i \(0.327089\pi\)
\(390\) 0 0
\(391\) 2323.79 0.300561
\(392\) 0 0
\(393\) −5950.36 −0.763756
\(394\) 0 0
\(395\) −515.439 −0.0656572
\(396\) 0 0
\(397\) −7087.28 −0.895970 −0.447985 0.894041i \(-0.647858\pi\)
−0.447985 + 0.894041i \(0.647858\pi\)
\(398\) 0 0
\(399\) 330.659 0.0414878
\(400\) 0 0
\(401\) −2459.84 −0.306331 −0.153166 0.988201i \(-0.548947\pi\)
−0.153166 + 0.988201i \(0.548947\pi\)
\(402\) 0 0
\(403\) −13999.9 −1.73048
\(404\) 0 0
\(405\) 90.9716 0.0111615
\(406\) 0 0
\(407\) −2726.33 −0.332038
\(408\) 0 0
\(409\) −1755.37 −0.212218 −0.106109 0.994354i \(-0.533839\pi\)
−0.106109 + 0.994354i \(0.533839\pi\)
\(410\) 0 0
\(411\) −5270.24 −0.632510
\(412\) 0 0
\(413\) 13.0305 0.00155252
\(414\) 0 0
\(415\) 703.175 0.0831747
\(416\) 0 0
\(417\) −4360.15 −0.512032
\(418\) 0 0
\(419\) −10632.1 −1.23965 −0.619823 0.784742i \(-0.712796\pi\)
−0.619823 + 0.784742i \(0.712796\pi\)
\(420\) 0 0
\(421\) −4067.83 −0.470912 −0.235456 0.971885i \(-0.575658\pi\)
−0.235456 + 0.971885i \(0.575658\pi\)
\(422\) 0 0
\(423\) −256.858 −0.0295245
\(424\) 0 0
\(425\) −2748.97 −0.313752
\(426\) 0 0
\(427\) 548.575 0.0621720
\(428\) 0 0
\(429\) 2346.06 0.264030
\(430\) 0 0
\(431\) 13789.1 1.54106 0.770532 0.637401i \(-0.219990\pi\)
0.770532 + 0.637401i \(0.219990\pi\)
\(432\) 0 0
\(433\) 15964.2 1.77180 0.885901 0.463873i \(-0.153541\pi\)
0.885901 + 0.463873i \(0.153541\pi\)
\(434\) 0 0
\(435\) −1021.58 −0.112600
\(436\) 0 0
\(437\) −13147.6 −1.43921
\(438\) 0 0
\(439\) 7233.47 0.786412 0.393206 0.919450i \(-0.371366\pi\)
0.393206 + 0.919450i \(0.371366\pi\)
\(440\) 0 0
\(441\) −3080.08 −0.332586
\(442\) 0 0
\(443\) −8411.08 −0.902082 −0.451041 0.892503i \(-0.648947\pi\)
−0.451041 + 0.892503i \(0.648947\pi\)
\(444\) 0 0
\(445\) −1764.04 −0.187918
\(446\) 0 0
\(447\) 2235.45 0.236540
\(448\) 0 0
\(449\) −1821.18 −0.191418 −0.0957090 0.995409i \(-0.530512\pi\)
−0.0957090 + 0.995409i \(0.530512\pi\)
\(450\) 0 0
\(451\) 4933.08 0.515055
\(452\) 0 0
\(453\) 7942.58 0.823786
\(454\) 0 0
\(455\) 70.0154 0.00721400
\(456\) 0 0
\(457\) −8637.94 −0.884170 −0.442085 0.896973i \(-0.645761\pi\)
−0.442085 + 0.896973i \(0.645761\pi\)
\(458\) 0 0
\(459\) 599.829 0.0609970
\(460\) 0 0
\(461\) 1775.51 0.179379 0.0896894 0.995970i \(-0.471413\pi\)
0.0896894 + 0.995970i \(0.471413\pi\)
\(462\) 0 0
\(463\) −14687.5 −1.47427 −0.737134 0.675746i \(-0.763822\pi\)
