Properties

Label 405.4.a.k.1.6
Level $405$
Weight $4$
Character 405.1
Self dual yes
Analytic conductor $23.896$
Analytic rank $1$
Dimension $6$
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] = [405,4,Mod(1,405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(405, 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("405.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 405.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: \(23.8957735523\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 38x^{4} + 42x^{3} + 393x^{2} - 72x - 432 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(5.11734\) of defining polynomial
Character \(\chi\) \(=\) 405.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.11734 q^{2} +8.95250 q^{4} -5.00000 q^{5} -20.0229 q^{7} +3.92177 q^{8} +O(q^{10})\) \(q+4.11734 q^{2} +8.95250 q^{4} -5.00000 q^{5} -20.0229 q^{7} +3.92177 q^{8} -20.5867 q^{10} +1.67827 q^{11} +11.1123 q^{13} -82.4412 q^{14} -55.4727 q^{16} -8.98682 q^{17} -50.6419 q^{19} -44.7625 q^{20} +6.91000 q^{22} -214.491 q^{23} +25.0000 q^{25} +45.7531 q^{26} -179.255 q^{28} +76.9088 q^{29} -273.537 q^{31} -259.774 q^{32} -37.0018 q^{34} +100.115 q^{35} -137.283 q^{37} -208.510 q^{38} -19.6088 q^{40} +53.3457 q^{41} +295.270 q^{43} +15.0247 q^{44} -883.134 q^{46} +194.983 q^{47} +57.9173 q^{49} +102.934 q^{50} +99.4829 q^{52} +450.173 q^{53} -8.39133 q^{55} -78.5252 q^{56} +316.660 q^{58} -481.145 q^{59} +675.854 q^{61} -1126.25 q^{62} -625.798 q^{64} -55.5615 q^{65} +894.589 q^{67} -80.4545 q^{68} +412.206 q^{70} -721.947 q^{71} +915.175 q^{73} -565.241 q^{74} -453.372 q^{76} -33.6038 q^{77} +59.6642 q^{79} +277.364 q^{80} +219.643 q^{82} -742.875 q^{83} +44.9341 q^{85} +1215.73 q^{86} +6.58177 q^{88} -1540.72 q^{89} -222.501 q^{91} -1920.23 q^{92} +802.812 q^{94} +253.210 q^{95} -1121.57 q^{97} +238.465 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{2} + 34 q^{4} - 30 q^{5} + 40 q^{7} - 66 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{2} + 34 q^{4} - 30 q^{5} + 40 q^{7} - 66 q^{8} + 20 q^{10} - 88 q^{11} + 20 q^{13} - 180 q^{14} + 58 q^{16} - 124 q^{17} - 46 q^{19} - 170 q^{20} - 74 q^{22} - 210 q^{23} + 150 q^{25} - 4 q^{26} + 352 q^{28} - 296 q^{29} - 104 q^{31} - 722 q^{32} - 428 q^{34} - 200 q^{35} - 204 q^{37} + 20 q^{38} + 330 q^{40} - 344 q^{41} + 512 q^{43} - 716 q^{44} - 186 q^{46} - 238 q^{47} + 68 q^{49} - 100 q^{50} - 468 q^{52} - 850 q^{53} + 440 q^{55} - 2316 q^{56} + 890 q^{58} - 1840 q^{59} - 364 q^{61} - 1038 q^{62} - 990 q^{64} - 100 q^{65} + 88 q^{67} - 236 q^{68} + 900 q^{70} - 1364 q^{71} + 836 q^{73} - 1316 q^{74} - 2106 q^{76} - 840 q^{77} - 680 q^{79} - 290 q^{80} + 1742 q^{82} - 2148 q^{83} + 620 q^{85} - 2872 q^{86} + 1296 q^{88} - 3000 q^{89} - 3058 q^{91} - 1002 q^{92} - 3662 q^{94} + 230 q^{95} - 612 q^{97} - 1982 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.11734 1.45570 0.727850 0.685736i \(-0.240520\pi\)
0.727850 + 0.685736i \(0.240520\pi\)
\(3\) 0 0
\(4\) 8.95250 1.11906
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) −20.0229 −1.08114 −0.540568 0.841300i \(-0.681791\pi\)
−0.540568 + 0.841300i \(0.681791\pi\)
\(8\) 3.92177 0.173319
\(9\) 0 0
\(10\) −20.5867 −0.651009
\(11\) 1.67827 0.0460015 0.0230007 0.999735i \(-0.492678\pi\)
0.0230007 + 0.999735i \(0.492678\pi\)
\(12\) 0 0
\(13\) 11.1123 0.237077 0.118538 0.992949i \(-0.462179\pi\)
