Properties

Label 693.4.a.m.1.3
Level $693$
Weight $4$
Character 693.1
Self dual yes
Analytic conductor $40.888$
Analytic rank $1$
Dimension $4$
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] = [693,4,Mod(1,693)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(693, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("693.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 693.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: \(40.8883236340\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.509800.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 10x^{2} + 5x + 15 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.11082\) of defining polynomial
Character \(\chi\) \(=\) 693.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.65527 q^{2} +5.36103 q^{4} +10.0822 q^{5} -7.00000 q^{7} -9.64616 q^{8} +O(q^{10})\) \(q+3.65527 q^{2} +5.36103 q^{4} +10.0822 q^{5} -7.00000 q^{7} -9.64616 q^{8} +36.8533 q^{10} -11.0000 q^{11} -84.5724 q^{13} -25.5869 q^{14} -78.1476 q^{16} +38.2525 q^{17} -127.283 q^{19} +54.0511 q^{20} -40.2080 q^{22} -140.378 q^{23} -23.3486 q^{25} -309.135 q^{26} -37.5272 q^{28} +116.806 q^{29} +338.709 q^{31} -208.482 q^{32} +139.823 q^{34} -70.5756 q^{35} -75.3416 q^{37} -465.256 q^{38} -97.2548 q^{40} +22.4446 q^{41} +181.844 q^{43} -58.9713 q^{44} -513.121 q^{46} -300.530 q^{47} +49.0000 q^{49} -85.3455 q^{50} -453.395 q^{52} +31.8596 q^{53} -110.905 q^{55} +67.5231 q^{56} +426.958 q^{58} +68.3030 q^{59} -145.315 q^{61} +1238.07 q^{62} -136.877 q^{64} -852.679 q^{65} -668.020 q^{67} +205.073 q^{68} -257.973 q^{70} -727.608 q^{71} -416.982 q^{73} -275.394 q^{74} -682.370 q^{76} +77.0000 q^{77} +458.805 q^{79} -787.902 q^{80} +82.0412 q^{82} -355.737 q^{83} +385.671 q^{85} +664.690 q^{86} +106.108 q^{88} +1245.97 q^{89} +592.007 q^{91} -752.571 q^{92} -1098.52 q^{94} -1283.30 q^{95} -935.338 q^{97} +179.108 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 22 q^{4} + 18 q^{5} - 28 q^{7} + 60 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 22 q^{4} + 18 q^{5} - 28 q^{7} + 60 q^{8} - 92 q^{10} - 44 q^{11} - 134 q^{13} - 28 q^{14} - 6 q^{16} + 74 q^{17} - 164 q^{19} - 116 q^{20} - 44 q^{22} - 194 q^{23} + 38 q^{25} - 734 q^{26} - 154 q^{28} + 108 q^{29} - 412 q^{31} + 4 q^{32} - 346 q^{34} - 126 q^{35} + 286 q^{37} - 224 q^{38} - 540 q^{40} + 18 q^{41} - 496 q^{43} - 242 q^{44} - 284 q^{46} - 62 q^{47} + 196 q^{49} - 212 q^{50} - 822 q^{52} + 828 q^{53} - 198 q^{55} - 420 q^{56} + 1388 q^{58} + 1224 q^{59} - 350 q^{61} + 878 q^{62} - 718 q^{64} + 396 q^{65} - 1498 q^{67} - 1058 q^{68} + 644 q^{70} - 2326 q^{71} - 1630 q^{73} + 1156 q^{74} - 3152 q^{76} + 308 q^{77} - 1020 q^{79} - 3072 q^{80} + 2118 q^{82} + 1920 q^{83} + 2008 q^{85} - 1056 q^{86} - 660 q^{88} - 1550 q^{89} + 938 q^{91} - 2592 q^{92} - 1042 q^{94} - 2332 q^{95} - 2202 q^{97} + 196 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.65527 1.29233 0.646167 0.763196i \(-0.276371\pi\)
0.646167 + 0.763196i \(0.276371\pi\)
\(3\) 0 0
\(4\) 5.36103 0.670129
\(5\) 10.0822 0.901782 0.450891 0.892579i \(-0.351106\pi\)
0.450891 + 0.892579i \(0.351106\pi\)
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) −9.64616 −0.426304
\(9\) 0 0
\(10\) 36.8533 1.16540
\(11\) −11.0000 −0.301511
\(12\) 0 0
\(13\) −84.5724 −1.80432 −0.902161 0.431400i \(-0.858020\pi\)
−0.902161 + 0.431400i \(0.858020\pi\)
\(14\) −25.5869 −0.488457
\(15\) 0 0
\(16\) −78.1476 −1.22106
\(17\) 38.2525 0.545741 0.272871 0.962051i \(-0.412027\pi\)
0.272871 + 0.962051i \(0.412027\pi\)
\(18\) 0 0
\(19\) −127.283 −1.53688 −0.768442 0.639919i \(-0.778968\pi\)
−0.768442 + 0.639919i \(0.778968\pi\)
\(20\) 54.0511 0.604310
\(21\) 0 0
\(22\) −40.2080 −0.389654
\(23\) −140.378 −1.27265 −0.636323 0.771423i \(-0.719545\pi\)
−0.636323 + 0.771423i \(0.719545\pi\)
\(24\) 0 0
\(25\) −23.3486 −0.186789
\(26\) −309.135 −2.33179
\(27\) 0 0
\(28\) −37.5272 −0.253285
\(29\) 116.806 0.747943 0.373971 0.927440i \(-0.377996\pi\)
0.373971 + 0.927440i \(0.377996\pi\)
\(30\) 0 0
\(31\) 338.709 1.96238 0.981192 0.193034i \(-0.0618327\pi\)
0.981192 + 0.193034i \(0.0618327\pi\)
\(32\) −208.482 −1.15171
\(33\) 0 0
\(34\) 139.823 0.705280
\(35\) −70.5756 −0.340842
\(36\) 0 0
\(37\) −75.3416 −0.334759 −0.167379 0.985893i \(-0.553531\pi\)
−0.167379 + 0.985893i \(0.553531\pi\)
\(38\) −465.256 −1.98617
\(39\) 0 0
\(40\) −97.2548 −0.384434
\(41\) 22.4446 0.0854941 0.0427471 0.999086i \(-0.486389\pi\)
0.0427471 + 0.999086i \(0.486389\pi\)
\(42\) 0 0
\(43\) 181.844 0.644906 0.322453 0.946585i \(-0.395492\pi\)
0.322453 + 0.946585i \(0.395492\pi\)
\(44\) −58.9713 −0.202051
