Properties

Label 975.4.a.l.1.3
Level $975$
Weight $4$
Character 975.1
Self dual yes
Analytic conductor $57.527$
Analytic rank $0$
Dimension $3$
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] = [975,4,Mod(1,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 975.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: \(57.5268622556\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.3144.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 16x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(4.73549\) of defining polynomial
Character \(\chi\) \(=\) 975.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.73549 q^{2} -3.00000 q^{3} +5.95388 q^{4} -11.2065 q^{6} -36.4129 q^{7} -7.64325 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+3.73549 q^{2} -3.00000 q^{3} +5.95388 q^{4} -11.2065 q^{6} -36.4129 q^{7} -7.64325 q^{8} +9.00000 q^{9} +19.1943 q^{11} -17.8616 q^{12} -13.0000 q^{13} -136.020 q^{14} -76.1823 q^{16} +83.8839 q^{17} +33.6194 q^{18} +46.8492 q^{19} +109.239 q^{21} +71.7000 q^{22} -103.905 q^{23} +22.9298 q^{24} -48.5614 q^{26} -27.0000 q^{27} -216.798 q^{28} +108.341 q^{29} -147.532 q^{31} -223.432 q^{32} -57.5828 q^{33} +313.347 q^{34} +53.5849 q^{36} +160.012 q^{37} +175.005 q^{38} +39.0000 q^{39} +231.490 q^{41} +408.060 q^{42} +340.314 q^{43} +114.280 q^{44} -388.135 q^{46} -119.653 q^{47} +228.547 q^{48} +982.902 q^{49} -251.652 q^{51} -77.4005 q^{52} +732.879 q^{53} -100.858 q^{54} +278.313 q^{56} -140.548 q^{57} +404.706 q^{58} -229.782 q^{59} +108.943 q^{61} -551.104 q^{62} -327.716 q^{63} -225.170 q^{64} -215.100 q^{66} -10.3955 q^{67} +499.435 q^{68} +311.714 q^{69} -869.201 q^{71} -68.7893 q^{72} +1099.07 q^{73} +597.724 q^{74} +278.934 q^{76} -698.920 q^{77} +145.684 q^{78} +140.410 q^{79} +81.0000 q^{81} +864.729 q^{82} +159.474 q^{83} +650.395 q^{84} +1271.24 q^{86} -325.023 q^{87} -146.707 q^{88} +1067.93 q^{89} +473.368 q^{91} -618.636 q^{92} +442.596 q^{93} -446.964 q^{94} +670.297 q^{96} -858.881 q^{97} +3671.62 q^{98} +172.748 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} - 9 q^{3} + 10 q^{4} + 6 q^{6} - 30 q^{7} + 6 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} - 9 q^{3} + 10 q^{4} + 6 q^{6} - 30 q^{7} + 6 q^{8} + 27 q^{9} - 16 q^{11} - 30 q^{12} - 39 q^{13} - 176 q^{14} - 110 q^{16} + 146 q^{17} - 18 q^{18} + 94 q^{19} + 90 q^{21} + 56 q^{22} + 48 q^{23} - 18 q^{24} + 26 q^{26} - 81 q^{27} - 80 q^{28} - 2 q^{29} + 302 q^{31} - 154 q^{32} + 48 q^{33} + 164 q^{34} + 90 q^{36} - 374 q^{37} - 312 q^{38} + 117 q^{39} + 480 q^{41} + 528 q^{42} + 260 q^{43} + 712 q^{44} - 1104 q^{46} + 24 q^{47} + 330 q^{48} + 447 q^{49} - 438 q^{51} - 130 q^{52} + 678 q^{53} + 54 q^{54} + 96 q^{56} - 282 q^{57} + 628 q^{58} - 1788 q^{59} + 230 q^{61} - 1952 q^{62} - 270 q^{63} - 750 q^{64} - 168 q^{66} - 74 q^{67} + 460 q^{68} - 144 q^{69} - 948 q^{71} + 54 q^{72} + 222 q^{73} + 1724 q^{74} + 2392 q^{76} - 112 q^{77} - 78 q^{78} - 24 q^{79} + 243 q^{81} - 564 q^{82} + 796 q^{83} + 240 q^{84} + 1800 q^{86} + 6 q^{87} - 1608 q^{88} + 1436 q^{89} + 390 q^{91} + 1296 q^{92} - 906 q^{93} - 1920 q^{94} + 462 q^{96} - 3242 q^{97} + 5070 q^{98} - 144 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.73549 1.32069 0.660347 0.750960i \(-0.270409\pi\)
0.660347 + 0.750960i \(0.270409\pi\)
\(3\) −3.00000 −0.577350
\(4\) 5.95388 0.744235
\(5\) 0 0
\(6\) −11.2065 −0.762504
\(7\) −36.4129 −1.96611 −0.983057 0.183301i \(-0.941322\pi\)
−0.983057 + 0.183301i \(0.941322\pi\)
\(8\) −7.64325 −0.337787
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 19.1943 0.526117 0.263059 0.964780i \(-0.415269\pi\)
0.263059 + 0.964780i \(0.415269\pi\)
\(12\) −17.8616 −0.429684
\(13\) −13.0000 −0.277350
\(14\) −136.020 −2.59664
\(15\) 0 0
\(16\) −76.1823 −1.19035
\(17\) 83.8839 1.19676 0.598378 0.801214i \(-0.295812\pi\)
0.598378 + 0.801214i \(0.295812\pi\)
\(18\) 33.6194 0.440232
\(19\) 46.8492 0.565681 0.282840 0.959167i \(-0.408723\pi\)
0.282840 + 0.959167i \(0.408723\pi\)
\(20\) 0 0
\(21\) 109.239 1.13514
\(22\) 71.7000 0.694840
\(23\) −103.905 −0.941983 −0.470991 0.882138i \(-0.656104\pi\)
−0.470991 + 0.882138i \(0.656104\pi\)
\(24\) 22.9298 0.195022
\(25\) 0 0
\(26\) −48.5614 −0.366295
\(27\) −27.0000 −0.192450
\(28\) −216.798 −1.46325
\(29\) 108.341 0.693738 0.346869 0.937914i \(-0.387245\pi\)
0.346869 + 0.937914i \(0.387245\pi\)
\(30\) 0 0
\(31\) −147.532 −0.854759 −0.427379 0.904072i \(-0.640563\pi\)
−0.427379 + 0.904072i \(0.640563\pi\)
\(32\) −223.432 −1.23430
\(33\) −57.5828 −0.303754
\(34\) 313.347 1.58055
\(35\) 0 0
\(36\) 53.5849 0.248078
\(37\) 160.012 0.710969 0.355484 0.934682i \(-0.384316\pi\)
0.355484 + 0.934682i \(0.384316\pi\)
\(38\) 175.005 0.747092
\(39\) 39.0000 0.160128
\(40\) 0 0
\(41\) 231.490 0.881772 0.440886 0.897563i \(-0.354664\pi\)
0.440886 + 0.897563i \(0.354664\pi\)
\(42\) 408.060 1.49917
\(43\) 340.314 1.20692 0.603458 0.797395i \(-0.293789\pi\)