0.118538 + 0.992949i \(0.462179\pi\)
\(14\) −82.4412 −1.57381
\(15\) 0 0
\(16\) −55.4727 −0.866762
\(17\) −8.98682 −0.128213 −0.0641066 0.997943i \(-0.520420\pi\)
−0.0641066 + 0.997943i \(0.520420\pi\)
\(18\) 0 0
\(19\) −50.6419 −0.611477 −0.305738 0.952116i \(-0.598903\pi\)
−0.305738 + 0.952116i \(0.598903\pi\)
\(20\) −44.7625 −0.500460
\(21\) 0 0
\(22\) 6.91000 0.0669644
\(23\) −214.491 −1.94454 −0.972272 0.233853i \(-0.924867\pi\)
−0.972272 + 0.233853i \(0.924867\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 45.7531 0.345113
\(27\) 0 0
\(28\) −179.255 −1.20986
\(29\) 76.9088 0.492469 0.246235 0.969210i \(-0.420807\pi\)
0.246235 + 0.969210i \(0.420807\pi\)
\(30\) 0 0
\(31\) −273.537 −1.58480 −0.792399 0.610003i \(-0.791168\pi\)
−0.792399 + 0.610003i \(0.791168\pi\)
\(32\) −259.774 −1.43506
\(33\) 0 0
\(34\) −37.0018 −0.186640
\(35\) 100.115 0.483499
\(36\) 0 0
\(37\) −137.283 −0.609978 −0.304989 0.952356i \(-0.598653\pi\)
−0.304989 + 0.952356i \(0.598653\pi\)
\(38\) −208.510 −0.890126
\(39\) 0 0
\(40\) −19.6088 −0.0775107
\(41\) 53.3457 0.203200 0.101600 0.994825i \(-0.467604\pi\)
0.101600 + 0.994825i \(0.467604\pi\)
\(42\) 0 0
\(43\) 295.270 1.04717 0.523584 0.851974i \(-0.324595\pi\)
0.523584 + 0.851974i \(0.324595\pi\)
\(44\) 15.0247 0.0514785
\(45\) 0 0
\(46\) −883.134 −2.83067
\(47\) 194.983 0.605132 0.302566 0.953128i \(-0.402157\pi\)
0.302566 + 0.953128i \(0.402157\pi\)
\(48\) 0 0
\(49\) 57.9173 0.168855
\(50\) 102.934 0.291140
\(51\) 0 0
\(52\) 99.4829 0.265304
\(53\) 450.173 1.16672 0.583359 0.812214i \(-0.301738\pi\)
0.583359 + 0.812214i \(0.301738\pi\)
\(54\) 0 0
\(55\) −8.39133 −0.0205725
\(56\) −78.5252 −0.187382
\(57\) 0 0
\(58\) 316.660 0.716888
\(59\) −481.145 −1.06169 −0.530845 0.847469i \(-0.678125\pi\)
−0.530845 + 0.847469i \(0.678125\pi\)
\(60\) 0 0
\(61\) 675.854 1.41859 0.709297 0.704910i \(-0.249012\pi\)
0.709297 + 0.704910i \(0.249012\pi\)
\(62\) −1126.25 −2.30699
\(63\) 0 0
\(64\) −625.798 −1.22226
\(65\) −55.5615 −0.106024
\(66\) 0 0
\(67\) 894.589 1.63122 0.815608 0.578605i \(-0.196403\pi\)
0.815608 + 0.578605i \(0.196403\pi\)
\(68\) −80.4545 −0.143479
\(69\) 0 0
\(70\) 412.206 0.703829
\(71\) −721.947 −1.20675 −0.603376 0.797457i \(-0.706178\pi\)
−0.603376 + 0.797457i \(0.706178\pi\)
\(72\) 0 0
\(73\) 915.175 1.46730 0.733651 0.679526i \(-0.237815\pi\)
0.733651 + 0.679526i \(0.237815\pi\)
\(74\) −565.241 −0.887945
\(75\) 0 0
\(76\) −453.372 −0.684280
\(77\) −33.6038 −0.0497339
\(78\) 0 0
\(79\) 59.6642 0.0849715 0.0424858 0.999097i \(-0.486472\pi\)
0.0424858 + 0.999097i \(0.486472\pi\)
\(80\) 277.364 0.387628
\(81\) 0 0
\(82\) 219.643 0.295798
\(83\) −742.875 −0.982423 −0.491212 0.871040i \(-0.663446\pi\)
−0.491212 + 0.871040i \(0.663446\pi\)
\(84\) 0 0
\(85\) 44.9341 0.0573387
\(86\) 1215.73 1.52436
\(87\) 0 0
\(88\) 6.58177 0.00797294
\(89\) −1540.72 −1.83501 −0.917505 0.397723i \(-0.869800\pi\)
−0.917505 + 0.397723i \(0.869800\pi\)
\(90\) 0 0
\(91\) −222.501 −0.256312
\(92\) −1920.23 −2.17607
\(93\) 0 0
\(94\) 802.812 0.880891
\(95\) 253.210 0.273461
\(96\) 0 0
\(97\) −1121.57 −1.17400 −0.586999 0.809588i \(-0.699691\pi\)
−0.586999 + 0.809588i \(0.699691\pi\)
\(98\) 238.465 0.245802
\(99\) 0 0
\(100\) 223.812 0.223812
\(101\) 1758.47 1.73242 0.866211 0.499679i \(-0.166549\pi\)
0.866211 + 0.499679i \(0.166549\pi\)
\(102\) 0 0
\(103\) 1056.48 1.01066 0.505332 0.862925i \(-0.331370\pi\)