\(45\) 0 0
\(46\) −513.121 −1.64468
\(47\) −300.530 −0.932698 −0.466349 0.884601i \(-0.654431\pi\)
−0.466349 + 0.884601i \(0.654431\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) −85.3455 −0.241394
\(51\) 0 0
\(52\) −453.395 −1.20913
\(53\) 31.8596 0.0825708 0.0412854 0.999147i \(-0.486855\pi\)
0.0412854 + 0.999147i \(0.486855\pi\)
\(54\) 0 0
\(55\) −110.905 −0.271898
\(56\) 67.5231 0.161128
\(57\) 0 0
\(58\) 426.958 0.966593
\(59\) 68.3030 0.150717 0.0753584 0.997157i \(-0.475990\pi\)
0.0753584 + 0.997157i \(0.475990\pi\)
\(60\) 0 0
\(61\) −145.315 −0.305012 −0.152506 0.988303i \(-0.548734\pi\)
−0.152506 + 0.988303i \(0.548734\pi\)
\(62\) 1238.07 2.53606
\(63\) 0 0
\(64\) −136.877 −0.267337
\(65\) −852.679 −1.62710
\(66\) 0 0
\(67\) −668.020 −1.21808 −0.609042 0.793138i \(-0.708446\pi\)
−0.609042 + 0.793138i \(0.708446\pi\)
\(68\) 205.073 0.365717
\(69\) 0 0
\(70\) −257.973 −0.440481
\(71\) −727.608 −1.21621 −0.608107 0.793855i \(-0.708071\pi\)
−0.608107 + 0.793855i \(0.708071\pi\)
\(72\) 0 0
\(73\) −416.982 −0.668548 −0.334274 0.942476i \(-0.608491\pi\)
−0.334274 + 0.942476i \(0.608491\pi\)
\(74\) −275.394 −0.432621
\(75\) 0 0
\(76\) −682.370 −1.02991
\(77\) 77.0000 0.113961
\(78\) 0 0
\(79\) 458.805 0.653413 0.326707 0.945126i \(-0.394061\pi\)
0.326707 + 0.945126i \(0.394061\pi\)
\(80\) −787.902 −1.10113
\(81\) 0 0
\(82\) 82.0412 0.110487
\(83\) −355.737 −0.470449 −0.235224 0.971941i \(-0.575582\pi\)
−0.235224 + 0.971941i \(0.575582\pi\)
\(84\) 0 0
\(85\) 385.671 0.492140
\(86\) 664.690 0.833435
\(87\) 0 0
\(88\) 106.108 0.128536
\(89\) 1245.97 1.48396 0.741980 0.670422i \(-0.233887\pi\)
0.741980 + 0.670422i \(0.233887\pi\)
\(90\) 0 0
\(91\) 592.007 0.681969
\(92\) −752.571 −0.852837
\(93\) 0 0
\(94\) −1098.52 −1.20536
\(95\) −1283.30 −1.38594
\(96\) 0 0
\(97\) −935.338 −0.979063 −0.489532 0.871986i \(-0.662832\pi\)
−0.489532 + 0.871986i \(0.662832\pi\)
\(98\) 179.108 0.184619
\(99\) 0 0
\(100\) −125.173 −0.125173
\(101\) 533.395 0.525493 0.262747 0.964865i \(-0.415372\pi\)
0.262747 + 0.964865i \(0.415372\pi\)
\(102\) 0 0
\(103\) −738.096 −0.706086 −0.353043 0.935607i \(-0.614853\pi\)
−0.353043 + 0.935607i \(0.614853\pi\)
\(104\) 815.800 0.769190
\(105\) 0 0
\(106\) 116.456 0.106709
\(107\) 2039.07 1.84228 0.921141 0.389229i \(-0.127259\pi\)
0.921141 + 0.389229i \(0.127259\pi\)
\(108\) 0 0
\(109\) 1488.69 1.30817 0.654085 0.756421i \(-0.273054\pi\)
0.654085 + 0.756421i \(0.273054\pi\)
\(110\) −405.387 −0.351383
\(111\) 0 0
\(112\) 547.033 0.461516
\(113\) 532.743 0.443507 0.221753 0.975103i \(-0.428822\pi\)
0.221753 + 0.975103i \(0.428822\pi\)
\(114\) 0 0
\(115\) −1415.32 −1.14765
\(116\) 626.201 0.501218
\(117\) 0 0
\(118\) 249.666 0.194776
\(119\) −267.768 −0.206271
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) −531.167 −0.394177
\(123\) 0 0
\(124\) 1815.83 1.31505
\(125\) −1495.68 −1.07023
\(126\) 0 0
\(127\) −2257.44 −1.57729 −0.788645 0.614849i \(-0.789217\pi\)
−0.788645 + 0.614849i \(0.789217\pi\)
\(128\) 1167.53 0.806220
\(129\) 0 0
\(130\) −3116.78 −2.10276
\(131\) −1174.21 −0.783142 −0.391571 0.920148i \(-0.628068\pi\)
−0.391571 + 0.920148i \(0.628068\pi\)
\(132\) 0 0
\(133\) 890.984 0.580888
\(134\) −2441.79 −1.57417
\(135\) 0 0
\(136\) −368.990 −0.232652
\(137\) −2690.08 −1.67758 −0.838792 0.544451i \(-0.816738\pi\)
−0.838792 + 0.544451i \(0.816738\pi\)
\(138\) 0 0
\(139\) 17.7500 0.0108312 0.00541559 0.999985i \(-0.498276\pi\)
0.00541559 + 0.999985i \(0.498276\pi\)
\(140\) −378.358 −0.228408
\(141\) 0 0
\(142\) −2659.61 −1.57175
\(143\) 930.297 0.544023
\(144\) 0 0
\(145\) 1177.67 0.674482
\(146\) −1524.18 −0.863988
\(147\) 0 0
\(148\) −403.908 −0.224332
\(149\) −1517.86 −0.834550 −0.417275 0.908780i \(-0.637015\pi\)
−0.417275 + 0.908780i \(0.637015\pi\)
\(150\) 0 0
\(151\) 1948.86 1.05031 0.525153 0.851008i \(-0.324008\pi\)
0.525153 + 0.851008i \(0.324008\pi\)
\(152\) 1227.80 0.655180
\(153\) 0 0
\(154\) 281.456 0.147275
\(155\) 3414.94 1.76964
\(156\) 0 0
\(157\) −1554.20 −0.790055 −0.395027 0.918669i \(-0.629265\pi\)
−0.395027 + 0.918669i \(0.629265\pi\)
\(158\) 1677.06 0.844428
\(159\) 0 0
\(160\) −2101.96 −1.03859
\(161\) 982.647 0.481015
\(162\) 0 0
\(163\) −3472.71 −1.66873 −0.834367 0.551209i \(-0.814167\pi\)
−0.834367 + 0.551209i \(0.814167\pi\)
\(164\) 120.326 0.0572921
\(165\) 0 0
\(166\) −1300.32 −0.607977
\(167\) 2228.90 1.03280 0.516400 0.856347i \(-0.327272\pi\)
0.516400 + 0.856347i \(0.327272\pi\)
\(168\) 0 0
\(169\) 4955.50 2.25558
\(170\) 1409.73 0.636009
\(171\) 0 0
\(172\) 974.872 0.432170
\(173\) 1008.04 0.443005 0.221502 0.975160i \(-0.428904\pi\)