0.603458 + 0.797395i \(0.293789\pi\)
\(44\) 114.280 0.391555
\(45\) 0 0
\(46\) −388.135 −1.24407
\(47\) −119.653 −0.371346 −0.185673 0.982612i \(-0.559446\pi\)
−0.185673 + 0.982612i \(0.559446\pi\)
\(48\) 228.547 0.687248
\(49\) 982.902 2.86560
\(50\) 0 0
\(51\) −251.652 −0.690947
\(52\) −77.4005 −0.206414
\(53\) 732.879 1.89941 0.949705 0.313146i \(-0.101383\pi\)
0.949705 + 0.313146i \(0.101383\pi\)
\(54\) −100.858 −0.254168
\(55\) 0 0
\(56\) 278.313 0.664128
\(57\) −140.548 −0.326596
\(58\) 404.706 0.916216
\(59\) −229.782 −0.507035 −0.253518 0.967331i \(-0.581588\pi\)
−0.253518 + 0.967331i \(0.581588\pi\)
\(60\) 0 0
\(61\) 108.943 0.228668 0.114334 0.993442i \(-0.463527\pi\)
0.114334 + 0.993442i \(0.463527\pi\)
\(62\) −551.104 −1.12888
\(63\) −327.716 −0.655371
\(64\) −225.170 −0.439786
\(65\) 0 0
\(66\) −215.100 −0.401166
\(67\) −10.3955 −0.0189555 −0.00947774 0.999955i \(-0.503017\pi\)
−0.00947774 + 0.999955i \(0.503017\pi\)
\(68\) 499.435 0.890667
\(69\) 311.714 0.543854
\(70\) 0 0
\(71\) −869.201 −1.45289 −0.726445 0.687224i \(-0.758829\pi\)
−0.726445 + 0.687224i \(0.758829\pi\)
\(72\) −68.7893 −0.112596
\(73\) 1099.07 1.76214 0.881072 0.472982i \(-0.156822\pi\)
0.881072 + 0.472982i \(0.156822\pi\)
\(74\) 597.724 0.938973
\(75\) 0 0
\(76\) 278.934 0.421000
\(77\) −698.920 −1.03441
\(78\) 145.684 0.211480
\(79\) 140.410 0.199967 0.0999835 0.994989i \(-0.468121\pi\)
0.0999835 + 0.994989i \(0.468121\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 864.729 1.16455
\(83\) 159.474 0.210898 0.105449 0.994425i \(-0.466372\pi\)
0.105449 + 0.994425i \(0.466372\pi\)
\(84\) 650.395 0.844808
\(85\) 0 0
\(86\) 1271.24 1.59397
\(87\) −325.023 −0.400530
\(88\) −146.707 −0.177716
\(89\) 1067.93 1.27192 0.635959 0.771723i \(-0.280605\pi\)
0.635959 + 0.771723i \(0.280605\pi\)
\(90\) 0 0
\(91\) 473.368 0.545302
\(92\) −618.636 −0.701057
\(93\) 442.596 0.493495
\(94\) −446.964 −0.490434
\(95\) 0 0
\(96\) 670.297 0.712624
\(97\) −858.881 −0.899032 −0.449516 0.893272i \(-0.648404\pi\)
−0.449516 + 0.893272i \(0.648404\pi\)
\(98\) 3671.62 3.78459
\(99\) 172.748 0.175372
\(100\) 0 0
\(101\) −1574.16 −1.55084 −0.775421 0.631444i \(-0.782462\pi\)
−0.775421 + 0.631444i \(0.782462\pi\)
\(102\) −940.042 −0.912530
\(103\) 129.724 0.124098 0.0620489 0.998073i \(-0.480237\pi\)
0.0620489 + 0.998073i \(0.480237\pi\)
\(104\) 99.3623 0.0936853
\(105\) 0 0
\(106\) 2737.66 2.50854
\(107\) 1957.43 1.76853 0.884263 0.466990i \(-0.154661\pi\)
0.884263 + 0.466990i \(0.154661\pi\)
\(108\) −160.755 −0.143228
\(109\) 1228.77 1.07977 0.539886 0.841738i \(-0.318467\pi\)
0.539886 + 0.841738i \(0.318467\pi\)
\(110\) 0 0
\(111\) −480.037 −0.410478
\(112\) 2774.02 2.34036
\(113\) −1629.50 −1.35655 −0.678275 0.734808i \(-0.737272\pi\)
−0.678275 + 0.734808i \(0.737272\pi\)
\(114\) −525.014 −0.431334
\(115\) 0 0
\(116\) 645.049 0.516304
\(117\) −117.000 −0.0924500
\(118\) −858.349 −0.669639
\(119\) −3054.46 −2.35296
\(120\) 0 0
\(121\) −962.580 −0.723201
\(122\) 406.956 0.302000
\(123\) −694.470 −0.509092
\(124\) −878.388 −0.636142
\(125\) 0 0
\(126\) −1224.18 −0.865546
\(127\) −276.112 −0.192921 −0.0964607 0.995337i \(-0.530752\pi\)
−0.0964607 + 0.995337i \(0.530752\pi\)
\(128\) 946.337 0.653478
\(129\) −1020.94 −0.696813
\(130\) 0 0
\(131\) −96.2240 −0.0641765 −0.0320883 0.999485i \(-0.510216\pi\)
−0.0320883 + 0.999485i \(0.510216\pi\)
\(132\) −342.841 −0.226064
\(133\) −1705.92 −1.11219
\(134\) −38.8324 −0.0250344
\(135\) 0 0
\(136\) −641.146 −0.404249
\(137\) −2618.38 −1.63287 −0.816435 0.577438i \(-0.804053\pi\)
−0.816435 + 0.577438i \(0.804053\pi\)
\(138\) 1164.40 0.718265
\(139\) 1963.34 1.19805 0.599023 0.800732i \(-0.295556\pi\)
0.599023 + 0.800732i \(0.295556\pi\)
\(140\) 0 0
\(141\) 358.960 0.214396
\(142\) −3246.89 −1.91882
\(143\) −249.526 −0.145919
\(144\) −685.641 −0.396783
\(145\) 0 0
\(146\) 4105.57 2.32725
\(147\) −2948.71 −1.65446
\(148\) 952.694 0.529128
\(149\) −301.111 −0.165557 −0.0827784 0.996568i \(-0.526379\pi\)
−0.0827784 + 0.996568i \(0.526379\pi\)
\(150\) 0 0
\(151\) 342.973 0.184839 0.0924197 0.995720i \(-0.470540\pi\)
0.0924197 + 0.995720i \(0.470540\pi\)
\(152\) −358.080 −0.191080
\(153\) 754.955 0.398918
\(154\) −2610.81 −1.36614
\(155\) 0 0
\(156\) 232.201 0.119173
\(157\) 1286.97 0.654211 0.327106 0.944988i \(-0.393927\pi\)
0.327106 + 0.944988i \(0.393927\pi\)
\(158\) 524.501 0.264095
\(159\) −2198.64 −1.09662
\(160\) 0 0
\(161\) 3783.47 1.85205
\(162\) 302.575 0.146744
\(163\) −532.561 −0.255910 −0.127955 0.991780i \(-0.540841\pi\)
−0.127955 + 0.991780i \(0.540841\pi\)
\(164\) 1378.26 0.656246
\(165\) 0 0
\(166\) 595.714 0.278532
\(167\) −41.9542 −0.0194402 −0.00972011 0.999953i \(-0.503094\pi\)
−0.00972011 + 0.999953i \(0.503094\pi\)
\(168\) −834.940 −0.383435
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) 421.643 0.188560
\(172\) 2026.19 0.898229
\(173\) 1066.50 0.468694 0.234347 0.972153i \(-0.424705\pi\)