0.505332 + 0.862925i \(0.331370\pi\)
\(104\) 43.5799 0.0410900
\(105\) 0 0
\(106\) 1853.52 1.69839
\(107\) −1470.03 −1.32816 −0.664081 0.747661i \(-0.731177\pi\)
−0.664081 + 0.747661i \(0.731177\pi\)
\(108\) 0 0
\(109\) 918.889 0.807464 0.403732 0.914877i \(-0.367713\pi\)
0.403732 + 0.914877i \(0.367713\pi\)
\(110\) −34.5500 −0.0299474
\(111\) 0 0
\(112\) 1110.73 0.937087
\(113\) −987.217 −0.821855 −0.410927 0.911668i \(-0.634795\pi\)
−0.410927 + 0.911668i \(0.634795\pi\)
\(114\) 0 0
\(115\) 1072.46 0.869627
\(116\) 688.526 0.551104
\(117\) 0 0
\(118\) −1981.04 −1.54550
\(119\) 179.942 0.138616
\(120\) 0 0
\(121\) −1328.18 −0.997884
\(122\) 2782.72 2.06505
\(123\) 0 0
\(124\) −2448.84 −1.77349
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 1480.10 1.03415 0.517077 0.855939i \(-0.327020\pi\)
0.517077 + 0.855939i \(0.327020\pi\)
\(128\) −498.428 −0.344182
\(129\) 0 0
\(130\) −228.766 −0.154339
\(131\) 506.595 0.337874 0.168937 0.985627i \(-0.445967\pi\)
0.168937 + 0.985627i \(0.445967\pi\)
\(132\) 0 0
\(133\) 1014.00 0.661089
\(134\) 3683.33 2.37456
\(135\) 0 0
\(136\) −35.2442 −0.0222218
\(137\) −975.723 −0.608479 −0.304240 0.952596i \(-0.598402\pi\)
−0.304240 + 0.952596i \(0.598402\pi\)
\(138\) 0 0
\(139\) −1888.83 −1.15258 −0.576288 0.817247i \(-0.695499\pi\)
−0.576288 + 0.817247i \(0.695499\pi\)
\(140\) 896.276 0.541065
\(141\) 0 0
\(142\) −2972.50 −1.75667
\(143\) 18.6494 0.0109059
\(144\) 0 0
\(145\) −384.544 −0.220239
\(146\) 3768.09 2.13595
\(147\) 0 0
\(148\) −1229.03 −0.682604
\(149\) −2582.29 −1.41980 −0.709898 0.704305i \(-0.751259\pi\)
−0.709898 + 0.704305i \(0.751259\pi\)
\(150\) 0 0
\(151\) 2537.92 1.36777 0.683885 0.729590i \(-0.260289\pi\)
0.683885 + 0.729590i \(0.260289\pi\)
\(152\) −198.606 −0.105981
\(153\) 0 0
\(154\) −138.358 −0.0723976
\(155\) 1367.69 0.708743
\(156\) 0 0
\(157\) 105.035 0.0533930 0.0266965 0.999644i \(-0.491501\pi\)
0.0266965 + 0.999644i \(0.491501\pi\)
\(158\) 245.658 0.123693
\(159\) 0 0
\(160\) 1298.87 0.641780
\(161\) 4294.74 2.10232
\(162\) 0 0
\(163\) 2228.88 1.07104 0.535519 0.844523i \(-0.320116\pi\)
0.535519 + 0.844523i \(0.320116\pi\)
\(164\) 477.578 0.227394
\(165\) 0 0
\(166\) −3058.67 −1.43011
\(167\) −2813.80 −1.30382 −0.651911 0.758296i \(-0.726032\pi\)
−0.651911 + 0.758296i \(0.726032\pi\)
\(168\) 0 0
\(169\) −2073.52 −0.943795
\(170\) 185.009 0.0834679
\(171\) 0 0
\(172\) 2643.40 1.17185
\(173\) 3284.45 1.44342 0.721711 0.692195i \(-0.243356\pi\)
0.721711 + 0.692195i \(0.243356\pi\)
\(174\) 0 0
\(175\) −500.573 −0.216227
\(176\) −93.0981 −0.0398723
\(177\) 0 0
\(178\) −6343.67 −2.67123
\(179\) −1268.62 −0.529726 −0.264863 0.964286i \(-0.585327\pi\)
−0.264863 + 0.964286i \(0.585327\pi\)
\(180\) 0 0
\(181\) 1080.10 0.443554 0.221777 0.975097i \(-0.428814\pi\)
0.221777 + 0.975097i \(0.428814\pi\)
\(182\) −916.111 −0.373114
\(183\) 0 0
\(184\) −841.185 −0.337027
\(185\) 686.415 0.272790
\(186\) 0 0
\(187\) −15.0823 −0.00589800
\(188\) 1745.59 0.677181
\(189\) 0 0
\(190\) 1042.55 0.398077
\(191\) −286.714 −0.108617 −0.0543087 0.998524i \(-0.517296\pi\)
−0.0543087 + 0.998524i \(0.517296\pi\)
\(192\) 0 0
\(193\) −4650.70 −1.73453 −0.867267 0.497843i \(-0.834125\pi\)
−0.867267 + 0.497843i \(0.834125\pi\)
\(194\) −4617.87 −1.70899
\(195\) 0 0
\(196\) 518.505 0.188959
\(197\) −1694.82 −0.612947 −0.306474 0.951879i \(-0.599149\pi\)
−0.306474 + 0.951879i \(0.599149\pi\)