0.221502 + 0.975160i \(0.428904\pi\)
\(174\) 0 0
\(175\) 163.440 0.0705995
\(176\) 859.624 0.368162
\(177\) 0 0
\(178\) 4554.36 1.91777
\(179\) 746.246 0.311603 0.155802 0.987788i \(-0.450204\pi\)
0.155802 + 0.987788i \(0.450204\pi\)
\(180\) 0 0
\(181\) −2787.76 −1.14482 −0.572410 0.819968i \(-0.693991\pi\)
−0.572410 + 0.819968i \(0.693991\pi\)
\(182\) 2163.95 0.881333
\(183\) 0 0
\(184\) 1354.11 0.542534
\(185\) −759.611 −0.301880
\(186\) 0 0
\(187\) −420.778 −0.164547
\(188\) −1611.15 −0.625028
\(189\) 0 0
\(190\) −4690.82 −1.79109
\(191\) 911.917 0.345466 0.172733 0.984969i \(-0.444740\pi\)
0.172733 + 0.984969i \(0.444740\pi\)
\(192\) 0 0
\(193\) 4454.66 1.66142 0.830709 0.556707i \(-0.187935\pi\)
0.830709 + 0.556707i \(0.187935\pi\)
\(194\) −3418.92 −1.26528
\(195\) 0 0
\(196\) 262.690 0.0957327
\(197\) 1377.46 0.498171 0.249086 0.968481i \(-0.419870\pi\)
0.249086 + 0.968481i \(0.419870\pi\)
\(198\) 0 0
\(199\) −94.3572 −0.0336121 −0.0168060 0.999859i \(-0.505350\pi\)
−0.0168060 + 0.999859i \(0.505350\pi\)
\(200\) 225.224 0.0796288
\(201\) 0 0
\(202\) 1949.71 0.679113
\(203\) −817.643 −0.282696
\(204\) 0 0
\(205\) 226.292 0.0770971
\(206\) −2697.95 −0.912499
\(207\) 0 0
\(208\) 6609.13 2.20318
\(209\) 1400.12 0.463388
\(210\) 0 0
\(211\) 1174.19 0.383101 0.191550 0.981483i \(-0.438648\pi\)
0.191550 + 0.981483i \(0.438648\pi\)
\(212\) 170.800 0.0553330
\(213\) 0 0
\(214\) 7453.35 2.38084
\(215\) 1833.40 0.581565
\(216\) 0 0
\(217\) −2370.96 −0.741712
\(218\) 5441.56 1.69059
\(219\) 0 0
\(220\) −594.563 −0.182206
\(221\) −3235.11 −0.984692
\(222\) 0 0
\(223\) 88.5875 0.0266021 0.0133010 0.999912i \(-0.495766\pi\)
0.0133010 + 0.999912i \(0.495766\pi\)
\(224\) 1459.37 0.435305
\(225\) 0 0
\(226\) 1947.32 0.573159
\(227\) −883.312 −0.258271 −0.129135 0.991627i \(-0.541220\pi\)
−0.129135 + 0.991627i \(0.541220\pi\)
\(228\) 0 0
\(229\) 1240.26 0.357898 0.178949 0.983858i \(-0.442730\pi\)
0.178949 + 0.983858i \(0.442730\pi\)
\(230\) −5173.40 −1.48315
\(231\) 0 0
\(232\) −1126.73 −0.318851
\(233\) 5479.93 1.54078 0.770392 0.637571i \(-0.220061\pi\)
0.770392 + 0.637571i \(0.220061\pi\)
\(234\) 0 0
\(235\) −3030.01 −0.841090
\(236\) 366.174 0.101000
\(237\) 0 0
\(238\) −978.764 −0.266571
\(239\) −594.006 −0.160766 −0.0803830 0.996764i \(-0.525614\pi\)
−0.0803830 + 0.996764i \(0.525614\pi\)
\(240\) 0 0
\(241\) 308.785 0.0825336 0.0412668 0.999148i \(-0.486861\pi\)
0.0412668 + 0.999148i \(0.486861\pi\)
\(242\) 442.288 0.117485
\(243\) 0 0
\(244\) −779.039 −0.204397
\(245\) 494.029 0.128826
\(246\) 0 0
\(247\) 10764.7 2.77303
\(248\) −3267.24 −0.836573
\(249\) 0 0
\(250\) −5467.14 −1.38309
\(251\) −3487.59 −0.877031 −0.438515 0.898724i \(-0.644495\pi\)
−0.438515 + 0.898724i \(0.644495\pi\)
\(252\) 0 0
\(253\) 1544.16 0.383717
\(254\) −8251.58 −2.03839
\(255\) 0 0
\(256\) 5362.66 1.30924
\(257\) −451.445 −0.109574 −0.0547868 0.998498i \(-0.517448\pi\)
−0.0547868 + 0.998498i \(0.517448\pi\)
\(258\) 0 0
\(259\) 527.391 0.126527
\(260\) −4571.24 −1.09037
\(261\) 0 0
\(262\) −4292.08 −1.01208
\(263\) 5878.61 1.37829 0.689146 0.724622i \(-0.257986\pi\)
0.689146 + 0.724622i \(0.257986\pi\)
\(264\) 0 0
\(265\) 321.216 0.0744609
\(266\) 3256.79 0.750701
\(267\) 0 0
\(268\) −3581.27 −0.816272
\(269\) 52.8516 0.0119792 0.00598962 0.999982i \(-0.498093\pi\)
0.00598962 + 0.999982i \(0.498093\pi\)
\(270\) 0 0
\(271\) −6822.19 −1.52922 −0.764610 0.644493i \(-0.777069\pi\)
−0.764610 + 0.644493i \(0.777069\pi\)
\(272\) −2989.34 −0.666381
\(273\) 0 0
\(274\) −9832.98 −2.16800
\(275\) 256.835 0.0563189
\(276\) 0 0
\(277\) 469.032 0.101738 0.0508689 0.998705i \(-0.483801\pi\)
0.0508689 + 0.998705i \(0.483801\pi\)
\(278\) 64.8810 0.0139975
\(279\) 0 0
\(280\) 680.784 0.145302
\(281\) 2305.05 0.489352 0.244676 0.969605i \(-0.421318\pi\)
0.244676 + 0.969605i \(0.421318\pi\)
\(282\) 0 0
\(283\) −7370.80 −1.54823 −0.774114 0.633046i \(-0.781805\pi\)
−0.774114 + 0.633046i \(0.781805\pi\)
\(284\) −3900.73 −0.815020
\(285\) 0 0
\(286\) 3400.49 0.703060
\(287\) −157.112 −0.0323137
\(288\) 0 0
\(289\) −3449.74 −0.702167
\(290\) 4304.69 0.871656
\(291\) 0 0
\(292\) −2235.45 −0.448013
\(293\) −1758.90 −0.350702 −0.175351 0.984506i \(-0.556106\pi\)
−0.175351 + 0.984506i \(0.556106\pi\)
\(294\) 0 0
\(295\) 688.646 0.135914
\(296\) 726.757 0.142709
\(297\) 0 0
\(298\) −5548.20 −1.07852
\(299\) 11872.1 2.29626
\(300\) 0 0
\(301\) −1272.91 −0.243752
\(302\) 7123.63 1.35735
\(303\) 0 0
\(304\) 9946.89 1.87662
\(305\) −1465.10 −0.275054
\(306\) 0 0
\(307\) 3468.10 0.644739 0.322369 0.946614i \(-0.395521\pi\)
0.322369 + 0.946614i \(0.395521\pi\)