0.234347 + 0.972153i \(0.424705\pi\)
\(174\) −1214.12 −0.528977
\(175\) 0 0
\(176\) −1462.26 −0.626263
\(177\) 689.346 0.292737
\(178\) 3989.25 1.67982
\(179\) 3174.61 1.32559 0.662797 0.748799i \(-0.269369\pi\)
0.662797 + 0.748799i \(0.269369\pi\)
\(180\) 0 0
\(181\) −2725.43 −1.11923 −0.559613 0.828754i \(-0.689050\pi\)
−0.559613 + 0.828754i \(0.689050\pi\)
\(182\) 1768.26 0.720177
\(183\) −326.829 −0.132021
\(184\) 794.169 0.318190
\(185\) 0 0
\(186\) 1653.31 0.651757
\(187\) 1610.09 0.629634
\(188\) −712.402 −0.276368
\(189\) 983.149 0.378379
\(190\) 0 0
\(191\) 784.888 0.297343 0.148672 0.988887i \(-0.452500\pi\)
0.148672 + 0.988887i \(0.452500\pi\)
\(192\) 675.511 0.253910
\(193\) 1255.87 0.468391 0.234195 0.972190i \(-0.424754\pi\)
0.234195 + 0.972190i \(0.424754\pi\)
\(194\) −3208.34 −1.18735
\(195\) 0 0
\(196\) 5852.08 2.13268
\(197\) 2777.35 1.00446 0.502229 0.864734i \(-0.332513\pi\)
0.502229 + 0.864734i \(0.332513\pi\)
\(198\) 645.300 0.231613
\(199\) 1490.43 0.530924 0.265462 0.964121i \(-0.414476\pi\)
0.265462 + 0.964121i \(0.414476\pi\)
\(200\) 0 0
\(201\) 31.1866 0.0109440
\(202\) −5880.27 −2.04819
\(203\) −3945.01 −1.36397
\(204\) −1498.30 −0.514227
\(205\) 0 0
\(206\) 484.582 0.163895
\(207\) −935.141 −0.313994
\(208\) 990.370 0.330143
\(209\) 899.236 0.297614
\(210\) 0 0
\(211\) 2305.63 0.752255 0.376127 0.926568i \(-0.377255\pi\)
0.376127 + 0.926568i \(0.377255\pi\)
\(212\) 4363.48 1.41361
\(213\) 2607.60 0.838827
\(214\) 7311.97 2.33568
\(215\) 0 0
\(216\) 206.368 0.0650072
\(217\) 5372.07 1.68055
\(218\) 4590.07 1.42605
\(219\) −3297.21 −1.01737
\(220\) 0 0
\(221\) −1090.49 −0.331920
\(222\) −1793.17 −0.542116
\(223\) −1241.98 −0.372956 −0.186478 0.982459i \(-0.559707\pi\)
−0.186478 + 0.982459i \(0.559707\pi\)
\(224\) 8135.83 2.42678
\(225\) 0 0
\(226\) −6086.97 −1.79159
\(227\) 1724.76 0.504300 0.252150 0.967688i \(-0.418862\pi\)
0.252150 + 0.967688i \(0.418862\pi\)
\(228\) −836.803 −0.243064
\(229\) −3273.72 −0.944688 −0.472344 0.881414i \(-0.656592\pi\)
−0.472344 + 0.881414i \(0.656592\pi\)
\(230\) 0 0
\(231\) 2096.76 0.597215
\(232\) −828.076 −0.234336
\(233\) 2129.52 0.598752 0.299376 0.954135i \(-0.403222\pi\)
0.299376 + 0.954135i \(0.403222\pi\)
\(234\) −437.052 −0.122098
\(235\) 0 0
\(236\) −1368.10 −0.377354
\(237\) −421.231 −0.115451
\(238\) −11409.9 −3.10754
\(239\) −5082.38 −1.37553 −0.687765 0.725933i \(-0.741408\pi\)
−0.687765 + 0.725933i \(0.741408\pi\)
\(240\) 0 0
\(241\) 4765.65 1.27379 0.636893 0.770953i \(-0.280219\pi\)
0.636893 + 0.770953i \(0.280219\pi\)
\(242\) −3595.71 −0.955127
\(243\) −243.000 −0.0641500
\(244\) 648.634 0.170183
\(245\) 0 0
\(246\) −2594.19 −0.672355
\(247\) −609.039 −0.156892
\(248\) 1127.62 0.288727
\(249\) −478.422 −0.121762
\(250\) 0 0
\(251\) −4339.96 −1.09138 −0.545689 0.837988i \(-0.683732\pi\)
−0.545689 + 0.837988i \(0.683732\pi\)
\(252\) −1951.18 −0.487750
\(253\) −1994.37 −0.495594
\(254\) −1031.41 −0.254790
\(255\) 0 0
\(256\) 5336.40 1.30283
\(257\) −4359.49 −1.05812 −0.529062 0.848583i \(-0.677456\pi\)
−0.529062 + 0.848583i \(0.677456\pi\)
\(258\) −3813.72 −0.920278
\(259\) −5826.51 −1.39785
\(260\) 0 0
\(261\) 975.068 0.231246
\(262\) −359.444 −0.0847576
\(263\) −608.077 −0.142569 −0.0712844 0.997456i \(-0.522710\pi\)
−0.0712844 + 0.997456i \(0.522710\pi\)
\(264\) 440.120 0.102604
\(265\) 0 0
\(266\) −6372.43 −1.46887
\(267\) −3203.80 −0.734342
\(268\) −61.8938 −0.0141073
\(269\) 3454.29 0.782942 0.391471 0.920190i \(-0.371966\pi\)
0.391471 + 0.920190i \(0.371966\pi\)
\(270\) 0 0
\(271\) 3703.72 0.830204 0.415102 0.909775i \(-0.363746\pi\)
0.415102 + 0.909775i \(0.363746\pi\)
\(272\) −6390.47 −1.42456
\(273\) −1420.10 −0.314830
\(274\) −9780.92 −2.15652
\(275\) 0 0
\(276\) 1855.91 0.404755
\(277\) 3566.89 0.773696 0.386848 0.922144i \(-0.373564\pi\)
0.386848 + 0.922144i \(0.373564\pi\)
\(278\) 7334.04 1.58225
\(279\) −1327.79 −0.284920
\(280\) 0 0
\(281\) −117.474 −0.0249392 −0.0124696 0.999922i \(-0.503969\pi\)
−0.0124696 + 0.999922i \(0.503969\pi\)
\(282\) 1340.89 0.283152
\(283\) 1737.62 0.364984 0.182492 0.983207i \(-0.441584\pi\)
0.182492 + 0.983207i \(0.441584\pi\)
\(284\) −5175.12 −1.08129
\(285\) 0 0
\(286\) −932.100 −0.192714
\(287\) −8429.23 −1.73366
\(288\) −2010.89 −0.411434
\(289\) 2123.51 0.432223
\(290\) 0 0
\(291\) 2576.64 0.519057
\(292\) 6543.74 1.31145
\(293\) −1904.05 −0.379643 −0.189822 0.981819i \(-0.560791\pi\)
−0.189822 + 0.981819i \(0.560791\pi\)
\(294\) −11014.9 −2.18503
\(295\) 0 0
\(296\) −1223.01 −0.240156
\(297\) −518.245 −0.101251
\(298\) −1124.80 −0.218650
\(299\) 1350.76 0.261259
\(300\) 0 0
\(301\) −12391.8 −2.37293
\(302\) 1281.17 0.244117
\(303\) 4722.49 0.895379
\(304\) −3569.08 −0.673358
\(305\) 0 0
\(306\) 2820.13 0.526850
\(307\) −2862.39 −0.532134 −0.266067 0.963955i \(-0.585724\pi\)
−0.266067 + 0.963955i \(0.585724\pi\)
\(308\) −4161.29 −0.769842
\(309\) −389.172 −0.0716479
\(310\) 0 0
