Properties

Label 1521.4.a.s.1.2
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
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] = [1521,4,Mod(1,1521)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1521, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1521.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.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: \(89.7419051187\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{14}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.74166\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.74166 q^{2} +14.4833 q^{4} +4.51669 q^{5} +7.48331 q^{7} +30.7417 q^{8} +O(q^{10})\) \(q+4.74166 q^{2} +14.4833 q^{4} +4.51669 q^{5} +7.48331 q^{7} +30.7417 q^{8} +21.4166 q^{10} -66.8999 q^{11} +35.4833 q^{14} +29.8999 q^{16} -96.9666 q^{17} -31.4833 q^{19} +65.4166 q^{20} -317.216 q^{22} -183.600 q^{23} -104.600 q^{25} +108.383 q^{28} -112.200 q^{29} +77.2831 q^{31} -104.158 q^{32} -459.783 q^{34} +33.7998 q^{35} -54.7664 q^{37} -149.283 q^{38} +138.850 q^{40} +451.716 q^{41} -113.434 q^{43} -968.932 q^{44} -870.566 q^{46} -42.2670 q^{47} -287.000 q^{49} -495.975 q^{50} +530.999 q^{53} -302.166 q^{55} +230.049 q^{56} -532.015 q^{58} +219.666 q^{59} +822.865 q^{61} +366.450 q^{62} -733.082 q^{64} +872.082 q^{67} -1404.40 q^{68} +160.267 q^{70} -100.299 q^{71} +165.634 q^{73} -259.684 q^{74} -455.983 q^{76} -500.633 q^{77} -545.266 q^{79} +135.048 q^{80} +2141.88 q^{82} -454.534 q^{83} -437.968 q^{85} -537.864 q^{86} -2056.61 q^{88} -230.915 q^{89} -2659.13 q^{92} -200.415 q^{94} -142.200 q^{95} +1089.16 q^{97} -1360.86 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 14 q^{4} + 24 q^{5} + 54 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 14 q^{4} + 24 q^{5} + 54 q^{8} - 32 q^{10} - 44 q^{11} + 56 q^{14} - 30 q^{16} - 164 q^{17} - 48 q^{19} + 56 q^{20} - 380 q^{22} - 8 q^{23} + 150 q^{25} + 112 q^{28} - 404 q^{29} - 40 q^{31} - 126 q^{32} - 276 q^{34} - 112 q^{35} + 100 q^{37} - 104 q^{38} + 592 q^{40} + 200 q^{41} - 616 q^{43} - 980 q^{44} - 1352 q^{46} - 324 q^{47} - 574 q^{49} - 1194 q^{50} + 164 q^{53} + 144 q^{55} + 56 q^{56} + 268 q^{58} + 140 q^{59} + 628 q^{61} + 688 q^{62} - 194 q^{64} + 472 q^{67} - 1372 q^{68} + 560 q^{70} + 428 q^{71} + 900 q^{73} - 684 q^{74} - 448 q^{76} - 672 q^{77} - 432 q^{79} - 1032 q^{80} + 2832 q^{82} - 1388 q^{83} - 1744 q^{85} + 840 q^{86} - 1524 q^{88} + 960 q^{89} - 2744 q^{92} + 572 q^{94} - 464 q^{95} + 532 q^{97} - 574 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.74166 1.67643 0.838215 0.545341i \(-0.183600\pi\)
0.838215 + 0.545341i \(0.183600\pi\)
\(3\) 0 0
\(4\) 14.4833 1.81041
\(5\) 4.51669 0.403985 0.201992 0.979387i \(-0.435258\pi\)
0.201992 + 0.979387i \(0.435258\pi\)
\(6\) 0 0
\(7\) 7.48331 0.404061 0.202031 0.979379i \(-0.435246\pi\)
0.202031 + 0.979379i \(0.435246\pi\)
\(8\) 30.7417 1.35860
\(9\) 0 0
\(10\) 21.4166 0.677252
\(11\) −66.8999 −1.83373 −0.916867 0.399193i \(-0.869290\pi\)
−0.916867 + 0.399193i \(0.869290\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 35.4833 0.677380
\(15\) 0 0
\(16\) 29.8999 0.467186
\(17\) −96.9666 −1.38340 −0.691702 0.722183i \(-0.743139\pi\)
−0.691702 + 0.722183i \(0.743139\pi\)
\(18\) 0 0
\(19\) −31.4833 −0.380146 −0.190073 0.981770i \(-0.560872\pi\)
−0.190073 + 0.981770i \(0.560872\pi\)
\(20\) 65.4166 0.731380
\(21\) 0 0
\(22\) −317.216 −3.07413
\(23\) −183.600 −1.66448 −0.832242 0.554412i \(-0.812943\pi\)
−0.832242 + 0.554412i \(0.812943\pi\)
\(24\) 0 0
\(25\) −104.600 −0.836796
\(26\) 0 0
\(27\) 0 0
\(28\) 108.383 0.731518
\(29\) −112.200 −0.718450 −0.359225 0.933251i \(-0.616959\pi\)
−0.359225 + 0.933251i \(0.616959\pi\)
\(30\) 0 0
\(31\) 77.2831 0.447757 0.223878 0.974617i \(-0.428128\pi\)
0.223878 + 0.974617i \(0.428128\pi\)
\(32\) −104.158 −0.575398
\(33\) 0 0
\(34\) −459.783 −2.31918
\(35\) 33.7998 0.163234
\(36\) 0 0
\(37\) −54.7664 −0.243339 −0.121669 0.992571i \(-0.538825\pi\)
−0.121669 + 0.992571i \(0.538825\pi\)
\(38\) −149.283 −0.637287
\(39\) 0 0
\(40\) 138.850 0.548854
\(41\) 451.716 1.72064 0.860319 0.509756i \(-0.170264\pi\)
0.860319 + 0.509756i \(0.170264\pi\)
\(42\) 0 0
\(43\) −113.434 −0.402291 −0.201145 0.979561i \(-0.564466\pi\)
−0.201145 + 0.979561i \(0.564466\pi\)
\(44\) −968.932 −3.31982
\(45\) 0 0
\(46\) −870.566 −2.79039
\(47\) −42.2670 −0.131176 −0.0655880 0.997847i \(-0.520892\pi\)
−0.0655880 + 0.997847i \(0.520892\pi\)
\(48\) 0 0
\(49\) −287.000 −0.836735
\(50\) −495.975 −1.40283
\(51\) 0 0
\(52\) 0 0
\(53\) 530.999 1.37619 0.688097 0.725618i \(-0.258446\pi\)
