Properties

Label 1521.4.a.bl.1.2
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $4$

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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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 82x^{10} + 2137x^{8} - 22488x^{6} + 89784x^{4} - 67392x^{2} + 11664 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 117)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.17158\) 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)
 
\(f(q)\) \(=\) \(q-4.90363 q^{2} +16.0456 q^{4} +5.61325 q^{5} +3.21345 q^{7} -39.4526 q^{8} +O(q^{10})\) \(q-4.90363 q^{2} +16.0456 q^{4} +5.61325 q^{5} +3.21345 q^{7} -39.4526 q^{8} -27.5253 q^{10} +43.6466 q^{11} -15.7576 q^{14} +65.0962 q^{16} +65.8757 q^{17} +3.06427 q^{19} +90.0680 q^{20} -214.027 q^{22} -163.764 q^{23} -93.4914 q^{25} +51.5617 q^{28} +155.449 q^{29} -240.463 q^{31} -3.58708 q^{32} -323.030 q^{34} +18.0379 q^{35} +317.662 q^{37} -15.0261 q^{38} -221.457 q^{40} +127.910 q^{41} -379.001 q^{43} +700.336 q^{44} +803.039 q^{46} +79.4626 q^{47} -332.674 q^{49} +458.447 q^{50} -571.746 q^{53} +244.999 q^{55} -126.779 q^{56} -762.263 q^{58} -105.538 q^{59} -724.638 q^{61} +1179.14 q^{62} -503.180 q^{64} -302.222 q^{67} +1057.01 q^{68} -88.4512 q^{70} -1003.39 q^{71} -277.988 q^{73} -1557.70 q^{74} +49.1681 q^{76} +140.256 q^{77} -959.137 q^{79} +365.402 q^{80} -627.224 q^{82} +410.957 q^{83} +369.777 q^{85} +1858.48 q^{86} -1721.97 q^{88} -226.199 q^{89} -2627.69 q^{92} -389.655 q^{94} +17.2005 q^{95} +888.458 q^{97} +1631.31 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 32 q^{4} - 112 q^{10} + 184 q^{16} - 584 q^{22} + 92 q^{25} - 448 q^{40} - 1620 q^{43} - 2136 q^{49} - 920 q^{55} - 2588 q^{61} + 184 q^{64} - 6380 q^{79} - 2536 q^{82} - 10280 q^{88} - 3320 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.90363 −1.73370 −0.866848 0.498573i \(-0.833857\pi\)
−0.866848 + 0.498573i \(0.833857\pi\)
\(3\) 0 0
\(4\) 16.0456 2.00570
\(5\) 5.61325 0.502065 0.251032 0.967979i \(-0.419230\pi\)
0.251032 + 0.967979i \(0.419230\pi\)
\(6\) 0 0
\(7\) 3.21345 0.173510 0.0867550 0.996230i \(-0.472350\pi\)
0.0867550 + 0.996230i \(0.472350\pi\)
\(8\) −39.4526 −1.74357
\(9\) 0 0
\(10\) −27.5253 −0.870427
\(11\) 43.6466 1.19636 0.598179 0.801362i \(-0.295891\pi\)
0.598179 + 0.801362i \(0.295891\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −15.7576 −0.300813
\(15\) 0 0
\(16\) 65.0962 1.01713
\(17\) 65.8757 0.939835 0.469918 0.882710i \(-0.344284\pi\)
0.469918 + 0.882710i \(0.344284\pi\)
\(18\) 0 0
\(19\) 3.06427 0.0369996 0.0184998 0.999829i \(-0.494111\pi\)
0.0184998 + 0.999829i \(0.494111\pi\)
\(20\) 90.0680 1.00699
\(21\) 0 0
\(22\) −214.027 −2.07412
\(23\) −163.764 −1.48466 −0.742330 0.670034i \(-0.766279\pi\)
−0.742330 + 0.670034i \(0.766279\pi\)
\(24\) 0 0
\(25\) −93.4914 −0.747931
\(26\) 0 0
\(27\) 0 0
\(28\) 51.5617 0.348009
\(29\) 155.449 0.995382 0.497691 0.867354i \(-0.334181\pi\)
0.497691 + 0.867354i \(0.334181\pi\)
\(30\) 0 0
\(31\) −240.463 −1.39318 −0.696588 0.717471i \(-0.745299\pi\)
−0.696588 + 0.717471i \(0.745299\pi\)
\(32\) −3.58708 −0.0198160
\(33\) 0 0
\(34\) −323.030 −1.62939
\(35\) 18.0379 0.0871132
\(36\) 0 0
\(37\) 317.662 1.41144 0.705721 0.708490i \(-0.250623\pi\)
0.705721 + 0.708490i \(0.250623\pi\)
\(38\) −15.0261 −0.0641460
\(39\) 0 0
\(40\) −221.457 −0.875387
\(41\) 127.910 0.487224 0.243612 0.969873i \(-0.421668\pi\)
0.243612 + 0.969873i \(0.421668\pi\)
\(42\) 0 0
\(43\) −379.001 −1.34412 −0.672060 0.740497i \(-0.734590\pi\)
−0.672060 + 0.740497i \(0.734590\pi\)
\(44\) 700.336 2.39954
\(45\) 0 0
\(46\) 803.039 2.57395
\(47\) 79.4626 0.246613 0.123306 0.992369i \(-0.460650\pi\)
0.123306 + 0.992369i \(0.460650\pi\)
\(48\) 0 0
\(49\) −332.674 −0.969894
\(50\) 458.447 1.29668
\(51\) 0 0
\(52\) 0 0
\(53\) −571.746 −1.48180 −0.740899 0.671616i \(-0.765600\pi\)
−0.740899 + 0.671616i \(0.765600\pi\)
\(54\) 0 0
\(55\) 244.999 0.600649
\(56\) −126.779 −0.302528
\(57\) 0 0
\(58\) −762.263 −1.72569
\(59\) −105.538 −0.232879 −0.116440 0.993198i \(-0.537148\pi\)
−0.116440 + 0.993198i \(0.537148\pi\)
\(60\) 0 0
\(61\) −724.638 −1.52099 −0.760495 0.649344i \(-0.775043\pi\)
−0.760495 + 0.649344i \(0.775043\pi\)
\(62\) 1179.14 2.41534
\(63\) 0 0
\(64\) −503.180 −0.982774
\(65\) 0 0
\(66\) 0 0
\(67\) −302.222 −0.551078 −0.275539 0.961290i \(-0.588856\pi\)
−0.275539 + 0.961290i \(0.588856\pi\)
\(68\) 1057.01 1.88503
\(69\) 0 0