−0.737134 + 0.675746i \(0.763822\pi\)
\(464\) 0 0
\(465\) 663.500 0.0661701
\(466\) 0 0
\(467\) 12523.4 1.24093 0.620463 0.784236i \(-0.286945\pi\)
0.620463 + 0.784236i \(0.286945\pi\)
\(468\) 0 0
\(469\) 586.416 0.0577360
\(470\) 0 0
\(471\) 7612.65 0.744739
\(472\) 0 0
\(473\) −2160.08 −0.209980
\(474\) 0 0
\(475\) 15553.1 1.50237
\(476\) 0 0
\(477\) −346.006 −0.0332128
\(478\) 0 0
\(479\) −8938.18 −0.852601 −0.426301 0.904582i \(-0.640183\pi\)
−0.426301 + 0.904582i \(0.640183\pi\)
\(480\) 0 0
\(481\) −17620.2 −1.67030
\(482\) 0 0
\(483\) −275.170 −0.0259228
\(484\) 0 0
\(485\) 929.170 0.0869926
\(486\) 0 0
\(487\) −7493.41 −0.697246 −0.348623 0.937263i \(-0.613351\pi\)
−0.348623 + 0.937263i \(0.613351\pi\)
\(488\) 0 0
\(489\) 254.591 0.0235440
\(490\) 0 0
\(491\) −2890.76 −0.265699 −0.132849 0.991136i \(-0.542413\pi\)
−0.132849 + 0.991136i \(0.542413\pi\)
\(492\) 0 0
\(493\) −6735.88 −0.615352
\(494\) 0 0
\(495\) −111.187 −0.0100960
\(496\) 0 0
\(497\) −157.178 −0.0141859
\(498\) 0 0
\(499\) −14757.4 −1.32391 −0.661957 0.749542i \(-0.730274\pi\)
−0.661957 + 0.749542i \(0.730274\pi\)
\(500\) 0 0
\(501\) 12399.1 1.10569
\(502\) 0 0
\(503\) −4214.22 −0.373564 −0.186782 0.982401i \(-0.559806\pi\)
−0.186782 + 0.982401i \(0.559806\pi\)
\(504\) 0 0
\(505\) 154.823 0.0136426
\(506\) 0 0
\(507\) 8571.56 0.750841
\(508\) 0 0
\(509\) −15335.5 −1.33543 −0.667716 0.744416i \(-0.732728\pi\)
−0.667716 + 0.744416i \(0.732728\pi\)
\(510\) 0 0
\(511\) −923.875 −0.0799800
\(512\) 0 0
\(513\) −3393.72 −0.292078
\(514\) 0 0
\(515\) −700.090 −0.0599023
\(516\) 0 0
\(517\) 313.937 0.0267059
\(518\) 0 0
\(519\) −3531.88 −0.298713
\(520\) 0 0
\(521\) −6502.03 −0.546755 −0.273377 0.961907i \(-0.588141\pi\)
−0.273377 + 0.961907i \(0.588141\pi\)
\(522\) 0 0
\(523\) −6816.32 −0.569898 −0.284949 0.958543i \(-0.591977\pi\)
−0.284949 + 0.958543i \(0.591977\pi\)
\(524\) 0 0
\(525\) 325.517 0.0270604
\(526\) 0 0
\(527\) 4374.85 0.361616
\(528\) 0 0
\(529\) −1225.76 −0.100745
\(530\) 0 0
\(531\) −133.739 −0.0109299
\(532\) 0 0
\(533\) 31882.4 2.59096
\(534\) 0 0
\(535\) −1118.69 −0.0904022
\(536\) 0 0
\(537\) 723.261 0.0581211
\(538\) 0 0
\(539\) 3764.54 0.300835
\(540\) 0 0
\(541\) −6746.57 −0.536151 −0.268076 0.963398i \(-0.586388\pi\)
−0.268076 + 0.963398i \(0.586388\pi\)
\(542\) 0 0
\(543\) 6377.56 0.504028
\(544\) 0 0
\(545\) 637.003 0.0500664