0.688097 + 0.725618i \(0.258446\pi\)
\(54\) 0 0
\(55\) −302.166 −0.740800
\(56\) 230.049 0.548958
\(57\) 0 0
\(58\) −532.015 −1.20443
\(59\) 219.666 0.484714 0.242357 0.970187i \(-0.422080\pi\)
0.242357 + 0.970187i \(0.422080\pi\)
\(60\) 0 0
\(61\) 822.865 1.72717 0.863583 0.504207i \(-0.168215\pi\)
0.863583 + 0.504207i \(0.168215\pi\)
\(62\) 366.450 0.750632
\(63\) 0 0
\(64\) −733.082 −1.43180
\(65\) 0 0
\(66\) 0 0
\(67\) 872.082 1.59018 0.795088 0.606495i \(-0.207425\pi\)
0.795088 + 0.606495i \(0.207425\pi\)
\(68\) −1404.40 −2.50453
\(69\) 0 0
\(70\) 160.267 0.273651
\(71\) −100.299 −0.167653 −0.0838263 0.996480i \(-0.526714\pi\)
−0.0838263 + 0.996480i \(0.526714\pi\)
\(72\) 0 0
\(73\) 165.634 0.265562 0.132781 0.991145i \(-0.457609\pi\)
0.132781 + 0.991145i \(0.457609\pi\)
\(74\) −259.684 −0.407941
\(75\) 0 0
\(76\) −455.983 −0.688221
\(77\) −500.633 −0.740940
\(78\) 0 0
\(79\) −545.266 −0.776547 −0.388273 0.921544i \(-0.626928\pi\)
−0.388273 + 0.921544i \(0.626928\pi\)
\(80\) 135.048 0.188736
\(81\) 0 0
\(82\) 2141.88 2.88453
\(83\) −454.534 −0.601103 −0.300552 0.953766i \(-0.597171\pi\)
−0.300552 + 0.953766i \(0.597171\pi\)
\(84\) 0 0
\(85\) −437.968 −0.558874
\(86\) −537.864 −0.674412
\(87\) 0 0
\(88\) −2056.61 −2.49132
\(89\) −230.915 −0.275022 −0.137511 0.990500i \(-0.543910\pi\)
−0.137511 + 0.990500i \(0.543910\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −2659.13 −3.01341
\(93\) 0 0
\(94\) −200.415 −0.219907
\(95\) −142.200 −0.153573
\(96\) 0 0
\(97\) 1089.16 1.14008 0.570041 0.821616i \(-0.306927\pi\)
0.570041 + 0.821616i \(0.306927\pi\)
\(98\) −1360.86 −1.40273
\(99\) 0 0
\(100\) −1514.95 −1.51495
\(101\) 77.2336 0.0760894 0.0380447 0.999276i \(-0.487887\pi\)
0.0380447 + 0.999276i \(0.487887\pi\)
\(102\) 0 0
\(103\) 1351.36 1.29275 0.646377 0.763018i \(-0.276283\pi\)
0.646377 + 0.763018i \(0.276283\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 2517.81 2.30709
\(107\) −1133.67 −1.02426 −0.512129 0.858908i \(-0.671143\pi\)
−0.512129 + 0.858908i \(0.671143\pi\)
\(108\) 0 0
\(109\) −1017.66 −0.894262 −0.447131 0.894469i \(-0.647554\pi\)
−0.447131 + 0.894469i \(0.647554\pi\)
\(110\) −1432.77 −1.24190
\(111\) 0 0
\(112\) 223.750 0.188772
\(113\) −1570.06 −1.30707 −0.653536 0.756895i \(-0.726715\pi\)
−0.653536 + 0.756895i \(0.726715\pi\)
\(114\) 0 0
\(115\) −829.261 −0.672426
\(116\) −1625.03 −1.30069
\(117\) 0 0
\(118\) 1041.58 0.812588
\(119\) −725.632 −0.558979
\(120\) 0 0
\(121\) 3144.60 2.36258
\(122\) 3901.75 2.89547
\(123\) 0 0
\(124\) 1119.32 0.810625
\(125\) −1037.03 −0.742037
\(126\) 0 0
\(127\) −1248.16 −0.872099 −0.436050 0.899923i \(-0.643623\pi\)
−0.436050 + 0.899923i \(0.643623\pi\)
\(128\) −2642.76 −1.82491
\(129\) 0 0
\(130\) 0 0
\(131\) 1274.80 0.850227 0.425113 0.905140i \(-0.360234\pi\)
0.425113 + 0.905140i \(0.360234\pi\)
\(132\) 0 0
\(133\) −235.600 −0.153602
\(134\) 4135.11 2.66582
\(135\) 0 0
\(136\) −2980.91 −1.87950
\(137\) 874.915 0.545613 0.272807 0.962069i \(-0.412048\pi\)
0.272807 + 0.962069i \(0.412048\pi\)
\(138\) 0 0
\(139\) 310.334 0.189368 0.0946840 0.995507i \(-0.469816\pi\)
0.0946840 + 0.995507i \(0.469816\pi\)
\(140\) 489.533 0.295522
\(141\) 0 0
\(142\) −475.585 −0.281058
\(143\) 0 0
\(144\) 0 0
\(145\) −506.773 −0.290243
\(146\) 785.380 0.445195
\(147\) 0 0
\(148\) −793.199 −0.440544
\(149\) 5.08064 0.00279344 0.00139672 0.999999i \(-0.499555\pi\)
0.00139672 + 0.999999i \(0.499555\pi\)
\(150\) 0 0
\(151\) −6.54894 −0.00352944 −0.00176472 0.999998i \(-0.500562\pi\)
−0.00176472 + 0.999998i \(0.500562\pi\)
\(152\) −967.849 −0.516467
\(153\) 0 0
\(154\) −2373.83 −1.24213
\(155\) 349.063 0.180887
\(156\) 0 0
\(157\) −2297.60 −1.16795 −0.583975 0.811772i \(-0.698503\pi\)
−0.583975 + 0.811772i \(0.698503\pi\)
\(158\) −2585.46 −1.30183
\(159\) 0 0
\(160\) −470.450 −0.232452
\(161\) −1373.93 −0.672553
\(162\) 0 0
\(163\) 1085.49 0.521606 0.260803 0.965392i \(-0.416013\pi\)
0.260803 + 0.965392i \(0.416013\pi\)
\(164\) 6542.34 3.11507
\(165\) 0 0
\(166\) −2155.24 −1.00771
\(167\) −109.066 −0.0505374 −0.0252687 0.999681i \(-0.508044\pi\)
−0.0252687 + 0.999681i \(0.508044\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −2076.69 −0.936912
\(171\) 0 0
\(172\) −1642.90 −0.728313
\(173\) 1889.17 0.830236 0.415118 0.909768i \(-0.363740\pi\)
0.415118 + 0.909768i \(0.363740\pi\)
\(174\) 0 0
\(175\) −782.751 −0.338117
\(176\) −2000.30 −0.856694
\(177\) 0 0
\(178\) −1094.92 −0.461054
\(179\) −3427.86 −1.43134 −0.715672 0.698437i \(-0.753879\pi\)
−0.715672 + 0.698437i \(0.753879\pi\)