\(311\) 4201.55 0.766071 0.383036 0.923734i \(-0.374879\pi\)
0.383036 + 0.923734i \(0.374879\pi\)
\(312\) −298.087 −0.0540892
\(313\) −3427.74 −0.619002 −0.309501 0.950899i \(-0.600162\pi\)
−0.309501 + 0.950899i \(0.600162\pi\)
\(314\) 4807.45 0.864014
\(315\) 0 0
\(316\) 835.987 0.148823
\(317\) −1676.09 −0.296966 −0.148483 0.988915i \(-0.547439\pi\)
−0.148483 + 0.988915i \(0.547439\pi\)
\(318\) −8212.99 −1.44831
\(319\) 2079.52 0.364987
\(320\) 0 0
\(321\) −5872.30 −1.02106
\(322\) 14133.1 2.44599
\(323\) 3929.89 0.676982
\(324\) 482.264 0.0826928
\(325\) 0 0
\(326\) −1989.37 −0.337979
\(327\) −3686.32 −0.623406
\(328\) −1769.34 −0.297851
\(329\) 4356.93 0.730108
\(330\) 0 0
\(331\) 11156.6 1.85264 0.926319 0.376740i \(-0.122955\pi\)
0.926319 + 0.376740i \(0.122955\pi\)
\(332\) 949.490 0.156958
\(333\) 1440.11 0.236990
\(334\) −156.720 −0.0256746
\(335\) 0 0
\(336\) −8322.07 −1.35121
\(337\) −1636.44 −0.264517 −0.132259 0.991215i \(-0.542223\pi\)
−0.132259 + 0.991215i \(0.542223\pi\)
\(338\) 631.298 0.101592
\(339\) 4888.49 0.783205
\(340\) 0 0
\(341\) −2831.77 −0.449703
\(342\) 1575.04 0.249031
\(343\) −23300.7 −3.66799
\(344\) −2601.10 −0.407681
\(345\) 0 0
\(346\) 3983.88 0.619002
\(347\) 2977.87 0.460693 0.230347 0.973109i \(-0.426014\pi\)
0.230347 + 0.973109i \(0.426014\pi\)
\(348\) −1935.15 −0.298088
\(349\) 9847.29 1.51035 0.755177 0.655521i \(-0.227551\pi\)
0.755177 + 0.655521i \(0.227551\pi\)
\(350\) 0 0
\(351\) 351.000 0.0533761
\(352\) −4288.62 −0.649387
\(353\) −4687.34 −0.706747 −0.353374 0.935482i \(-0.614966\pi\)
−0.353374 + 0.935482i \(0.614966\pi\)
\(354\) 2575.05 0.386616
\(355\) 0 0
\(356\) 6358.35 0.946606
\(357\) 9163.38 1.35848
\(358\) 11858.7 1.75070
\(359\) −2069.88 −0.304301 −0.152151 0.988357i \(-0.548620\pi\)
−0.152151 + 0.988357i \(0.548620\pi\)
\(360\) 0 0
\(361\) −4664.16 −0.680005
\(362\) −10180.8 −1.47816
\(363\) 2887.74 0.417540
\(364\) 2818.38 0.405833
\(365\) 0 0
\(366\) −1220.87 −0.174360
\(367\) 7299.16 1.03818 0.519092 0.854719i \(-0.326270\pi\)
0.519092 + 0.854719i \(0.326270\pi\)
\(368\) 7915.70 1.12129
\(369\) 2083.41 0.293924
\(370\) 0 0
\(371\) −26686.3 −3.73446
\(372\) 2635.16 0.367277
\(373\) 8964.32 1.24438 0.622192 0.782865i \(-0.286242\pi\)
0.622192 + 0.782865i \(0.286242\pi\)
\(374\) 6014.48 0.831554
\(375\) 0 0
\(376\) 914.541 0.125436
\(377\) −1408.43 −0.192408
\(378\) 3672.54 0.499723
\(379\) −4399.26 −0.596239 −0.298120 0.954529i \(-0.596359\pi\)
−0.298120 + 0.954529i \(0.596359\pi\)
\(380\) 0 0
\(381\) 828.337 0.111383
\(382\) 2931.94 0.392700
\(383\) −3529.74 −0.470917 −0.235459 0.971884i \(-0.575659\pi\)
−0.235459 + 0.971884i \(0.575659\pi\)
\(384\) −2839.01 −0.377286
\(385\) 0 0
\(386\) 4691.28 0.618601
\(387\) 3062.82 0.402305
\(388\) −5113.68 −0.669092
\(389\) −3034.77 −0.395549 −0.197775 0.980248i \(-0.563371\pi\)
−0.197775 + 0.980248i \(0.563371\pi\)
\(390\) 0 0
\(391\) −8715.93 −1.12732
\(392\) −7512.57 −0.967964
\(393\) 288.672 0.0370523
\(394\) 10374.8 1.32658
\(395\) 0 0
\(396\) 1028.52 0.130518
\(397\) 3997.36 0.505344 0.252672 0.967552i \(-0.418691\pi\)
0.252672 + 0.967552i \(0.418691\pi\)
\(398\) 5567.49 0.701188
\(399\) 5117.75 0.642125
\(400\) 0 0
\(401\) −9092.88 −1.13236 −0.566181 0.824281i \(-0.691580\pi\)
−0.566181 + 0.824281i \(0.691580\pi\)
\(402\) 116.497 0.0144536
\(403\) 1917.92 0.237067
\(404\) −9372.38 −1.15419
\(405\) 0 0
\(406\) −14736.5 −1.80138
\(407\) 3071.32 0.374053
\(408\) 1923.44 0.233393
\(409\) 7143.54 0.863631 0.431816 0.901962i \(-0.357873\pi\)
0.431816 + 0.901962i \(0.357873\pi\)
\(410\) 0 0
\(411\) 7855.13 0.942738
\(412\) 772.361 0.0923579
\(413\) 8367.04 0.996889
\(414\) −3493.21 −0.414691
\(415\) 0 0
\(416\) 2904.62 0.342333
\(417\) −5890.02 −0.691692
\(418\) 3359.09 0.393058
\(419\) 8213.84 0.957691 0.478845 0.877899i \(-0.341055\pi\)
0.478845 + 0.877899i \(0.341055\pi\)
\(420\) 0 0
\(421\) −7997.40 −0.925818 −0.462909 0.886406i \(-0.653194\pi\)
−0.462909 + 0.886406i \(0.653194\pi\)
\(422\) 8612.64 0.993499
\(423\) −1076.88 −0.123782
\(424\) −5601.58 −0.641597
\(425\) 0 0
\(426\) 9740.68 1.10783
\(427\) −3966.94 −0.449587
\(428\) 11654.3 1.31620
\(429\) 748.577 0.0842462
\(430\) 0 0
\(431\) −13694.8 −1.53053 −0.765263 0.643718i \(-0.777391\pi\)
−0.765263 + 0.643718i \(0.777391\pi\)
\(432\) 2056.92 0.229083
\(433\) −6716.57 −0.745445 −0.372722 0.927943i \(-0.621576\pi\)
−0.372722 + 0.927943i \(0.621576\pi\)
\(434\) 20067.3 2.21950
\(435\) 0 0
\(436\) 7315.97 0.803604
\(437\) −4867.84 −0.532862
\(438\) −12316.7 −1.34364
\(439\) −5933.32 −0.645061 −0.322531 0.946559i \(-0.604534\pi\)
−0.322531 + 0.946559i \(0.604534\pi\)
\(440\) 0 0
\(441\) 8846.12 0.955201
\(442\) −4073.52 −0.438365
\(443\) 6923.40 0.742530 0.371265 0.928527i \(-0.378924\pi\)
0.371265 + 0.928527i \(0.378924\pi\)
\(444\) −2858.08 −0.305492
\(445\) 0 0
\(446\) −4639.41 −0.492561
\(447\) 903.333 0.0955843
\(448\) 8199.11 0.864669
