Properties

Label 1521.4.a.bh.1.8
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 4x^{8} - 46x^{7} + 145x^{6} + 680x^{5} - 1501x^{4} - 3203x^{3} + 4784x^{2} + 3584x - 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 13^{2} \)
Twist minimal: no (minimal twist has level 169)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-3.82555\) 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)
 
\(f(q)\) \(=\) \(q+4.82555 q^{2} +15.2860 q^{4} -12.7712 q^{5} -26.1871 q^{7} +35.1589 q^{8} +O(q^{10})\) \(q+4.82555 q^{2} +15.2860 q^{4} -12.7712 q^{5} -26.1871 q^{7} +35.1589 q^{8} -61.6281 q^{10} +42.3430 q^{11} -126.367 q^{14} +47.3733 q^{16} +27.3331 q^{17} +13.1196 q^{19} -195.220 q^{20} +204.329 q^{22} +28.7976 q^{23} +38.1033 q^{25} -400.296 q^{28} +141.628 q^{29} +56.0144 q^{31} -52.6687 q^{32} +131.897 q^{34} +334.441 q^{35} +313.982 q^{37} +63.3095 q^{38} -449.021 q^{40} +352.375 q^{41} -320.676 q^{43} +647.255 q^{44} +138.964 q^{46} +339.339 q^{47} +342.765 q^{49} +183.869 q^{50} -349.461 q^{53} -540.771 q^{55} -920.710 q^{56} +683.432 q^{58} +258.742 q^{59} +650.732 q^{61} +270.301 q^{62} -633.142 q^{64} +894.996 q^{67} +417.813 q^{68} +1613.86 q^{70} +741.469 q^{71} -820.643 q^{73} +1515.14 q^{74} +200.546 q^{76} -1108.84 q^{77} -199.372 q^{79} -605.013 q^{80} +1700.40 q^{82} +541.696 q^{83} -349.076 q^{85} -1547.44 q^{86} +1488.73 q^{88} +380.615 q^{89} +440.199 q^{92} +1637.50 q^{94} -167.553 q^{95} +1430.50 q^{97} +1654.03 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 5 q^{2} + 37 q^{4} + 30 q^{5} - 38 q^{7} + 60 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 5 q^{2} + 37 q^{4} + 30 q^{5} - 38 q^{7} + 60 q^{8} - 147 q^{10} + 181 q^{11} + 147 q^{14} + 269 q^{16} + 55 q^{17} - 161 q^{19} + 370 q^{20} + 340 q^{22} + 204 q^{23} + 307 q^{25} - 344 q^{28} - 280 q^{29} - 706 q^{31} + 680 q^{32} - 216 q^{34} - 20 q^{35} - 298 q^{37} + 739 q^{38} + 13 q^{40} + 1201 q^{41} - 533 q^{43} + 355 q^{44} + 840 q^{46} + 956 q^{47} + 403 q^{49} - 1156 q^{50} + 278 q^{53} - 250 q^{55} - 250 q^{56} + 2877 q^{58} + 1377 q^{59} - 136 q^{61} - 2035 q^{62} + 284 q^{64} + 931 q^{67} + 1536 q^{68} + 4854 q^{70} + 2046 q^{71} + 45 q^{73} + 1990 q^{74} + 3608 q^{76} + 718 q^{77} + 412 q^{79} - 787 q^{80} + 2757 q^{82} + 3709 q^{83} + 2106 q^{85} + 125 q^{86} - 636 q^{88} + 1663 q^{89} - 4010 q^{92} - 2531 q^{94} + 1614 q^{95} + 1087 q^{97} - 282 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.82555 1.70609 0.853046 0.521836i \(-0.174753\pi\)
0.853046 + 0.521836i \(0.174753\pi\)
\(3\) 0 0
\(4\) 15.2860 1.91075
\(5\) −12.7712 −1.14229 −0.571145 0.820849i \(-0.693501\pi\)
−0.571145 + 0.820849i \(0.693501\pi\)
\(6\) 0 0
\(7\) −26.1871 −1.41397 −0.706986 0.707228i \(-0.749945\pi\)
−0.706986 + 0.707228i \(0.749945\pi\)
\(8\) 35.1589 1.55382
\(9\) 0 0
\(10\) −61.6281 −1.94885
\(11\) 42.3430 1.16063 0.580314 0.814393i \(-0.302930\pi\)
0.580314 + 0.814393i \(0.302930\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −126.367 −2.41236
\(15\) 0 0
\(16\) 47.3733 0.740208
\(17\) 27.3331 0.389955 0.194978 0.980808i \(-0.437537\pi\)
0.194978 + 0.980808i \(0.437537\pi\)
\(18\) 0 0
\(19\) 13.1196 0.158413 0.0792065 0.996858i \(-0.474761\pi\)
0.0792065 + 0.996858i \(0.474761\pi\)
\(20\) −195.220 −2.18263
\(21\) 0 0
\(22\) 204.329 1.98014
\(23\) 28.7976 0.261074 0.130537 0.991443i \(-0.458330\pi\)
0.130537 + 0.991443i \(0.458330\pi\)
\(24\) 0 0
\(25\) 38.1033 0.304826
\(26\) 0 0
\(27\) 0 0
\(28\) −400.296 −2.70174
\(29\) 141.628 0.906882 0.453441 0.891286i \(-0.350196\pi\)
0.453441 + 0.891286i \(0.350196\pi\)
\(30\) 0 0
\(31\) 56.0144 0.324532 0.162266 0.986747i \(-0.448120\pi\)
0.162266 + 0.986747i \(0.448120\pi\)
\(32\) −52.6687 −0.290956
\(33\) 0 0
\(34\) 131.897 0.665300
\(35\) 334.441 1.61516
\(36\) 0 0
\(37\) 313.982 1.39509 0.697545 0.716541i \(-0.254276\pi\)
0.697545 + 0.716541i \(0.254276\pi\)
\(38\) 63.3095 0.270267
\(39\) 0 0
\(40\) −449.021 −1.77491
\(41\) 352.375 1.34224 0.671118 0.741351i \(-0.265814\pi\)
0.671118 + 0.741351i \(0.265814\pi\)
\(42\) 0 0
\(43\) −320.676 −1.13727 −0.568636 0.822589i \(-0.692529\pi\)
−0.568636 + 0.822589i \(0.692529\pi\)
\(44\) 647.255 2.21767
\(45\) 0 0
\(46\) 138.964 0.445417
\(47\) 339.339 1.05314 0.526571 0.850131i \(-0.323477\pi\)
0.526571 + 0.850131i \(0.323477\pi\)
\(48\) 0 0
\(49\) 342.765 0.999315
\(50\) 183.869 0.520061
\(51\) 0 0
\(52\) 0 0
\(53\) −349.461 −0.905701 −0.452851 0.891586i \(-0.649593\pi\)
−0.452851 + 0.891586i \(0.649593\pi\)
\(54\) 0 0
\(55\) −540.771 −1.32577