\(546\) 0 0
\(547\) 14961.8 1.16951 0.584754 0.811210i \(-0.301191\pi\)
0.584754 + 0.811210i \(0.301191\pi\)
\(548\) 0 0
\(549\) −5630.30 −0.437696
\(550\) 0 0
\(551\) 38110.3 2.94655
\(552\) 0 0
\(553\) 402.443 0.0309469
\(554\) 0 0
\(555\) 835.080 0.0638688
\(556\) 0 0
\(557\) −3962.80 −0.301453 −0.150727 0.988575i \(-0.548161\pi\)
−0.150727 + 0.988575i \(0.548161\pi\)
\(558\) 0 0
\(559\) −13960.6 −1.05630
\(560\) 0 0
\(561\) −733.125 −0.0551739
\(562\) 0 0
\(563\) −3231.60 −0.241910 −0.120955 0.992658i \(-0.538596\pi\)
−0.120955 + 0.992658i \(0.538596\pi\)
\(564\) 0 0
\(565\) −494.890 −0.0368499
\(566\) 0 0
\(567\) −71.0284 −0.00526087
\(568\) 0 0
\(569\) −18512.3 −1.36393 −0.681965 0.731385i \(-0.738874\pi\)
−0.681965 + 0.731385i \(0.738874\pi\)
\(570\) 0 0
\(571\) 1859.16 0.136258 0.0681292 0.997677i \(-0.478297\pi\)
0.0681292 + 0.997677i \(0.478297\pi\)
\(572\) 0 0
\(573\) −10044.8 −0.732333
\(574\) 0 0
\(575\) −12943.1 −0.938722
\(576\) 0 0
\(577\) 5922.28 0.427292 0.213646 0.976911i \(-0.431466\pi\)
0.213646 + 0.976911i \(0.431466\pi\)
\(578\) 0 0
\(579\) −4312.83 −0.309560
\(580\) 0 0
\(581\) −549.022 −0.0392036
\(582\) 0 0
\(583\) 422.896 0.0300421
\(584\) 0 0
\(585\) −718.602 −0.0507873
\(586\) 0 0
\(587\) −2908.32 −0.204496 −0.102248 0.994759i \(-0.532604\pi\)
−0.102248 + 0.994759i \(0.532604\pi\)
\(588\) 0 0
\(589\) −24752.0 −1.73156
\(590\) 0 0
\(591\) −9828.26 −0.684062
\(592\) 0 0
\(593\) −10853.0 −0.751565 −0.375782 0.926708i \(-0.622626\pi\)
−0.375782 + 0.926708i \(0.622626\pi\)
\(594\) 0 0
\(595\) −21.8792 −0.00150750
\(596\) 0 0
\(597\) 13438.8 0.921298
\(598\) 0 0
\(599\) 15577.7 1.06258 0.531290 0.847190i \(-0.321707\pi\)
0.531290 + 0.847190i \(0.321707\pi\)
\(600\) 0 0
\(601\) −1353.97 −0.0918961 −0.0459481 0.998944i \(-0.514631\pi\)
−0.0459481 + 0.998944i \(0.514631\pi\)
\(602\) 0 0
\(603\) −6018.68 −0.406467
\(604\) 0 0
\(605\) 135.896 0.00913215
\(606\) 0 0
\(607\) 16849.6 1.12670 0.563348 0.826220i \(-0.309513\pi\)
0.563348 + 0.826220i \(0.309513\pi\)
\(608\) 0 0
\(609\) 797.625 0.0530729
\(610\) 0 0
\(611\) 2028.97 0.134343
\(612\) 0 0
\(613\) −6436.00 −0.424058 −0.212029 0.977263i \(-0.568007\pi\)
−0.212029 + 0.977263i \(0.568007\pi\)
\(614\) 0 0
\(615\) −1511.01 −0.0990729
\(616\) 0 0
\(617\) −4556.21 −0.297287 −0.148644 0.988891i \(-0.547491\pi\)
−0.148644 + 0.988891i \(0.547491\pi\)