\(449\) 8886.78 0.934061 0.467030 0.884241i \(-0.345324\pi\)
0.467030 + 0.884241i \(0.345324\pi\)
\(450\) 0 0
\(451\) 4443.28 0.463916
\(452\) −9701.84 −1.00959
\(453\) −1028.92 −0.106717
\(454\) 6442.81 0.666026
\(455\) 0 0
\(456\) 1074.24 0.110320
\(457\) −10965.0 −1.12237 −0.561184 0.827691i \(-0.689654\pi\)
−0.561184 + 0.827691i \(0.689654\pi\)
\(458\) −12228.9 −1.24764
\(459\) −2264.87 −0.230316
\(460\) 0 0
\(461\) 10069.2 1.01729 0.508644 0.860977i \(-0.330147\pi\)
0.508644 + 0.860977i \(0.330147\pi\)
\(462\) 7832.42 0.788739
\(463\) −5599.72 −0.562076 −0.281038 0.959697i \(-0.590679\pi\)
−0.281038 + 0.959697i \(0.590679\pi\)
\(464\) −8253.66 −0.825790
\(465\) 0 0
\(466\) 7954.78 0.790769
\(467\) 13247.8 1.31271 0.656355 0.754452i \(-0.272097\pi\)
0.656355 + 0.754452i \(0.272097\pi\)
\(468\) −696.604 −0.0688046
\(469\) 378.532 0.0372686
\(470\) 0 0
\(471\) −3860.90 −0.377709
\(472\) 1756.28 0.171270
\(473\) 6532.08 0.634979
\(474\) −1573.50 −0.152476
\(475\) 0 0
\(476\) −18185.9 −1.75115
\(477\) 6595.92 0.633137
\(478\) −18985.2 −1.81666
\(479\) 16725.4 1.59541 0.797707 0.603045i \(-0.206046\pi\)
0.797707 + 0.603045i \(0.206046\pi\)
\(480\) 0 0
\(481\) −2080.16 −0.197187
\(482\) 17802.0 1.68228
\(483\) −11350.4 −1.06928
\(484\) −5731.09 −0.538231
\(485\) 0 0
\(486\) −907.724 −0.0847226
\(487\) 5305.86 0.493699 0.246850 0.969054i \(-0.420605\pi\)
0.246850 + 0.969054i \(0.420605\pi\)
\(488\) −832.679 −0.0772410
\(489\) 1597.68 0.147750
\(490\) 0 0
\(491\) 16200.2 1.48901 0.744506 0.667616i \(-0.232685\pi\)
0.744506 + 0.667616i \(0.232685\pi\)
\(492\) −4134.79 −0.378884
\(493\) 9088.05 0.830234
\(494\) −2275.06 −0.207206
\(495\) 0 0
\(496\) 11239.3 1.01746
\(497\) 31650.2 2.85655
\(498\) −1787.14 −0.160811
\(499\) 4392.70 0.394076 0.197038 0.980396i \(-0.436868\pi\)
0.197038 + 0.980396i \(0.436868\pi\)
\(500\) 0 0
\(501\) 125.863 0.0112238
\(502\) −16211.9 −1.44138
\(503\) 14955.2 1.32568 0.662841 0.748760i \(-0.269350\pi\)
0.662841 + 0.748760i \(0.269350\pi\)
\(504\) 2504.82 0.221376
\(505\) 0 0
\(506\) −7449.96 −0.654528
\(507\) −507.000 −0.0444116
\(508\) −1643.94 −0.143579
\(509\) 13403.4 1.16719 0.583593 0.812047i \(-0.301647\pi\)
0.583593 + 0.812047i \(0.301647\pi\)
\(510\) 0 0
\(511\) −40020.4 −3.46458
\(512\) 12363.4 1.06716
\(513\) −1264.93 −0.108865
\(514\) −16284.8 −1.39746
\(515\) 0 0
\(516\) −6078.57 −0.518593
\(517\) −2296.66 −0.195371
\(518\) −21764.9 −1.84613
\(519\) −3199.49 −0.270601
\(520\) 0 0
\(521\) 19643.0 1.65178 0.825888 0.563834i \(-0.190674\pi\)
0.825888 + 0.563834i \(0.190674\pi\)
\(522\) 3642.35 0.305405
\(523\) −14657.4 −1.22548 −0.612738 0.790286i \(-0.709932\pi\)
−0.612738 + 0.790286i \(0.709932\pi\)
\(524\) −572.906 −0.0477624
\(525\) 0 0
\(526\) −2271.46 −0.188290
\(527\) −12375.6 −1.02294
\(528\) 4386.79 0.361573
\(529\) −1370.83 −0.112668
\(530\) 0 0
\(531\) −2068.04 −0.169012
\(532\) −10156.8 −0.827733
\(533\) −3009.37 −0.244560
\(534\) −11967.8 −0.969842
\(535\) 0 0
\(536\) 79.4558 0.00640292
\(537\) −9523.82 −0.765332
\(538\) 12903.4 1.03403
\(539\) 18866.1 1.50764
\(540\) 0 0
\(541\) 13921.3 1.10633 0.553164 0.833072i \(-0.313420\pi\)
0.553164 + 0.833072i \(0.313420\pi\)
\(542\) 13835.2 1.09645
\(543\) 8176.30 0.646185
\(544\) −18742.4 −1.47716
\(545\) 0 0
\(546\) −5304.79 −0.415795
\(547\) 2324.11 0.181667 0.0908335 0.995866i \(-0.471047\pi\)
0.0908335 + 0.995866i \(0.471047\pi\)
\(548\) −15589.5 −1.21524
\(549\) 980.488 0.0762226
\(550\) 0 0
\(551\) 5075.68 0.392434
\(552\) −2382.51 −0.183707
\(553\) −5112.75 −0.393158
\(554\) 13324.1 1.02182
\(555\) 0 0
\(556\) 11689.5 0.891628
\(557\) 16962.8 1.29037 0.645185 0.764027i \(-0.276780\pi\)
0.645185 + 0.764027i \(0.276780\pi\)
\(558\) −4959.94 −0.376292
\(559\) −4424.08 −0.334738
\(560\) 0 0
\(561\) −4830.27 −0.363519
\(562\) −438.823 −0.0329370
\(563\) 389.000 0.0291197 0.0145599 0.999894i \(-0.495365\pi\)
0.0145599 + 0.999894i \(0.495365\pi\)
\(564\) 2137.21 0.159561
\(565\) 0 0
\(566\) 6490.85 0.482033
\(567\) −2949.45 −0.218457
\(568\) 6643.53 0.490768
\(569\) 2217.56 0.163383 0.0816914 0.996658i \(-0.473968\pi\)
0.0816914 + 0.996658i \(0.473968\pi\)
\(570\) 0 0
\(571\) −17087.3 −1.25233 −0.626167 0.779689i \(-0.715377\pi\)
−0.626167 + 0.779689i \(0.715377\pi\)
\(572\) −1485.65 −0.108598
\(573\) −2354.67 −0.171671
\(574\) −31487.3 −2.28964
\(575\) 0 0
\(576\) −2026.53 −0.146595
\(577\) 3977.26 0.286959 0.143480 0.989653i \(-0.454171\pi\)
0.143480 + 0.989653i \(0.454171\pi\)
\(578\) 7932.35 0.570835
\(579\) −3767.61 −0.270425
\(580\) 0 0
\(581\) −5806.92 −0.414650
\(582\) 9625.02 0.685516
\(583\) 14067.1 0.999312
\(584\) −8400.48 −0.595230
\(585\) 0 0
\(586\) −7112.54 −0.501393
\(587\) 16880.3 1.18693 0.593463 0.804861i \(-0.297760\pi\)
0.593463 + 0.804861i \(0.297760\pi\)
\(588\) −17556.2 −1.23131
\(589\) −6911.75 −0.483521
\(590\) 0 0
\(591\) −8332.06 −0.579924
\(592\) −12190.1 −0.846301