\(56\) −920.710 −2.19705
\(57\) 0 0
\(58\) 683.432 1.54722
\(59\) 258.742 0.570937 0.285469 0.958388i \(-0.407851\pi\)
0.285469 + 0.958388i \(0.407851\pi\)
\(60\) 0 0
\(61\) 650.732 1.36586 0.682932 0.730482i \(-0.260705\pi\)
0.682932 + 0.730482i \(0.260705\pi\)
\(62\) 270.301 0.553681
\(63\) 0 0
\(64\) −633.142 −1.23661
\(65\) 0 0
\(66\) 0 0
\(67\) 894.996 1.63196 0.815979 0.578082i \(-0.196199\pi\)
0.815979 + 0.578082i \(0.196199\pi\)
\(68\) 417.813 0.745106
\(69\) 0 0
\(70\) 1613.86 2.75562
\(71\) 741.469 1.23938 0.619691 0.784846i \(-0.287258\pi\)
0.619691 + 0.784846i \(0.287258\pi\)
\(72\) 0 0
\(73\) −820.643 −1.31574 −0.657870 0.753131i \(-0.728542\pi\)
−0.657870 + 0.753131i \(0.728542\pi\)
\(74\) 1515.14 2.38015
\(75\) 0 0
\(76\) 200.546 0.302687
\(77\) −1108.84 −1.64109
\(78\) 0 0
\(79\) −199.372 −0.283938 −0.141969 0.989871i \(-0.545343\pi\)
−0.141969 + 0.989871i \(0.545343\pi\)
\(80\) −605.013 −0.845532
\(81\) 0 0
\(82\) 1700.40 2.28998
\(83\) 541.696 0.716372 0.358186 0.933650i \(-0.383395\pi\)
0.358186 + 0.933650i \(0.383395\pi\)
\(84\) 0 0
\(85\) −349.076 −0.445442
\(86\) −1547.44 −1.94029
\(87\) 0 0
\(88\) 1488.73 1.80340
\(89\) 380.615 0.453315 0.226658 0.973974i \(-0.427220\pi\)
0.226658 + 0.973974i \(0.427220\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 440.199 0.498847
\(93\) 0 0
\(94\) 1637.50 1.79676
\(95\) −167.553 −0.180954
\(96\) 0 0
\(97\) 1430.50 1.49737 0.748687 0.662924i \(-0.230685\pi\)
0.748687 + 0.662924i \(0.230685\pi\)
\(98\) 1654.03 1.70492
\(99\) 0 0
\(100\) 582.446 0.582446
\(101\) 801.195 0.789326 0.394663 0.918826i \(-0.370861\pi\)
0.394663 + 0.918826i \(0.370861\pi\)
\(102\) 0 0
\(103\) −745.805 −0.713459 −0.356730 0.934208i \(-0.616108\pi\)
−0.356730 + 0.934208i \(0.616108\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −1686.34 −1.54521
\(107\) 134.546 0.121561 0.0607806 0.998151i \(-0.480641\pi\)
0.0607806 + 0.998151i \(0.480641\pi\)
\(108\) 0 0
\(109\) −293.245 −0.257686 −0.128843 0.991665i \(-0.541126\pi\)
−0.128843 + 0.991665i \(0.541126\pi\)
\(110\) −2609.52 −2.26189
\(111\) 0 0
\(112\) −1240.57 −1.04663
\(113\) −1705.38 −1.41972 −0.709861 0.704342i \(-0.751242\pi\)
−0.709861 + 0.704342i \(0.751242\pi\)
\(114\) 0 0
\(115\) −367.779 −0.298223
\(116\) 2164.92 1.73282
\(117\) 0 0
\(118\) 1248.57 0.974071
\(119\) −715.774 −0.551386
\(120\) 0 0
\(121\) 461.932 0.347056
\(122\) 3140.14 2.33029
\(123\) 0 0
\(124\) 856.235 0.620098
\(125\) 1109.77 0.794090
\(126\) 0 0
\(127\) 1493.18 1.04329 0.521646 0.853162i \(-0.325318\pi\)
0.521646 + 0.853162i \(0.325318\pi\)
\(128\) −2633.91 −1.81881
\(129\) 0 0
\(130\) 0 0
\(131\) 459.112 0.306205 0.153102 0.988210i \(-0.451074\pi\)
0.153102 + 0.988210i \(0.451074\pi\)
\(132\) 0 0
\(133\) −343.565 −0.223991
\(134\) 4318.85 2.78427
\(135\) 0 0
\(136\) 961.000 0.605920
\(137\) −2503.06 −1.56096 −0.780478 0.625183i \(-0.785024\pi\)
−0.780478 + 0.625183i \(0.785024\pi\)
\(138\) 0 0
\(139\) 1803.73 1.10065 0.550326 0.834950i \(-0.314503\pi\)
0.550326 + 0.834950i \(0.314503\pi\)
\(140\) 5112.25 3.08617
\(141\) 0 0
\(142\) 3578.00 2.11450
\(143\) 0 0
\(144\) 0 0
\(145\) −1808.75 −1.03592
\(146\) −3960.06 −2.24477
\(147\) 0 0
\(148\) 4799.52 2.66566
\(149\) 2925.44 1.60846 0.804232 0.594316i \(-0.202577\pi\)
0.804232 + 0.594316i \(0.202577\pi\)
\(150\) 0 0
\(151\) 1769.28 0.953524 0.476762 0.879032i \(-0.341810\pi\)
0.476762 + 0.879032i \(0.341810\pi\)
\(152\) 461.271 0.246145
\(153\) 0 0
\(154\) −5350.78 −2.79986
\(155\) −715.371 −0.370709
\(156\) 0 0
\(157\) −1157.24 −0.588265 −0.294132 0.955765i \(-0.595031\pi\)
−0.294132 + 0.955765i \(0.595031\pi\)
\(158\) −962.079 −0.484423
\(159\) 0 0
\(160\) 672.641 0.332356
\(161\) −754.126 −0.369152
\(162\) 0 0
\(163\) −2909.21 −1.39795 −0.698977 0.715144i \(-0.746361\pi\)
−0.698977 + 0.715144i \(0.746361\pi\)
\(164\) 5386.39 2.56467
\(165\) 0 0
\(166\) 2613.98 1.22220
\(167\) −1272.16 −0.589476 −0.294738 0.955578i \(-0.595232\pi\)
−0.294738 + 0.955578i \(0.595232\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −1684.48 −0.759965
\(171\) 0 0
\(172\) −4901.85 −2.17304
\(173\) −2371.80 −1.04234 −0.521169 0.853453i \(-0.674504\pi\)
−0.521169 + 0.853453i \(0.674504\pi\)
\(174\) 0 0
\(175\) −997.815 −0.431015
\(176\) 2005.93 0.859105
\(177\) 0 0
\(178\) 1836.68 0.773397
\(179\) −727.183 −0.303643 −0.151822 0.988408i \(-0.548514\pi\)
−0.151822 + 0.988408i \(0.548514\pi\)
\(180\) 0 0
\(181\) 3874.20 1.59098 0.795488 0.605969i \(-0.207215\pi\)
0.795488 + 0.605969i \(0.207215\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1012.49 0.405662