\(180\) 0 0
\(181\) 208.403 0.0855826 0.0427913 0.999084i \(-0.486375\pi\)
0.0427913 + 0.999084i \(0.486375\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −5644.15 −2.26137
\(185\) −247.363 −0.0983052
\(186\) 0 0
\(187\) 6487.06 2.53679
\(188\) −612.166 −0.237483
\(189\) 0 0
\(190\) −674.265 −0.257454
\(191\) −957.735 −0.362824 −0.181412 0.983407i \(-0.558067\pi\)
−0.181412 + 0.983407i \(0.558067\pi\)
\(192\) 0 0
\(193\) 512.730 0.191228 0.0956142 0.995418i \(-0.469518\pi\)
0.0956142 + 0.995418i \(0.469518\pi\)
\(194\) 5164.45 1.91127
\(195\) 0 0
\(196\) −4156.71 −1.51484
\(197\) −3870.35 −1.39975 −0.699876 0.714265i \(-0.746761\pi\)
−0.699876 + 0.714265i \(0.746761\pi\)
\(198\) 0 0
\(199\) 2305.83 0.821388 0.410694 0.911773i \(-0.365286\pi\)
0.410694 + 0.911773i \(0.365286\pi\)
\(200\) −3215.56 −1.13687
\(201\) 0 0
\(202\) 366.215 0.127558
\(203\) −839.630 −0.290298
\(204\) 0 0
\(205\) 2040.26 0.695111
\(206\) 6407.70 2.16721
\(207\) 0 0
\(208\) 0 0
\(209\) 2106.23 0.697086
\(210\) 0 0
\(211\) −3672.40 −1.19819 −0.599096 0.800677i \(-0.704473\pi\)
−0.599096 + 0.800677i \(0.704473\pi\)
\(212\) 7690.62 2.49148
\(213\) 0 0
\(214\) −5375.46 −1.71710
\(215\) −512.345 −0.162519
\(216\) 0 0
\(217\) 578.334 0.180921
\(218\) −4825.41 −1.49917
\(219\) 0 0
\(220\) −4376.36 −1.34116
\(221\) 0 0
\(222\) 0 0
\(223\) −5087.05 −1.52760 −0.763798 0.645455i \(-0.776668\pi\)
−0.763798 + 0.645455i \(0.776668\pi\)
\(224\) −779.449 −0.232496
\(225\) 0 0
\(226\) −7444.71 −2.19122
\(227\) −2625.83 −0.767763 −0.383882 0.923382i \(-0.625413\pi\)
−0.383882 + 0.923382i \(0.625413\pi\)
\(228\) 0 0
\(229\) −1678.73 −0.484425 −0.242213 0.970223i \(-0.577873\pi\)
−0.242213 + 0.970223i \(0.577873\pi\)
\(230\) −3932.07 −1.12727
\(231\) 0 0
\(232\) −3449.22 −0.976088
\(233\) −648.506 −0.182339 −0.0911696 0.995835i \(-0.529061\pi\)
−0.0911696 + 0.995835i \(0.529061\pi\)
\(234\) 0 0
\(235\) −190.907 −0.0529931
\(236\) 3181.50 0.877533
\(237\) 0 0
\(238\) −3440.70 −0.937089
\(239\) 5219.69 1.41269 0.706347 0.707866i \(-0.250342\pi\)
0.706347 + 0.707866i \(0.250342\pi\)
\(240\) 0 0
\(241\) −6103.56 −1.63139 −0.815695 0.578483i \(-0.803645\pi\)
−0.815695 + 0.578483i \(0.803645\pi\)
\(242\) 14910.6 3.96070
\(243\) 0 0
\(244\) 11917.8 3.12689
\(245\) −1296.29 −0.338028
\(246\) 0 0
\(247\) 0 0
\(248\) 2375.81 0.608323
\(249\) 0 0
\(250\) −4917.24 −1.24397
\(251\) −6423.40 −1.61530 −0.807652 0.589660i \(-0.799262\pi\)
−0.807652 + 0.589660i \(0.799262\pi\)
\(252\) 0 0
\(253\) 12282.8 3.05222
\(254\) −5918.36 −1.46201
\(255\) 0 0
\(256\) −6666.39 −1.62754
\(257\) 1230.23 0.298597 0.149299 0.988792i \(-0.452298\pi\)
0.149299 + 0.988792i \(0.452298\pi\)
\(258\) 0 0
\(259\) −409.834 −0.0983238
\(260\) 0 0
\(261\) 0 0
\(262\) 6044.66 1.42534
\(263\) 514.992 0.120744 0.0603722 0.998176i \(-0.480771\pi\)
0.0603722 + 0.998176i \(0.480771\pi\)
\(264\) 0 0
\(265\) 2398.35 0.555961
\(266\) −1117.13 −0.257503
\(267\) 0 0
\(268\) 12630.6 2.87888
\(269\) 5132.60 1.16335 0.581673 0.813423i \(-0.302398\pi\)
0.581673 + 0.813423i \(0.302398\pi\)
\(270\) 0 0
\(271\) 4300.00 0.963862 0.481931 0.876209i \(-0.339936\pi\)
0.481931 + 0.876209i \(0.339936\pi\)
\(272\) −2899.29 −0.646306
\(273\) 0 0
\(274\) 4148.55 0.914682
\(275\) 6997.70 1.53446
\(276\) 0 0
\(277\) −1812.80 −0.393215 −0.196607 0.980482i \(-0.562992\pi\)
−0.196607 + 0.980482i \(0.562992\pi\)
\(278\) 1471.50 0.317462
\(279\) 0 0
\(280\) 1039.06 0.221771
\(281\) −4073.08 −0.864696 −0.432348 0.901707i \(-0.642315\pi\)
−0.432348 + 0.901707i \(0.642315\pi\)
\(282\) 0 0
\(283\) 6346.29 1.33303 0.666516 0.745491i \(-0.267785\pi\)
0.666516 + 0.745491i \(0.267785\pi\)
\(284\) −1452.67 −0.303520
\(285\) 0 0
\(286\) 0 0
\(287\) 3380.33 0.695243
\(288\) 0 0
\(289\) 4489.53 0.913806
\(290\) −2402.94 −0.486572
\(291\) 0 0
\(292\) 2398.93 0.480777
\(293\) −8390.97 −1.67306 −0.836529 0.547923i \(-0.815419\pi\)
−0.836529 + 0.547923i \(0.815419\pi\)
\(294\) 0 0
\(295\) 992.164 0.195817
\(296\) −1683.61 −0.330601
\(297\) 0 0
\(298\) 24.0907 0.00468300
\(299\) 0 0
\(300\) 0 0
\(301\) −848.861 −0.162550
\(302\) −31.0528 −0.00591685
\(303\) 0 0
\(304\) −941.348 −0.177599
\(305\) 3716.62 0.697748
\(306\) 0 0
\(307\) −4005.27 −0.744603 −0.372301 0.928112i \(-0.621431\pi\)
−0.372301 + 0.928112i \(0.621431\pi\)
\(308\) −7250.82 −1.34141
\(309\) 0 0
\(310\) 1655.14 0.303244
\(311\) −5836.53 −1.06418 −0.532088 0.846689i \(-0.678593\pi\)
−0.532088 + 0.846689i \(0.678593\pi\)
\(312\) 0 0
\(313\) 1763.19 0.318407 0.159204 0.987246i \(-0.449107\pi\)
0.159204 + 0.987246i \(0.449107\pi\)