\(198\) 0 0
\(199\) −3249.77 −1.15764 −0.578819 0.815456i \(-0.696486\pi\)
−0.578819 + 0.815456i \(0.696486\pi\)
\(200\) 98.0442 0.0346639
\(201\) 0 0
\(202\) 7240.23 2.52189
\(203\) −1539.94 −0.532426
\(204\) 0 0
\(205\) −266.729 −0.0908738
\(206\) 4349.90 1.47122
\(207\) 0 0
\(208\) −616.430 −0.205489
\(209\) −84.9906 −0.0281288
\(210\) 0 0
\(211\) −1078.56 −0.351900 −0.175950 0.984399i \(-0.556300\pi\)
−0.175950 + 0.984399i \(0.556300\pi\)
\(212\) 4030.18 1.30563
\(213\) 0 0
\(214\) −6052.62 −1.93340
\(215\) −1476.35 −0.468308
\(216\) 0 0
\(217\) 5477.01 1.71338
\(218\) 3783.38 1.17543
\(219\) 0 0
\(220\) −75.1234 −0.0230219
\(221\) −99.8643 −0.0303964
\(222\) 0 0
\(223\) −5143.28 −1.54448 −0.772241 0.635330i \(-0.780864\pi\)
−0.772241 + 0.635330i \(0.780864\pi\)
\(224\) 5201.44 1.55150
\(225\) 0 0
\(226\) −4064.71 −1.19637
\(227\) −2909.66 −0.850754 −0.425377 0.905016i \(-0.639858\pi\)
−0.425377 + 0.905016i \(0.639858\pi\)
\(228\) 0 0
\(229\) 1725.82 0.498015 0.249007 0.968502i \(-0.419896\pi\)
0.249007 + 0.968502i \(0.419896\pi\)
\(230\) 4415.67 1.26592
\(231\) 0 0
\(232\) 301.619 0.0853544
\(233\) −1087.48 −0.305765 −0.152882 0.988244i \(-0.548856\pi\)
−0.152882 + 0.988244i \(0.548856\pi\)
\(234\) 0 0
\(235\) −974.916 −0.270623
\(236\) −4307.45 −1.18810
\(237\) 0 0
\(238\) 740.884 0.201783
\(239\) 319.345 0.0864297 0.0432149 0.999066i \(-0.486240\pi\)
0.0432149 + 0.999066i \(0.486240\pi\)
\(240\) 0 0
\(241\) 3645.83 0.974475 0.487238 0.873269i \(-0.338005\pi\)
0.487238 + 0.873269i \(0.338005\pi\)
\(242\) −5468.58 −1.45262
\(243\) 0 0
\(244\) 6050.58 1.58750
\(245\) −289.587 −0.0755143
\(246\) 0 0
\(247\) −562.748 −0.144967
\(248\) −1072.75 −0.274676
\(249\) 0 0
\(250\) −514.668 −0.130202
\(251\) 572.874 0.144062 0.0720309 0.997402i \(-0.477052\pi\)
0.0720309 + 0.997402i \(0.477052\pi\)
\(252\) 0 0
\(253\) −359.973 −0.0894519
\(254\) 6094.07 1.50542
\(255\) 0 0
\(256\) 2954.18 0.721236
\(257\) 3677.66 0.892630 0.446315 0.894876i \(-0.352736\pi\)
0.446315 + 0.894876i \(0.352736\pi\)
\(258\) 0 0
\(259\) 2748.81 0.659469
\(260\) −497.414 −0.118647
\(261\) 0 0
\(262\) 2085.83 0.491842
\(263\) −2001.71 −0.469319 −0.234659 0.972078i \(-0.575397\pi\)
−0.234659 + 0.972078i \(0.575397\pi\)
\(264\) 0 0
\(265\) −2250.87 −0.521772
\(266\) 4174.98 0.962348
\(267\) 0 0
\(268\) 8008.81 1.82543
\(269\) 20.1629 0.00457009 0.00228504 0.999997i \(-0.499273\pi\)
0.00228504 + 0.999997i \(0.499273\pi\)
\(270\) 0 0
\(271\) 4733.12 1.06095 0.530474 0.847701i \(-0.322014\pi\)
0.530474 + 0.847701i \(0.322014\pi\)
\(272\) 498.524 0.111130
\(273\) 0 0
\(274\) −4017.38 −0.885763
\(275\) 41.9567 0.00920030
\(276\) 0 0
\(277\) 5029.12 1.09087 0.545434 0.838154i \(-0.316365\pi\)
0.545434 + 0.838154i \(0.316365\pi\)
\(278\) −7776.94 −1.67780
\(279\) 0 0
\(280\) 392.626 0.0837996
\(281\) −5808.31 −1.23308 −0.616539 0.787325i \(-0.711466\pi\)
−0.616539 + 0.787325i \(0.711466\pi\)
\(282\) 0 0
\(283\) 213.023 0.0447453 0.0223726 0.999750i \(-0.492878\pi\)
0.0223726 + 0.999750i \(0.492878\pi\)
\(284\) −6463.23 −1.35043
\(285\) 0 0
\(286\) 76.7859 0.0158757
\(287\) −1068.14 −0.219687
\(288\) 0 0
\(289\) −4832.24 −0.983561
\(290\) −1583.30 −0.320602
\(291\) 0 0
\(292\) 8193.10 1.64200
\(293\) 3593.09 0.716419 0.358210 0.933641i \(-0.383387\pi\)
0.358210 + 0.933641i \(0.383387\pi\)
\(294\) 0 0
\(295\) 2405.72 0.474802
\(296\) −538.392 −0.105721