\(618\) 0 0
\(619\) −15419.6 −1.00124 −0.500620 0.865667i \(-0.666895\pi\)
−0.500620 + 0.865667i \(0.666895\pi\)
\(620\) 0 0
\(621\) 2824.21 0.182499
\(622\) 0 0
\(623\) 1377.32 0.0885734
\(624\) 0 0
\(625\) 15153.6 0.969829
\(626\) 0 0
\(627\) 4147.87 0.264195
\(628\) 0 0
\(629\) 5506.18 0.349039
\(630\) 0 0
\(631\) 10148.5 0.640265 0.320132 0.947373i \(-0.396273\pi\)
0.320132 + 0.947373i \(0.396273\pi\)
\(632\) 0 0
\(633\) 12979.8 0.815006
\(634\) 0 0
\(635\) −598.651 −0.0374122
\(636\) 0 0
\(637\) 24330.2 1.51334
\(638\) 0 0
\(639\) 1613.20 0.0998704
\(640\) 0 0
\(641\) −24451.5 −1.50667 −0.753336 0.657635i \(-0.771557\pi\)
−0.753336 + 0.657635i \(0.771557\pi\)
\(642\) 0 0
\(643\) 24688.2 1.51416 0.757082 0.653320i \(-0.226624\pi\)
0.757082 + 0.653320i \(0.226624\pi\)
\(644\) 0 0
\(645\) 661.637 0.0403906
\(646\) 0 0
\(647\) 7484.57 0.454789 0.227395 0.973803i \(-0.426979\pi\)
0.227395 + 0.973803i \(0.426979\pi\)
\(648\) 0 0
\(649\) 163.459 0.00988646
\(650\) 0 0
\(651\) −518.045 −0.0311886
\(652\) 0 0
\(653\) 12122.3 0.726465 0.363233 0.931699i \(-0.381673\pi\)
0.363233 + 0.931699i \(0.381673\pi\)
\(654\) 0 0
\(655\) −2227.63 −0.132887
\(656\) 0 0
\(657\) 9482.18 0.563067
\(658\) 0 0
\(659\) 31197.8 1.84415 0.922074 0.387013i \(-0.126493\pi\)
0.922074 + 0.387013i \(0.126493\pi\)
\(660\) 0 0
\(661\) 3983.93 0.234428 0.117214 0.993107i \(-0.462604\pi\)
0.117214 + 0.993107i \(0.462604\pi\)
\(662\) 0 0
\(663\) −4738.17 −0.277550
\(664\) 0 0
\(665\) 123.788 0.00721850
\(666\) 0 0
\(667\) −31714.9 −1.84109
\(668\) 0 0
\(669\) −2696.91 −0.155857
\(670\) 0 0
\(671\) 6881.48 0.395911
\(672\) 0 0
\(673\) 159.587 0.00914060 0.00457030 0.999990i \(-0.498545\pi\)
0.00457030 + 0.999990i \(0.498545\pi\)
\(674\) 0 0
\(675\) −3340.94 −0.190508
\(676\) 0 0
\(677\) −23774.6 −1.34968 −0.674838 0.737965i \(-0.735787\pi\)
−0.674838 + 0.737965i \(0.735787\pi\)
\(678\) 0 0
\(679\) −725.474 −0.0410031
\(680\) 0 0
\(681\) −1804.59 −0.101545
\(682\) 0 0
\(683\) 17460.9 0.978215 0.489108 0.872223i \(-0.337323\pi\)
0.489108 + 0.872223i \(0.337323\pi\)
\(684\) 0 0
\(685\) −1973.01 −0.110051
\(686\) 0 0
\(687\) 9431.83 0.523794
\(688\) 0 0
\(689\) 2733.17 0.151125
\(690\) 0 0
\(691\) −25778.2 −1.41917 −0.709587 0.704617i \(-0.751119\pi\)
−0.709587 + 0.704617i \(0.751119\pi\)
\(692\) 0 0
\(693\) 86.8125 0.00475864
\(694\) 0 0