\(70\) −88.4512 −0.151028
\(71\) −1003.39 −1.67719 −0.838594 0.544757i \(-0.816622\pi\)
−0.838594 + 0.544757i \(0.816622\pi\)
\(72\) 0 0
\(73\) −277.988 −0.445699 −0.222850 0.974853i \(-0.571536\pi\)
−0.222850 + 0.974853i \(0.571536\pi\)
\(74\) −1557.70 −2.44701
\(75\) 0 0
\(76\) 49.1681 0.0742101
\(77\) 140.256 0.207580
\(78\) 0 0
\(79\) −959.137 −1.36597 −0.682983 0.730434i \(-0.739318\pi\)
−0.682983 + 0.730434i \(0.739318\pi\)
\(80\) 365.402 0.510664
\(81\) 0 0
\(82\) −627.224 −0.844699
\(83\) 410.957 0.543475 0.271737 0.962371i \(-0.412402\pi\)
0.271737 + 0.962371i \(0.412402\pi\)
\(84\) 0 0
\(85\) 369.777 0.471858
\(86\) 1858.48 2.33029
\(87\) 0 0
\(88\) −1721.97 −2.08594
\(89\) −226.199 −0.269405 −0.134703 0.990886i \(-0.543008\pi\)
−0.134703 + 0.990886i \(0.543008\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −2627.69 −2.97778
\(93\) 0 0
\(94\) −389.655 −0.427552
\(95\) 17.2005 0.0185762
\(96\) 0 0
\(97\) 888.458 0.929992 0.464996 0.885313i \(-0.346056\pi\)
0.464996 + 0.885313i \(0.346056\pi\)
\(98\) 1631.31 1.68150
\(99\) 0 0
\(100\) −1500.12 −1.50012
\(101\) 1860.36 1.83280 0.916400 0.400264i \(-0.131082\pi\)
0.916400 + 0.400264i \(0.131082\pi\)
\(102\) 0 0
\(103\) 328.512 0.314265 0.157132 0.987578i \(-0.449775\pi\)
0.157132 + 0.987578i \(0.449775\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 2803.63 2.56899
\(107\) −328.774 −0.297045 −0.148522 0.988909i \(-0.547452\pi\)
−0.148522 + 0.988909i \(0.547452\pi\)
\(108\) 0 0
\(109\) −201.673 −0.177218 −0.0886092 0.996066i \(-0.528242\pi\)
−0.0886092 + 0.996066i \(0.528242\pi\)
\(110\) −1201.39 −1.04134
\(111\) 0 0
\(112\) 209.183 0.176482
\(113\) −289.984 −0.241411 −0.120705 0.992688i \(-0.538516\pi\)
−0.120705 + 0.992688i \(0.538516\pi\)
\(114\) 0 0
\(115\) −919.250 −0.745395
\(116\) 2494.27 1.99644
\(117\) 0 0
\(118\) 517.520 0.403742
\(119\) 211.688 0.163071
\(120\) 0 0
\(121\) 574.026 0.431274
\(122\) 3553.36 2.63693
\(123\) 0 0
\(124\) −3858.37 −2.79429
\(125\) −1226.45 −0.877574
\(126\) 0 0
\(127\) 752.567 0.525823 0.262911 0.964820i \(-0.415317\pi\)
0.262911 + 0.964820i \(0.415317\pi\)
\(128\) 2496.11 1.72365
\(129\) 0 0
\(130\) 0 0
\(131\) −2200.41 −1.46756 −0.733782 0.679385i \(-0.762247\pi\)
−0.733782 + 0.679385i \(0.762247\pi\)
\(132\) 0 0
\(133\) 9.84689 0.00641980
\(134\) 1481.98 0.955402
\(135\) 0 0
\(136\) −2598.97 −1.63867
\(137\) 364.076 0.227044 0.113522 0.993535i \(-0.463787\pi\)
0.113522 + 0.993535i \(0.463787\pi\)
\(138\) 0 0
\(139\) 2453.59 1.49720 0.748600 0.663022i \(-0.230727\pi\)
0.748600 + 0.663022i \(0.230727\pi\)
\(140\) 289.429 0.174723
\(141\) 0 0
\(142\) 4920.25 2.90773
\(143\) 0 0
\(144\) 0 0
\(145\) 872.573 0.499746
\(146\) 1363.15 0.772706
\(147\) 0 0
\(148\) 5097.08 2.83093
\(149\) −65.1448 −0.0358179 −0.0179090 0.999840i \(-0.505701\pi\)
−0.0179090 + 0.999840i \(0.505701\pi\)
\(150\) 0 0
\(151\) 250.204 0.134843 0.0674216 0.997725i \(-0.478523\pi\)
0.0674216 + 0.997725i \(0.478523\pi\)
\(152\) −120.894 −0.0645116
\(153\) 0 0
\(154\) −687.764 −0.359881
\(155\) −1349.78 −0.699464
\(156\) 0 0
\(157\) 1768.99 0.899242 0.449621 0.893219i \(-0.351559\pi\)
0.449621 + 0.893219i \(0.351559\pi\)
\(158\) 4703.25 2.36817
\(159\) 0 0
\(160\) −20.1352 −0.00994890
\(161\) −526.248 −0.257603
\(162\) 0 0
\(163\) −2707.32 −1.30094 −0.650470 0.759532i \(-0.725428\pi\)
−0.650470 + 0.759532i \(0.725428\pi\)
\(164\) 2052.39 0.977226
\(165\) 0 0
\(166\) −2015.18 −0.942220
\(167\) 3492.29 1.61821 0.809107 0.587661i \(-0.199951\pi\)
0.809107 + 0.587661i \(0.199951\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −1813.25 −0.818058
\(171\) 0 0
\(172\) −6081.30 −2.69590
\(173\) 3981.96 1.74996 0.874979 0.484160i \(-0.160875\pi\)
0.874979 + 0.484160i \(0.160875\pi\)
\(174\) 0 0
\(175\) −300.430 −0.129774
\(176\) 2841.23 1.21685
\(177\) 0 0
\(178\) 1109.20 0.467066
\(179\) −2255.05 −0.941624 −0.470812 0.882234i \(-0.656039\pi\)
−0.470812 + 0.882234i \(0.656039\pi\)
\(180\) 0 0
\(181\) −890.108 −0.365532 −0.182766 0.983156i \(-0.558505\pi\)
−0.182766 + 0.983156i \(0.558505\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 6460.92 2.58862
\(185\) 1783.12 0.708635
\(186\) 0 0
\(187\) 2875.25 1.12438
\(188\) 1275.02 0.494631
\(189\) 0 0
\(190\) −84.3451 −0.0322055
\(191\) −799.774 −0.302982 −0.151491 0.988459i \(-0.548407\pi\)
−0.151491 + 0.988459i \(0.548407\pi\)
\(192\) 0 0
\(193\) −2693.40 −1.00454 −0.502268 0.864712i \(-0.667501\pi\)
−0.502268 + 0.864712i \(0.667501\pi\)