\(185\) −4009.92 −1.59360
\(186\) 0 0
\(187\) 1157.36 0.452593
\(188\) 5187.13 2.01229
\(189\) 0 0
\(190\) −808.537 −0.308723
\(191\) −2003.77 −0.759098 −0.379549 0.925172i \(-0.623921\pi\)
−0.379549 + 0.925172i \(0.623921\pi\)
\(192\) 0 0
\(193\) 1026.25 0.382751 0.191376 0.981517i \(-0.438705\pi\)
0.191376 + 0.981517i \(0.438705\pi\)
\(194\) 6902.96 2.55466
\(195\) 0 0
\(196\) 5239.50 1.90944
\(197\) −1308.19 −0.473120 −0.236560 0.971617i \(-0.576020\pi\)
−0.236560 + 0.971617i \(0.576020\pi\)
\(198\) 0 0
\(199\) −285.120 −0.101566 −0.0507830 0.998710i \(-0.516172\pi\)
−0.0507830 + 0.998710i \(0.516172\pi\)
\(200\) 1339.67 0.473644
\(201\) 0 0
\(202\) 3866.21 1.34666
\(203\) −3708.82 −1.28231
\(204\) 0 0
\(205\) −4500.24 −1.53322
\(206\) −3598.92 −1.21723
\(207\) 0 0
\(208\) 0 0
\(209\) 555.525 0.183859
\(210\) 0 0
\(211\) −2813.91 −0.918092 −0.459046 0.888412i \(-0.651809\pi\)
−0.459046 + 0.888412i \(0.651809\pi\)
\(212\) −5341.86 −1.73057
\(213\) 0 0
\(214\) 649.259 0.207395
\(215\) 4095.42 1.29909
\(216\) 0 0
\(217\) −1466.86 −0.458879
\(218\) −1415.07 −0.439635
\(219\) 0 0
\(220\) −8266.21 −2.53322
\(221\) 0 0
\(222\) 0 0
\(223\) −1509.86 −0.453399 −0.226699 0.973965i \(-0.572794\pi\)
−0.226699 + 0.973965i \(0.572794\pi\)
\(224\) 1379.24 0.411403
\(225\) 0 0
\(226\) −8229.40 −2.42217
\(227\) −643.122 −0.188042 −0.0940209 0.995570i \(-0.529972\pi\)
−0.0940209 + 0.995570i \(0.529972\pi\)
\(228\) 0 0
\(229\) −154.009 −0.0444420 −0.0222210 0.999753i \(-0.507074\pi\)
−0.0222210 + 0.999753i \(0.507074\pi\)
\(230\) −1774.74 −0.508795
\(231\) 0 0
\(232\) 4979.47 1.40913
\(233\) 261.668 0.0735726 0.0367863 0.999323i \(-0.488288\pi\)
0.0367863 + 0.999323i \(0.488288\pi\)
\(234\) 0 0
\(235\) −4333.76 −1.20299
\(236\) 3955.12 1.09092
\(237\) 0 0
\(238\) −3454.01 −0.940714
\(239\) −2495.09 −0.675288 −0.337644 0.941274i \(-0.609630\pi\)
−0.337644 + 0.941274i \(0.609630\pi\)
\(240\) 0 0
\(241\) −3835.70 −1.02522 −0.512612 0.858620i \(-0.671322\pi\)
−0.512612 + 0.858620i \(0.671322\pi\)
\(242\) 2229.08 0.592110
\(243\) 0 0
\(244\) 9947.07 2.60982
\(245\) −4377.52 −1.14151
\(246\) 0 0
\(247\) 0 0
\(248\) 1969.40 0.504263
\(249\) 0 0
\(250\) 5355.28 1.35479
\(251\) 3977.59 1.00025 0.500126 0.865953i \(-0.333287\pi\)
0.500126 + 0.865953i \(0.333287\pi\)
\(252\) 0 0
\(253\) 1219.38 0.303010
\(254\) 7205.41 1.77995
\(255\) 0 0
\(256\) −7644.95 −1.86644
\(257\) 121.486 0.0294868 0.0147434 0.999891i \(-0.495307\pi\)
0.0147434 + 0.999891i \(0.495307\pi\)
\(258\) 0 0
\(259\) −8222.29 −1.97262
\(260\) 0 0
\(261\) 0 0
\(262\) 2215.47 0.522413
\(263\) −1175.42 −0.275586 −0.137793 0.990461i \(-0.544001\pi\)
−0.137793 + 0.990461i \(0.544001\pi\)
\(264\) 0 0
\(265\) 4463.03 1.03457
\(266\) −1657.89 −0.382150
\(267\) 0 0
\(268\) 13680.9 3.11826
\(269\) 8319.66 1.88572 0.942860 0.333190i \(-0.108125\pi\)
0.942860 + 0.333190i \(0.108125\pi\)
\(270\) 0 0
\(271\) −3928.56 −0.880602 −0.440301 0.897850i \(-0.645128\pi\)
−0.440301 + 0.897850i \(0.645128\pi\)
\(272\) 1294.86 0.288648
\(273\) 0 0
\(274\) −12078.7 −2.66313
\(275\) 1613.41 0.353790
\(276\) 0 0
\(277\) −6022.61 −1.30637 −0.653183 0.757200i \(-0.726567\pi\)
−0.653183 + 0.757200i \(0.726567\pi\)
\(278\) 8704.01 1.87781
\(279\) 0 0
\(280\) 11758.6 2.50967
\(281\) −2183.71 −0.463592 −0.231796 0.972764i \(-0.574460\pi\)
−0.231796 + 0.972764i \(0.574460\pi\)
\(282\) 0 0
\(283\) −8133.25 −1.70838 −0.854190 0.519961i \(-0.825946\pi\)
−0.854190 + 0.519961i \(0.825946\pi\)
\(284\) 11334.1 2.36815
\(285\) 0 0
\(286\) 0 0
\(287\) −9227.67 −1.89788
\(288\) 0 0
\(289\) −4165.90 −0.847935
\(290\) −8728.24 −1.76738
\(291\) 0 0
\(292\) −12544.3 −2.51405
\(293\) −3182.55 −0.634562 −0.317281 0.948332i \(-0.602770\pi\)
−0.317281 + 0.948332i \(0.602770\pi\)
\(294\) 0 0
\(295\) −3304.44 −0.652176
\(296\) 11039.3 2.16772
\(297\) 0 0
\(298\) 14116.9 2.74419
\(299\) 0 0
\(300\) 0 0
\(301\) 8397.59 1.60807
\(302\) 8537.77 1.62680
\(303\) 0 0
\(304\) 621.520 0.117259
\(305\) −8310.62 −1.56021
\(306\) 0 0
\(307\) −1230.21 −0.228703 −0.114352 0.993440i \(-0.536479\pi\)
−0.114352 + 0.993440i \(0.536479\pi\)
\(308\) −16949.7 −3.13572
\(309\) 0 0
\(310\) −3452.06 −0.632464
\(311\) 5881.49 1.07237 0.536187 0.844099i \(-0.319864\pi\)
0.536187 + 0.844099i \(0.319864\pi\)
\(312\) 0 0
\(313\) 6800.24 1.22803 0.614013 0.789296i \(-0.289554\pi\)
0.614013 + 0.789296i \(0.289554\pi\)
\(314\) −5584.31 −1.00363
\(315\) 0 0
\(316\) −3047.59 −0.542533
\(317\) 2212.50 0.392008 0.196004 0.980603i \(-0.437204\pi\)