\(314\) −10894.4 −1.95798
\(315\) 0 0
\(316\) −7897.26 −1.40587
\(317\) −6106.35 −1.08191 −0.540957 0.841050i \(-0.681938\pi\)
−0.540957 + 0.841050i \(0.681938\pi\)
\(318\) 0 0
\(319\) 7506.18 1.31745
\(320\) −3311.10 −0.578425
\(321\) 0 0
\(322\) −6514.72 −1.12749
\(323\) 3052.83 0.525895
\(324\) 0 0
\(325\) 0 0
\(326\) 5147.00 0.874436
\(327\) 0 0
\(328\) 13886.5 2.33766
\(329\) −316.297 −0.0530031
\(330\) 0 0
\(331\) −7490.38 −1.24383 −0.621916 0.783084i \(-0.713646\pi\)
−0.621916 + 0.783084i \(0.713646\pi\)
\(332\) −6583.16 −1.08825
\(333\) 0 0
\(334\) −517.152 −0.0847224
\(335\) 3938.92 0.642406
\(336\) 0 0
\(337\) 9462.46 1.52953 0.764767 0.644307i \(-0.222854\pi\)
0.764767 + 0.644307i \(0.222854\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −6343.22 −1.01179
\(341\) −5170.23 −0.821066
\(342\) 0 0
\(343\) −4714.49 −0.742153
\(344\) −3487.14 −0.546553
\(345\) 0 0
\(346\) 8957.79 1.39183
\(347\) −11460.3 −1.77297 −0.886487 0.462753i \(-0.846862\pi\)
−0.886487 + 0.462753i \(0.846862\pi\)
\(348\) 0 0
\(349\) 3673.96 0.563503 0.281751 0.959487i \(-0.409085\pi\)
0.281751 + 0.959487i \(0.409085\pi\)
\(350\) −3711.54 −0.566829
\(351\) 0 0
\(352\) 6968.17 1.05513
\(353\) 3388.65 0.510935 0.255467 0.966818i \(-0.417771\pi\)
0.255467 + 0.966818i \(0.417771\pi\)
\(354\) 0 0
\(355\) −453.020 −0.0677290
\(356\) −3344.41 −0.497903
\(357\) 0 0
\(358\) −16253.7 −2.39955
\(359\) −9673.98 −1.42221 −0.711105 0.703086i \(-0.751805\pi\)
−0.711105 + 0.703086i \(0.751805\pi\)
\(360\) 0 0
\(361\) −5867.80 −0.855489
\(362\) 988.174 0.143473
\(363\) 0 0
\(364\) 0 0
\(365\) 748.117 0.107283
\(366\) 0 0
\(367\) −8715.98 −1.23970 −0.619851 0.784720i \(-0.712807\pi\)
−0.619851 + 0.784720i \(0.712807\pi\)
\(368\) −5489.61 −0.777624
\(369\) 0 0
\(370\) −1172.91 −0.164802
\(371\) 3973.63 0.556067
\(372\) 0 0
\(373\) 4667.99 0.647987 0.323994 0.946059i \(-0.394974\pi\)
0.323994 + 0.946059i \(0.394974\pi\)
\(374\) 30759.4 4.25276
\(375\) 0 0
\(376\) −1299.36 −0.178216
\(377\) 0 0
\(378\) 0 0
\(379\) −10862.7 −1.47225 −0.736123 0.676848i \(-0.763345\pi\)
−0.736123 + 0.676848i \(0.763345\pi\)
\(380\) −2059.53 −0.278031
\(381\) 0 0
\(382\) −4541.25 −0.608248
\(383\) 10054.2 1.34137 0.670686 0.741742i \(-0.266000\pi\)
0.670686 + 0.741742i \(0.266000\pi\)
\(384\) 0 0
\(385\) −2261.20 −0.299329
\(386\) 2431.19 0.320581
\(387\) 0 0
\(388\) 15774.7 2.06402
\(389\) −6418.50 −0.836584 −0.418292 0.908313i \(-0.637371\pi\)
−0.418292 + 0.908313i \(0.637371\pi\)
\(390\) 0 0
\(391\) 17803.0 2.30265
\(392\) −8822.86 −1.13679
\(393\) 0 0
\(394\) −18351.9 −2.34658
\(395\) −2462.79 −0.313713
\(396\) 0 0
\(397\) 12019.9 1.51955 0.759775 0.650186i \(-0.225309\pi\)
0.759775 + 0.650186i \(0.225309\pi\)
\(398\) 10933.5 1.37700
\(399\) 0 0
\(400\) −3127.52 −0.390939
\(401\) 3599.80 0.448293 0.224147 0.974555i \(-0.428041\pi\)
0.224147 + 0.974555i \(0.428041\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 1118.60 0.137753
\(405\) 0 0
\(406\) −3981.24 −0.486664
\(407\) 3663.87 0.446219
\(408\) 0 0
\(409\) 48.0968 0.00581475 0.00290737 0.999996i \(-0.499075\pi\)
0.00290737 + 0.999996i \(0.499075\pi\)
\(410\) 9674.20 1.16530
\(411\) 0 0
\(412\) 19572.2 2.34042
\(413\) 1643.83 0.195854
\(414\) 0 0
\(415\) −2052.99 −0.242837
\(416\) 0 0
\(417\) 0 0
\(418\) 9987.02 1.16862
\(419\) −723.462 −0.0843518 −0.0421759 0.999110i \(-0.513429\pi\)
−0.0421759 + 0.999110i \(0.513429\pi\)
\(420\) 0 0
\(421\) 14845.5 1.71859 0.859295 0.511481i \(-0.170903\pi\)
0.859295 + 0.511481i \(0.170903\pi\)
\(422\) −17413.3 −2.00868
\(423\) 0 0
\(424\) 16323.8 1.86970
\(425\) 10142.7 1.15763
\(426\) 0 0
\(427\) 6157.76 0.697880
\(428\) −16419.2 −1.85433
\(429\) 0 0
\(430\) −2429.36 −0.272452
\(431\) 1103.11 0.123283 0.0616417 0.998098i \(-0.480366\pi\)
0.0616417 + 0.998098i \(0.480366\pi\)
\(432\) 0 0
\(433\) 8893.53 0.987057 0.493528 0.869730i \(-0.335707\pi\)
0.493528 + 0.869730i \(0.335707\pi\)
\(434\) 2742.26 0.303301
\(435\) 0 0
\(436\) −14739.1 −1.61898
\(437\) 5780.32 0.632747
\(438\) 0 0
\(439\) −10901.7 −1.18521 −0.592607 0.805492i \(-0.701901\pi\)
−0.592607 + 0.805492i \(0.701901\pi\)
\(440\) −9289.08 −1.00645
\(441\) 0 0
\(442\) 0 0
\(443\) 3781.37 0.405550 0.202775 0.979225i \(-0.435004\pi\)
0.202775 + 0.979225i \(0.435004\pi\)
\(444\) 0 0
\(445\) −1042.97 −0.111105
\(446\) −24121.0 −2.56091
\(447\) 0 0
\(448\) −5485.88 −0.578535
\(449\) 106.834 0.0112289 0.00561447 0.999984i \(-0.498213\pi\)
0.00561447 + 0.999984i \(0.498213\pi\)
\(450\) 0 0
\(451\) −30219.7 −3.15519
\(452\) −22739.7 −2.36634