\(194\) −4356.67 −1.61232
\(195\) 0 0
\(196\) −5337.95 −1.94532
\(197\) −2735.70 −0.989395 −0.494698 0.869065i \(-0.664721\pi\)
−0.494698 + 0.869065i \(0.664721\pi\)
\(198\) 0 0
\(199\) −1176.44 −0.419072 −0.209536 0.977801i \(-0.567195\pi\)
−0.209536 + 0.977801i \(0.567195\pi\)
\(200\) 3688.48 1.30407
\(201\) 0 0
\(202\) −9122.52 −3.17752
\(203\) 499.526 0.172709
\(204\) 0 0
\(205\) 717.992 0.244618
\(206\) −1610.90 −0.544839
\(207\) 0 0
\(208\) 0 0
\(209\) 133.745 0.0442648
\(210\) 0 0
\(211\) −2914.65 −0.950961 −0.475481 0.879726i \(-0.657726\pi\)
−0.475481 + 0.879726i \(0.657726\pi\)
\(212\) −9174.00 −2.97204
\(213\) 0 0
\(214\) 1612.19 0.514985
\(215\) −2127.43 −0.674835
\(216\) 0 0
\(217\) −772.716 −0.241730
\(218\) 988.932 0.307243
\(219\) 0 0
\(220\) 3931.16 1.20472
\(221\) 0 0
\(222\) 0 0
\(223\) −1481.39 −0.444848 −0.222424 0.974950i \(-0.571397\pi\)
−0.222424 + 0.974950i \(0.571397\pi\)
\(224\) −11.5269 −0.00343827
\(225\) 0 0
\(226\) 1421.98 0.418533
\(227\) −4.38339 −0.00128166 −0.000640828 1.00000i \(-0.500204\pi\)
−0.000640828 1.00000i \(0.500204\pi\)
\(228\) 0 0
\(229\) 4689.58 1.35326 0.676629 0.736324i \(-0.263440\pi\)
0.676629 + 0.736324i \(0.263440\pi\)
\(230\) 4507.66 1.29229
\(231\) 0 0
\(232\) −6132.85 −1.73552
\(233\) 4124.52 1.15968 0.579842 0.814729i \(-0.303114\pi\)
0.579842 + 0.814729i \(0.303114\pi\)
\(234\) 0 0
\(235\) 446.044 0.123816
\(236\) −1693.42 −0.467086
\(237\) 0 0
\(238\) −1038.04 −0.282715
\(239\) −3455.41 −0.935196 −0.467598 0.883941i \(-0.654880\pi\)
−0.467598 + 0.883941i \(0.654880\pi\)
\(240\) 0 0
\(241\) 4109.67 1.09845 0.549226 0.835674i \(-0.314923\pi\)
0.549226 + 0.835674i \(0.314923\pi\)
\(242\) −2814.81 −0.747698
\(243\) 0 0
\(244\) −11627.2 −3.05065
\(245\) −1867.38 −0.486950
\(246\) 0 0
\(247\) 0 0
\(248\) 9486.89 2.42911
\(249\) 0 0
\(250\) 6014.05 1.52145
\(251\) 5032.21 1.26546 0.632730 0.774372i \(-0.281934\pi\)
0.632730 + 0.774372i \(0.281934\pi\)
\(252\) 0 0
\(253\) −7147.75 −1.77619
\(254\) −3690.31 −0.911617
\(255\) 0 0
\(256\) −8214.54 −2.00550
\(257\) 5633.97 1.36746 0.683731 0.729735i \(-0.260356\pi\)
0.683731 + 0.729735i \(0.260356\pi\)
\(258\) 0 0
\(259\) 1020.79 0.244899
\(260\) 0 0
\(261\) 0 0
\(262\) 10790.0 2.54431
\(263\) −5112.62 −1.19870 −0.599350 0.800487i \(-0.704574\pi\)
−0.599350 + 0.800487i \(0.704574\pi\)
\(264\) 0 0
\(265\) −3209.35 −0.743959
\(266\) −48.2855 −0.0111300
\(267\) 0 0
\(268\) −4849.32 −1.10530
\(269\) −6562.49 −1.48744 −0.743721 0.668490i \(-0.766941\pi\)
−0.743721 + 0.668490i \(0.766941\pi\)
\(270\) 0 0
\(271\) 8161.31 1.82939 0.914694 0.404146i \(-0.132431\pi\)
0.914694 + 0.404146i \(0.132431\pi\)
\(272\) 4288.26 0.955934
\(273\) 0 0
\(274\) −1785.29 −0.393626
\(275\) −4080.58 −0.894794
\(276\) 0 0
\(277\) 2726.57 0.591421 0.295710 0.955278i \(-0.404444\pi\)
0.295710 + 0.955278i \(0.404444\pi\)
\(278\) −12031.5 −2.59569
\(279\) 0 0
\(280\) −711.642 −0.151888
\(281\) −1216.67 −0.258294 −0.129147 0.991625i \(-0.541224\pi\)
−0.129147 + 0.991625i \(0.541224\pi\)
\(282\) 0 0
\(283\) −3248.32 −0.682306 −0.341153 0.940008i \(-0.610817\pi\)
−0.341153 + 0.940008i \(0.610817\pi\)
\(284\) −16100.0 −3.36393
\(285\) 0 0
\(286\) 0 0
\(287\) 411.033 0.0845383
\(288\) 0 0
\(289\) −573.393 −0.116709
\(290\) −4278.77 −0.866408
\(291\) 0 0
\(292\) −4460.48 −0.893938
\(293\) −9222.59 −1.83887 −0.919435 0.393241i \(-0.871354\pi\)
−0.919435 + 0.393241i \(0.871354\pi\)
\(294\) 0 0
\(295\) −592.412 −0.116920
\(296\) −12532.6 −2.46095
\(297\) 0 0
\(298\) 319.446 0.0620974
\(299\) 0 0
\(300\) 0 0
\(301\) −1217.90 −0.233218
\(302\) −1226.91 −0.233777
\(303\) 0 0
\(304\) 199.473 0.0376334
\(305\) −4067.58 −0.763635
\(306\) 0 0
\(307\) −2993.33 −0.556477 −0.278238 0.960512i \(-0.589751\pi\)
−0.278238 + 0.960512i \(0.589751\pi\)
\(308\) 2250.49 0.416343
\(309\) 0 0
\(310\) 6618.82 1.21266
\(311\) −7261.04 −1.32391 −0.661955 0.749544i \(-0.730273\pi\)
−0.661955 + 0.749544i \(0.730273\pi\)
\(312\) 0 0
\(313\) −3903.02 −0.704829 −0.352415 0.935844i \(-0.614639\pi\)
−0.352415 + 0.935844i \(0.614639\pi\)
\(314\) −8674.49 −1.55901
\(315\) 0 0
\(316\) −15389.9 −2.73972
\(317\) −4592.24 −0.813646 −0.406823 0.913507i \(-0.633363\pi\)
−0.406823 + 0.913507i \(0.633363\pi\)
\(318\) 0 0
\(319\) 6784.81 1.19083
\(320\) −2824.48 −0.493416
\(321\) 0 0
\(322\) 2580.52 0.446606
\(323\) 201.861 0.0347735
\(324\) 0 0
\(325\) 0 0
\(326\) 13275.7 2.25543
\(327\) 0 0
\(328\) −5046.39 −0.849512