\(308\) 412.799 0.0763682
\(309\) 0 0
\(310\) 12482.5 2.28697
\(311\) −1983.98 −0.361741 −0.180870 0.983507i \(-0.557891\pi\)
−0.180870 + 0.983507i \(0.557891\pi\)
\(312\) 0 0
\(313\) −10094.2 −1.82287 −0.911436 0.411443i \(-0.865025\pi\)
−0.911436 + 0.411443i \(0.865025\pi\)
\(314\) −5681.03 −1.02102
\(315\) 0 0
\(316\) 2459.67 0.437871
\(317\) 3051.34 0.540633 0.270316 0.962772i \(-0.412872\pi\)
0.270316 + 0.962772i \(0.412872\pi\)
\(318\) 0 0
\(319\) −1284.87 −0.225513
\(320\) −1380.02 −0.241080
\(321\) 0 0
\(322\) 3591.84 0.621632
\(323\) −4868.91 −0.838741
\(324\) 0 0
\(325\) 1974.65 0.337027
\(326\) −12693.7 −2.15656
\(327\) 0 0
\(328\) −216.504 −0.0364465
\(329\) 2103.71 0.352527
\(330\) 0 0
\(331\) −26.9826 −0.00448066 −0.00224033 0.999997i \(-0.500713\pi\)
−0.00224033 + 0.999997i \(0.500713\pi\)
\(332\) −1907.12 −0.315261
\(333\) 0 0
\(334\) 8147.25 1.33472
\(335\) −6735.13 −1.09845
\(336\) 0 0
\(337\) 6818.62 1.10218 0.551089 0.834447i \(-0.314213\pi\)
0.551089 + 0.834447i \(0.314213\pi\)
\(338\) 18113.7 2.91496
\(339\) 0 0
\(340\) 2067.59 0.329797
\(341\) −3725.80 −0.591681
\(342\) 0 0
\(343\) −343.000 −0.0539949
\(344\) −1754.10 −0.274926
\(345\) 0 0
\(346\) 3684.66 0.572510
\(347\) −11907.0 −1.84208 −0.921038 0.389473i \(-0.872657\pi\)
−0.921038 + 0.389473i \(0.872657\pi\)
\(348\) 0 0
\(349\) −4352.72 −0.667609 −0.333805 0.942642i \(-0.608333\pi\)
−0.333805 + 0.942642i \(0.608333\pi\)
\(350\) 597.419 0.0912382
\(351\) 0 0
\(352\) 2293.30 0.347253
\(353\) 1326.31 0.199978 0.0999891 0.994989i \(-0.468119\pi\)
0.0999891 + 0.994989i \(0.468119\pi\)
\(354\) 0 0
\(355\) −7335.91 −1.09676
\(356\) 6679.68 0.994444
\(357\) 0 0
\(358\) 2727.73 0.402696
\(359\) 8292.92 1.21917 0.609587 0.792719i \(-0.291335\pi\)
0.609587 + 0.792719i \(0.291335\pi\)
\(360\) 0 0
\(361\) 9342.06 1.36201
\(362\) −10190.0 −1.47949
\(363\) 0 0
\(364\) 3173.77 0.457007
\(365\) −4204.10 −0.602885
\(366\) 0 0
\(367\) −11027.5 −1.56848 −0.784241 0.620457i \(-0.786947\pi\)
−0.784241 + 0.620457i \(0.786947\pi\)
\(368\) 10970.2 1.55397
\(369\) 0 0
\(370\) −2776.59 −0.390130
\(371\) −223.017 −0.0312088
\(372\) 0 0
\(373\) 8245.72 1.14463 0.572316 0.820034i \(-0.306045\pi\)
0.572316 + 0.820034i \(0.306045\pi\)
\(374\) −1538.06 −0.212650
\(375\) 0 0
\(376\) 2898.96 0.397613
\(377\) −9878.58 −1.34953
\(378\) 0 0
\(379\) 10163.4 1.37747 0.688734 0.725014i \(-0.258167\pi\)
0.688734 + 0.725014i \(0.258167\pi\)
\(380\) −6879.81 −0.928755
\(381\) 0 0
\(382\) 3333.31 0.446458
\(383\) −14338.9 −1.91301 −0.956506 0.291714i \(-0.905774\pi\)
−0.956506 + 0.291714i \(0.905774\pi\)
\(384\) 0 0
\(385\) 776.332 0.102768
\(386\) 16283.0 2.14711
\(387\) 0 0
\(388\) −5014.37 −0.656098
\(389\) 2382.91 0.310587 0.155294 0.987868i \(-0.450368\pi\)
0.155294 + 0.987868i \(0.450368\pi\)
\(390\) 0 0
\(391\) −5369.82 −0.694535
\(392\) −472.662 −0.0609006
\(393\) 0 0
\(394\) 5034.98 0.643804
\(395\) 4625.78 0.589236
\(396\) 0 0
\(397\) −9868.22 −1.24754 −0.623768 0.781609i \(-0.714399\pi\)
−0.623768 + 0.781609i \(0.714399\pi\)
\(398\) −344.901 −0.0434380
\(399\) 0 0
\(400\) 1824.64 0.228080
\(401\) 5879.12 0.732143 0.366072 0.930587i \(-0.380702\pi\)
0.366072 + 0.930587i \(0.380702\pi\)
\(402\) 0 0
\(403\) −28645.4 −3.54077
\(404\) 2859.55 0.352148
\(405\) 0 0
\(406\) −2988.71 −0.365338
\(407\) 828.758 0.100934
\(408\) 0 0
\(409\) 5680.84 0.686796 0.343398 0.939190i \(-0.388422\pi\)
0.343398 + 0.939190i \(0.388422\pi\)
\(410\) 827.158 0.0996352
\(411\) 0 0
\(412\) −3956.96 −0.473168
\(413\) −478.121 −0.0569656
\(414\) 0 0
\(415\) −3586.62 −0.424242
\(416\) 17631.8 2.07805
\(417\) 0 0
\(418\) 5117.81 0.598853
\(419\) 1098.50 0.128079 0.0640395 0.997947i \(-0.479602\pi\)
0.0640395 + 0.997947i \(0.479602\pi\)
\(420\) 0 0
\(421\) −5265.06 −0.609509 −0.304754 0.952431i \(-0.598574\pi\)
−0.304754 + 0.952431i \(0.598574\pi\)
\(422\) 4291.97 0.495095
\(423\) 0 0
\(424\) −307.323 −0.0352003
\(425\) −893.143 −0.101938
\(426\) 0 0
\(427\) 1017.21 0.115284
\(428\) 10931.5 1.23457
\(429\) 0 0
\(430\) 6701.56 0.751577
\(431\) −4273.45 −0.477598 −0.238799 0.971069i \(-0.576754\pi\)
−0.238799 + 0.971069i \(0.576754\pi\)
\(432\) 0 0
\(433\) −8560.19 −0.950061 −0.475031 0.879969i \(-0.657563\pi\)
−0.475031 + 0.879969i \(0.657563\pi\)
\(434\) −8666.52 −0.958539
\(435\) 0 0
\(436\) 7980.90 0.876642
\(437\) 17867.8 1.95591
\(438\) 0 0
\(439\) 12664.2 1.37684 0.688419 0.725314i \(-0.258305\pi\)
0.688419 + 0.725314i \(0.258305\pi\)
\(440\) 1069.80 0.115911
\(441\) 0 0
\(442\) −11825.2 −1.27255
\(443\) −12368.9 −1.32656 −0.663279 0.748372i \(-0.730836\pi\)