\(593\) −2423.25 −0.167810 −0.0839048 0.996474i \(-0.526739\pi\)
−0.0839048 + 0.996474i \(0.526739\pi\)
\(594\) −1935.90 −0.133722
\(595\) 0 0
\(596\) −1792.78 −0.123213
\(597\) −4471.29 −0.306529
\(598\) 5045.75 0.345044
\(599\) −3900.55 −0.266064 −0.133032 0.991112i \(-0.542471\pi\)
−0.133032 + 0.991112i \(0.542471\pi\)
\(600\) 0 0
\(601\) 28653.4 1.94476 0.972378 0.233413i \(-0.0749893\pi\)
0.972378 + 0.233413i \(0.0749893\pi\)
\(602\) −46289.5 −3.13392
\(603\) −93.5599 −0.00631850
\(604\) 2042.02 0.137564
\(605\) 0 0
\(606\) 17640.8 1.18252
\(607\) −214.736 −0.0143589 −0.00717946 0.999974i \(-0.502285\pi\)
−0.00717946 + 0.999974i \(0.502285\pi\)
\(608\) −10467.6 −0.698220
\(609\) 11835.0 0.787487
\(610\) 0 0
\(611\) 1555.49 0.102993
\(612\) 4494.91 0.296889
\(613\) −26438.5 −1.74199 −0.870996 0.491290i \(-0.836525\pi\)
−0.870996 + 0.491290i \(0.836525\pi\)
\(614\) −10692.4 −0.702787
\(615\) 0 0
\(616\) 5342.02 0.349409
\(617\) −6700.96 −0.437229 −0.218615 0.975811i \(-0.570154\pi\)
−0.218615 + 0.975811i \(0.570154\pi\)
\(618\) −1453.75 −0.0946250
\(619\) −27319.1 −1.77391 −0.886953 0.461860i \(-0.847182\pi\)
−0.886953 + 0.461860i \(0.847182\pi\)
\(620\) 0 0
\(621\) 2805.42 0.181285
\(622\) 15694.9 1.01175
\(623\) −38886.6 −2.50074
\(624\) −2971.11 −0.190608
\(625\) 0 0
\(626\) −12804.3 −0.817513
\(627\) −2697.71 −0.171828
\(628\) 7662.45 0.486887
\(629\) 13422.4 0.850855
\(630\) 0 0
\(631\) −7126.87 −0.449629 −0.224815 0.974402i \(-0.572178\pi\)
−0.224815 + 0.974402i \(0.572178\pi\)
\(632\) −1073.19 −0.0675463
\(633\) −6916.88 −0.434315
\(634\) −6261.00 −0.392202
\(635\) 0 0
\(636\) −13090.4 −0.816147
\(637\) −12777.7 −0.794775
\(638\) 7768.04 0.482037
\(639\) −7822.81 −0.484297
\(640\) 0 0
\(641\) 23615.0 1.45513 0.727565 0.686039i \(-0.240652\pi\)
0.727565 + 0.686039i \(0.240652\pi\)
\(642\) −21935.9 −1.34851
\(643\) 8144.41 0.499509 0.249755 0.968309i \(-0.419650\pi\)
0.249755 + 0.968309i \(0.419650\pi\)
\(644\) 22526.3 1.37836
\(645\) 0 0
\(646\) 14680.1 0.894086
\(647\) −9682.00 −0.588313 −0.294157 0.955757i \(-0.595039\pi\)
−0.294157 + 0.955757i \(0.595039\pi\)
\(648\) −619.103 −0.0375319
\(649\) −4410.50 −0.266760
\(650\) 0 0
\(651\) −16116.2 −0.970268
\(652\) −3170.80 −0.190457
\(653\) 18193.6 1.09030 0.545152 0.838337i \(-0.316472\pi\)
0.545152 + 0.838337i \(0.316472\pi\)
\(654\) −13770.2 −0.823329
\(655\) 0 0
\(656\) −17635.5 −1.04962
\(657\) 9891.64 0.587381
\(658\) 16275.3 0.964250
\(659\) 9300.88 0.549789 0.274895 0.961474i \(-0.411357\pi\)
0.274895 + 0.961474i \(0.411357\pi\)
\(660\) 0 0
\(661\) −5437.29 −0.319949 −0.159974 0.987121i \(-0.551141\pi\)
−0.159974 + 0.987121i \(0.551141\pi\)
\(662\) 41675.4 2.44677
\(663\) 3271.47 0.191634
\(664\) −1218.90 −0.0712388
\(665\) 0 0
\(666\) 5379.51 0.312991
\(667\) −11257.1 −0.653489
\(668\) −249.791 −0.0144681
\(669\) 3725.94 0.215326
\(670\) 0 0
\(671\) 2091.08 0.120306
\(672\) −24407.5 −1.40110
\(673\) 8682.75 0.497319 0.248659 0.968591i \(-0.420010\pi\)
0.248659 + 0.968591i \(0.420010\pi\)
\(674\) −6112.89 −0.349347
\(675\) 0 0
\(676\) 1006.21 0.0572489
\(677\) −13300.1 −0.755041 −0.377521 0.926001i \(-0.623223\pi\)
−0.377521 + 0.926001i \(0.623223\pi\)
\(678\) 18260.9 1.03437
\(679\) 31274.4 1.76760
\(680\) 0 0
\(681\) −5174.27 −0.291158
\(682\) −10578.0 −0.593921
\(683\) 504.175 0.0282455 0.0141228 0.999900i \(-0.495504\pi\)
0.0141228 + 0.999900i \(0.495504\pi\)
\(684\) 2510.41 0.140333
\(685\) 0 0
\(686\) −87039.6 −4.84429
\(687\) 9821.16 0.545416
\(688\) −25925.9 −1.43665
\(689\) −9527.43 −0.526802
\(690\) 0 0
\(691\) 13443.8 0.740124 0.370062 0.929007i \(-0.379336\pi\)
0.370062 + 0.929007i \(0.379336\pi\)
\(692\) 6349.79 0.348819
\(693\) −6290.28 −0.344802
\(694\) 11123.8 0.608435
\(695\) 0 0
\(696\) 2484.23 0.135294
\(697\) 19418.3 1.05527
\(698\) 36784.4 1.99472
\(699\) −6388.55 −0.345690
\(700\) 0 0
\(701\) 28735.6 1.54826 0.774128 0.633030i \(-0.218189\pi\)
0.774128 + 0.633030i \(0.218189\pi\)
\(702\) 1311.16 0.0704935
\(703\) 7496.44 0.402181
\(704\) −4321.98 −0.231379
\(705\) 0 0
\(706\) −17509.5 −0.933398
\(707\) 57319.9 3.04913
\(708\) 4104.29 0.217865
\(709\) 17610.2 0.932812 0.466406 0.884571i \(-0.345549\pi\)
0.466406 + 0.884571i \(0.345549\pi\)
\(710\) 0 0
\(711\) 1263.69 0.0666557
\(712\) −8162.48 −0.429638
\(713\) 15329.3 0.805168
\(714\) 34229.7 1.79414
\(715\) 0 0
\(716\) 18901.2 0.986553
\(717\) 15247.1 0.794163
\(718\) −7732.02 −0.401889
\(719\) −9226.04 −0.478544 −0.239272 0.970953i \(-0.576909\pi\)
−0.239272 + 0.970953i \(0.576909\pi\)
\(720\) 0 0
\(721\) −4723.63 −0.243990
\(722\) −17422.9 −0.898079
\(723\) −14296.9 −0.735420
\(724\) −16226.9 −0.832967
\(725\) 0 0
\(726\) 10787.1 0.551443
\(727\) −33246.0 −1.69604 −0.848022 0.529961i \(-0.822207\pi\)
−0.848022 + 0.529961i \(0.822207\pi\)
\(728\) −3618.07 −0.184196
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 28546.9 1.44438
\(732\) −1945.90 −0.0982549
\(733\) 4423.26 0.222888 0.111444 0.993771i \(-0.464452\pi\)