\(695\) −1632.30 −0.0890889
\(696\) 0 0
\(697\) −9962.99 −0.541428
\(698\) 0 0
\(699\) 5097.54 0.275832
\(700\) 0 0
\(701\) −5531.04 −0.298009 −0.149005 0.988837i \(-0.547607\pi\)
−0.149005 + 0.988837i \(0.547607\pi\)
\(702\) 0 0
\(703\) −31152.9 −1.67134
\(704\) 0 0
\(705\) −96.1595 −0.00513699
\(706\) 0 0
\(707\) −120.882 −0.00643031
\(708\) 0 0
\(709\) −5591.04 −0.296158 −0.148079 0.988976i \(-0.547309\pi\)
−0.148079 + 0.988976i \(0.547309\pi\)
\(710\) 0 0
\(711\) −4130.47 −0.217869
\(712\) 0 0
\(713\) 20598.3 1.08193
\(714\) 0 0
\(715\) 878.292 0.0459388
\(716\) 0 0
\(717\) 21612.1 1.12569
\(718\) 0 0
\(719\) −16219.3 −0.841276 −0.420638 0.907228i \(-0.638194\pi\)
−0.420638 + 0.907228i \(0.638194\pi\)
\(720\) 0 0
\(721\) 546.614 0.0282344
\(722\) 0 0
\(723\) 10042.6 0.516583
\(724\) 0 0
\(725\) 37517.6 1.92189
\(726\) 0 0
\(727\) −34227.8 −1.74613 −0.873067 0.487601i \(-0.837872\pi\)
−0.873067 + 0.487601i \(0.837872\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 4362.56 0.220732
\(732\) 0 0
\(733\) −12844.2 −0.647220 −0.323610 0.946191i \(-0.604896\pi\)
−0.323610 + 0.946191i \(0.604896\pi\)
\(734\) 0 0
\(735\) −1153.08 −0.0578669
\(736\) 0 0
\(737\) 7356.17 0.367663
\(738\) 0 0
\(739\) 11145.6 0.554800 0.277400 0.960755i \(-0.410527\pi\)
0.277400 + 0.960755i \(0.410527\pi\)
\(740\) 0 0
\(741\) 26807.6 1.32902
\(742\) 0 0
\(743\) −15771.0 −0.778711 −0.389356 0.921088i \(-0.627302\pi\)
−0.389356 + 0.921088i \(0.627302\pi\)
\(744\) 0 0
\(745\) 836.884 0.0411558
\(746\) 0 0
\(747\) 5634.89 0.275997
\(748\) 0 0
\(749\) 873.446 0.0426102
\(750\) 0 0
\(751\) 18340.4 0.891146 0.445573 0.895246i \(-0.353000\pi\)
0.445573 + 0.895246i \(0.353000\pi\)
\(752\) 0 0
\(753\) −152.602 −0.00738529
\(754\) 0 0
\(755\) 2973.45 0.143331
\(756\) 0 0
\(757\) −3727.76 −0.178980 −0.0894900 0.995988i \(-0.528524\pi\)
−0.0894900 + 0.995988i \(0.528524\pi\)
\(758\) 0 0
\(759\) −3451.81 −0.165076
\(760\) 0 0
\(761\) 36537.0 1.74043 0.870214 0.492674i \(-0.163980\pi\)
0.870214 + 0.492674i \(0.163980\pi\)
\(762\) 0 0
\(763\) −497.357 −0.0235983
\(764\) 0 0
\(765\) 224.557 0.0106129
\(766\) 0 0
\(767\) 1056.43 0.0497333
\(768\) 0 0
\(769\) −4108.97 −0.192683 −0.0963416 0.995348i \(-0.530714\pi\)
−0.0963416 + 0.995348i \(0.530714\pi\)
\(770\) 0 0
\(771\) 12649.6 0.590874
\(772\) 0 0
\(773\) 22430.1 1.04367 0.521833 0.853048i \(-0.325248\pi\)