\(329\) 255.349 0.0427898
\(330\) 0 0
\(331\) 5115.67 0.849495 0.424747 0.905312i \(-0.360363\pi\)
0.424747 + 0.905312i \(0.360363\pi\)
\(332\) 6594.05 1.09005
\(333\) 0 0
\(334\) −17124.9 −2.80549
\(335\) −1696.45 −0.276677
\(336\) 0 0
\(337\) −6808.37 −1.10052 −0.550260 0.834993i \(-0.685471\pi\)
−0.550260 + 0.834993i \(0.685471\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 5933.29 0.946405
\(341\) −10495.4 −1.66674
\(342\) 0 0
\(343\) −2171.24 −0.341796
\(344\) 14952.6 2.34357
\(345\) 0 0
\(346\) −19526.1 −3.03389
\(347\) 10918.2 1.68910 0.844551 0.535475i \(-0.179867\pi\)
0.844551 + 0.535475i \(0.179867\pi\)
\(348\) 0 0
\(349\) −10239.2 −1.57047 −0.785234 0.619199i \(-0.787457\pi\)
−0.785234 + 0.619199i \(0.787457\pi\)
\(350\) 1473.20 0.224988
\(351\) 0 0
\(352\) −156.564 −0.0237070
\(353\) −864.031 −0.130277 −0.0651384 0.997876i \(-0.520749\pi\)
−0.0651384 + 0.997876i \(0.520749\pi\)
\(354\) 0 0
\(355\) −5632.27 −0.842057
\(356\) −3629.50 −0.540346
\(357\) 0 0
\(358\) 11058.0 1.63249
\(359\) −11430.0 −1.68036 −0.840182 0.542305i \(-0.817552\pi\)
−0.840182 + 0.542305i \(0.817552\pi\)
\(360\) 0 0
\(361\) −6849.61 −0.998631
\(362\) 4364.76 0.633720
\(363\) 0 0
\(364\) 0 0
\(365\) −1560.42 −0.223770
\(366\) 0 0
\(367\) 9212.03 1.31026 0.655128 0.755518i \(-0.272615\pi\)
0.655128 + 0.755518i \(0.272615\pi\)
\(368\) −10660.4 −1.51009
\(369\) 0 0
\(370\) −8743.75 −1.22856
\(371\) −1837.28 −0.257107
\(372\) 0 0
\(373\) −1118.13 −0.155213 −0.0776065 0.996984i \(-0.524728\pi\)
−0.0776065 + 0.996984i \(0.524728\pi\)
\(374\) −14099.2 −1.94933
\(375\) 0 0
\(376\) −3135.00 −0.429988
\(377\) 0 0
\(378\) 0 0
\(379\) −8067.99 −1.09347 −0.546735 0.837306i \(-0.684129\pi\)
−0.546735 + 0.837306i \(0.684129\pi\)
\(380\) 275.993 0.0372583
\(381\) 0 0
\(382\) 3921.79 0.525279
\(383\) 2023.73 0.269995 0.134997 0.990846i \(-0.456897\pi\)
0.134997 + 0.990846i \(0.456897\pi\)
\(384\) 0 0
\(385\) 787.293 0.104219
\(386\) 13207.5 1.74156
\(387\) 0 0
\(388\) 14255.8 1.86528
\(389\) −9697.21 −1.26393 −0.631964 0.774998i \(-0.717751\pi\)
−0.631964 + 0.774998i \(0.717751\pi\)
\(390\) 0 0
\(391\) −10788.1 −1.39534
\(392\) 13124.8 1.69108
\(393\) 0 0
\(394\) 13414.9 1.71531
\(395\) −5383.88 −0.685803
\(396\) 0 0
\(397\) −3396.25 −0.429353 −0.214676 0.976685i \(-0.568870\pi\)
−0.214676 + 0.976685i \(0.568870\pi\)
\(398\) 5768.80 0.726543
\(399\) 0 0
\(400\) −6085.94 −0.760742
\(401\) 6649.91 0.828131 0.414066 0.910247i \(-0.364108\pi\)
0.414066 + 0.910247i \(0.364108\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 29850.6 3.67604
\(405\) 0 0
\(406\) −2449.49 −0.299424
\(407\) 13864.9 1.68859
\(408\) 0 0
\(409\) 2515.44 0.304109 0.152055 0.988372i \(-0.451411\pi\)
0.152055 + 0.988372i \(0.451411\pi\)
\(410\) −3520.77 −0.424093
\(411\) 0 0
\(412\) 5271.17 0.630320
\(413\) −339.141 −0.0404069
\(414\) 0 0
\(415\) 2306.81 0.272860
\(416\) 0 0
\(417\) 0 0
\(418\) −655.837 −0.0767417
\(419\) 11907.4 1.38835 0.694173 0.719808i \(-0.255770\pi\)
0.694173 + 0.719808i \(0.255770\pi\)
\(420\) 0 0
\(421\) −2382.88 −0.275854 −0.137927 0.990442i \(-0.544044\pi\)
−0.137927 + 0.990442i \(0.544044\pi\)
\(422\) 14292.4 1.64868
\(423\) 0 0
\(424\) 22556.9 2.58363
\(425\) −6158.81 −0.702932
\(426\) 0 0
\(427\) −2328.59 −0.263907
\(428\) −5275.37 −0.595782
\(429\) 0 0
\(430\) 10432.1 1.16996
\(431\) 286.545 0.0320241 0.0160121 0.999872i \(-0.494903\pi\)
0.0160121 + 0.999872i \(0.494903\pi\)
\(432\) 0 0
\(433\) −848.354 −0.0941555 −0.0470777 0.998891i \(-0.514991\pi\)
−0.0470777 + 0.998891i \(0.514991\pi\)
\(434\) 3789.11 0.419086
\(435\) 0 0
\(436\) −3235.97 −0.355447
\(437\) −501.818 −0.0549318
\(438\) 0 0
\(439\) 69.6932 0.00757694 0.00378847 0.999993i \(-0.498794\pi\)
0.00378847 + 0.999993i \(0.498794\pi\)
\(440\) −9665.87 −1.04728
\(441\) 0 0
\(442\) 0 0
\(443\) −7331.02 −0.786247 −0.393123 0.919486i \(-0.628605\pi\)
−0.393123 + 0.919486i \(0.628605\pi\)
\(444\) 0 0
\(445\) −1269.71 −0.135259
\(446\) 7264.18 0.771231
\(447\) 0 0
\(448\) −1616.94 −0.170521
\(449\) −14213.2 −1.49390 −0.746950 0.664880i \(-0.768483\pi\)
−0.746950 + 0.664880i \(0.768483\pi\)
\(450\) 0 0
\(451\) 5582.84 0.582895
\(452\) −4652.97 −0.484197
\(453\) 0 0
\(454\) 21.4945 0.00222200
\(455\) 0 0
\(456\) 0 0
\(457\) 5903.20 0.604246 0.302123 0.953269i \(-0.402305\pi\)
0.302123 + 0.953269i \(0.402305\pi\)
\(458\) −22996.0 −2.34614
\(459\) 0 0
\(460\) −14749.9 −1.49504
\(461\) 563.806 0.0569610 0.0284805 0.999594i \(-0.490933\pi\)