0.111444 + 0.993771i \(0.464452\pi\)
\(734\) 27266.0 1.37112
\(735\) 0 0
\(736\) 23215.6 1.16269
\(737\) −199.535 −0.00997281
\(738\) 7782.56 0.388184
\(739\) −4529.56 −0.225470 −0.112735 0.993625i \(-0.535961\pi\)
−0.112735 + 0.993625i \(0.535961\pi\)
\(740\) 0 0
\(741\) 1827.12 0.0905814
\(742\) −99686.4 −4.93208
\(743\) −10851.5 −0.535803 −0.267901 0.963446i \(-0.586330\pi\)
−0.267901 + 0.963446i \(0.586330\pi\)
\(744\) −3382.87 −0.166696
\(745\) 0 0
\(746\) 33486.1 1.64345
\(747\) 1435.27 0.0702994
\(748\) 9586.29 0.468595
\(749\) −71275.9 −3.47712
\(750\) 0 0
\(751\) −33022.6 −1.60454 −0.802272 0.596958i \(-0.796376\pi\)
−0.802272 + 0.596958i \(0.796376\pi\)
\(752\) 9115.48 0.442031
\(753\) 13019.9 0.630107
\(754\) −5261.18 −0.254113
\(755\) 0 0
\(756\) 5853.55 0.281603
\(757\) 3443.77 0.165345 0.0826724 0.996577i \(-0.473654\pi\)
0.0826724 + 0.996577i \(0.473654\pi\)
\(758\) −16433.4 −0.787450
\(759\) 5983.12 0.286131
\(760\) 0 0
\(761\) 19562.6 0.931858 0.465929 0.884822i \(-0.345720\pi\)
0.465929 + 0.884822i \(0.345720\pi\)
\(762\) 3094.24 0.147103
\(763\) −44743.2 −2.12295
\(764\) 4673.13 0.221293
\(765\) 0 0
\(766\) −13185.3 −0.621938
\(767\) 2987.17 0.140626
\(768\) −16009.2 −0.752190
\(769\) −17061.1 −0.800049 −0.400025 0.916504i \(-0.630998\pi\)
−0.400025 + 0.916504i \(0.630998\pi\)
\(770\) 0 0
\(771\) 13078.5 0.610908
\(772\) 7477.29 0.348593
\(773\) −10798.4 −0.502448 −0.251224 0.967929i \(-0.580833\pi\)
−0.251224 + 0.967929i \(0.580833\pi\)
\(774\) 11441.1 0.531322
\(775\) 0 0
\(776\) 6564.64 0.303682
\(777\) 17479.5 0.807046
\(778\) −11336.3 −0.522400
\(779\) 10845.1 0.498802
\(780\) 0 0
\(781\) −16683.7 −0.764391
\(782\) −32558.2 −1.48885
\(783\) −2925.20 −0.133510
\(784\) −74879.8 −3.41107
\(785\) 0 0
\(786\) 1078.33 0.0489348
\(787\) 35607.0 1.61277 0.806386 0.591390i \(-0.201420\pi\)
0.806386 + 0.591390i \(0.201420\pi\)
\(788\) 16536.0 0.747553
\(789\) 1824.23 0.0823122
\(790\) 0 0
\(791\) 59334.8 2.66713
\(792\) −1320.36 −0.0592386
\(793\) −1416.26 −0.0634210
\(794\) 14932.1 0.667405
\(795\) 0 0
\(796\) 8873.85 0.395132
\(797\) −22155.3 −0.984668 −0.492334 0.870406i \(-0.663856\pi\)
−0.492334 + 0.870406i \(0.663856\pi\)
\(798\) 19117.3 0.848051
\(799\) −10037.0 −0.444410
\(800\) 0 0
\(801\) 9611.40 0.423973
\(802\) −33966.4 −1.49550
\(803\) 21095.9 0.927094
\(804\) 185.681 0.00814488
\(805\) 0 0
\(806\) 7164.35 0.313094
\(807\) −10362.9 −0.452032
\(808\) 12031.7 0.523855
\(809\) −22524.6 −0.978889 −0.489445 0.872034i \(-0.662800\pi\)
−0.489445 + 0.872034i \(0.662800\pi\)
\(810\) 0 0
\(811\) 4452.39 0.192780 0.0963900 0.995344i \(-0.469270\pi\)
0.0963900 + 0.995344i \(0.469270\pi\)
\(812\) −23488.1 −1.01511
\(813\) −11111.2 −0.479318
\(814\) 11472.9 0.494010
\(815\) 0 0
\(816\) 19171.4 0.822468
\(817\) 15943.4 0.682729
\(818\) 26684.6 1.14059
\(819\) 4260.31 0.181767
\(820\) 0 0
\(821\) −7097.27 −0.301701 −0.150851 0.988557i \(-0.548201\pi\)
−0.150851 + 0.988557i \(0.548201\pi\)
\(822\) 29342.8 1.24507
\(823\) 12193.8 0.516463 0.258231 0.966083i \(-0.416860\pi\)
0.258231 + 0.966083i \(0.416860\pi\)
\(824\) −991.512 −0.0419187
\(825\) 0 0
\(826\) 31255.0 1.31659
\(827\) −7427.97 −0.312329 −0.156164 0.987731i \(-0.549913\pi\)
−0.156164 + 0.987731i \(0.549913\pi\)
\(828\) −5567.72 −0.233686
\(829\) 16966.2 0.710810 0.355405 0.934712i \(-0.384343\pi\)
0.355405 + 0.934712i \(0.384343\pi\)
\(830\) 0 0
\(831\) −10700.7 −0.446693
\(832\) 2927.21 0.121975
\(833\) 82449.7 3.42943
\(834\) −22002.1 −0.913514
\(835\) 0 0
\(836\) 5353.94 0.221495
\(837\) 3983.36 0.164498
\(838\) 30682.7 1.26482
\(839\) 12025.7 0.494844 0.247422 0.968908i \(-0.420417\pi\)
0.247422 + 0.968908i \(0.420417\pi\)
\(840\) 0 0
\(841\) −12651.3 −0.518728
\(842\) −29874.2 −1.22272
\(843\) 352.422 0.0143986
\(844\) 13727.4 0.559855
\(845\) 0 0
\(846\) −4022.68 −0.163478
\(847\) 35050.4 1.42189
\(848\) −55832.5 −2.26096
\(849\) −5212.85 −0.210724
\(850\) 0 0
\(851\) −16626.0 −0.669720
\(852\) 15525.4 0.624284
\(853\) 22187.2 0.890593 0.445297 0.895383i \(-0.353098\pi\)
0.445297 + 0.895383i \(0.353098\pi\)
\(854\) −14818.5 −0.593767
\(855\) 0 0
\(856\) −14961.2 −0.597385
\(857\) −5746.19 −0.229038 −0.114519 0.993421i \(-0.536533\pi\)
−0.114519 + 0.993421i \(0.536533\pi\)
\(858\) 2796.30 0.111264
\(859\) 8305.66 0.329902 0.164951 0.986302i \(-0.447253\pi\)
0.164951 + 0.986302i \(0.447253\pi\)
\(860\) 0 0
\(861\) 25287.7 1.00093
\(862\) −51156.9 −2.02136
\(863\) −38086.4 −1.50229 −0.751146 0.660137i \(-0.770498\pi\)
−0.751146 + 0.660137i \(0.770498\pi\)
\(864\) 6032.67 0.237541
\(865\) 0 0
\(866\) −25089.7 −0.984505
\(867\) −6370.53 −0.249544
\(868\) 31984.7 1.25073
\(869\) 2695.07 0.105206
\(870\) 0 0
\(871\) 135.142 0.00525731
\(872\) −9391.82 −0.364733
\(873\) −7729.93 −0.299677
\(874\) −18183.8 −0.703748
\(875\) 0 0
\(876\) −19631.2 −0.757166
\(877\) 2098.53 0.0808009 0.0404005 0.999184i \(-0.487137\pi\)