0.521833 + 0.853048i \(0.325248\pi\)
\(774\) 0 0
\(775\) −24367.1 −1.12941
\(776\) 0 0
\(777\) −652.011 −0.0301039
\(778\) 0 0
\(779\) 56368.6 2.59257
\(780\) 0 0
\(781\) −1971.69 −0.0903362
\(782\) 0 0
\(783\) −8186.42 −0.373638
\(784\) 0 0
\(785\) 2849.94 0.129578
\(786\) 0 0
\(787\) 12213.2 0.553181 0.276590 0.960988i \(-0.410795\pi\)
0.276590 + 0.960988i \(0.410795\pi\)
\(788\) 0 0
\(789\) 6001.14 0.270781
\(790\) 0 0
\(791\) 386.398 0.0173688
\(792\) 0 0
\(793\) 44474.9 1.99161
\(794\) 0 0
\(795\) −129.534 −0.00577873
\(796\) 0 0
\(797\) −30552.4 −1.35787 −0.678934 0.734199i \(-0.737558\pi\)
−0.678934 + 0.734199i \(0.737558\pi\)
\(798\) 0 0
\(799\) −634.036 −0.0280733
\(800\) 0 0
\(801\) −14136.1 −0.623565
\(802\) 0 0
\(803\) −11589.3 −0.509313
\(804\) 0 0
\(805\) −103.015 −0.00451032
\(806\) 0 0
\(807\) 6915.53 0.301658
\(808\) 0 0
\(809\) 17537.6 0.762163 0.381081 0.924541i \(-0.375552\pi\)
0.381081 + 0.924541i \(0.375552\pi\)
\(810\) 0 0
\(811\) −42391.0 −1.83545 −0.917725 0.397217i \(-0.869976\pi\)
−0.917725 + 0.397217i \(0.869976\pi\)
\(812\) 0 0
\(813\) 14004.6 0.604134
\(814\) 0 0
\(815\) 95.3108 0.00409643
\(816\) 0 0
\(817\) −24682.5 −1.05695
\(818\) 0 0
\(819\) 561.068 0.0239381
\(820\) 0 0
\(821\) 3922.86 0.166759 0.0833793 0.996518i \(-0.473429\pi\)
0.0833793 + 0.996518i \(0.473429\pi\)
\(822\) 0 0
\(823\) −10990.0 −0.465477 −0.232739 0.972539i \(-0.574769\pi\)
−0.232739 + 0.972539i \(0.574769\pi\)
\(824\) 0 0
\(825\) 4083.37 0.172321
\(826\) 0 0
\(827\) 41723.5 1.75438 0.877188 0.480146i \(-0.159416\pi\)
0.877188 + 0.480146i \(0.159416\pi\)
\(828\) 0 0
\(829\) 37425.8 1.56798 0.783989 0.620775i \(-0.213182\pi\)
0.783989 + 0.620775i \(0.213182\pi\)
\(830\) 0 0
\(831\) −8801.88 −0.367429
\(832\) 0 0
\(833\) −7602.97 −0.316239
\(834\) 0 0
\(835\) 4641.83 0.192380
\(836\) 0 0
\(837\) 5316.95 0.219571
\(838\) 0 0
\(839\) −43490.7 −1.78959 −0.894794 0.446479i \(-0.852678\pi\)
−0.894794 + 0.446479i \(0.852678\pi\)
\(840\) 0 0
\(841\) 67541.7 2.76935
\(842\) 0 0
\(843\) −13186.2 −0.538739
\(844\) 0 0
\(845\) 3208.92 0.130639
\(846\) 0 0
\(847\) −106.104 −0.00430435
\(848\) 0 0
\(849\) 4581.20 0.185190
\(850\) 0 0
\(851\) 25925.0 1.04430
\(852\) 0 0
\(853\) 6522.01 0.261793 0.130896 0.991396i \(-0.458214\pi\)
0.130896 + 0.991396i \(0.458214\pi\)
\(854\) 0 0
\(855\) −1270.50 −0.0508189
\(856\) 0 0