0.196004 + 0.980603i \(0.437204\pi\)
\(318\) 0 0
\(319\) 5996.94 1.05255
\(320\) 8085.97 1.41256
\(321\) 0 0
\(322\) −3639.07 −0.629807
\(323\) 358.600 0.0617740
\(324\) 0 0
\(325\) 0 0
\(326\) −14038.5 −2.38504
\(327\) 0 0
\(328\) 12389.1 2.08559
\(329\) −8886.31 −1.48911
\(330\) 0 0
\(331\) 1744.14 0.289627 0.144814 0.989459i \(-0.453742\pi\)
0.144814 + 0.989459i \(0.453742\pi\)
\(332\) 8280.35 1.36881
\(333\) 0 0
\(334\) −6138.87 −1.00570
\(335\) −11430.2 −1.86417
\(336\) 0 0
\(337\) 8805.54 1.42335 0.711674 0.702510i \(-0.247937\pi\)
0.711674 + 0.702510i \(0.247937\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −5335.96 −0.851127
\(341\) 2371.82 0.376661
\(342\) 0 0
\(343\) 6.15700 0.000969232 0
\(344\) −11274.6 −1.76711
\(345\) 0 0
\(346\) −11445.2 −1.77832
\(347\) 1943.15 0.300616 0.150308 0.988639i \(-0.451973\pi\)
0.150308 + 0.988639i \(0.451973\pi\)
\(348\) 0 0
\(349\) 5316.91 0.815495 0.407748 0.913095i \(-0.366314\pi\)
0.407748 + 0.913095i \(0.366314\pi\)
\(350\) −4815.01 −0.735352
\(351\) 0 0
\(352\) −2230.15 −0.337692
\(353\) 5013.67 0.755951 0.377976 0.925816i \(-0.376620\pi\)
0.377976 + 0.925816i \(0.376620\pi\)
\(354\) 0 0
\(355\) −9469.44 −1.41573
\(356\) 5818.07 0.866171
\(357\) 0 0
\(358\) −3509.06 −0.518043
\(359\) 8143.55 1.19722 0.598608 0.801042i \(-0.295721\pi\)
0.598608 + 0.801042i \(0.295721\pi\)
\(360\) 0 0
\(361\) −6686.88 −0.974905
\(362\) 18695.1 2.71435
\(363\) 0 0
\(364\) 0 0
\(365\) 10480.6 1.50296
\(366\) 0 0
\(367\) 6517.85 0.927054 0.463527 0.886083i \(-0.346584\pi\)
0.463527 + 0.886083i \(0.346584\pi\)
\(368\) 1364.24 0.193249
\(369\) 0 0
\(370\) −19350.1 −2.71882
\(371\) 9151.38 1.28064
\(372\) 0 0
\(373\) 7247.57 1.00607 0.503036 0.864266i \(-0.332216\pi\)
0.503036 + 0.864266i \(0.332216\pi\)
\(374\) 5584.93 0.772165
\(375\) 0 0
\(376\) 11930.8 1.63639
\(377\) 0 0
\(378\) 0 0
\(379\) 6549.25 0.887632 0.443816 0.896118i \(-0.353624\pi\)
0.443816 + 0.896118i \(0.353624\pi\)
\(380\) −2561.21 −0.345757
\(381\) 0 0
\(382\) −9669.31 −1.29509
\(383\) 8030.33 1.07136 0.535679 0.844421i \(-0.320056\pi\)
0.535679 + 0.844421i \(0.320056\pi\)
\(384\) 0 0
\(385\) 14161.2 1.87460
\(386\) 4952.22 0.653009
\(387\) 0 0
\(388\) 21866.6 2.86110
\(389\) −481.389 −0.0627440 −0.0313720 0.999508i \(-0.509988\pi\)
−0.0313720 + 0.999508i \(0.509988\pi\)
\(390\) 0 0
\(391\) 787.126 0.101807
\(392\) 12051.2 1.55275
\(393\) 0 0
\(394\) −6312.74 −0.807186
\(395\) 2546.21 0.324339
\(396\) 0 0
\(397\) −2802.07 −0.354237 −0.177118 0.984190i \(-0.556678\pi\)
−0.177118 + 0.984190i \(0.556678\pi\)
\(398\) −1375.86 −0.173281
\(399\) 0 0
\(400\) 1805.08 0.225635
\(401\) −4376.79 −0.545053 −0.272527 0.962148i \(-0.587859\pi\)
−0.272527 + 0.962148i \(0.587859\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 12247.1 1.50820
\(405\) 0 0
\(406\) −17897.1 −2.18773
\(407\) 13295.0 1.61918
\(408\) 0 0
\(409\) 12969.0 1.56791 0.783954 0.620819i \(-0.213200\pi\)
0.783954 + 0.620819i \(0.213200\pi\)
\(410\) −21716.2 −2.61582
\(411\) 0 0
\(412\) −11400.4 −1.36324
\(413\) −6775.70 −0.807289
\(414\) 0 0
\(415\) −6918.10 −0.818304
\(416\) 0 0
\(417\) 0 0
\(418\) 2680.71 0.313679
\(419\) −7679.05 −0.895336 −0.447668 0.894200i \(-0.647745\pi\)
−0.447668 + 0.894200i \(0.647745\pi\)
\(420\) 0 0
\(421\) −2963.10 −0.343022 −0.171511 0.985182i \(-0.554865\pi\)
−0.171511 + 0.985182i \(0.554865\pi\)
\(422\) −13578.7 −1.56635
\(423\) 0 0
\(424\) −12286.7 −1.40730
\(425\) 1041.48 0.118869
\(426\) 0 0
\(427\) −17040.8 −1.93129
\(428\) 2056.67 0.232273
\(429\) 0 0
\(430\) 19762.7 2.21637
\(431\) 14202.6 1.58728 0.793639 0.608389i \(-0.208184\pi\)
0.793639 + 0.608389i \(0.208184\pi\)
\(432\) 0 0
\(433\) 10118.2 1.12298 0.561488 0.827485i \(-0.310229\pi\)
0.561488 + 0.827485i \(0.310229\pi\)
\(434\) −7078.39 −0.782889
\(435\) 0 0
\(436\) −4482.53 −0.492372
\(437\) 377.814 0.0413576
\(438\) 0 0
\(439\) −16184.3 −1.75954 −0.879768 0.475404i \(-0.842302\pi\)
−0.879768 + 0.475404i \(0.842302\pi\)
\(440\) −19012.9 −2.06001
\(441\) 0 0
\(442\) 0 0
\(443\) 10025.2 1.07519 0.537596 0.843203i \(-0.319333\pi\)
0.537596 + 0.843203i \(0.319333\pi\)
\(444\) 0 0
\(445\) −4860.90 −0.517818
\(446\) −7285.93 −0.773540
\(447\) 0 0
\(448\) 16580.2 1.74852
\(449\) −10428.3 −1.09608 −0.548041 0.836451i \(-0.684626\pi\)
−0.548041 + 0.836451i \(0.684626\pi\)
\(450\) 0 0
\(451\) 14920.6 1.55784
\(452\) −26068.4 −2.71273
\(453\) 0 0
\(454\) −3103.42 −0.320816
\(455\) 0 0
\(456\) 0 0
\(457\) 12263.8 1.25531 0.627656 0.778491i \(-0.284015\pi\)
0.627656 + 0.778491i \(0.284015\pi\)