−0.663279 + 0.748372i \(0.730836\pi\)
\(444\) 0 0
\(445\) 12562.2 1.33821
\(446\) 323.812 0.0343788
\(447\) 0 0
\(448\) 958.136 0.101044
\(449\) 2092.69 0.219956 0.109978 0.993934i \(-0.464922\pi\)
0.109978 + 0.993934i \(0.464922\pi\)
\(450\) 0 0
\(451\) −246.891 −0.0257775
\(452\) 2856.05 0.297207
\(453\) 0 0
\(454\) −3228.75 −0.333772
\(455\) 5968.75 0.614988
\(456\) 0 0
\(457\) 7825.71 0.801031 0.400515 0.916290i \(-0.368831\pi\)
0.400515 + 0.916290i \(0.368831\pi\)
\(458\) 4533.49 0.462525
\(459\) 0 0
\(460\) −7587.60 −0.769073
\(461\) −4775.60 −0.482477 −0.241238 0.970466i \(-0.577554\pi\)
−0.241238 + 0.970466i \(0.577554\pi\)
\(462\) 0 0
\(463\) 11518.3 1.15615 0.578077 0.815982i \(-0.303803\pi\)
0.578077 + 0.815982i \(0.303803\pi\)
\(464\) −9128.12 −0.913280
\(465\) 0 0
\(466\) 20030.7 1.99121
\(467\) 7420.17 0.735256 0.367628 0.929973i \(-0.380170\pi\)
0.367628 + 0.929973i \(0.380170\pi\)
\(468\) 0 0
\(469\) 4676.14 0.460392
\(470\) −11075.5 −1.08697
\(471\) 0 0
\(472\) −658.861 −0.0642512
\(473\) −2000.29 −0.194447
\(474\) 0 0
\(475\) 2971.89 0.287073
\(476\) −1435.51 −0.138228
\(477\) 0 0
\(478\) −2171.26 −0.207764
\(479\) −10159.2 −0.969076 −0.484538 0.874770i \(-0.661012\pi\)
−0.484538 + 0.874770i \(0.661012\pi\)
\(480\) 0 0
\(481\) 6371.82 0.604013
\(482\) 1128.69 0.106661
\(483\) 0 0
\(484\) 648.685 0.0609208
\(485\) −9430.29 −0.882902
\(486\) 0 0
\(487\) −12344.9 −1.14866 −0.574331 0.818623i \(-0.694738\pi\)
−0.574331 + 0.818623i \(0.694738\pi\)
\(488\) 1401.73 0.130028
\(489\) 0 0
\(490\) 1805.81 0.166486
\(491\) −15344.9 −1.41040 −0.705200 0.709009i \(-0.749143\pi\)
−0.705200 + 0.709009i \(0.749143\pi\)
\(492\) 0 0
\(493\) 4468.13 0.408183
\(494\) 39347.8 3.58369
\(495\) 0 0
\(496\) −26469.3 −2.39618
\(497\) 5093.26 0.459686
\(498\) 0 0
\(499\) −5022.76 −0.450601 −0.225300 0.974289i \(-0.572336\pi\)
−0.225300 + 0.974289i \(0.572336\pi\)
\(500\) −8018.41 −0.717188
\(501\) 0 0
\(502\) −12748.1 −1.13342
\(503\) −8735.90 −0.774383 −0.387191 0.921999i \(-0.626555\pi\)
−0.387191 + 0.921999i \(0.626555\pi\)
\(504\) 0 0
\(505\) 5377.82 0.473881
\(506\) 5644.33 0.495891
\(507\) 0 0
\(508\) −12102.2 −1.05699
\(509\) 21805.5 1.89884 0.949420 0.314008i \(-0.101672\pi\)
0.949420 + 0.314008i \(0.101672\pi\)
\(510\) 0 0
\(511\) 2918.87 0.252687
\(512\) 10261.7 0.885760
\(513\) 0 0
\(514\) −1650.16 −0.141606
\(515\) −7441.66 −0.636735
\(516\) 0 0
\(517\) 3305.83 0.281219
\(518\) 1927.76 0.163515
\(519\) 0 0
\(520\) 8225.08 0.693642
\(521\) 4732.28 0.397936 0.198968 0.980006i \(-0.436241\pi\)
0.198968 + 0.980006i \(0.436241\pi\)
\(522\) 0 0
\(523\) −7511.23 −0.627998 −0.313999 0.949423i \(-0.601669\pi\)
−0.313999 + 0.949423i \(0.601669\pi\)
\(524\) −6295.00 −0.524806
\(525\) 0 0
\(526\) 21487.9 1.78121
\(527\) 12956.5 1.07095
\(528\) 0 0
\(529\) 7539.02 0.619629
\(530\) 1174.13 0.0962283
\(531\) 0 0
\(532\) 4776.59 0.389270
\(533\) −1898.20 −0.154259
\(534\) 0 0
\(535\) 20558.4 1.66134
\(536\) 6443.82 0.519274
\(537\) 0 0
\(538\) 193.187 0.0154812
\(539\) −539.000 −0.0430730
\(540\) 0 0
\(541\) −598.410 −0.0475557 −0.0237779 0.999717i \(-0.507569\pi\)
−0.0237779 + 0.999717i \(0.507569\pi\)
\(542\) −24937.0 −1.97626
\(543\) 0 0
\(544\) −7974.95 −0.628535
\(545\) 15009.3 1.17968
\(546\) 0 0
\(547\) −14042.3 −1.09763 −0.548816 0.835943i \(-0.684921\pi\)
−0.548816 + 0.835943i \(0.684921\pi\)
\(548\) −14421.6 −1.12420
\(549\) 0 0
\(550\) 938.801 0.0727829
\(551\) −14867.5 −1.14950
\(552\) 0 0
\(553\) −3211.64 −0.246967
\(554\) 1714.44 0.131479
\(555\) 0 0
\(556\) 95.1581 0.00725828
\(557\) −4965.87 −0.377757 −0.188878 0.982000i \(-0.560485\pi\)
−0.188878 + 0.982000i \(0.560485\pi\)
\(558\) 0 0
\(559\) −15379.0 −1.16362
\(560\) 5515.32 0.416187
\(561\) 0 0
\(562\) 8425.60 0.632406
\(563\) 15362.4 1.14999 0.574997 0.818155i \(-0.305003\pi\)
0.574997 + 0.818155i \(0.305003\pi\)
\(564\) 0 0
\(565\) 5371.24 0.399947
\(566\) −26942.3 −2.00083
\(567\) 0 0
\(568\) 7018.62 0.518477
\(569\) −17735.3 −1.30668 −0.653341 0.757064i \(-0.726633\pi\)
−0.653341 + 0.757064i \(0.726633\pi\)
\(570\) 0 0
\(571\) −19818.8 −1.45252 −0.726262 0.687418i \(-0.758744\pi\)
−0.726262 + 0.687418i \(0.758744\pi\)
\(572\) 4987.35 0.364566
\(573\) 0 0
\(574\) −574.288 −0.0417602
\(575\) 3277.63 0.237716
\(576\) 0 0
\(577\) 6579.03 0.474677 0.237339 0.971427i \(-0.423725\pi\)
0.237339 + 0.971427i \(0.423725\pi\)
\(578\) −12609.8 −0.907434
\(579\) 0 0
\(580\) 6313.50 0.451989
\(581\) 2490.16 0.177813
\(582\) 0 0
\(583\) −350.455 −0.0248960
\(584\) 4022.27 0.285005
\(585\) 0 0
\(586\) −6429.25 −0.453225
\(587\) −13901.5 −0.977470 −0.488735 0.872432i \(-0.662541\pi\)