0.0404005 + 0.999184i \(0.487137\pi\)
\(878\) −22163.8 −0.851929
\(879\) 5712.14 0.219187
\(880\) 0 0
\(881\) 14555.3 0.556619 0.278309 0.960491i \(-0.410226\pi\)
0.278309 + 0.960491i \(0.410226\pi\)
\(882\) 33044.6 1.26153
\(883\) −2122.88 −0.0809066 −0.0404533 0.999181i \(-0.512880\pi\)
−0.0404533 + 0.999181i \(0.512880\pi\)
\(884\) −6492.65 −0.247027
\(885\) 0 0
\(886\) 25862.3 0.980656
\(887\) 12487.3 0.472696 0.236348 0.971668i \(-0.424049\pi\)
0.236348 + 0.971668i \(0.424049\pi\)
\(888\) 3669.04 0.138654
\(889\) 10054.1 0.379305
\(890\) 0 0
\(891\) 1554.74 0.0584575
\(892\) −7394.61 −0.277567
\(893\) −5605.66 −0.210063
\(894\) 3374.39 0.126238
\(895\) 0 0
\(896\) −34458.9 −1.28481
\(897\) −4052.28 −0.150838
\(898\) 33196.5 1.23361
\(899\) −15983.7 −0.592978
\(900\) 0 0
\(901\) 61476.8 2.27313
\(902\) 16597.8 0.612691
\(903\) 37175.5 1.37001
\(904\) 12454.7 0.458226
\(905\) 0 0
\(906\) −3843.52 −0.140941
\(907\) −29679.8 −1.08655 −0.543275 0.839555i \(-0.682816\pi\)
−0.543275 + 0.839555i \(0.682816\pi\)
\(908\) 10269.0 0.375318
\(909\) −14167.5 −0.516948
\(910\) 0 0
\(911\) 24800.0 0.901934 0.450967 0.892541i \(-0.351079\pi\)
0.450967 + 0.892541i \(0.351079\pi\)
\(912\) 10707.2 0.388763
\(913\) 3060.99 0.110957
\(914\) −40959.7 −1.48230
\(915\) 0 0
\(916\) −19491.3 −0.703070
\(917\) 3503.80 0.126178
\(918\) −8460.38 −0.304177
\(919\) −6597.90 −0.236828 −0.118414 0.992964i \(-0.537781\pi\)
−0.118414 + 0.992964i \(0.537781\pi\)
\(920\) 0 0
\(921\) 8587.17 0.307228
\(922\) 37613.4 1.34353
\(923\) 11299.6 0.402959
\(924\) 12483.9 0.444468
\(925\) 0 0
\(926\) −20917.7 −0.742331
\(927\) 1167.51 0.0413659
\(928\) −24206.8 −0.856281
\(929\) 15056.0 0.531724 0.265862 0.964011i \(-0.414344\pi\)
0.265862 + 0.964011i \(0.414344\pi\)
\(930\) 0 0
\(931\) 46048.1 1.62102
\(932\) 12678.9 0.445612
\(933\) −12604.7 −0.442291
\(934\) 49487.1 1.73369
\(935\) 0 0
\(936\) 894.261 0.0312284
\(937\) 35777.0 1.24737 0.623683 0.781677i \(-0.285635\pi\)
0.623683 + 0.781677i \(0.285635\pi\)
\(938\) 1414.00 0.0492205
\(939\) 10283.2 0.357381
\(940\) 0 0
\(941\) −22973.6 −0.795873 −0.397937 0.917413i \(-0.630274\pi\)
−0.397937 + 0.917413i \(0.630274\pi\)
\(942\) −14422.4 −0.498838
\(943\) −24052.9 −0.830615
\(944\) 17505.3 0.603549
\(945\) 0 0
\(946\) 24400.5 0.838614
\(947\) 51038.3 1.75134 0.875671 0.482908i \(-0.160420\pi\)
0.875671 + 0.482908i \(0.160420\pi\)
\(948\) −2507.96 −0.0859227
\(949\) −14287.9 −0.488731
\(950\) 0 0
\(951\) 5028.26 0.171454
\(952\) 23346.0 0.794799
\(953\) 22586.5 0.767733 0.383866 0.923389i \(-0.374592\pi\)
0.383866 + 0.923389i \(0.374592\pi\)
\(954\) 24639.0 0.836180
\(955\) 0 0
\(956\) −30259.9 −1.02372
\(957\) −6238.57 −0.210726
\(958\) 62477.6 2.10706
\(959\) 95342.8 3.21041
\(960\) 0 0
\(961\) −8025.32 −0.269387
\(962\) −7770.41 −0.260424
\(963\) 17616.9 0.589509
\(964\) 28374.1 0.947996
\(965\) 0 0
\(966\) −42399.4 −1.41219
\(967\) 36678.7 1.21976 0.609880 0.792494i \(-0.291218\pi\)
0.609880 + 0.792494i \(0.291218\pi\)
\(968\) 7357.24 0.244288
\(969\) −11789.7 −0.390855
\(970\) 0 0
\(971\) 32635.2 1.07859 0.539296 0.842116i \(-0.318690\pi\)
0.539296 + 0.842116i \(0.318690\pi\)
\(972\) −1446.79 −0.0477427
\(973\) −71491.0 −2.35549
\(974\) 19820.0 0.652026
\(975\) 0 0
\(976\) −8299.54 −0.272194
\(977\) −44432.6 −1.45499 −0.727496 0.686112i \(-0.759316\pi\)
−0.727496 + 0.686112i \(0.759316\pi\)
\(978\) 5968.12 0.195133
\(979\) 20498.2 0.669178
\(980\) 0 0
\(981\) 11059.0 0.359924
\(982\) 60515.6 1.96653
\(983\) −484.485 −0.0157199 −0.00785996 0.999969i \(-0.502502\pi\)
−0.00785996 + 0.999969i \(0.502502\pi\)
\(984\) 5308.01 0.171965
\(985\) 0 0
\(986\) 33948.3 1.09649
\(987\) −13070.8 −0.421528
\(988\) −3626.15 −0.116764
\(989\) −35360.2 −1.13689
\(990\) 0 0
\(991\) −48017.1 −1.53917 −0.769583 0.638546i \(-0.779536\pi\)
−0.769583 + 0.638546i \(0.779536\pi\)
\(992\) 32963.4 1.05503
\(993\) −33469.8 −1.06962
\(994\) 118229. 3.77263
\(995\) 0 0
\(996\) −2848.47 −0.0906197
\(997\) 26561.9 0.843755 0.421877 0.906653i \(-0.361371\pi\)
0.421877 + 0.906653i \(0.361371\pi\)
\(998\) 16408.9 0.520455
\(999\) −4320.33 −0.136826
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 975.4.a.l.1.3 3
5.4 even 2 39.4.a.c.1.1 3
15.14 odd 2 117.4.a.f.1.3 3
20.19 odd 2 624.4.a.t.1.2 3
35.34 odd 2 1911.4.a.k.1.1 3
40.19 odd 2 2496.4.a.bp.1.2 3
40.29 even 2 2496.4.a.bl.1.2 3
60.59 even 2 1872.4.a.bk.1.2 3
65.34 odd 4 507.4.b.g.337.2 6
65.44 odd 4 507.4.b.g.337.5 6
65.64 even 2 507.4.a.h.1.3 3
195.194 odd 2 1521.4.a.u.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.c.1.1 3 5.4 even 2
117.4.a.f.1.3 3 15.14 odd 2
507.4.a.h.1.3 3 65.64 even 2
507.4.b.g.337.2 6 65.34 odd 4
507.4.b.g.337.5 6 65.44 odd 4
624.4.a.t.1.2 3 20.19 odd 2
975.4.a.l.1.3 3 1.1 even 1 trivial
1521.4.a.u.1.1 3 195.194 odd 2
1872.4.a.bk.1.2 3 60.59 even 2
1911.4.a.k.1.1 3 35.34 odd 2
2496.4.a.bl.1.2 3 40.29 even 2
2496.4.a.bp.1.2 3 40.19 odd 2