\(857\) −29468.6 −1.17459 −0.587297 0.809371i \(-0.699808\pi\)
−0.587297 + 0.809371i \(0.699808\pi\)
\(858\) 0 0
\(859\) 16576.4 0.658416 0.329208 0.944257i \(-0.393218\pi\)
0.329208 + 0.944257i \(0.393218\pi\)
\(860\) 0 0
\(861\) 1179.76 0.0466971
\(862\) 0 0
\(863\) 36972.2 1.45834 0.729170 0.684332i \(-0.239906\pi\)
0.729170 + 0.684332i \(0.239906\pi\)
\(864\) 0 0
\(865\) −1322.22 −0.0519733
\(866\) 0 0
\(867\) −13258.4 −0.519351
\(868\) 0 0
\(869\) 5048.35 0.197070
\(870\) 0 0
\(871\) 47542.8 1.84951
\(872\) 0 0
\(873\) 7445.90 0.288666
\(874\) 0 0
\(875\) 244.969 0.00946453
\(876\) 0 0
\(877\) −44872.7 −1.72776 −0.863878 0.503701i \(-0.831972\pi\)
−0.863878 + 0.503701i \(0.831972\pi\)
\(878\) 0 0
\(879\) 13923.4 0.534272
\(880\) 0 0
\(881\) −33961.3 −1.29873 −0.649367 0.760475i \(-0.724966\pi\)
−0.649367 + 0.760475i \(0.724966\pi\)
\(882\) 0 0
\(883\) 30892.1 1.17735 0.588676 0.808369i \(-0.299650\pi\)
0.588676 + 0.808369i \(0.299650\pi\)
\(884\) 0 0
\(885\) −50.0676 −0.00190170
\(886\) 0 0
\(887\) 7879.82 0.298284 0.149142 0.988816i \(-0.452349\pi\)
0.149142 + 0.988816i \(0.452349\pi\)
\(888\) 0 0
\(889\) 467.413 0.0176339
\(890\) 0 0
\(891\) −891.000 −0.0335013
\(892\) 0 0
\(893\) 3587.25 0.134426
\(894\) 0 0
\(895\) 270.766 0.0101125
\(896\) 0 0
\(897\) −22309.0 −0.830408
\(898\) 0 0
\(899\) −59707.6 −2.21508
\(900\) 0 0
\(901\) −854.092 −0.0315804
\(902\) 0 0
\(903\) −516.590 −0.0190377
\(904\) 0 0
\(905\) 2387.56 0.0876962
\(906\) 0 0
\(907\) 12109.4 0.443316 0.221658 0.975124i \(-0.428853\pi\)
0.221658 + 0.975124i \(0.428853\pi\)
\(908\) 0 0
\(909\) 1240.67 0.0452700
\(910\) 0 0
\(911\) −27148.2 −0.987334 −0.493667 0.869651i \(-0.664344\pi\)
−0.493667 + 0.869651i \(0.664344\pi\)
\(912\) 0 0
\(913\) −6887.08 −0.249649
\(914\) 0 0
\(915\) −2107.81 −0.0761552
\(916\) 0 0
\(917\) 1739.28 0.0626348
\(918\) 0 0
\(919\) −2030.55 −0.0728854 −0.0364427 0.999336i \(-0.511603\pi\)
−0.0364427 + 0.999336i \(0.511603\pi\)
\(920\) 0 0
\(921\) −17755.9 −0.635263
\(922\) 0 0
\(923\) −12743.0 −0.454432
\(924\) 0 0
\(925\) −30668.4 −1.09013
\(926\) 0 0
\(927\) −5610.17 −0.198773
\(928\) 0 0
\(929\) 16071.8 0.567598 0.283799 0.958884i \(-0.408405\pi\)
0.283799 + 0.958884i \(0.408405\pi\)
\(930\) 0 0
\(931\) 43016.1 1.51428
\(932\) 0 0
\(933\) −13867.4 −0.486601
\(934\) 0 0
\(935\) −274.459 −0.00959975
\(936\) 0 0