−0.488735 + 0.872432i \(0.662541\pi\)
\(588\) 0 0
\(589\) −43112.0 −3.01596
\(590\) 2517.19 0.175646
\(591\) 0 0
\(592\) 5887.76 0.408760
\(593\) −23928.0 −1.65700 −0.828502 0.559986i \(-0.810807\pi\)
−0.828502 + 0.559986i \(0.810807\pi\)
\(594\) 0 0
\(595\) −2699.70 −0.186011
\(596\) −8137.30 −0.559256
\(597\) 0 0
\(598\) 43395.9 2.96754
\(599\) −24078.9 −1.64247 −0.821233 0.570594i \(-0.806713\pi\)
−0.821233 + 0.570594i \(0.806713\pi\)
\(600\) 0 0
\(601\) −11806.6 −0.801336 −0.400668 0.916223i \(-0.631222\pi\)
−0.400668 + 0.916223i \(0.631222\pi\)
\(602\) −4652.83 −0.315009
\(603\) 0 0
\(604\) 10447.9 0.703840
\(605\) 1219.95 0.0819802
\(606\) 0 0
\(607\) 1957.71 0.130908 0.0654539 0.997856i \(-0.479150\pi\)
0.0654539 + 0.997856i \(0.479150\pi\)
\(608\) 26536.2 1.77004
\(609\) 0 0
\(610\) −5355.35 −0.355462
\(611\) 25416.6 1.68289
\(612\) 0 0
\(613\) −16029.2 −1.05614 −0.528069 0.849201i \(-0.677084\pi\)
−0.528069 + 0.849201i \(0.677084\pi\)
\(614\) 12676.9 0.833218
\(615\) 0 0
\(616\) −742.754 −0.0485819
\(617\) 7153.99 0.466789 0.233394 0.972382i \(-0.425017\pi\)
0.233394 + 0.972382i \(0.425017\pi\)
\(618\) 0 0
\(619\) −11035.9 −0.716590 −0.358295 0.933608i \(-0.616642\pi\)
−0.358295 + 0.933608i \(0.616642\pi\)
\(620\) 18307.6 1.18589
\(621\) 0 0
\(622\) −7252.00 −0.467490
\(623\) −8721.78 −0.560884
\(624\) 0 0
\(625\) −12161.3 −0.778321
\(626\) −36897.1 −2.35576
\(627\) 0 0
\(628\) −8332.11 −0.529438
\(629\) −2882.01 −0.182692
\(630\) 0 0
\(631\) −4311.46 −0.272007 −0.136004 0.990708i \(-0.543426\pi\)
−0.136004 + 0.990708i \(0.543426\pi\)
\(632\) −4425.71 −0.278553
\(633\) 0 0
\(634\) 11153.5 0.698678
\(635\) −22760.1 −1.42237
\(636\) 0 0
\(637\) −4144.05 −0.257760
\(638\) −4696.54 −0.291439
\(639\) 0 0
\(640\) 11771.3 0.727035
\(641\) −2692.13 −0.165886 −0.0829429 0.996554i \(-0.526432\pi\)
−0.0829429 + 0.996554i \(0.526432\pi\)
\(642\) 0 0
\(643\) 19694.3 1.20788 0.603941 0.797029i \(-0.293596\pi\)
0.603941 + 0.797029i \(0.293596\pi\)
\(644\) 5268.00 0.322342
\(645\) 0 0
\(646\) −17797.2 −1.08393
\(647\) 21225.5 1.28974 0.644870 0.764292i \(-0.276911\pi\)
0.644870 + 0.764292i \(0.276911\pi\)
\(648\) 0 0
\(649\) −751.333 −0.0454428
\(650\) 7217.88 0.435552
\(651\) 0 0
\(652\) −18617.3 −1.11827
\(653\) 12929.7 0.774850 0.387425 0.921901i \(-0.373365\pi\)
0.387425 + 0.921901i \(0.373365\pi\)
\(654\) 0 0
\(655\) −11838.7 −0.706224
\(656\) −1753.99 −0.104393
\(657\) 0 0
\(658\) 7689.64 0.455582
\(659\) 20835.3 1.23161 0.615803 0.787900i \(-0.288832\pi\)
0.615803 + 0.787900i \(0.288832\pi\)
\(660\) 0 0
\(661\) −1451.06 −0.0853850 −0.0426925 0.999088i \(-0.513594\pi\)
−0.0426925 + 0.999088i \(0.513594\pi\)
\(662\) −98.6288 −0.00579051
\(663\) 0 0
\(664\) 3431.50 0.200554
\(665\) 8983.10 0.523834
\(666\) 0 0
\(667\) −16397.0 −0.951867
\(668\) 11949.2 0.692109
\(669\) 0 0
\(670\) −24618.7 −1.41956
\(671\) 1598.47 0.0919645
\(672\) 0 0
\(673\) −28986.0 −1.66022 −0.830111 0.557598i \(-0.811723\pi\)
−0.830111 + 0.557598i \(0.811723\pi\)
\(674\) 24923.9 1.42438
\(675\) 0 0
\(676\) 26566.6 1.51153
\(677\) −24818.6 −1.40895 −0.704474 0.709730i \(-0.748817\pi\)
−0.704474 + 0.709730i \(0.748817\pi\)
\(678\) 0 0
\(679\) 6547.36 0.370051
\(680\) −3720.24 −0.209801
\(681\) 0 0
\(682\) −13618.8 −0.764650
\(683\) 7450.70 0.417413 0.208706 0.977978i \(-0.433075\pi\)
0.208706 + 0.977978i \(0.433075\pi\)
\(684\) 0 0
\(685\) −27122.0 −1.51282
\(686\) −1253.76 −0.0697795
\(687\) 0 0
\(688\) −14210.7 −0.787467
\(689\) −2694.44 −0.148984
\(690\) 0 0
\(691\) −28469.1 −1.56731 −0.783657 0.621193i \(-0.786648\pi\)
−0.783657 + 0.621193i \(0.786648\pi\)
\(692\) 5404.13 0.296870
\(693\) 0 0
\(694\) −43523.3 −2.38058
\(695\) 178.959 0.00976736
\(696\) 0 0
\(697\) 858.563 0.0466577
\(698\) −15910.4 −0.862775
\(699\) 0 0
\(700\) 876.208 0.0473108
\(701\) −20045.7 −1.08005 −0.540027 0.841648i \(-0.681586\pi\)
−0.540027 + 0.841648i \(0.681586\pi\)
\(702\) 0 0
\(703\) 9589.73 0.514486
\(704\) 1505.64 0.0806052
\(705\) 0 0
\(706\) 4848.02 0.258439
\(707\) −3733.77 −0.198618
\(708\) 0 0
\(709\) 15326.9 0.811868 0.405934 0.913902i \(-0.366946\pi\)
0.405934 + 0.913902i \(0.366946\pi\)
\(710\) −26814.8 −1.41738
\(711\) 0 0
\(712\) −12018.8 −0.632618
\(713\) −47547.3 −2.49742
\(714\) 0 0
\(715\) 9379.47 0.490591
\(716\) 4000.64 0.208814
\(717\) 0 0
\(718\) 30312.9 1.57558
\(719\) 11023.4 0.571770 0.285885 0.958264i \(-0.407712\pi\)
0.285885 + 0.958264i \(0.407712\pi\)
\(720\) 0 0
\(721\) 5166.68 0.266875
\(722\) 34147.8 1.76018
\(723\) 0 0
\(724\) −14945.3 −0.767177
\(725\) −2727.26 −0.139707
\(726\) 0 0