\(297\) 0 0
\(298\) −10632.2 −2.06680
\(299\) −2383.49 −0.461006
\(300\) 0 0
\(301\) −5912.17 −1.13213
\(302\) 10449.5 1.99106
\(303\) 0 0
\(304\) 2809.25 0.530004
\(305\) −3379.27 −0.634415
\(306\) 0 0
\(307\) 2403.23 0.446774 0.223387 0.974730i \(-0.428289\pi\)
0.223387 + 0.974730i \(0.428289\pi\)
\(308\) −300.838 −0.0556553
\(309\) 0 0
\(310\) 5631.23 1.03172
\(311\) 1608.16 0.293217 0.146608 0.989195i \(-0.453164\pi\)
0.146608 + 0.989195i \(0.453164\pi\)
\(312\) 0 0
\(313\) 2780.54 0.502127 0.251063 0.967971i \(-0.419220\pi\)
0.251063 + 0.967971i \(0.419220\pi\)
\(314\) 432.465 0.0777242
\(315\) 0 0
\(316\) 534.144 0.0950885
\(317\) −5535.70 −0.980807 −0.490403 0.871496i \(-0.663150\pi\)
−0.490403 + 0.871496i \(0.663150\pi\)
\(318\) 0 0
\(319\) 129.074 0.0226543
\(320\) 3128.99 0.546612
\(321\) 0 0
\(322\) 17682.9 3.06034
\(323\) 455.110 0.0783994
\(324\) 0 0
\(325\) 277.807 0.0474153
\(326\) 9177.06 1.55911
\(327\) 0 0
\(328\) 209.209 0.0352185
\(329\) −3904.13 −0.654230
\(330\) 0 0
\(331\) −9616.75 −1.59693 −0.798466 0.602040i \(-0.794355\pi\)
−0.798466 + 0.602040i \(0.794355\pi\)
\(332\) −6650.59 −1.09939
\(333\) 0 0
\(334\) −11585.4 −1.89797
\(335\) −4472.95 −0.729502
\(336\) 0 0
\(337\) −2084.46 −0.336936 −0.168468 0.985707i \(-0.553882\pi\)
−0.168468 + 0.985707i \(0.553882\pi\)
\(338\) −8537.38 −1.37388
\(339\) 0 0
\(340\) 402.273 0.0641656
\(341\) −459.068 −0.0729030
\(342\) 0 0
\(343\) 5708.19 0.898581
\(344\) 1157.98 0.181494
\(345\) 0 0
\(346\) 13523.2 2.10119
\(347\) 6113.42 0.945780 0.472890 0.881121i \(-0.343211\pi\)
0.472890 + 0.881121i \(0.343211\pi\)
\(348\) 0 0
\(349\) 2189.06 0.335753 0.167877 0.985808i \(-0.446309\pi\)
0.167877 + 0.985808i \(0.446309\pi\)
\(350\) −2061.03 −0.314762
\(351\) 0 0
\(352\) −435.971 −0.0660151
\(353\) 2500.20 0.376976 0.188488 0.982076i \(-0.439641\pi\)
0.188488 + 0.982076i \(0.439641\pi\)
\(354\) 0 0
\(355\) 3609.74 0.539676
\(356\) −13793.3 −2.05349
\(357\) 0 0
\(358\) −5223.33 −0.771122
\(359\) −841.607 −0.123728 −0.0618639 0.998085i \(-0.519704\pi\)
−0.0618639 + 0.998085i \(0.519704\pi\)
\(360\) 0 0
\(361\) −4294.40 −0.626096
\(362\) 4447.14 0.645681
\(363\) 0 0
\(364\) −1991.94 −0.286829
\(365\) −4575.87 −0.656198
\(366\) 0 0
\(367\) −5614.42 −0.798557 −0.399278 0.916830i \(-0.630739\pi\)
−0.399278 + 0.916830i \(0.630739\pi\)
\(368\) 11898.4 1.68546
\(369\) 0 0
\(370\) 2826.21 0.397101
\(371\) −9013.78 −1.26138
\(372\) 0 0
\(373\) 1319.17 0.183121 0.0915607 0.995799i \(-0.470814\pi\)
0.0915607 + 0.995799i \(0.470814\pi\)
\(374\) −62.0989 −0.00858572
\(375\) 0 0
\(376\) 764.678 0.104881
\(377\) 854.634 0.116753
\(378\) 0 0
\(379\) −5002.07 −0.677939 −0.338970 0.940797i \(-0.610078\pi\)
−0.338970 + 0.940797i \(0.610078\pi\)
\(380\) 2266.86 0.306020
\(381\) 0 0
\(382\) −1180.50 −0.158114
\(383\) −2242.45 −0.299174 −0.149587 0.988749i \(-0.547794\pi\)
−0.149587 + 0.988749i \(0.547794\pi\)
\(384\) 0 0
\(385\) 168.019 0.0222417
\(386\) −19148.5 −2.52496
\(387\) 0 0
\(388\) −10040.8 −1.31378
\(389\) −11393.8 −1.48506 −0.742529 0.669814i \(-0.766374\pi\)
−0.742529 + 0.669814i \(0.766374\pi\)
\(390\) 0 0
\(391\) 1927.59 0.249316
\(392\) 227.138 0.0292659
\(393\) 0 0
\(394\) −6978.13 −0.892267
\(395\) −298.321 −0.0380004
\(396\) 0 0
\(397\) −14926.0 −1.88694 −0.943472 0.331454i \(-0.892461\pi\)
−0.943472 + 0.331454i \(0.892461\pi\)
\(398\) −13380.4 −1.68517
\(399\) 0 0