\(453\) 0 0
\(454\) −12450.8 −1.28710
\(455\) 0 0
\(456\) 0 0
\(457\) 1237.64 0.126684 0.0633419 0.997992i \(-0.479824\pi\)
0.0633419 + 0.997992i \(0.479824\pi\)
\(458\) −7959.95 −0.812105
\(459\) 0 0
\(460\) −12010.5 −1.21737
\(461\) 8790.90 0.888141 0.444071 0.895992i \(-0.353534\pi\)
0.444071 + 0.895992i \(0.353534\pi\)
\(462\) 0 0
\(463\) 3861.55 0.387606 0.193803 0.981040i \(-0.437918\pi\)
0.193803 + 0.981040i \(0.437918\pi\)
\(464\) −3354.77 −0.335650
\(465\) 0 0
\(466\) −3074.99 −0.305679
\(467\) 8991.38 0.890945 0.445473 0.895296i \(-0.353036\pi\)
0.445473 + 0.895296i \(0.353036\pi\)
\(468\) 0 0
\(469\) 6526.06 0.642528
\(470\) −905.214 −0.0888391
\(471\) 0 0
\(472\) 6752.91 0.658533
\(473\) 7588.71 0.737694
\(474\) 0 0
\(475\) 3293.14 0.318105
\(476\) −10509.6 −1.01198
\(477\) 0 0
\(478\) 24750.0 2.36828
\(479\) 4179.82 0.398707 0.199354 0.979928i \(-0.436116\pi\)
0.199354 + 0.979928i \(0.436116\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −28941.0 −2.73491
\(483\) 0 0
\(484\) 45544.2 4.27725
\(485\) 4919.41 0.460575
\(486\) 0 0
\(487\) 18443.8 1.71616 0.858078 0.513519i \(-0.171658\pi\)
0.858078 + 0.513519i \(0.171658\pi\)
\(488\) 25296.2 2.34653
\(489\) 0 0
\(490\) −6146.56 −0.566680
\(491\) −8093.26 −0.743877 −0.371939 0.928257i \(-0.621307\pi\)
−0.371939 + 0.928257i \(0.621307\pi\)
\(492\) 0 0
\(493\) 10879.7 0.993907
\(494\) 0 0
\(495\) 0 0
\(496\) 2310.76 0.209185
\(497\) −750.571 −0.0677418
\(498\) 0 0
\(499\) 10941.6 0.981591 0.490796 0.871275i \(-0.336706\pi\)
0.490796 + 0.871275i \(0.336706\pi\)
\(500\) −15019.6 −1.34340
\(501\) 0 0
\(502\) −30457.5 −2.70794
\(503\) 9260.11 0.820851 0.410425 0.911894i \(-0.365380\pi\)
0.410425 + 0.911894i \(0.365380\pi\)
\(504\) 0 0
\(505\) 348.840 0.0307389
\(506\) 58240.8 5.11684
\(507\) 0 0
\(508\) −18077.5 −1.57886
\(509\) −9996.40 −0.870497 −0.435248 0.900310i \(-0.643339\pi\)
−0.435248 + 0.900310i \(0.643339\pi\)
\(510\) 0 0
\(511\) 1239.49 0.107303
\(512\) −10467.7 −0.903538
\(513\) 0 0
\(514\) 5833.32 0.500577
\(515\) 6103.68 0.522253
\(516\) 0 0
\(517\) 2827.66 0.240542
\(518\) −1943.29 −0.164833
\(519\) 0 0
\(520\) 0 0
\(521\) −11427.9 −0.960973 −0.480486 0.877002i \(-0.659540\pi\)
−0.480486 + 0.877002i \(0.659540\pi\)
\(522\) 0 0
\(523\) −4810.47 −0.402193 −0.201097 0.979571i \(-0.564451\pi\)
−0.201097 + 0.979571i \(0.564451\pi\)
\(524\) 18463.3 1.53926
\(525\) 0 0
\(526\) 2441.92 0.202419
\(527\) −7493.88 −0.619428
\(528\) 0 0
\(529\) 21541.8 1.77051
\(530\) 11372.2 0.932030
\(531\) 0 0
\(532\) −3412.26 −0.278083
\(533\) 0 0
\(534\) 0 0
\(535\) −5120.41 −0.413785
\(536\) 26809.2 2.16042
\(537\) 0 0
\(538\) 24337.0 1.95027
\(539\) 19200.3 1.53435
\(540\) 0 0
\(541\) −2411.99 −0.191681 −0.0958406 0.995397i \(-0.530554\pi\)
−0.0958406 + 0.995397i \(0.530554\pi\)
\(542\) 20389.1 1.61585
\(543\) 0 0
\(544\) 10099.9 0.796008
\(545\) −4596.47 −0.361268
\(546\) 0 0
\(547\) −4396.34 −0.343646 −0.171823 0.985128i \(-0.554966\pi\)
−0.171823 + 0.985128i \(0.554966\pi\)
\(548\) 12671.7 0.987786
\(549\) 0 0
\(550\) 33180.7 2.57242
\(551\) 3532.43 0.273116
\(552\) 0 0
\(553\) −4080.40 −0.313772
\(554\) −8595.67 −0.659197
\(555\) 0 0
\(556\) 4494.66 0.342835
\(557\) −17488.0 −1.33032 −0.665160 0.746701i \(-0.731637\pi\)
−0.665160 + 0.746701i \(0.731637\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 1010.61 0.0762608
\(561\) 0 0
\(562\) −19313.2 −1.44960
\(563\) 6881.77 0.515154 0.257577 0.966258i \(-0.417076\pi\)
0.257577 + 0.966258i \(0.417076\pi\)
\(564\) 0 0
\(565\) −7091.49 −0.528037
\(566\) 30092.0 2.23473
\(567\) 0 0
\(568\) −3083.36 −0.227773
\(569\) −14733.5 −1.08552 −0.542758 0.839889i \(-0.682620\pi\)
−0.542758 + 0.839889i \(0.682620\pi\)
\(570\) 0 0
\(571\) 4488.51 0.328964 0.164482 0.986380i \(-0.447405\pi\)
0.164482 + 0.986380i \(0.447405\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 16028.4 1.16553
\(575\) 19204.4 1.39284
\(576\) 0 0
\(577\) 10552.2 0.761338 0.380669 0.924711i \(-0.375694\pi\)
0.380669 + 0.924711i \(0.375694\pi\)
\(578\) 21287.8 1.53193
\(579\) 0 0
\(580\) −7339.75 −0.525460
\(581\) −3401.42 −0.242882
\(582\) 0 0
\(583\) −35523.8 −2.52357
\(584\) 5091.86 0.360793
\(585\) 0 0
\(586\) −39787.1 −2.80476
\(587\) −1637.20 −0.115118 −0.0575591 0.998342i \(-0.518332\pi\)
−0.0575591 + 0.998342i \(0.518332\pi\)
\(588\) 0 0
\(589\) −2433.13 −0.170213
\(590\) 4704.50 0.328273
\(591\) 0 0
\(592\) −1637.51 −0.113685
\(593\) 14024.0 0.971155 0.485577 0.874194i \(-0.338609\pi\)
0.485577 + 0.874194i \(0.338609\pi\)
\(594\) 0 0
\(595\) −3277.45 −0.225819