\(727\) 28755.9 1.46698 0.733492 0.679699i \(-0.237889\pi\)
0.733492 + 0.679699i \(0.237889\pi\)
\(728\) −5710.60 −0.290726
\(729\) 0 0
\(730\) −15367.2 −0.779129
\(731\) 6956.00 0.351952
\(732\) 0 0
\(733\) 21500.9 1.08343 0.541714 0.840563i \(-0.317775\pi\)
0.541714 + 0.840563i \(0.317775\pi\)
\(734\) −40308.7 −2.02700
\(735\) 0 0
\(736\) 29266.3 1.46572
\(737\) 7348.21 0.367266
\(738\) 0 0
\(739\) −2598.63 −0.129353 −0.0646767 0.997906i \(-0.520602\pi\)
−0.0646767 + 0.997906i \(0.520602\pi\)
\(740\) −4072.30 −0.202298
\(741\) 0 0
\(742\) −815.189 −0.0403322
\(743\) −29920.5 −1.47736 −0.738678 0.674058i \(-0.764550\pi\)
−0.738678 + 0.674058i \(0.764550\pi\)
\(744\) 0 0
\(745\) −15303.4 −0.752583
\(746\) 30140.4 1.47925
\(747\) 0 0
\(748\) −2255.80 −0.110268
\(749\) −14273.5 −0.696317
\(750\) 0 0
\(751\) 17763.1 0.863096 0.431548 0.902090i \(-0.357968\pi\)
0.431548 + 0.902090i \(0.357968\pi\)
\(752\) 23485.7 1.13888
\(753\) 0 0
\(754\) −36108.9 −1.74404
\(755\) 19648.9 0.947147
\(756\) 0 0
\(757\) 6472.04 0.310740 0.155370 0.987856i \(-0.450343\pi\)
0.155370 + 0.987856i \(0.450343\pi\)
\(758\) 37150.1 1.78015
\(759\) 0 0
\(760\) 12378.9 0.590830
\(761\) 33720.2 1.60625 0.803126 0.595809i \(-0.203169\pi\)
0.803126 + 0.595809i \(0.203169\pi\)
\(762\) 0 0
\(763\) −10420.8 −0.494441
\(764\) 4888.81 0.231507
\(765\) 0 0
\(766\) −52412.6 −2.47225
\(767\) −5776.55 −0.271941
\(768\) 0 0
\(769\) −9361.49 −0.438991 −0.219495 0.975614i \(-0.570441\pi\)
−0.219495 + 0.975614i \(0.570441\pi\)
\(770\) 2837.71 0.132810
\(771\) 0 0
\(772\) 23881.6 1.11336
\(773\) −34886.0 −1.62324 −0.811618 0.584188i \(-0.801413\pi\)
−0.811618 + 0.584188i \(0.801413\pi\)
\(774\) 0 0
\(775\) −7908.38 −0.366551
\(776\) 9022.42 0.417379
\(777\) 0 0
\(778\) 8710.20 0.401383
\(779\) −2856.83 −0.131395
\(780\) 0 0
\(781\) 8003.69 0.366702
\(782\) −19628.2 −0.897572
\(783\) 0 0
\(784\) −3829.23 −0.174437
\(785\) −15669.8 −0.712458
\(786\) 0 0
\(787\) −9526.64 −0.431497 −0.215748 0.976449i \(-0.569219\pi\)
−0.215748 + 0.976449i \(0.569219\pi\)
\(788\) 7384.58 0.333839
\(789\) 0 0
\(790\) 16908.5 0.761490
\(791\) −3729.20 −0.167630
\(792\) 0 0
\(793\) 12289.7 0.550339
\(794\) −36071.1 −1.61223
\(795\) 0 0
\(796\) −505.852 −0.0225244
\(797\) 33267.9 1.47856 0.739278 0.673400i \(-0.235167\pi\)
0.739278 + 0.673400i \(0.235167\pi\)
\(798\) 0 0
\(799\) −11496.0 −0.509012
\(800\) 4867.75 0.215126
\(801\) 0 0
\(802\) 21489.8 0.946174
\(803\) 4586.80 0.201575
\(804\) 0 0
\(805\) 9907.27 0.433771
\(806\) −104707. −4.57586
\(807\) 0 0
\(808\) −5145.22 −0.224020
\(809\) −17949.6 −0.780069 −0.390035 0.920800i \(-0.627537\pi\)
−0.390035 + 0.920800i \(0.627537\pi\)
\(810\) 0 0
\(811\) 10877.9 0.470991 0.235495 0.971875i \(-0.424329\pi\)
0.235495 + 0.971875i \(0.424329\pi\)
\(812\) −4383.41 −0.189443
\(813\) 0 0
\(814\) 3029.34 0.130440
\(815\) −35012.7 −1.50484
\(816\) 0 0
\(817\) −23145.7 −0.991147
\(818\) 20765.0 0.887570
\(819\) 0 0
\(820\) 1213.16 0.0516650
\(821\) 4519.04 0.192102 0.0960509 0.995376i \(-0.469379\pi\)
0.0960509 + 0.995376i \(0.469379\pi\)
\(822\) 0 0
\(823\) 36987.7 1.56660 0.783300 0.621644i \(-0.213535\pi\)
0.783300 + 0.621644i \(0.213535\pi\)
\(824\) 7119.80 0.301007
\(825\) 0 0
\(826\) −1747.66 −0.0736186
\(827\) −15325.3 −0.644392 −0.322196 0.946673i \(-0.604421\pi\)
−0.322196 + 0.946673i \(0.604421\pi\)
\(828\) 0 0
\(829\) 26546.3 1.11217 0.556087 0.831124i \(-0.312302\pi\)
0.556087 + 0.831124i \(0.312302\pi\)
\(830\) −13110.1 −0.548263
\(831\) 0 0
\(832\) 11576.0 0.482362
\(833\) 1874.37 0.0779630
\(834\) 0 0
\(835\) 22472.3 0.931361
\(836\) 7506.07 0.310530
\(837\) 0 0
\(838\) 4015.31 0.165521
\(839\) −11906.5 −0.489938 −0.244969 0.969531i \(-0.578778\pi\)
−0.244969 + 0.969531i \(0.578778\pi\)
\(840\) 0 0
\(841\) −10745.3 −0.440581
\(842\) −19245.2 −0.787689
\(843\) 0 0
\(844\) 6294.85 0.256727
\(845\) 49962.5 2.03404
\(846\) 0 0
\(847\) −847.000 −0.0343604
\(848\) −2489.75 −0.100824
\(849\) 0 0
\(850\) −3264.68 −0.131738
\(851\) 10576.3 0.426030
\(852\) 0 0
\(853\) −6859.53 −0.275341 −0.137670 0.990478i \(-0.543961\pi\)
−0.137670 + 0.990478i \(0.543961\pi\)
\(854\) 3718.17 0.148985
\(855\) 0 0
\(856\) −19669.2 −0.785372
\(857\) −5193.59 −0.207013 −0.103506 0.994629i \(-0.533006\pi\)
−0.103506 + 0.994629i \(0.533006\pi\)
\(858\) 0 0
\(859\) 5265.73 0.209155 0.104578 0.994517i \(-0.466651\pi\)
0.104578 + 0.994517i \(0.466651\pi\)
\(860\) 9828.89 0.389724
\(861\) 0 0
\(862\) −15620.6 −0.617216
\(863\) 16016.7 0.631767 0.315884 0.948798i \(-0.397699\pi\)
0.315884 + 0.948798i \(0.397699\pi\)
\(864\) 0 0
\(865\) 10163.3 0.399494