0.0284805 + 0.999594i \(0.490933\pi\)
\(462\) 0 0
\(463\) 17543.8 1.76097 0.880487 0.474070i \(-0.157216\pi\)
0.880487 + 0.474070i \(0.157216\pi\)
\(464\) 10119.1 1.01243
\(465\) 0 0
\(466\) −20225.1 −2.01054
\(467\) −9125.81 −0.904266 −0.452133 0.891951i \(-0.649337\pi\)
−0.452133 + 0.891951i \(0.649337\pi\)
\(468\) 0 0
\(469\) −971.174 −0.0956176
\(470\) −2187.23 −0.214659
\(471\) 0 0
\(472\) 4163.75 0.406043
\(473\) −16542.1 −1.60805
\(474\) 0 0
\(475\) −286.483 −0.0276732
\(476\) 3396.66 0.327071
\(477\) 0 0
\(478\) 16944.0 1.62134
\(479\) 4970.15 0.474096 0.237048 0.971498i \(-0.423820\pi\)
0.237048 + 0.971498i \(0.423820\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −20152.3 −1.90438
\(483\) 0 0
\(484\) 9210.59 0.865006
\(485\) 4987.14 0.466916
\(486\) 0 0
\(487\) 5716.80 0.531936 0.265968 0.963982i \(-0.414308\pi\)
0.265968 + 0.963982i \(0.414308\pi\)
\(488\) 28588.8 2.65196
\(489\) 0 0
\(490\) 9156.95 0.844222
\(491\) 6044.21 0.555543 0.277771 0.960647i \(-0.410404\pi\)
0.277771 + 0.960647i \(0.410404\pi\)
\(492\) 0 0
\(493\) 10240.3 0.935496
\(494\) 0 0
\(495\) 0 0
\(496\) −15653.2 −1.41704
\(497\) −3224.34 −0.291009
\(498\) 0 0
\(499\) −1101.16 −0.0987870 −0.0493935 0.998779i \(-0.515729\pi\)
−0.0493935 + 0.998779i \(0.515729\pi\)
\(500\) −19679.1 −1.76015
\(501\) 0 0
\(502\) −24676.1 −2.19392
\(503\) 1341.97 0.118957 0.0594787 0.998230i \(-0.481056\pi\)
0.0594787 + 0.998230i \(0.481056\pi\)
\(504\) 0 0
\(505\) 10442.7 0.920184
\(506\) 35049.9 3.07937
\(507\) 0 0
\(508\) 12075.4 1.05464
\(509\) −13473.6 −1.17329 −0.586647 0.809843i \(-0.699553\pi\)
−0.586647 + 0.809843i \(0.699553\pi\)
\(510\) 0 0
\(511\) −893.300 −0.0773332
\(512\) 20312.2 1.75328
\(513\) 0 0
\(514\) −27626.9 −2.37076
\(515\) 1844.02 0.157781
\(516\) 0 0
\(517\) 3468.27 0.295038
\(518\) −5005.58 −0.424581
\(519\) 0 0
\(520\) 0 0
\(521\) −10374.9 −0.872423 −0.436211 0.899844i \(-0.643680\pi\)
−0.436211 + 0.899844i \(0.643680\pi\)
\(522\) 0 0
\(523\) −22295.4 −1.86407 −0.932037 0.362364i \(-0.881970\pi\)
−0.932037 + 0.362364i \(0.881970\pi\)
\(524\) −35306.9 −2.94349
\(525\) 0 0
\(526\) 25070.4 2.07818
\(527\) −15840.7 −1.30936
\(528\) 0 0
\(529\) 14651.7 1.20422
\(530\) 15737.5 1.28980
\(531\) 0 0
\(532\) 157.999 0.0128762
\(533\) 0 0
\(534\) 0 0
\(535\) −1845.49 −0.149136
\(536\) 11923.4 0.960846
\(537\) 0 0
\(538\) 32180.0 2.57877
\(539\) −14520.1 −1.16034
\(540\) 0 0
\(541\) 10201.9 0.810747 0.405373 0.914151i \(-0.367141\pi\)
0.405373 + 0.914151i \(0.367141\pi\)
\(542\) −40020.1 −3.17160
\(543\) 0 0
\(544\) −236.301 −0.0186238
\(545\) −1132.04 −0.0889751
\(546\) 0 0
\(547\) −9312.12 −0.727893 −0.363947 0.931420i \(-0.618571\pi\)
−0.363947 + 0.931420i \(0.618571\pi\)
\(548\) 5841.81 0.455383
\(549\) 0 0
\(550\) 20009.7 1.55130
\(551\) 476.337 0.0368288
\(552\) 0 0
\(553\) −3082.14 −0.237009
\(554\) −13370.1 −1.02534
\(555\) 0 0
\(556\) 39369.3 3.00293
\(557\) −144.324 −0.0109788 −0.00548940 0.999985i \(-0.501747\pi\)
−0.00548940 + 0.999985i \(0.501747\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 1174.20 0.0886054
\(561\) 0 0
\(562\) 5966.11 0.447802
\(563\) 14667.0 1.09794 0.548968 0.835843i \(-0.315021\pi\)
0.548968 + 0.835843i \(0.315021\pi\)
\(564\) 0 0
\(565\) −1627.75 −0.121204
\(566\) 15928.6 1.18291
\(567\) 0 0
\(568\) 39586.3 2.92430
\(569\) 18754.1 1.38174 0.690871 0.722978i \(-0.257227\pi\)
0.690871 + 0.722978i \(0.257227\pi\)
\(570\) 0 0
\(571\) −13886.4 −1.01773 −0.508867 0.860845i \(-0.669935\pi\)
−0.508867 + 0.860845i \(0.669935\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −2015.55 −0.146564
\(575\) 15310.5 1.11042
\(576\) 0 0
\(577\) 9338.81 0.673795 0.336898 0.941541i \(-0.390622\pi\)
0.336898 + 0.941541i \(0.390622\pi\)
\(578\) 2811.71 0.202338
\(579\) 0 0
\(580\) 14000.9 1.00234
\(581\) 1320.59 0.0942983
\(582\) 0 0
\(583\) −24954.8 −1.77276
\(584\) 10967.3 0.777110
\(585\) 0 0
\(586\) 45224.1 3.18804
\(587\) 3724.13 0.261859 0.130930 0.991392i \(-0.458204\pi\)
0.130930 + 0.991392i \(0.458204\pi\)
\(588\) 0 0
\(589\) −736.845 −0.0515470
\(590\) 2904.97 0.202704
\(591\) 0 0
\(592\) 20678.6 1.43562
\(593\) 19910.2 1.37877 0.689387 0.724393i \(-0.257880\pi\)
0.689387 + 0.724393i \(0.257880\pi\)
\(594\) 0 0
\(595\) 1188.26 0.0818721
\(596\) −1045.29 −0.0718400
\(597\) 0 0
\(598\) 0 0
\(599\) −14756.0 −1.00653 −0.503265 0.864132i \(-0.667868\pi\)
−0.503265 + 0.864132i \(0.667868\pi\)
\(600\) 0 0