\(458\) −743.180 −0.0758221
\(459\) 0 0
\(460\) −5621.87 −0.569828
\(461\) 4047.98 0.408966 0.204483 0.978870i \(-0.434449\pi\)
0.204483 + 0.978870i \(0.434449\pi\)
\(462\) 0 0
\(463\) 10473.9 1.05133 0.525663 0.850693i \(-0.323817\pi\)
0.525663 + 0.850693i \(0.323817\pi\)
\(464\) 6709.37 0.671281
\(465\) 0 0
\(466\) 1262.69 0.125522
\(467\) −4906.17 −0.486146 −0.243073 0.970008i \(-0.578155\pi\)
−0.243073 + 0.970008i \(0.578155\pi\)
\(468\) 0 0
\(469\) −23437.4 −2.30754
\(470\) −20912.8 −2.05242
\(471\) 0 0
\(472\) 9097.07 0.887132
\(473\) −13578.4 −1.31995
\(474\) 0 0
\(475\) 499.901 0.0482884
\(476\) −10941.3 −1.05356
\(477\) 0 0
\(478\) −12040.2 −1.15210
\(479\) −1249.42 −0.119180 −0.0595902 0.998223i \(-0.518979\pi\)
−0.0595902 + 0.998223i \(0.518979\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −18509.4 −1.74913
\(483\) 0 0
\(484\) 7061.08 0.663137
\(485\) −18269.2 −1.71044
\(486\) 0 0
\(487\) −19290.2 −1.79492 −0.897458 0.441100i \(-0.854588\pi\)
−0.897458 + 0.441100i \(0.854588\pi\)
\(488\) 22879.0 2.12230
\(489\) 0 0
\(490\) −21123.9 −1.94751
\(491\) 8381.17 0.770339 0.385170 0.922846i \(-0.374143\pi\)
0.385170 + 0.922846i \(0.374143\pi\)
\(492\) 0 0
\(493\) 3871.12 0.353644
\(494\) 0 0
\(495\) 0 0
\(496\) 2653.59 0.240221
\(497\) −19416.9 −1.75245
\(498\) 0 0
\(499\) −199.025 −0.0178549 −0.00892743 0.999960i \(-0.502842\pi\)
−0.00892743 + 0.999960i \(0.502842\pi\)
\(500\) 16964.0 1.51731
\(501\) 0 0
\(502\) 19194.1 1.70652
\(503\) 7567.90 0.670847 0.335424 0.942067i \(-0.391121\pi\)
0.335424 + 0.942067i \(0.391121\pi\)
\(504\) 0 0
\(505\) −10232.2 −0.901639
\(506\) 5884.17 0.516963
\(507\) 0 0
\(508\) 22824.7 1.99347
\(509\) 5998.22 0.522331 0.261166 0.965294i \(-0.415893\pi\)
0.261166 + 0.965294i \(0.415893\pi\)
\(510\) 0 0
\(511\) 21490.3 1.86042
\(512\) −15819.8 −1.36552
\(513\) 0 0
\(514\) 586.238 0.0503071
\(515\) 9524.81 0.814977
\(516\) 0 0
\(517\) 14368.6 1.22231
\(518\) −39677.1 −3.36547
\(519\) 0 0
\(520\) 0 0
\(521\) 2863.99 0.240833 0.120416 0.992723i \(-0.461577\pi\)
0.120416 + 0.992723i \(0.461577\pi\)
\(522\) 0 0
\(523\) −2529.09 −0.211452 −0.105726 0.994395i \(-0.533717\pi\)
−0.105726 + 0.994395i \(0.533717\pi\)
\(524\) 7017.98 0.585080
\(525\) 0 0
\(526\) −5672.03 −0.470176
\(527\) 1531.05 0.126553
\(528\) 0 0
\(529\) −11337.7 −0.931840
\(530\) 21536.6 1.76508
\(531\) 0 0
\(532\) −5251.73 −0.427991
\(533\) 0 0
\(534\) 0 0
\(535\) −1718.31 −0.138858
\(536\) 31467.0 2.53576
\(537\) 0 0
\(538\) 40147.0 3.21721
\(539\) 14513.7 1.15983
\(540\) 0 0
\(541\) −22053.6 −1.75260 −0.876302 0.481762i \(-0.839997\pi\)
−0.876302 + 0.481762i \(0.839997\pi\)
\(542\) −18957.5 −1.50239
\(543\) 0 0
\(544\) −1439.60 −0.113460
\(545\) 3745.08 0.294352
\(546\) 0 0
\(547\) −2821.80 −0.220569 −0.110285 0.993900i \(-0.535176\pi\)
−0.110285 + 0.993900i \(0.535176\pi\)
\(548\) −38261.8 −2.98259
\(549\) 0 0
\(550\) 7785.59 0.603597
\(551\) 1858.10 0.143662
\(552\) 0 0
\(553\) 5220.97 0.401479
\(554\) −29062.4 −2.22878
\(555\) 0 0
\(556\) 27571.8 2.10307
\(557\) 1837.17 0.139754 0.0698772 0.997556i \(-0.477739\pi\)
0.0698772 + 0.997556i \(0.477739\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 15843.6 1.19556
\(561\) 0 0
\(562\) −10537.6 −0.790930
\(563\) 24877.9 1.86230 0.931152 0.364631i \(-0.118805\pi\)
0.931152 + 0.364631i \(0.118805\pi\)
\(564\) 0 0
\(565\) 21779.7 1.62173
\(566\) −39247.4 −2.91465
\(567\) 0 0
\(568\) 26069.2 1.92577
\(569\) 7111.72 0.523969 0.261985 0.965072i \(-0.415623\pi\)
0.261985 + 0.965072i \(0.415623\pi\)
\(570\) 0 0
\(571\) −11919.4 −0.873578 −0.436789 0.899564i \(-0.643884\pi\)
−0.436789 + 0.899564i \(0.643884\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −44528.6 −3.23796
\(575\) 1097.28 0.0795823
\(576\) 0 0
\(577\) −11600.6 −0.836984 −0.418492 0.908220i \(-0.637441\pi\)
−0.418492 + 0.908220i \(0.637441\pi\)
\(578\) −20102.8 −1.44665
\(579\) 0 0
\(580\) −27648.6 −1.97939
\(581\) −14185.5 −1.01293
\(582\) 0 0
\(583\) −14797.2 −1.05118
\(584\) −28852.9 −2.04442
\(585\) 0 0
\(586\) −15357.6 −1.08262
\(587\) −10341.7 −0.727167 −0.363583 0.931562i \(-0.618447\pi\)
−0.363583 + 0.931562i \(0.618447\pi\)
\(588\) 0 0
\(589\) 734.888 0.0514101
\(590\) −15945.8 −1.11267
\(591\) 0 0
\(592\) 14874.4 1.03266
\(593\) −2782.21 −0.192668 −0.0963338 0.995349i \(-0.530712\pi\)
−0.0963338 + 0.995349i \(0.530712\pi\)
\(594\) 0 0
\(595\) 9141.29 0.629842
\(596\) 44718.2 3.07337
\(597\) 0 0
\(598\) 0 0
\(599\) −10560.6 −0.720359 −0.360179 0.932883i \(-0.617285\pi\)
−0.360179 + 0.932883i \(0.617285\pi\)