\(937\) 771.363 0.0268936 0.0134468 0.999910i \(-0.495720\pi\)
0.0134468 + 0.999910i \(0.495720\pi\)
\(938\) 0 0
\(939\) 29586.6 1.02824
\(940\) 0 0
\(941\) 16905.2 0.585645 0.292823 0.956167i \(-0.405405\pi\)
0.292823 + 0.956167i \(0.405405\pi\)
\(942\) 0 0
\(943\) −46909.3 −1.61991
\(944\) 0 0
\(945\) −26.5908 −0.000915343 0
\(946\) 0 0
\(947\) −32050.0 −1.09977 −0.549886 0.835240i \(-0.685329\pi\)
−0.549886 + 0.835240i \(0.685329\pi\)
\(948\) 0 0
\(949\) −74901.6 −2.56208
\(950\) 0 0
\(951\) 20393.0 0.695362
\(952\) 0 0
\(953\) 1653.20 0.0561934 0.0280967 0.999605i \(-0.491055\pi\)
0.0280967 + 0.999605i \(0.491055\pi\)
\(954\) 0 0
\(955\) −3760.45 −0.127419
\(956\) 0 0
\(957\) 10005.6 0.337969
\(958\) 0 0
\(959\) 1540.48 0.0518714
\(960\) 0 0
\(961\) 8988.15 0.301707
\(962\) 0 0
\(963\) −8964.61 −0.299980
\(964\) 0 0
\(965\) −1614.59 −0.0538605
\(966\) 0 0
\(967\) −47148.3 −1.56793 −0.783964 0.620807i \(-0.786805\pi\)
−0.783964 + 0.620807i \(0.786805\pi\)
\(968\) 0 0
\(969\) −8377.16 −0.277723
\(970\) 0 0
\(971\) 47880.7 1.58245 0.791227 0.611522i \(-0.209443\pi\)
0.791227 + 0.611522i \(0.209443\pi\)
\(972\) 0 0
\(973\) 1274.46 0.0419912
\(974\) 0 0
\(975\) 26390.8 0.866853
\(976\) 0 0
\(977\) −42111.0 −1.37897 −0.689483 0.724302i \(-0.742162\pi\)
−0.689483 + 0.724302i \(0.742162\pi\)
\(978\) 0 0
\(979\) 17277.5 0.564036
\(980\) 0 0
\(981\) 5104.62 0.166135
\(982\) 0 0
\(983\) 1243.50 0.0403474 0.0201737 0.999796i \(-0.493578\pi\)
0.0201737 + 0.999796i \(0.493578\pi\)
\(984\) 0 0
\(985\) −3679.39 −0.119020
\(986\) 0 0
\(987\) 75.0790 0.00242127
\(988\) 0 0
\(989\) 20540.5 0.660414
\(990\) 0 0
\(991\) −3354.93 −0.107541 −0.0537703 0.998553i \(-0.517124\pi\)
−0.0537703 + 0.998553i \(0.517124\pi\)
\(992\) 0 0
\(993\) 17542.7 0.560624
\(994\) 0 0
\(995\) 5031.07 0.160297
\(996\) 0 0
\(997\) −15908.8 −0.505353 −0.252677 0.967551i \(-0.581311\pi\)
−0.252677 + 0.967551i \(0.581311\pi\)
\(998\) 0 0
\(999\) 6691.91 0.211935
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 528.4.a.r.1.2 2
3.2 odd 2 1584.4.a.bf.1.1 2
4.3 odd 2 264.4.a.e.1.2 2
8.3 odd 2 2112.4.a.bl.1.1 2
8.5 even 2 2112.4.a.be.1.1 2
12.11 even 2 792.4.a.h.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
264.4.a.e.1.2 2 4.3 odd 2
528.4.a.r.1.2 2 1.1 even 1 trivial
792.4.a.h.1.1 2 12.11 even 2
1584.4.a.bf.1.1 2 3.2 odd 2
2112.4.a.be.1.1 2 8.5 even 2
2112.4.a.bl.1.1 2 8.3 odd 2