\(596\) 73.5845 0.00505728
\(597\) 0 0
\(598\) 0 0
\(599\) −365.860 −0.0249560 −0.0124780 0.999922i \(-0.503972\pi\)
−0.0124780 + 0.999922i \(0.503972\pi\)
\(600\) 0 0
\(601\) −10128.9 −0.687465 −0.343733 0.939068i \(-0.611691\pi\)
−0.343733 + 0.939068i \(0.611691\pi\)
\(602\) −4025.01 −0.272503
\(603\) 0 0
\(604\) −94.8504 −0.00638975
\(605\) 14203.1 0.954446
\(606\) 0 0
\(607\) 2324.97 0.155465 0.0777327 0.996974i \(-0.475232\pi\)
0.0777327 + 0.996974i \(0.475232\pi\)
\(608\) 3279.25 0.218735
\(609\) 0 0
\(610\) 17623.0 1.16973
\(611\) 0 0
\(612\) 0 0
\(613\) −633.133 −0.0417162 −0.0208581 0.999782i \(-0.506640\pi\)
−0.0208581 + 0.999782i \(0.506640\pi\)
\(614\) −18991.6 −1.24827
\(615\) 0 0
\(616\) −15390.3 −1.00664
\(617\) −2981.85 −0.194562 −0.0972810 0.995257i \(-0.531015\pi\)
−0.0972810 + 0.995257i \(0.531015\pi\)
\(618\) 0 0
\(619\) −15158.6 −0.984292 −0.492146 0.870513i \(-0.663787\pi\)
−0.492146 + 0.870513i \(0.663787\pi\)
\(620\) 5055.60 0.327480
\(621\) 0 0
\(622\) −27674.8 −1.78402
\(623\) −1728.01 −0.111126
\(624\) 0 0
\(625\) 8391.01 0.537025
\(626\) 8360.45 0.533787
\(627\) 0 0
\(628\) −33276.8 −2.11447
\(629\) 5310.51 0.336636
\(630\) 0 0
\(631\) −5562.30 −0.350922 −0.175461 0.984486i \(-0.556142\pi\)
−0.175461 + 0.984486i \(0.556142\pi\)
\(632\) −16762.4 −1.05502
\(633\) 0 0
\(634\) −28954.2 −1.81375
\(635\) −5637.56 −0.352315
\(636\) 0 0
\(637\) 0 0
\(638\) 35591.7 2.20861
\(639\) 0 0
\(640\) −11936.5 −0.737237
\(641\) 24140.7 1.48752 0.743761 0.668446i \(-0.233040\pi\)
0.743761 + 0.668446i \(0.233040\pi\)
\(642\) 0 0
\(643\) 1749.69 0.107311 0.0536555 0.998560i \(-0.482913\pi\)
0.0536555 + 0.998560i \(0.482913\pi\)
\(644\) −19899.1 −1.21760
\(645\) 0 0
\(646\) 14475.5 0.881626
\(647\) −1489.51 −0.0905083 −0.0452542 0.998976i \(-0.514410\pi\)
−0.0452542 + 0.998976i \(0.514410\pi\)
\(648\) 0 0
\(649\) −14695.7 −0.888836
\(650\) 0 0
\(651\) 0 0
\(652\) 15721.4 0.944323
\(653\) −10668.2 −0.639327 −0.319663 0.947531i \(-0.603570\pi\)
−0.319663 + 0.947531i \(0.603570\pi\)
\(654\) 0 0
\(655\) 5757.86 0.343478
\(656\) 13506.3 0.803858
\(657\) 0 0
\(658\) −1499.77 −0.0888559
\(659\) 13933.1 0.823608 0.411804 0.911272i \(-0.364899\pi\)
0.411804 + 0.911272i \(0.364899\pi\)
\(660\) 0 0
\(661\) 2349.69 0.138264 0.0691320 0.997608i \(-0.477977\pi\)
0.0691320 + 0.997608i \(0.477977\pi\)
\(662\) −35516.8 −2.08520
\(663\) 0 0
\(664\) −13973.1 −0.816660
\(665\) −1064.13 −0.0620529
\(666\) 0 0
\(667\) 20599.9 1.19585
\(668\) −1579.63 −0.0914937
\(669\) 0 0
\(670\) 18677.0 1.07695
\(671\) −55049.6 −3.16716
\(672\) 0 0
\(673\) −32421.0 −1.85697 −0.928483 0.371374i \(-0.878887\pi\)
−0.928483 + 0.371374i \(0.878887\pi\)
\(674\) 44867.7 2.56415
\(675\) 0 0
\(676\) 0 0
\(677\) 1071.74 0.0608421 0.0304211 0.999537i \(-0.490315\pi\)
0.0304211 + 0.999537i \(0.490315\pi\)
\(678\) 0 0
\(679\) 8150.56 0.460663
\(680\) −13463.9 −0.759287
\(681\) 0 0
\(682\) −24515.5 −1.37646
\(683\) −305.487 −0.0171144 −0.00855721 0.999963i \(-0.502724\pi\)
−0.00855721 + 0.999963i \(0.502724\pi\)
\(684\) 0 0
\(685\) 3951.72 0.220419
\(686\) −22354.5 −1.24417
\(687\) 0 0
\(688\) −3391.66 −0.187944
\(689\) 0 0
\(690\) 0 0
\(691\) −2180.81 −0.120061 −0.0600303 0.998197i \(-0.519120\pi\)
−0.0600303 + 0.998197i \(0.519120\pi\)
\(692\) 27361.4 1.50307
\(693\) 0 0
\(694\) −54340.9 −2.97227
\(695\) 1401.68 0.0765018
\(696\) 0 0
\(697\) −43801.4 −2.38034
\(698\) 17420.6 0.944672
\(699\) 0 0
\(700\) −11336.8 −0.612132
\(701\) 15168.3 0.817259 0.408629 0.912700i \(-0.366007\pi\)
0.408629 + 0.912700i \(0.366007\pi\)
\(702\) 0 0
\(703\) 1724.23 0.0925043
\(704\) 49043.1 2.62554
\(705\) 0 0
\(706\) 16067.8 0.856545
\(707\) 577.963 0.0307448
\(708\) 0 0
\(709\) 8988.17 0.476104 0.238052 0.971252i \(-0.423491\pi\)
0.238052 + 0.971252i \(0.423491\pi\)
\(710\) −2148.07 −0.113543
\(711\) 0 0
\(712\) −7098.71 −0.373645
\(713\) −14189.1 −0.745284
\(714\) 0 0
\(715\) 0 0
\(716\) −49646.8 −2.59132
\(717\) 0 0
\(718\) −45870.7 −2.38423
\(719\) −8448.18 −0.438198 −0.219099 0.975703i \(-0.570312\pi\)
−0.219099 + 0.975703i \(0.570312\pi\)
\(720\) 0 0
\(721\) 10112.7 0.522352
\(722\) −27823.1 −1.43417
\(723\) 0 0
\(724\) 3018.36 0.154940
\(725\) 11736.1 0.601197
\(726\) 0 0
\(727\) −8624.18 −0.439963 −0.219982 0.975504i \(-0.570600\pi\)
−0.219982 + 0.975504i \(0.570600\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 3547.31 0.179852
\(731\) 10999.3 0.556530
\(732\) 0 0
\(733\) −31124.2 −1.56835 −0.784174 0.620541i \(-0.786913\pi\)
−0.784174 + 0.620541i \(0.786913\pi\)
\(734\) −41328.2 −2.07827
\(735\) 0 0