\(601\) −7939.05 −0.538837 −0.269418 0.963023i \(-0.586831\pi\)
−0.269418 + 0.963023i \(0.586831\pi\)
\(602\) 5972.14 0.404329
\(603\) 0 0
\(604\) 4014.67 0.270455
\(605\) 3222.15 0.216528
\(606\) 0 0
\(607\) −2563.88 −0.171441 −0.0857205 0.996319i \(-0.527319\pi\)
−0.0857205 + 0.996319i \(0.527319\pi\)
\(608\) −10.9918 −0.000733184 0
\(609\) 0 0
\(610\) 19945.9 1.32391
\(611\) 0 0
\(612\) 0 0
\(613\) −17593.4 −1.15920 −0.579602 0.814900i \(-0.696792\pi\)
−0.579602 + 0.814900i \(0.696792\pi\)
\(614\) 14678.2 0.964761
\(615\) 0 0
\(616\) −5533.47 −0.361932
\(617\) −24479.7 −1.59727 −0.798635 0.601816i \(-0.794444\pi\)
−0.798635 + 0.601816i \(0.794444\pi\)
\(618\) 0 0
\(619\) −2927.43 −0.190086 −0.0950431 0.995473i \(-0.530299\pi\)
−0.0950431 + 0.995473i \(0.530299\pi\)
\(620\) −21658.0 −1.40291
\(621\) 0 0
\(622\) 35605.5 2.29526
\(623\) −726.879 −0.0467445
\(624\) 0 0
\(625\) 4802.06 0.307332
\(626\) 19138.9 1.22196
\(627\) 0 0
\(628\) 28384.5 1.80361
\(629\) 20926.2 1.32652
\(630\) 0 0
\(631\) −10183.0 −0.642439 −0.321220 0.947005i \(-0.604093\pi\)
−0.321220 + 0.947005i \(0.604093\pi\)
\(632\) 37840.4 2.38166
\(633\) 0 0
\(634\) 22518.6 1.41061
\(635\) 4224.35 0.263997
\(636\) 0 0
\(637\) 0 0
\(638\) −33270.2 −2.06454
\(639\) 0 0
\(640\) 14011.3 0.865382
\(641\) 17103.4 1.05389 0.526946 0.849899i \(-0.323337\pi\)
0.526946 + 0.849899i \(0.323337\pi\)
\(642\) 0 0
\(643\) −21733.9 −1.33297 −0.666485 0.745518i \(-0.732202\pi\)
−0.666485 + 0.745518i \(0.732202\pi\)
\(644\) −8443.96 −0.516675
\(645\) 0 0
\(646\) −989.853 −0.0602867
\(647\) 22302.3 1.35517 0.677584 0.735446i \(-0.263027\pi\)
0.677584 + 0.735446i \(0.263027\pi\)
\(648\) 0 0
\(649\) −4606.38 −0.278607
\(650\) 0 0
\(651\) 0 0
\(652\) −43440.5 −2.60930
\(653\) −6977.15 −0.418127 −0.209063 0.977902i \(-0.567042\pi\)
−0.209063 + 0.977902i \(0.567042\pi\)
\(654\) 0 0
\(655\) −12351.5 −0.736812
\(656\) 8326.47 0.495570
\(657\) 0 0
\(658\) −1252.14 −0.0741845
\(659\) −10026.2 −0.592662 −0.296331 0.955085i \(-0.595763\pi\)
−0.296331 + 0.955085i \(0.595763\pi\)
\(660\) 0 0
\(661\) −4710.64 −0.277190 −0.138595 0.990349i \(-0.544259\pi\)
−0.138595 + 0.990349i \(0.544259\pi\)
\(662\) −25085.4 −1.47277
\(663\) 0 0
\(664\) −16213.3 −0.947589
\(665\) 55.2731 0.00322316
\(666\) 0 0
\(667\) −25456.9 −1.47780
\(668\) 56035.9 3.24565
\(669\) 0 0
\(670\) 8318.75 0.479674
\(671\) −31628.0 −1.81965
\(672\) 0 0
\(673\) −14727.2 −0.843523 −0.421762 0.906707i \(-0.638588\pi\)
−0.421762 + 0.906707i \(0.638588\pi\)
\(674\) 33385.7 1.90797
\(675\) 0 0
\(676\) 0 0
\(677\) −2118.31 −0.120256 −0.0601279 0.998191i \(-0.519151\pi\)
−0.0601279 + 0.998191i \(0.519151\pi\)
\(678\) 0 0
\(679\) 2855.02 0.161363
\(680\) −14588.7 −0.822720
\(681\) 0 0
\(682\) 51465.6 2.88962
\(683\) −7070.72 −0.396125 −0.198063 0.980189i \(-0.563465\pi\)
−0.198063 + 0.980189i \(0.563465\pi\)
\(684\) 0 0
\(685\) 2043.65 0.113991
\(686\) 10647.0 0.592571
\(687\) 0 0
\(688\) −24671.6 −1.36714
\(689\) 0 0
\(690\) 0 0
\(691\) 15584.0 0.857947 0.428974 0.903317i \(-0.358875\pi\)
0.428974 + 0.903317i \(0.358875\pi\)
\(692\) 63892.9 3.50989
\(693\) 0 0
\(694\) −53538.7 −2.92839
\(695\) 13772.6 0.751691
\(696\) 0 0
\(697\) 8426.17 0.457911
\(698\) 50209.4 2.72271
\(699\) 0 0
\(700\) −4820.57 −0.260287
\(701\) −13190.6 −0.710703 −0.355352 0.934733i \(-0.615639\pi\)
−0.355352 + 0.934733i \(0.615639\pi\)
\(702\) 0 0
\(703\) 973.404 0.0522228
\(704\) −21962.1 −1.17575
\(705\) 0 0
\(706\) 4236.89 0.225860
\(707\) 5978.17 0.318009
\(708\) 0 0
\(709\) 31565.9 1.67205 0.836025 0.548692i \(-0.184874\pi\)
0.836025 + 0.548692i \(0.184874\pi\)
\(710\) 27618.6 1.45987
\(711\) 0 0
\(712\) 8924.14 0.469728
\(713\) 39379.2 2.06839
\(714\) 0 0
\(715\) 0 0
\(716\) −36183.7 −1.88861
\(717\) 0 0
\(718\) 56048.3 2.91324
\(719\) −15535.3 −0.805800 −0.402900 0.915244i \(-0.631998\pi\)
−0.402900 + 0.915244i \(0.631998\pi\)
\(720\) 0 0
\(721\) 1055.66 0.0545281
\(722\) 33588.0 1.73132
\(723\) 0 0
\(724\) −14282.3 −0.733146
\(725\) −14533.1 −0.744477
\(726\) 0 0
\(727\) −27639.6 −1.41004 −0.705018 0.709189i \(-0.749061\pi\)
−0.705018 + 0.709189i \(0.749061\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 7651.71 0.387949
\(731\) −24967.0 −1.26325
\(732\) 0 0
\(733\) −1318.97 −0.0664629 −0.0332314 0.999448i \(-0.510580\pi\)
−0.0332314 + 0.999448i \(0.510580\pi\)
\(734\) −45172.4 −2.27158
\(735\) 0 0
\(736\) 587.434 0.0294200
\(737\) −13190.9 −0.659287
\(738\) 0 0