\(400\) −1386.82 −0.173352
\(401\) −7165.63 −0.892356 −0.446178 0.894944i \(-0.647215\pi\)
−0.446178 + 0.894944i \(0.647215\pi\)
\(402\) 0 0
\(403\) −3039.63 −0.375719
\(404\) 15742.7 1.93869
\(405\) 0 0
\(406\) −6340.46 −0.775053
\(407\) −230.397 −0.0280599
\(408\) 0 0
\(409\) −11324.1 −1.36905 −0.684524 0.728990i \(-0.739990\pi\)
−0.684524 + 0.728990i \(0.739990\pi\)
\(410\) −1098.21 −0.132285
\(411\) 0 0
\(412\) 9458.16 1.13100
\(413\) 9633.93 1.14783
\(414\) 0 0
\(415\) 3714.38 0.439353
\(416\) −2886.69 −0.340220
\(417\) 0 0
\(418\) −349.936 −0.0409471
\(419\) −12699.8 −1.48073 −0.740365 0.672205i \(-0.765347\pi\)
−0.740365 + 0.672205i \(0.765347\pi\)
\(420\) 0 0
\(421\) −17162.8 −1.98685 −0.993424 0.114494i \(-0.963475\pi\)
−0.993424 + 0.114494i \(0.963475\pi\)
\(422\) −4440.79 −0.512261
\(423\) 0 0
\(424\) 1765.48 0.202215
\(425\) −224.671 −0.0256426
\(426\) 0 0
\(427\) −13532.6 −1.53369
\(428\) −13160.5 −1.48630
\(429\) 0 0
\(430\) −6078.63 −0.681716
\(431\) −11130.9 −1.24399 −0.621994 0.783022i \(-0.713677\pi\)
−0.621994 + 0.783022i \(0.713677\pi\)
\(432\) 0 0
\(433\) 7675.55 0.851878 0.425939 0.904752i \(-0.359944\pi\)
0.425939 + 0.904752i \(0.359944\pi\)
\(434\) 22550.7 2.49417
\(435\) 0 0
\(436\) 8226.35 0.903603
\(437\) 10862.2 1.18904
\(438\) 0 0
\(439\) 3363.80 0.365707 0.182853 0.983140i \(-0.441467\pi\)
0.182853 + 0.983140i \(0.441467\pi\)
\(440\) −32.9089 −0.00356561
\(441\) 0 0
\(442\) −411.175 −0.0442480
\(443\) 7894.06 0.846633 0.423316 0.905982i \(-0.360866\pi\)
0.423316 + 0.905982i \(0.360866\pi\)
\(444\) 0 0
\(445\) 7703.60 0.820642
\(446\) −21176.6 −2.24830
\(447\) 0 0
\(448\) 12530.3 1.32143
\(449\) 2244.52 0.235914 0.117957 0.993019i \(-0.462366\pi\)
0.117957 + 0.993019i \(0.462366\pi\)
\(450\) 0 0
\(451\) 89.5283 0.00934750
\(452\) −8838.06 −0.919707
\(453\) 0 0
\(454\) −11980.1 −1.23844
\(455\) 1112.50 0.114626
\(456\) 0 0
\(457\) 14749.3 1.50972 0.754862 0.655884i \(-0.227704\pi\)
0.754862 + 0.655884i \(0.227704\pi\)
\(458\) 7105.79 0.724960
\(459\) 0 0
\(460\) 9601.16 0.973166
\(461\) 12479.5 1.26080 0.630400 0.776271i \(-0.282891\pi\)
0.630400 + 0.776271i \(0.282891\pi\)
\(462\) 0 0
\(463\) −9679.12 −0.971549 −0.485774 0.874084i \(-0.661462\pi\)
−0.485774 + 0.874084i \(0.661462\pi\)
\(464\) −4266.34 −0.426854
\(465\) 0 0
\(466\) −4477.53 −0.445102
\(467\) 6714.33 0.665315 0.332658 0.943048i \(-0.392055\pi\)
0.332658 + 0.943048i \(0.392055\pi\)
\(468\) 0 0
\(469\) −17912.3 −1.76357
\(470\) −4014.06 −0.393946
\(471\) 0 0
\(472\) −1886.94 −0.184011
\(473\) 495.542 0.0481713
\(474\) 0 0
\(475\) −1266.05 −0.122295
\(476\) 1610.93 0.155120
\(477\) 0 0
\(478\) 1314.85 0.125816
\(479\) 2127.48 0.202937 0.101469 0.994839i \(-0.467646\pi\)
0.101469 + 0.994839i \(0.467646\pi\)
\(480\) 0 0
\(481\) −1525.53 −0.144612
\(482\) 15011.1 1.41854
\(483\) 0 0
\(484\) −11890.6 −1.11669
\(485\) 5607.83 0.525028
\(486\) 0 0
\(487\) −6549.04 −0.609375 −0.304687 0.952452i \(-0.598552\pi\)
−0.304687 + 0.952452i \(0.598552\pi\)
\(488\) 2650.54 0.245870
\(489\) 0 0
\(490\) −1192.33 −0.109926
\(491\) 12933.7 1.18878 0.594389 0.804177i \(-0.297394\pi\)
0.594389 + 0.804177i \(0.297394\pi\)
\(492\) 0 0
\(493\) −691.166 −0.0631411
\(494\) −2317.03 −0.211028
\(495\) 0 0
\(496\) 15173.9 1.37364
\(497\) 14455.5 1.30466
\(498\) 0 0
\(499\) 5120.66 0.459383 0.229692 0.973263i \(-0.426228\pi\)
0.229692 + 0.973263i \(0.426228\pi\)