\(600\) 0 0
\(601\) 3846.85 0.261092 0.130546 0.991442i \(-0.458327\pi\)
0.130546 + 0.991442i \(0.458327\pi\)
\(602\) 40523.0 2.74351
\(603\) 0 0
\(604\) 27045.2 1.82194
\(605\) −5899.42 −0.396439
\(606\) 0 0
\(607\) 6495.50 0.434340 0.217170 0.976134i \(-0.430317\pi\)
0.217170 + 0.976134i \(0.430317\pi\)
\(608\) −690.993 −0.0460912
\(609\) 0 0
\(610\) −40103.4 −2.66186
\(611\) 0 0
\(612\) 0 0
\(613\) 3532.60 0.232758 0.116379 0.993205i \(-0.462871\pi\)
0.116379 + 0.993205i \(0.462871\pi\)
\(614\) −5936.46 −0.390189
\(615\) 0 0
\(616\) −38985.6 −2.54996
\(617\) 16297.9 1.06342 0.531710 0.846926i \(-0.321549\pi\)
0.531710 + 0.846926i \(0.321549\pi\)
\(618\) 0 0
\(619\) 9123.86 0.592438 0.296219 0.955120i \(-0.404274\pi\)
0.296219 + 0.955120i \(0.404274\pi\)
\(620\) −10935.1 −0.708332
\(621\) 0 0
\(622\) 28381.4 1.82957
\(623\) −9967.20 −0.640975
\(624\) 0 0
\(625\) −18936.0 −1.21191
\(626\) 32814.9 2.09512
\(627\) 0 0
\(628\) −17689.5 −1.12402
\(629\) 8582.09 0.544023
\(630\) 0 0
\(631\) −23250.8 −1.46688 −0.733438 0.679756i \(-0.762086\pi\)
−0.733438 + 0.679756i \(0.762086\pi\)
\(632\) −7009.68 −0.441187
\(633\) 0 0
\(634\) 10676.5 0.668801
\(635\) −19069.7 −1.19174
\(636\) 0 0
\(637\) 0 0
\(638\) 28938.6 1.79575
\(639\) 0 0
\(640\) 33638.2 2.07760
\(641\) 7478.44 0.460812 0.230406 0.973095i \(-0.425995\pi\)
0.230406 + 0.973095i \(0.425995\pi\)
\(642\) 0 0
\(643\) −8327.98 −0.510768 −0.255384 0.966840i \(-0.582202\pi\)
−0.255384 + 0.966840i \(0.582202\pi\)
\(644\) −11527.5 −0.705356
\(645\) 0 0
\(646\) 1730.44 0.105392
\(647\) −6278.75 −0.381519 −0.190760 0.981637i \(-0.561095\pi\)
−0.190760 + 0.981637i \(0.561095\pi\)
\(648\) 0 0
\(649\) 10955.9 0.662645
\(650\) 0 0
\(651\) 0 0
\(652\) −44470.1 −2.67114
\(653\) 27163.5 1.62786 0.813928 0.580966i \(-0.197325\pi\)
0.813928 + 0.580966i \(0.197325\pi\)
\(654\) 0 0
\(655\) −5863.41 −0.349774
\(656\) 16693.1 0.993533
\(657\) 0 0
\(658\) −42881.4 −2.54056
\(659\) 10352.0 0.611921 0.305961 0.952044i \(-0.401022\pi\)
0.305961 + 0.952044i \(0.401022\pi\)
\(660\) 0 0
\(661\) −6270.46 −0.368975 −0.184487 0.982835i \(-0.559063\pi\)
−0.184487 + 0.982835i \(0.559063\pi\)
\(662\) 8416.44 0.494130
\(663\) 0 0
\(664\) 19045.4 1.11311
\(665\) 4387.73 0.255863
\(666\) 0 0
\(667\) 4078.53 0.236764
\(668\) −19446.2 −1.12634
\(669\) 0 0
\(670\) −55156.9 −3.18044
\(671\) 27554.0 1.58526
\(672\) 0 0
\(673\) −30301.9 −1.73559 −0.867794 0.496923i \(-0.834463\pi\)
−0.867794 + 0.496923i \(0.834463\pi\)
\(674\) 42491.6 2.42836
\(675\) 0 0
\(676\) 0 0
\(677\) 10223.3 0.580375 0.290188 0.956970i \(-0.406282\pi\)
0.290188 + 0.956970i \(0.406282\pi\)
\(678\) 0 0
\(679\) −37460.7 −2.11724
\(680\) −12273.1 −0.692136
\(681\) 0 0
\(682\) 11445.3 0.642617
\(683\) 16978.0 0.951163 0.475582 0.879672i \(-0.342238\pi\)
0.475582 + 0.879672i \(0.342238\pi\)
\(684\) 0 0
\(685\) 31967.1 1.78307
\(686\) 29.7109 0.00165360
\(687\) 0 0
\(688\) −15191.5 −0.841818
\(689\) 0 0
\(690\) 0 0
\(691\) −7935.33 −0.436866 −0.218433 0.975852i \(-0.570094\pi\)
−0.218433 + 0.975852i \(0.570094\pi\)
\(692\) −36255.3 −1.99165
\(693\) 0 0
\(694\) 9376.79 0.512879
\(695\) −23035.8 −1.25726
\(696\) 0 0
\(697\) 9631.48 0.523412
\(698\) 25657.1 1.39131
\(699\) 0 0
\(700\) −15252.6 −0.823562
\(701\) −18557.3 −0.999857 −0.499928 0.866067i \(-0.666640\pi\)
−0.499928 + 0.866067i \(0.666640\pi\)
\(702\) 0 0
\(703\) 4119.33 0.221001
\(704\) −26809.1 −1.43524
\(705\) 0 0
\(706\) 24193.7 1.28972
\(707\) −20981.0 −1.11608
\(708\) 0 0
\(709\) −14163.7 −0.750253 −0.375127 0.926974i \(-0.622401\pi\)
−0.375127 + 0.926974i \(0.622401\pi\)
\(710\) −45695.3 −2.41537
\(711\) 0 0
\(712\) 13382.0 0.704370
\(713\) 1613.08 0.0847270
\(714\) 0 0
\(715\) 0 0
\(716\) −11115.7 −0.580186
\(717\) 0 0
\(718\) 39297.2 2.04256
\(719\) −24836.7 −1.28825 −0.644125 0.764920i \(-0.722778\pi\)
−0.644125 + 0.764920i \(0.722778\pi\)
\(720\) 0 0
\(721\) 19530.5 1.00881
\(722\) −32267.9 −1.66328
\(723\) 0 0
\(724\) 59220.9 3.03995
\(725\) 5396.48 0.276441
\(726\) 0 0
\(727\) −10641.0 −0.542852 −0.271426 0.962459i \(-0.587495\pi\)
−0.271426 + 0.962459i \(0.587495\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 50574.7 2.56418
\(731\) −8765.07 −0.443486
\(732\) 0 0
\(733\) 22429.1 1.13020 0.565102 0.825021i \(-0.308837\pi\)
0.565102 + 0.825021i \(0.308837\pi\)
\(734\) 31452.2 1.58164
\(735\) 0 0
\(736\) −1516.73 −0.0759612
\(737\) 37896.8 1.89409
\(738\) 0 0
\(739\) 33425.8 1.66386 0.831928 0.554884i \(-0.187237\pi\)
0.831928 + 0.554884i \(0.187237\pi\)
\(740\) −61295.6 −3.04496