\(739\) −2970.75 −0.147876 −0.0739382 0.997263i \(-0.523557\pi\)
−0.0739382 + 0.997263i \(0.523557\pi\)
\(740\) 28611.2 1.42131
\(741\) 0 0
\(742\) 9009.32 0.445745
\(743\) 28792.6 1.42167 0.710833 0.703361i \(-0.248318\pi\)
0.710833 + 0.703361i \(0.248318\pi\)
\(744\) 0 0
\(745\) −365.675 −0.0179829
\(746\) 5482.88 0.269092
\(747\) 0 0
\(748\) 46135.1 2.25517
\(749\) −1056.50 −0.0515402
\(750\) 0 0
\(751\) −7839.45 −0.380913 −0.190456 0.981696i \(-0.560997\pi\)
−0.190456 + 0.981696i \(0.560997\pi\)
\(752\) 5172.71 0.250837
\(753\) 0 0
\(754\) 0 0
\(755\) 1404.46 0.0677000
\(756\) 0 0
\(757\) −3192.42 −0.153277 −0.0766383 0.997059i \(-0.524419\pi\)
−0.0766383 + 0.997059i \(0.524419\pi\)
\(758\) 39562.5 1.89574
\(759\) 0 0
\(760\) −678.606 −0.0323890
\(761\) 25421.9 1.21096 0.605481 0.795860i \(-0.292981\pi\)
0.605481 + 0.795860i \(0.292981\pi\)
\(762\) 0 0
\(763\) −648.067 −0.0307492
\(764\) −12832.8 −0.607691
\(765\) 0 0
\(766\) −9923.64 −0.468089
\(767\) 0 0
\(768\) 0 0
\(769\) −33930.6 −1.59112 −0.795559 0.605876i \(-0.792823\pi\)
−0.795559 + 0.605876i \(0.792823\pi\)
\(770\) −3860.60 −0.180683
\(771\) 0 0
\(772\) −43217.2 −2.01480
\(773\) 40596.9 1.88897 0.944483 0.328560i \(-0.106563\pi\)
0.944483 + 0.328560i \(0.106563\pi\)
\(774\) 0 0
\(775\) 22481.2 1.04200
\(776\) −35052.0 −1.62151
\(777\) 0 0
\(778\) 47551.5 2.19127
\(779\) 391.952 0.0180271
\(780\) 0 0
\(781\) −43794.5 −2.00652
\(782\) 52900.7 2.41909
\(783\) 0 0
\(784\) −21655.8 −0.986507
\(785\) 9929.81 0.451478
\(786\) 0 0
\(787\) 39046.2 1.76855 0.884274 0.466969i \(-0.154654\pi\)
0.884274 + 0.466969i \(0.154654\pi\)
\(788\) −43896.0 −1.98443
\(789\) 0 0
\(790\) 26400.5 1.18897
\(791\) −931.850 −0.0418872
\(792\) 0 0
\(793\) 0 0
\(794\) 16654.0 0.744367
\(795\) 0 0
\(796\) −18876.6 −0.840532
\(797\) 8074.79 0.358876 0.179438 0.983769i \(-0.442572\pi\)
0.179438 + 0.983769i \(0.442572\pi\)
\(798\) 0 0
\(799\) 5234.65 0.231776
\(800\) 335.361 0.0148210
\(801\) 0 0
\(802\) −32608.7 −1.43573
\(803\) −12133.2 −0.533216
\(804\) 0 0
\(805\) −2953.96 −0.129334
\(806\) 0 0
\(807\) 0 0
\(808\) −73396.1 −3.19562
\(809\) 30128.0 1.30933 0.654663 0.755921i \(-0.272811\pi\)
0.654663 + 0.755921i \(0.272811\pi\)
\(810\) 0 0
\(811\) −707.812 −0.0306469 −0.0153235 0.999883i \(-0.504878\pi\)
−0.0153235 + 0.999883i \(0.504878\pi\)
\(812\) 8015.20 0.346402
\(813\) 0 0
\(814\) −67988.2 −2.92750
\(815\) −15196.9 −0.653156
\(816\) 0 0
\(817\) −1161.36 −0.0497319
\(818\) −12334.8 −0.527233
\(819\) 0 0
\(820\) 11520.6 0.490630
\(821\) −15620.3 −0.664009 −0.332004 0.943278i \(-0.607725\pi\)
−0.332004 + 0.943278i \(0.607725\pi\)
\(822\) 0 0
\(823\) 32710.1 1.38542 0.692710 0.721216i \(-0.256417\pi\)
0.692710 + 0.721216i \(0.256417\pi\)
\(824\) −12960.7 −0.547944
\(825\) 0 0
\(826\) 1663.02 0.0700532
\(827\) 28246.1 1.18768 0.593842 0.804582i \(-0.297611\pi\)
0.593842 + 0.804582i \(0.297611\pi\)
\(828\) 0 0
\(829\) −14099.2 −0.590694 −0.295347 0.955390i \(-0.595435\pi\)
−0.295347 + 0.955390i \(0.595435\pi\)
\(830\) −11311.7 −0.473055
\(831\) 0 0
\(832\) 0 0
\(833\) −21915.1 −0.911541
\(834\) 0 0
\(835\) 19603.1 0.812449
\(836\) 2146.02 0.0887819
\(837\) 0 0
\(838\) −58389.7 −2.40697
\(839\) −11485.3 −0.472605 −0.236302 0.971680i \(-0.575936\pi\)
−0.236302 + 0.971680i \(0.575936\pi\)
\(840\) 0 0
\(841\) −224.714 −0.00921373
\(842\) 11684.8 0.478247
\(843\) 0 0
\(844\) −46767.3 −1.90734
\(845\) 0 0
\(846\) 0 0
\(847\) 1844.60 0.0748304
\(848\) −37218.5 −1.50718
\(849\) 0 0
\(850\) 30200.5 1.21867
\(851\) −52021.7 −2.09551
\(852\) 0 0
\(853\) 11836.8 0.475128 0.237564 0.971372i \(-0.423651\pi\)
0.237564 + 0.971372i \(0.423651\pi\)
\(854\) 11418.5 0.457534
\(855\) 0 0
\(856\) 12971.0 0.517920
\(857\) 31156.2 1.24186 0.620931 0.783865i \(-0.286755\pi\)
0.620931 + 0.783865i \(0.286755\pi\)
\(858\) 0 0
\(859\) −1584.88 −0.0629515 −0.0314758 0.999505i \(-0.510021\pi\)
−0.0314758 + 0.999505i \(0.510021\pi\)
\(860\) −34135.9 −1.35352
\(861\) 0 0
\(862\) −1405.11 −0.0555200
\(863\) −19884.8 −0.784340 −0.392170 0.919893i \(-0.628276\pi\)
−0.392170 + 0.919893i \(0.628276\pi\)
\(864\) 0 0
\(865\) 22351.8 0.878592
\(866\) 4160.02 0.163237
\(867\) 0 0
\(868\) −12398.7 −0.484837
\(869\) −41863.1 −1.63419
\(870\) 0 0
\(871\) 0 0
\(872\) 7956.54 0.308994
\(873\) 0 0
\(874\) 2460.73 0.0952351
\(875\) −3941.13 −0.152268
\(876\) 0 0
\(877\) −5483.43 −0.211132 −0.105566 0.994412i \(-0.533665\pi\)
−0.105566 + 0.994412i \(0.533665\pi\)
\(878\) −341.750 −0.0131361