\(736\) 19123.4 0.957742
\(737\) −58342.2 −2.91596
\(738\) 0 0
\(739\) 17671.1 0.879626 0.439813 0.898089i \(-0.355045\pi\)
0.439813 + 0.898089i \(0.355045\pi\)
\(740\) −3582.63 −0.177973
\(741\) 0 0
\(742\) 18841.6 0.932206
\(743\) −21331.1 −1.05325 −0.526623 0.850099i \(-0.676542\pi\)
−0.526623 + 0.850099i \(0.676542\pi\)
\(744\) 0 0
\(745\) 22.9477 0.00112851
\(746\) 22134.0 1.08630
\(747\) 0 0
\(748\) 93954.1 4.59265
\(749\) −8483.58 −0.413863
\(750\) 0 0
\(751\) 11712.9 0.569122 0.284561 0.958658i \(-0.408152\pi\)
0.284561 + 0.958658i \(0.408152\pi\)
\(752\) −1263.78 −0.0612835
\(753\) 0 0
\(754\) 0 0
\(755\) −29.5795 −0.00142584
\(756\) 0 0
\(757\) −16610.9 −0.797537 −0.398768 0.917052i \(-0.630562\pi\)
−0.398768 + 0.917052i \(0.630562\pi\)
\(758\) −51507.4 −2.46812
\(759\) 0 0
\(760\) −4371.47 −0.208645
\(761\) −29365.5 −1.39882 −0.699408 0.714723i \(-0.746553\pi\)
−0.699408 + 0.714723i \(0.746553\pi\)
\(762\) 0 0
\(763\) −7615.50 −0.361336
\(764\) −13871.2 −0.656861
\(765\) 0 0
\(766\) 47673.5 2.24871
\(767\) 0 0
\(768\) 0 0
\(769\) 28599.9 1.34114 0.670572 0.741844i \(-0.266049\pi\)
0.670572 + 0.741844i \(0.266049\pi\)
\(770\) −10721.8 −0.501803
\(771\) 0 0
\(772\) 7426.03 0.346203
\(773\) 13491.8 0.627772 0.313886 0.949461i \(-0.398369\pi\)
0.313886 + 0.949461i \(0.398369\pi\)
\(774\) 0 0
\(775\) −8083.78 −0.374681
\(776\) 33482.7 1.54892
\(777\) 0 0
\(778\) −30434.3 −1.40247
\(779\) −14221.5 −0.654093
\(780\) 0 0
\(781\) 6710.01 0.307430
\(782\) 84415.9 3.86024
\(783\) 0 0
\(784\) −8581.27 −0.390911
\(785\) −10377.5 −0.471834
\(786\) 0 0
\(787\) −25876.7 −1.17205 −0.586025 0.810293i \(-0.699308\pi\)
−0.586025 + 0.810293i \(0.699308\pi\)
\(788\) −56055.5 −2.53413
\(789\) 0 0
\(790\) −11677.7 −0.525918
\(791\) −11749.3 −0.528137
\(792\) 0 0
\(793\) 0 0
\(794\) 56994.2 2.54742
\(795\) 0 0
\(796\) 33396.1 1.48705
\(797\) −21936.4 −0.974938 −0.487469 0.873140i \(-0.662080\pi\)
−0.487469 + 0.873140i \(0.662080\pi\)
\(798\) 0 0
\(799\) 4098.49 0.181469
\(800\) 10894.9 0.481491
\(801\) 0 0
\(802\) 17069.0 0.751531
\(803\) −11080.9 −0.486969
\(804\) 0 0
\(805\) −6205.62 −0.271701
\(806\) 0 0
\(807\) 0 0
\(808\) 2374.29 0.103375
\(809\) −5583.23 −0.242640 −0.121320 0.992613i \(-0.538713\pi\)
−0.121320 + 0.992613i \(0.538713\pi\)
\(810\) 0 0
\(811\) −12925.4 −0.559647 −0.279823 0.960052i \(-0.590276\pi\)
−0.279823 + 0.960052i \(0.590276\pi\)
\(812\) −12160.6 −0.525559
\(813\) 0 0
\(814\) 17372.8 0.748054
\(815\) 4902.80 0.210721
\(816\) 0 0
\(817\) 3571.27 0.152929
\(818\) 228.058 0.00974801
\(819\) 0 0
\(820\) 29549.7 1.25844
\(821\) −10153.2 −0.431608 −0.215804 0.976437i \(-0.569237\pi\)
−0.215804 + 0.976437i \(0.569237\pi\)
\(822\) 0 0
\(823\) 3282.93 0.139047 0.0695235 0.997580i \(-0.477852\pi\)
0.0695235 + 0.997580i \(0.477852\pi\)
\(824\) 41543.1 1.75634
\(825\) 0 0
\(826\) 7794.49 0.328335
\(827\) −17689.3 −0.743795 −0.371897 0.928274i \(-0.621293\pi\)
−0.371897 + 0.928274i \(0.621293\pi\)
\(828\) 0 0
\(829\) 38181.5 1.59964 0.799818 0.600243i \(-0.204930\pi\)
0.799818 + 0.600243i \(0.204930\pi\)
\(830\) −9734.56 −0.407098
\(831\) 0 0
\(832\) 0 0
\(833\) 27829.4 1.15754
\(834\) 0 0
\(835\) −492.615 −0.0204163
\(836\) 30505.2 1.26201
\(837\) 0 0
\(838\) −3430.41 −0.141410
\(839\) 43895.2 1.80623 0.903117 0.429395i \(-0.141273\pi\)
0.903117 + 0.429395i \(0.141273\pi\)
\(840\) 0 0
\(841\) −11800.1 −0.483829
\(842\) 70392.4 2.88109
\(843\) 0 0
\(844\) −53188.5 −2.16922
\(845\) 0 0
\(846\) 0 0
\(847\) 23532.0 0.954627
\(848\) 15876.8 0.642938
\(849\) 0 0
\(850\) 48093.0 1.94068
\(851\) 10055.1 0.405034
\(852\) 0 0
\(853\) 19955.2 0.800998 0.400499 0.916297i \(-0.368837\pi\)
0.400499 + 0.916297i \(0.368837\pi\)
\(854\) 29198.0 1.16995
\(855\) 0 0
\(856\) −34850.8 −1.39156
\(857\) −26030.4 −1.03755 −0.518776 0.854910i \(-0.673612\pi\)
−0.518776 + 0.854910i \(0.673612\pi\)
\(858\) 0 0
\(859\) −45617.4 −1.81193 −0.905964 0.423354i \(-0.860853\pi\)
−0.905964 + 0.423354i \(0.860853\pi\)
\(860\) −7420.45 −0.294227
\(861\) 0 0
\(862\) 5230.59 0.206676
\(863\) −2010.93 −0.0793195 −0.0396597 0.999213i \(-0.512627\pi\)
−0.0396597 + 0.999213i \(0.512627\pi\)
\(864\) 0 0
\(865\) 8532.78 0.335403
\(866\) 42170.1 1.65473
\(867\) 0 0
\(868\) 8376.19 0.327542
\(869\) 36478.2 1.42398
\(870\) 0 0
\(871\) 0 0
\(872\) −31284.7 −1.21495
\(873\) 0 0
\(874\) 27408.3 1.06076
\(875\) −7760.41 −0.299828
\(876\) 0 0
\(877\) 36767.3 1.41567 0.707836 0.706376i \(-0.249671\pi\)
0.707836 + 0.706376i \(0.249671\pi\)
\(878\) −51692.0 −1.98693
\(879\) 0 0
\(880\) −9034.72 −0.346091