\(741\) 0 0
\(742\) 44160.5 2.18488
\(743\) −13136.3 −0.648619 −0.324309 0.945951i \(-0.605132\pi\)
−0.324309 + 0.945951i \(0.605132\pi\)
\(744\) 0 0
\(745\) −37361.3 −1.83733
\(746\) 34973.5 1.71645
\(747\) 0 0
\(748\) 17691.5 0.864791
\(749\) −3523.37 −0.171884
\(750\) 0 0
\(751\) 20688.2 1.00522 0.502612 0.864512i \(-0.332373\pi\)
0.502612 + 0.864512i \(0.332373\pi\)
\(752\) 16075.6 0.779544
\(753\) 0 0
\(754\) 0 0
\(755\) −22595.8 −1.08920
\(756\) 0 0
\(757\) −6145.85 −0.295079 −0.147539 0.989056i \(-0.547135\pi\)
−0.147539 + 0.989056i \(0.547135\pi\)
\(758\) 31603.8 1.51438
\(759\) 0 0
\(760\) −5890.98 −0.281169
\(761\) 16488.5 0.785424 0.392712 0.919662i \(-0.371537\pi\)
0.392712 + 0.919662i \(0.371537\pi\)
\(762\) 0 0
\(763\) 7679.23 0.364360
\(764\) −30629.6 −1.45045
\(765\) 0 0
\(766\) 38750.8 1.82784
\(767\) 0 0
\(768\) 0 0
\(769\) −7801.71 −0.365848 −0.182924 0.983127i \(-0.558556\pi\)
−0.182924 + 0.983127i \(0.558556\pi\)
\(770\) 68335.8 3.19825
\(771\) 0 0
\(772\) 15687.2 0.731341
\(773\) 27301.3 1.27032 0.635161 0.772380i \(-0.280934\pi\)
0.635161 + 0.772380i \(0.280934\pi\)
\(774\) 0 0
\(775\) 2134.33 0.0989258
\(776\) 50294.8 2.32665
\(777\) 0 0
\(778\) −2322.97 −0.107047
\(779\) 4623.02 0.212628
\(780\) 0 0
\(781\) 31396.0 1.43846
\(782\) 3798.32 0.173693
\(783\) 0 0
\(784\) 16237.9 0.739700
\(785\) 14779.3 0.671969
\(786\) 0 0
\(787\) 26016.0 1.17836 0.589181 0.808001i \(-0.299450\pi\)
0.589181 + 0.808001i \(0.299450\pi\)
\(788\) −19997.0 −0.904013
\(789\) 0 0
\(790\) 12286.9 0.553352
\(791\) 44658.9 2.00745
\(792\) 0 0
\(793\) 0 0
\(794\) −13521.5 −0.604360
\(795\) 0 0
\(796\) −4358.34 −0.194067
\(797\) −8178.16 −0.363470 −0.181735 0.983348i \(-0.558171\pi\)
−0.181735 + 0.983348i \(0.558171\pi\)
\(798\) 0 0
\(799\) 9275.18 0.410678
\(800\) −2006.85 −0.0886910
\(801\) 0 0
\(802\) −21120.4 −0.929911
\(803\) −34748.5 −1.52708
\(804\) 0 0
\(805\) 9631.08 0.421678
\(806\) 0 0
\(807\) 0 0
\(808\) 28169.1 1.22647
\(809\) −16162.0 −0.702380 −0.351190 0.936304i \(-0.614223\pi\)
−0.351190 + 0.936304i \(0.614223\pi\)
\(810\) 0 0
\(811\) 24714.6 1.07009 0.535046 0.844823i \(-0.320294\pi\)
0.535046 + 0.844823i \(0.320294\pi\)
\(812\) −56692.9 −2.45016
\(813\) 0 0
\(814\) 64155.5 2.76247
\(815\) 37154.0 1.59687
\(816\) 0 0
\(817\) −4207.15 −0.180159
\(818\) 62582.5 2.67499
\(819\) 0 0
\(820\) −68790.6 −2.92960
\(821\) 28565.3 1.21429 0.607147 0.794589i \(-0.292314\pi\)
0.607147 + 0.794589i \(0.292314\pi\)
\(822\) 0 0
\(823\) 9276.01 0.392881 0.196441 0.980516i \(-0.437062\pi\)
0.196441 + 0.980516i \(0.437062\pi\)
\(824\) −26221.7 −1.10859
\(825\) 0 0
\(826\) −32696.5 −1.37731
\(827\) 36.2767 0.00152535 0.000762676 1.00000i \(-0.499757\pi\)
0.000762676 1.00000i \(0.499757\pi\)
\(828\) 0 0
\(829\) 16764.4 0.702355 0.351177 0.936309i \(-0.385781\pi\)
0.351177 + 0.936309i \(0.385781\pi\)
\(830\) −33383.7 −1.39610
\(831\) 0 0
\(832\) 0 0
\(833\) 9368.82 0.389688
\(834\) 0 0
\(835\) 16247.0 0.673353
\(836\) 8491.74 0.351307
\(837\) 0 0
\(838\) −37055.7 −1.52753
\(839\) 45316.6 1.86473 0.932363 0.361525i \(-0.117744\pi\)
0.932363 + 0.361525i \(0.117744\pi\)
\(840\) 0 0
\(841\) −4330.61 −0.177564
\(842\) −14298.6 −0.585227
\(843\) 0 0
\(844\) −43013.3 −1.75424
\(845\) 0 0
\(846\) 0 0
\(847\) −12096.7 −0.490728
\(848\) −16555.1 −0.670407
\(849\) 0 0
\(850\) 5025.72 0.202801
\(851\) 9041.93 0.364222
\(852\) 0 0
\(853\) −17557.7 −0.704766 −0.352383 0.935856i \(-0.614629\pi\)
−0.352383 + 0.935856i \(0.614629\pi\)
\(854\) −82231.3 −3.29496
\(855\) 0 0
\(856\) 4730.49 0.188884
\(857\) −37943.6 −1.51240 −0.756201 0.654339i \(-0.772947\pi\)
−0.756201 + 0.654339i \(0.772947\pi\)
\(858\) 0 0
\(859\) −12833.2 −0.509735 −0.254867 0.966976i \(-0.582032\pi\)
−0.254867 + 0.966976i \(0.582032\pi\)
\(860\) 62602.5 2.48224
\(861\) 0 0
\(862\) 68535.6 2.70804
\(863\) 24264.0 0.957076 0.478538 0.878067i \(-0.341167\pi\)
0.478538 + 0.878067i \(0.341167\pi\)
\(864\) 0 0
\(865\) 30290.7 1.19065
\(866\) 48825.8 1.91590
\(867\) 0 0
\(868\) −22422.3 −0.876801
\(869\) −8442.00 −0.329546
\(870\) 0 0
\(871\) 0 0
\(872\) −10310.2 −0.400397
\(873\) 0 0
\(874\) 1823.16 0.0705598
\(875\) −29061.8 −1.12282
\(876\) 0 0
\(877\) −4098.68 −0.157814 −0.0789069 0.996882i \(-0.525143\pi\)
−0.0789069 + 0.996882i \(0.525143\pi\)
\(878\) −78098.4 −3.00193
\(879\) 0 0
\(880\) −25618.1 −0.981347
\(881\) −46763.9 −1.78833 −0.894164 0.447740i \(-0.852229\pi\)
−0.894164 + 0.447740i \(0.852229\pi\)
\(882\) 0 0
\(883\) −27183.9 −1.03603 −0.518013 0.855373i \(-0.673328\pi\)