\(879\) 0 0
\(880\) 15948.5 0.610938
\(881\) 6581.12 0.251672 0.125836 0.992051i \(-0.459839\pi\)
0.125836 + 0.992051i \(0.459839\pi\)
\(882\) 0 0
\(883\) 6942.60 0.264595 0.132297 0.991210i \(-0.457765\pi\)
0.132297 + 0.991210i \(0.457765\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 35948.6 1.36311
\(887\) −6562.68 −0.248425 −0.124213 0.992256i \(-0.539640\pi\)
−0.124213 + 0.992256i \(0.539640\pi\)
\(888\) 0 0
\(889\) 2418.34 0.0912355
\(890\) 6226.20 0.234498
\(891\) 0 0
\(892\) −23769.7 −0.892231
\(893\) 243.495 0.00912458
\(894\) 0 0
\(895\) −12658.2 −0.472756
\(896\) 8021.11 0.299070
\(897\) 0 0
\(898\) 69696.2 2.58997
\(899\) −37379.7 −1.38674
\(900\) 0 0
\(901\) −37664.1 −1.39265
\(902\) −27376.2 −1.01056
\(903\) 0 0
\(904\) 11440.6 0.420918
\(905\) −4996.40 −0.183521
\(906\) 0 0
\(907\) −1652.33 −0.0604902 −0.0302451 0.999543i \(-0.509629\pi\)
−0.0302451 + 0.999543i \(0.509629\pi\)
\(908\) −70.3341 −0.00257061
\(909\) 0 0
\(910\) 0 0
\(911\) 506.240 0.0184110 0.00920552 0.999958i \(-0.497070\pi\)
0.00920552 + 0.999958i \(0.497070\pi\)
\(912\) 0 0
\(913\) 17936.9 0.650191
\(914\) −28947.1 −1.04758
\(915\) 0 0
\(916\) 75247.1 2.71423
\(917\) −7070.91 −0.254637
\(918\) 0 0
\(919\) −2469.29 −0.0886336 −0.0443168 0.999018i \(-0.514111\pi\)
−0.0443168 + 0.999018i \(0.514111\pi\)
\(920\) 36266.8 1.29965
\(921\) 0 0
\(922\) −2764.69 −0.0987531
\(923\) 0 0
\(924\) 0 0
\(925\) −29698.7 −1.05566
\(926\) −86028.5 −3.05299
\(927\) 0 0
\(928\) −557.606 −0.0197245
\(929\) 41645.7 1.47078 0.735388 0.677646i \(-0.237000\pi\)
0.735388 + 0.677646i \(0.237000\pi\)
\(930\) 0 0
\(931\) −1019.40 −0.0358857
\(932\) 66180.3 2.32597
\(933\) 0 0
\(934\) 44749.6 1.56772
\(935\) 16139.5 0.564512
\(936\) 0 0
\(937\) −33891.2 −1.18162 −0.590810 0.806811i \(-0.701192\pi\)
−0.590810 + 0.806811i \(0.701192\pi\)
\(938\) 4762.28 0.165772
\(939\) 0 0
\(940\) 7157.03 0.248337
\(941\) −28351.3 −0.982173 −0.491087 0.871111i \(-0.663400\pi\)
−0.491087 + 0.871111i \(0.663400\pi\)
\(942\) 0 0
\(943\) −20947.1 −0.723363
\(944\) −6870.13 −0.236868
\(945\) 0 0
\(946\) 81116.5 2.78787
\(947\) −3732.71 −0.128085 −0.0640427 0.997947i \(-0.520399\pi\)
−0.0640427 + 0.997947i \(0.520399\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 1404.81 0.0479768
\(951\) 0 0
\(952\) −8351.65 −0.284326
\(953\) −9652.09 −0.328082 −0.164041 0.986454i \(-0.552453\pi\)
−0.164041 + 0.986454i \(0.552453\pi\)
\(954\) 0 0
\(955\) −4489.33 −0.152117
\(956\) −55444.1 −1.87572
\(957\) 0 0
\(958\) −24371.8 −0.821938
\(959\) 1169.94 0.0393945
\(960\) 0 0
\(961\) 28031.5 0.940939
\(962\) 0 0
\(963\) 0 0
\(964\) 65942.1 2.20316
\(965\) −15118.8 −0.504342
\(966\) 0 0
\(967\) −36196.6 −1.20373 −0.601864 0.798598i \(-0.705575\pi\)
−0.601864 + 0.798598i \(0.705575\pi\)
\(968\) −22646.8 −0.751959
\(969\) 0 0
\(970\) −24455.1 −0.809490
\(971\) 5250.98 0.173545 0.0867723 0.996228i \(-0.472345\pi\)
0.0867723 + 0.996228i \(0.472345\pi\)
\(972\) 0 0
\(973\) 7884.48 0.259779
\(974\) −28033.1 −0.922215
\(975\) 0 0
\(976\) −47171.2 −1.54704
\(977\) 23174.8 0.758881 0.379440 0.925216i \(-0.376117\pi\)
0.379440 + 0.925216i \(0.376117\pi\)
\(978\) 0 0
\(979\) −9872.82 −0.322305
\(980\) −29963.2 −0.976674
\(981\) 0 0
\(982\) −29638.6 −0.963142
\(983\) 33285.5 1.08000 0.540001 0.841664i \(-0.318424\pi\)
0.540001 + 0.841664i \(0.318424\pi\)
\(984\) 0 0
\(985\) −15356.2 −0.496740
\(986\) −50214.6 −1.62186
\(987\) 0 0
\(988\) 0 0
\(989\) 62066.8 1.99556
\(990\) 0 0
\(991\) −15535.4 −0.497979 −0.248990 0.968506i \(-0.580098\pi\)
−0.248990 + 0.968506i \(0.580098\pi\)
\(992\) 862.559 0.0276071
\(993\) 0 0
\(994\) 15811.0 0.504520
\(995\) −6603.63 −0.210401
\(996\) 0 0
\(997\) 26302.4 0.835511 0.417756 0.908559i \(-0.362817\pi\)
0.417756 + 0.908559i \(0.362817\pi\)
\(998\) 5399.69 0.171267
\(999\) 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.bl.1.2 12
3.2 odd 2 inner 1521.4.a.bl.1.12 12
13.2 odd 12 117.4.q.f.82.1 yes 12
13.7 odd 12 117.4.q.f.10.1 12
13.12 even 2 inner 1521.4.a.bl.1.11 12
39.2 even 12 117.4.q.f.82.6 yes 12
39.20 even 12 117.4.q.f.10.6 yes 12
39.38 odd 2 inner 1521.4.a.bl.1.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.4.q.f.10.1 12 13.7 odd 12
117.4.q.f.10.6 yes 12 39.20 even 12
117.4.q.f.82.1 yes 12 13.2 odd 12
117.4.q.f.82.6 yes 12 39.2 even 12
1521.4.a.bl.1.1 12 39.38 odd 2 inner
1521.4.a.bl.1.2 12 1.1 even 1 trivial
1521.4.a.bl.1.11 12 13.12 even 2 inner
1521.4.a.bl.1.12 12 3.2 odd 2 inner