\(500\) −1119.06 −0.100092
\(501\) 0 0
\(502\) 2358.72 0.209711
\(503\) 10809.6 0.958204 0.479102 0.877759i \(-0.340962\pi\)
0.479102 + 0.877759i \(0.340962\pi\)
\(504\) 0 0
\(505\) −8792.36 −0.774762
\(506\) −1482.13 −0.130215
\(507\) 0 0
\(508\) 13250.6 1.15728
\(509\) 14756.9 1.28505 0.642524 0.766266i \(-0.277887\pi\)
0.642524 + 0.766266i \(0.277887\pi\)
\(510\) 0 0
\(511\) −18324.5 −1.58635
\(512\) 16150.8 1.39409
\(513\) 0 0
\(514\) 15142.2 1.29940
\(515\) −5282.42 −0.451983
\(516\) 0 0
\(517\) 327.234 0.0278370
\(518\) 11317.8 0.959989
\(519\) 0 0
\(520\) −217.899 −0.0183760
\(521\) −1742.68 −0.146542 −0.0732709 0.997312i \(-0.523344\pi\)
−0.0732709 + 0.997312i \(0.523344\pi\)
\(522\) 0 0
\(523\) 10304.8 0.861565 0.430782 0.902456i \(-0.358238\pi\)
0.430782 + 0.902456i \(0.358238\pi\)
\(524\) 4535.29 0.378102
\(525\) 0 0
\(526\) −8241.73 −0.683187
\(527\) 2458.23 0.203192
\(528\) 0 0
\(529\) 33839.5 2.78125
\(530\) −9267.59 −0.759544
\(531\) 0 0
\(532\) 9077.83 0.739800
\(533\) 592.794 0.0481740
\(534\) 0 0
\(535\) 7350.16 0.593972
\(536\) 3508.37 0.282721
\(537\) 0 0
\(538\) 83.0175 0.00665267
\(539\) 97.2007 0.00776759
\(540\) 0 0
\(541\) 3966.86 0.315247 0.157623 0.987499i \(-0.449617\pi\)
0.157623 + 0.987499i \(0.449617\pi\)
\(542\) 19487.9 1.54442
\(543\) 0 0
\(544\) 2334.55 0.183994
\(545\) −4594.44 −0.361109
\(546\) 0 0
\(547\) −5808.72 −0.454045 −0.227023 0.973889i \(-0.572899\pi\)
−0.227023 + 0.973889i \(0.572899\pi\)
\(548\) −8735.16 −0.680926
\(549\) 0 0
\(550\) 172.750 0.0133929
\(551\) −3894.81 −0.301133
\(552\) 0 0
\(553\) −1194.65 −0.0918658
\(554\) 20706.6 1.58798
\(555\) 0 0
\(556\) −16909.7 −1.28980
\(557\) −3563.91 −0.271109 −0.135554 0.990770i \(-0.543282\pi\)
−0.135554 + 0.990770i \(0.543282\pi\)
\(558\) 0 0
\(559\) 3281.13 0.248259
\(560\) −5553.63 −0.419078
\(561\) 0 0
\(562\) −23914.8 −1.79499
\(563\) 511.544 0.0382931 0.0191466 0.999817i \(-0.493905\pi\)
0.0191466 + 0.999817i \(0.493905\pi\)
\(564\) 0 0
\(565\) 4936.09 0.367545
\(566\) 877.089 0.0651357
\(567\) 0 0
\(568\) −2831.31 −0.209153
\(569\) −8116.54 −0.598002 −0.299001 0.954253i \(-0.596653\pi\)
−0.299001 + 0.954253i \(0.596653\pi\)
\(570\) 0 0
\(571\) 15078.1 1.10508 0.552539 0.833487i \(-0.313659\pi\)
0.552539 + 0.833487i \(0.313659\pi\)
\(572\) 166.959 0.0122044
\(573\) 0 0
\(574\) −4397.88 −0.319798
\(575\) −5362.28 −0.388909
\(576\) 0 0
\(577\) 27101.5 1.95537 0.977685 0.210077i \(-0.0673716\pi\)
0.977685 + 0.210077i \(0.0673716\pi\)
\(578\) −19896.0 −1.43177
\(579\) 0 0
\(580\) −3442.63 −0.246461
\(581\) 14874.5 1.06213
\(582\) 0 0
\(583\) 755.511 0.0536708
\(584\) 3589.10 0.254312
\(585\) 0 0
\(586\) 14794.0 1.04289
\(587\) −102.082 −0.00717782 −0.00358891 0.999994i \(-0.501142\pi\)
−0.00358891 + 0.999994i \(0.501142\pi\)
\(588\) 0 0
\(589\) 13852.4 0.969067
\(590\) 9905.19 0.691170
\(591\) 0 0
\(592\) 7615.47 0.528706
\(593\) −16467.2 −1.14035 −0.570174 0.821524i \(-0.693124\pi\)
−0.570174 + 0.821524i \(0.693124\pi\)
\(594\) 0 0
\(595\) −899.712 −0.0619909
\(596\) −23118.0 −1.58884
\(597\) 0 0
\(598\) −9813.65 −0.671087
\(599\) −23216.2 −1.58362 −0.791809 0.610769i \(-0.790860\pi\)
−0.791809 + 0.610769i \(0.790860\pi\)
\(600\) 0 0
\(601\) −1678.29 −0.113909 −0.0569543 0.998377i \(-0.518139\pi\)
−0.0569543 + 0.998377i \(0.518139\pi\)
\(602\) −24342.4 −1.64804
\(603\) 0 0
\(604\) 22720.8 1.53062