−0.518013 + 0.855373i \(0.673328\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 48377.0 1.83438
\(887\) 21182.0 0.801827 0.400914 0.916116i \(-0.368693\pi\)
0.400914 + 0.916116i \(0.368693\pi\)
\(888\) 0 0
\(889\) −39102.0 −1.47518
\(890\) −23456.5 −0.883444
\(891\) 0 0
\(892\) −23079.7 −0.866331
\(893\) 4452.00 0.166831
\(894\) 0 0
\(895\) 9286.99 0.346849
\(896\) 68974.5 2.57174
\(897\) 0 0
\(898\) −50322.3 −1.87002
\(899\) 7933.19 0.294312
\(900\) 0 0
\(901\) −9551.85 −0.353183
\(902\) 72000.2 2.65781
\(903\) 0 0
\(904\) −59959.2 −2.20599
\(905\) −49478.1 −1.81736
\(906\) 0 0
\(907\) −50810.3 −1.86012 −0.930060 0.367409i \(-0.880245\pi\)
−0.930060 + 0.367409i \(0.880245\pi\)
\(908\) −9830.75 −0.359300
\(909\) 0 0
\(910\) 0 0
\(911\) 7309.10 0.265819 0.132910 0.991128i \(-0.457568\pi\)
0.132910 + 0.991128i \(0.457568\pi\)
\(912\) 0 0
\(913\) 22937.1 0.831441
\(914\) 59179.7 2.14168
\(915\) 0 0
\(916\) −2354.18 −0.0849174
\(917\) −12022.8 −0.432964
\(918\) 0 0
\(919\) 3509.95 0.125988 0.0629939 0.998014i \(-0.479935\pi\)
0.0629939 + 0.998014i \(0.479935\pi\)
\(920\) −12930.7 −0.463384
\(921\) 0 0
\(922\) 19533.8 0.697733
\(923\) 0 0
\(924\) 0 0
\(925\) 11963.7 0.425260
\(926\) 50542.4 1.79366
\(927\) 0 0
\(928\) −7459.34 −0.263863
\(929\) −6624.68 −0.233960 −0.116980 0.993134i \(-0.537321\pi\)
−0.116980 + 0.993134i \(0.537321\pi\)
\(930\) 0 0
\(931\) 4496.95 0.158304
\(932\) 3999.84 0.140579
\(933\) 0 0
\(934\) −23675.0 −0.829410
\(935\) −14780.9 −0.516992
\(936\) 0 0
\(937\) 36109.1 1.25895 0.629473 0.777022i \(-0.283271\pi\)
0.629473 + 0.777022i \(0.283271\pi\)
\(938\) −113098. −3.93687
\(939\) 0 0
\(940\) −66245.8 −2.29862
\(941\) −47894.5 −1.65921 −0.829604 0.558352i \(-0.811434\pi\)
−0.829604 + 0.558352i \(0.811434\pi\)
\(942\) 0 0
\(943\) 10147.5 0.350423
\(944\) 12257.4 0.422612
\(945\) 0 0
\(946\) −65523.4 −2.25195
\(947\) −34541.3 −1.18526 −0.592631 0.805474i \(-0.701911\pi\)
−0.592631 + 0.805474i \(0.701911\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 2412.30 0.0823845
\(951\) 0 0
\(952\) −25165.8 −0.856753
\(953\) −37217.0 −1.26503 −0.632517 0.774546i \(-0.717978\pi\)
−0.632517 + 0.774546i \(0.717978\pi\)
\(954\) 0 0
\(955\) 25590.5 0.867110
\(956\) −38139.9 −1.29030
\(957\) 0 0
\(958\) −6029.14 −0.203332
\(959\) 65548.0 2.20715
\(960\) 0 0
\(961\) −26653.4 −0.894679
\(962\) 0 0
\(963\) 0 0
\(964\) −58632.4 −1.95894
\(965\) −13106.4 −0.437213
\(966\) 0 0
\(967\) −19683.1 −0.654567 −0.327283 0.944926i \(-0.606133\pi\)
−0.327283 + 0.944926i \(0.606133\pi\)
\(968\) 16241.0 0.539262
\(969\) 0 0
\(970\) −88159.0 −2.91816
\(971\) −33750.5 −1.11545 −0.557726 0.830025i \(-0.688326\pi\)
−0.557726 + 0.830025i \(0.688326\pi\)
\(972\) 0 0
\(973\) −47234.6 −1.55629
\(974\) −93086.1 −3.06229
\(975\) 0 0
\(976\) 30827.3 1.01102
\(977\) 23945.2 0.784110 0.392055 0.919942i \(-0.371764\pi\)
0.392055 + 0.919942i \(0.371764\pi\)
\(978\) 0 0
\(979\) 16116.4 0.526130
\(980\) −66914.6 −2.18113
\(981\) 0 0
\(982\) 40443.8 1.31427
\(983\) −53283.1 −1.72886 −0.864429 0.502756i \(-0.832320\pi\)
−0.864429 + 0.502756i \(0.832320\pi\)
\(984\) 0 0
\(985\) 16707.1 0.540440
\(986\) 18680.3 0.603348
\(987\) 0 0
\(988\) 0 0
\(989\) −9234.71 −0.296913
\(990\) 0 0
\(991\) −46669.4 −1.49597 −0.747983 0.663717i \(-0.768978\pi\)
−0.747983 + 0.663717i \(0.768978\pi\)
\(992\) −2950.20 −0.0944245
\(993\) 0 0
\(994\) −93697.4 −2.98984
\(995\) 3641.32 0.116018
\(996\) 0 0
\(997\) −28293.1 −0.898748 −0.449374 0.893344i \(-0.648353\pi\)
−0.449374 + 0.893344i \(0.648353\pi\)
\(998\) −960.405 −0.0304620
\(999\) 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.bh.1.8 9
3.2 odd 2 169.4.a.k.1.2 9
13.12 even 2 1521.4.a.bg.1.2 9
39.2 even 12 169.4.e.h.147.3 36
39.5 even 4 169.4.b.g.168.16 18
39.8 even 4 169.4.b.g.168.3 18
39.11 even 12 169.4.e.h.147.16 36
39.17 odd 6 169.4.c.k.146.2 18
39.20 even 12 169.4.e.h.23.3 36
39.23 odd 6 169.4.c.k.22.2 18
39.29 odd 6 169.4.c.l.22.8 18
39.32 even 12 169.4.e.h.23.16 36
39.35 odd 6 169.4.c.l.146.8 18
39.38 odd 2 169.4.a.l.1.8 yes 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.4.a.k.1.2 9 3.2 odd 2
169.4.a.l.1.8 yes 9 39.38 odd 2
169.4.b.g.168.3 18 39.8 even 4
169.4.b.g.168.16 18 39.5 even 4
169.4.c.k.22.2 18 39.23 odd 6
169.4.c.k.146.2 18 39.17 odd 6
169.4.c.l.22.8 18 39.29 odd 6
169.4.c.l.146.8 18 39.35 odd 6
169.4.e.h.23.3 36 39.20 even 12
169.4.e.h.23.16 36 39.32 even 12
169.4.e.h.147.3 36 39.2 even 12
169.4.e.h.147.16 36 39.11 even 12
1521.4.a.bg.1.2 9 13.12 even 2
1521.4.a.bh.1.8 9 1.1 even 1 trivial