\(881\) −35401.2 −1.35380 −0.676899 0.736076i \(-0.736677\pi\)
−0.676899 + 0.736076i \(0.736677\pi\)
\(882\) 0 0
\(883\) −11928.0 −0.454596 −0.227298 0.973825i \(-0.572989\pi\)
−0.227298 + 0.973825i \(0.572989\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 17930.0 0.679875
\(887\) 32939.3 1.24689 0.623447 0.781866i \(-0.285732\pi\)
0.623447 + 0.781866i \(0.285732\pi\)
\(888\) 0 0
\(889\) −9340.40 −0.352381
\(890\) −4945.41 −0.186259
\(891\) 0 0
\(892\) −73677.3 −2.76558
\(893\) 1330.70 0.0498660
\(894\) 0 0
\(895\) −15482.6 −0.578241
\(896\) −19776.6 −0.737376
\(897\) 0 0
\(898\) 506.569 0.0188245
\(899\) −8671.18 −0.321691
\(900\) 0 0
\(901\) −51489.2 −1.90383
\(902\) −143292. −5.28946
\(903\) 0 0
\(904\) −48266.4 −1.77579
\(905\) 941.289 0.0345740
\(906\) 0 0
\(907\) 1500.89 0.0549461 0.0274731 0.999623i \(-0.491254\pi\)
0.0274731 + 0.999623i \(0.491254\pi\)
\(908\) −38030.7 −1.38997
\(909\) 0 0
\(910\) 0 0
\(911\) −19728.2 −0.717481 −0.358740 0.933437i \(-0.616794\pi\)
−0.358740 + 0.933437i \(0.616794\pi\)
\(912\) 0 0
\(913\) 30408.3 1.10226
\(914\) 5868.48 0.212376
\(915\) 0 0
\(916\) −24313.5 −0.877011
\(917\) 9539.72 0.343543
\(918\) 0 0
\(919\) 19992.0 0.717602 0.358801 0.933414i \(-0.383186\pi\)
0.358801 + 0.933414i \(0.383186\pi\)
\(920\) −25492.9 −0.913560
\(921\) 0 0
\(922\) 41683.4 1.48891
\(923\) 0 0
\(924\) 0 0
\(925\) 5728.54 0.203625
\(926\) 18310.2 0.649794
\(927\) 0 0
\(928\) 11686.6 0.413395
\(929\) −1923.62 −0.0679354 −0.0339677 0.999423i \(-0.510814\pi\)
−0.0339677 + 0.999423i \(0.510814\pi\)
\(930\) 0 0
\(931\) 9035.71 0.318081
\(932\) −9392.52 −0.330110
\(933\) 0 0
\(934\) 42634.0 1.49361
\(935\) 29300.0 1.02483
\(936\) 0 0
\(937\) −10252.9 −0.357468 −0.178734 0.983897i \(-0.557200\pi\)
−0.178734 + 0.983897i \(0.557200\pi\)
\(938\) 30944.4 1.07715
\(939\) 0 0
\(940\) −2764.96 −0.0959394
\(941\) 11043.0 0.382563 0.191282 0.981535i \(-0.438736\pi\)
0.191282 + 0.981535i \(0.438736\pi\)
\(942\) 0 0
\(943\) −82934.8 −2.86398
\(944\) 6568.00 0.226451
\(945\) 0 0
\(946\) 35983.1 1.23669
\(947\) 32105.2 1.10167 0.550833 0.834615i \(-0.314310\pi\)
0.550833 + 0.834615i \(0.314310\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 15614.9 0.533280
\(951\) 0 0
\(952\) −22307.1 −0.759431
\(953\) 9473.37 0.322007 0.161003 0.986954i \(-0.448527\pi\)
0.161003 + 0.986954i \(0.448527\pi\)
\(954\) 0 0
\(955\) −4325.79 −0.146575
\(956\) 75598.4 2.55756
\(957\) 0 0
\(958\) 19819.3 0.668404
\(959\) 6547.26 0.220461
\(960\) 0 0
\(961\) −23818.3 −0.799514
\(962\) 0 0
\(963\) 0 0
\(964\) −88399.8 −2.95349
\(965\) 2315.84 0.0772534
\(966\) 0 0
\(967\) 5310.75 0.176610 0.0883052 0.996093i \(-0.471855\pi\)
0.0883052 + 0.996093i \(0.471855\pi\)
\(968\) 96670.1 3.20981
\(969\) 0 0
\(970\) 23326.2 0.772122
\(971\) −24271.2 −0.802164 −0.401082 0.916042i \(-0.631366\pi\)
−0.401082 + 0.916042i \(0.631366\pi\)
\(972\) 0 0
\(973\) 2322.32 0.0765163
\(974\) 87454.2 2.87702
\(975\) 0 0
\(976\) 24603.6 0.806907
\(977\) −49602.5 −1.62428 −0.812142 0.583460i \(-0.801699\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(978\) 0 0
\(979\) 15448.2 0.504317
\(980\) −18774.6 −0.611971
\(981\) 0 0
\(982\) −38375.5 −1.24706
\(983\) 47385.7 1.53751 0.768753 0.639545i \(-0.220877\pi\)
0.768753 + 0.639545i \(0.220877\pi\)
\(984\) 0 0
\(985\) −17481.2 −0.565478
\(986\) 51587.7 1.66621
\(987\) 0 0
\(988\) 0 0
\(989\) 20826.4 0.669607
\(990\) 0 0
\(991\) −8947.33 −0.286802 −0.143401 0.989665i \(-0.545804\pi\)
−0.143401 + 0.989665i \(0.545804\pi\)
\(992\) −8049.67 −0.257638
\(993\) 0 0
\(994\) −3558.95 −0.113564
\(995\) 10414.7 0.331828
\(996\) 0 0
\(997\) −14908.6 −0.473582 −0.236791 0.971561i \(-0.576096\pi\)
−0.236791 + 0.971561i \(0.576096\pi\)
\(998\) 51881.4 1.64557
\(999\) 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 1521.4.a.s.1.2 2
3.2 odd 2 507.4.a.f.1.1 2
13.12 even 2 117.4.a.c.1.1 2
39.5 even 4 507.4.b.f.337.4 4
39.8 even 4 507.4.b.f.337.1 4
39.38 odd 2 39.4.a.b.1.2 2
52.51 odd 2 1872.4.a.t.1.2 2
156.155 even 2 624.4.a.r.1.1 2
195.194 odd 2 975.4.a.j.1.1 2
273.272 even 2 1911.4.a.h.1.2 2
312.77 odd 2 2496.4.a.bc.1.2 2
312.155 even 2 2496.4.a.s.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.b.1.2 2 39.38 odd 2
117.4.a.c.1.1 2 13.12 even 2
507.4.a.f.1.1 2 3.2 odd 2
507.4.b.f.337.1 4 39.8 even 4
507.4.b.f.337.4 4 39.5 even 4
624.4.a.r.1.1 2 156.155 even 2
975.4.a.j.1.1 2 195.194 odd 2
1521.4.a.s.1.2 2 1.1 even 1 trivial
1872.4.a.t.1.2 2 52.51 odd 2
1911.4.a.h.1.2 2 273.272 even 2
2496.4.a.s.1.2 2 312.155 even 2
2496.4.a.bc.1.2 2 312.77 odd 2