\(866\) −31289.8 −1.22780
\(867\) 0 0
\(868\) −12710.8 −0.497042
\(869\) −5046.86 −0.197011
\(870\) 0 0
\(871\) 56496.0 2.19781
\(872\) −14360.1 −0.557678
\(873\) 0 0
\(874\) 65311.7 2.52769
\(875\) 10469.8 0.404507
\(876\) 0 0
\(877\) −11195.5 −0.431065 −0.215533 0.976497i \(-0.569149\pi\)
−0.215533 + 0.976497i \(0.569149\pi\)
\(878\) 46291.3 1.77933
\(879\) 0 0
\(880\) 8666.92 0.332002
\(881\) −45542.5 −1.74162 −0.870809 0.491621i \(-0.836405\pi\)
−0.870809 + 0.491621i \(0.836405\pi\)
\(882\) 0 0
\(883\) 10394.9 0.396167 0.198083 0.980185i \(-0.436528\pi\)
0.198083 + 0.980185i \(0.436528\pi\)
\(884\) −17343.5 −0.659871
\(885\) 0 0
\(886\) −45211.8 −1.71436
\(887\) 4020.51 0.152193 0.0760967 0.997100i \(-0.475754\pi\)
0.0760967 + 0.997100i \(0.475754\pi\)
\(888\) 0 0
\(889\) 15802.1 0.596159
\(890\) 45918.1 1.72941
\(891\) 0 0
\(892\) 474.920 0.0178268
\(893\) 38252.5 1.43345
\(894\) 0 0
\(895\) 7523.82 0.280998
\(896\) −8172.72 −0.304723
\(897\) 0 0
\(898\) 7649.36 0.284257
\(899\) 39563.3 1.46775
\(900\) 0 0
\(901\) 1218.71 0.0450623
\(902\) −902.453 −0.0333131
\(903\) 0 0
\(904\) −5138.93 −0.189069
\(905\) −28106.8 −1.03238
\(906\) 0 0
\(907\) −23907.5 −0.875231 −0.437615 0.899162i \(-0.644177\pi\)
−0.437615 + 0.899162i \(0.644177\pi\)
\(908\) −4735.46 −0.173075
\(909\) 0 0
\(910\) 21817.4 0.794770
\(911\) −40571.4 −1.47551 −0.737755 0.675069i \(-0.764114\pi\)
−0.737755 + 0.675069i \(0.764114\pi\)
\(912\) 0 0
\(913\) 3913.11 0.141846
\(914\) 28605.1 1.03520
\(915\) 0 0
\(916\) 6649.07 0.239838
\(917\) 8219.50 0.296000
\(918\) 0 0
\(919\) 20551.0 0.737667 0.368834 0.929495i \(-0.379757\pi\)
0.368834 + 0.929495i \(0.379757\pi\)
\(920\) 13652.5 0.489248
\(921\) 0 0
\(922\) −17456.1 −0.623521
\(923\) 61535.6 2.19444
\(924\) 0 0
\(925\) 1759.12 0.0625292
\(926\) 42102.4 1.49414
\(927\) 0 0
\(928\) −24351.9 −0.861413
\(929\) −34068.4 −1.20317 −0.601587 0.798807i \(-0.705465\pi\)
−0.601587 + 0.798807i \(0.705465\pi\)
\(930\) 0 0
\(931\) −6236.89 −0.219555
\(932\) 29378.1 1.03252
\(933\) 0 0
\(934\) 27122.8 0.950197
\(935\) −4242.38 −0.148386
\(936\) 0 0
\(937\) 15597.9 0.543824 0.271912 0.962322i \(-0.412344\pi\)
0.271912 + 0.962322i \(0.412344\pi\)
\(938\) 17092.6 0.594981
\(939\) 0 0
\(940\) −16244.0 −0.563639
\(941\) 22855.3 0.791775 0.395887 0.918299i \(-0.370437\pi\)
0.395887 + 0.918299i \(0.370437\pi\)
\(942\) 0 0
\(943\) −3150.73 −0.108804
\(944\) −5337.71 −0.184034
\(945\) 0 0
\(946\) −7311.59 −0.251290
\(947\) −36670.7 −1.25833 −0.629164 0.777272i \(-0.716603\pi\)
−0.629164 + 0.777272i \(0.716603\pi\)
\(948\) 0 0
\(949\) 35265.1 1.20628
\(950\) 10863.1 0.370994
\(951\) 0 0
\(952\) 2582.93 0.0879341
\(953\) 19922.8 0.677192 0.338596 0.940932i \(-0.390048\pi\)
0.338596 + 0.940932i \(0.390048\pi\)
\(954\) 0 0
\(955\) 9194.16 0.311535
\(956\) −3184.49 −0.107734
\(957\) 0 0
\(958\) −37134.8 −1.25237
\(959\) 18830.6 0.634068
\(960\) 0 0
\(961\) 84932.7 2.85095
\(962\) 23290.8 0.780587
\(963\) 0 0
\(964\) 1655.41 0.0553081
\(965\) 44913.0 1.49824
\(966\) 0 0
\(967\) 24523.8 0.815545 0.407772 0.913084i \(-0.366306\pi\)
0.407772 + 0.913084i \(0.366306\pi\)
\(968\) −1167.19 −0.0387549
\(969\) 0 0
\(970\) −34470.3 −1.14100
\(971\) −4493.82 −0.148520 −0.0742602 0.997239i \(-0.523660\pi\)
−0.0742602 + 0.997239i \(0.523660\pi\)
\(972\) 0 0
\(973\) −124.250 −0.00409380
\(974\) −45123.8 −1.48446
\(975\) 0 0
\(976\) 11356.0 0.372436
\(977\) 18285.3 0.598771 0.299385 0.954132i \(-0.403218\pi\)
0.299385 + 0.954132i \(0.403218\pi\)
\(978\) 0 0
\(979\) −13705.7 −0.447431
\(980\) 2648.51 0.0863300
\(981\) 0 0
\(982\) −56089.9 −1.82271
\(983\) 25850.8 0.838772 0.419386 0.907808i \(-0.362245\pi\)
0.419386 + 0.907808i \(0.362245\pi\)
\(984\) 0 0
\(985\) 13887.8 0.449242
\(986\) 16332.2 0.527509
\(987\) 0 0
\(988\) 57709.7 1.85829
\(989\) −25526.9 −0.820738
\(990\) 0 0
\(991\) −26842.1 −0.860412 −0.430206 0.902731i \(-0.641559\pi\)
−0.430206 + 0.902731i \(0.641559\pi\)
\(992\) −70614.6 −2.26010
\(993\) 0 0
\(994\) 18617.2 0.594068
\(995\) −951.331 −0.0303108
\(996\) 0 0
\(997\) −10468.1 −0.332526 −0.166263 0.986081i \(-0.553170\pi\)
−0.166263 + 0.986081i \(0.553170\pi\)
\(998\) −18359.6 −0.582327
\(999\) 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 693.4.a.m.1.3 4
3.2 odd 2 77.4.a.c.1.2 4
12.11 even 2 1232.4.a.w.1.1 4
15.14 odd 2 1925.4.a.q.1.3 4
21.20 even 2 539.4.a.f.1.2 4
33.32 even 2 847.4.a.e.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.4.a.c.1.2 4 3.2 odd 2
539.4.a.f.1.2 4 21.20 even 2
693.4.a.m.1.3 4 1.1 even 1 trivial
847.4.a.e.1.3 4 33.32 even 2
1232.4.a.w.1.1 4 12.11 even 2
1925.4.a.q.1.3 4 15.14 odd 2