\(605\) 6640.92 0.446267
\(606\) 0 0
\(607\) 19307.3 1.29103 0.645517 0.763746i \(-0.276642\pi\)
0.645517 + 0.763746i \(0.276642\pi\)
\(608\) 13155.5 0.877508
\(609\) 0 0
\(610\) −13913.6 −0.923517
\(611\) 2166.71 0.143463
\(612\) 0 0
\(613\) 13659.9 0.900032 0.450016 0.893020i \(-0.351418\pi\)
0.450016 + 0.893020i \(0.351418\pi\)
\(614\) 9894.93 0.650369
\(615\) 0 0
\(616\) −131.786 −0.00861984
\(617\) 14603.7 0.952872 0.476436 0.879209i \(-0.341928\pi\)
0.476436 + 0.879209i \(0.341928\pi\)
\(618\) 0 0
\(619\) −26388.3 −1.71346 −0.856732 0.515762i \(-0.827509\pi\)
−0.856732 + 0.515762i \(0.827509\pi\)
\(620\) 12244.2 0.793128
\(621\) 0 0
\(622\) 6621.35 0.426836
\(623\) 30849.7 1.98390
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 11448.5 0.730946
\(627\) 0 0
\(628\) 940.326 0.0597501
\(629\) 1233.74 0.0782072
\(630\) 0 0
\(631\) 14241.1 0.898459 0.449229 0.893416i \(-0.351699\pi\)
0.449229 + 0.893416i \(0.351699\pi\)
\(632\) 233.989 0.0147272
\(633\) 0 0
\(634\) −22792.4 −1.42776
\(635\) −7400.50 −0.462488
\(636\) 0 0
\(637\) 643.595 0.0400316
\(638\) 531.440 0.0329779
\(639\) 0 0
\(640\) 2492.14 0.153923
\(641\) −18960.5 −1.16832 −0.584160 0.811638i \(-0.698576\pi\)
−0.584160 + 0.811638i \(0.698576\pi\)
\(642\) 0 0
\(643\) 11054.3 0.677974 0.338987 0.940791i \(-0.389916\pi\)
0.338987 + 0.940791i \(0.389916\pi\)
\(644\) 38448.7 2.35262
\(645\) 0 0
\(646\) 1873.84 0.114126
\(647\) 19310.1 1.17335 0.586677 0.809821i \(-0.300436\pi\)
0.586677 + 0.809821i \(0.300436\pi\)
\(648\) 0 0
\(649\) −807.489 −0.0488393
\(650\) 1143.83 0.0690225
\(651\) 0 0
\(652\) 19954.0 1.19856
\(653\) 6934.26 0.415557 0.207778 0.978176i \(-0.433377\pi\)
0.207778 + 0.978176i \(0.433377\pi\)
\(654\) 0 0
\(655\) −2532.98 −0.151102
\(656\) −2959.23 −0.176126
\(657\) 0 0
\(658\) −16074.6 −0.952363
\(659\) −15878.7 −0.938613 −0.469307 0.883035i \(-0.655496\pi\)
−0.469307 + 0.883035i \(0.655496\pi\)
\(660\) 0 0
\(661\) −9418.32 −0.554206 −0.277103 0.960840i \(-0.589374\pi\)
−0.277103 + 0.960840i \(0.589374\pi\)
\(662\) −39595.4 −2.32465
\(663\) 0 0
\(664\) −2913.38 −0.170273
\(665\) −5070.00 −0.295648
\(666\) 0 0
\(667\) −16496.3 −0.957628
\(668\) −25190.5 −1.45906
\(669\) 0 0
\(670\) −18416.6 −1.06194
\(671\) 1134.26 0.0652574
\(672\) 0 0
\(673\) −18392.3 −1.05345 −0.526726 0.850035i \(-0.676581\pi\)
−0.526726 + 0.850035i \(0.676581\pi\)
\(674\) −8582.41 −0.490478
\(675\) 0 0
\(676\) −18563.2 −1.05617
\(677\) 10570.7 0.600095 0.300048 0.953924i \(-0.402997\pi\)
0.300048 + 0.953924i \(0.402997\pi\)
\(678\) 0 0
\(679\) 22457.0 1.26925
\(680\) 176.221 0.00993790
\(681\) 0 0
\(682\) −1890.14 −0.106125
\(683\) −23975.0 −1.34316 −0.671579 0.740933i \(-0.734384\pi\)
−0.671579 + 0.740933i \(0.734384\pi\)
\(684\) 0 0
\(685\) 4878.61 0.272120
\(686\) 23502.6 1.30806
\(687\) 0 0
\(688\) −16379.4 −0.907645
\(689\) 5002.46 0.276602
\(690\) 0 0
\(691\) 19641.0 1.08130 0.540650 0.841248i \(-0.318178\pi\)
0.540650 + 0.841248i \(0.318178\pi\)
\(692\) 29404.0 1.61528
\(693\) 0 0
\(694\) 25171.1 1.37677
\(695\) 9444.13 0.515448
\(696\) 0 0
\(697\) −479.408 −0.0260529
\(698\) 9013.12 0.488756
\(699\) 0 0
\(700\) −4481.38 −0.241972
\(701\) 36098.2 1.94495 0.972475 0.233007i \(-0.0748566\pi\)
0.972475 + 0.233007i \(0.0748566\pi\)
\(702\) 0 0
\(703\) 6952.28 0.372987
\(704\) −1050.26 −0.0562258
\(705\) 0 0
\(706\) 10294.2 0.548764
\(707\) −35209.8 −1.87298
\(708\) 0 0
\(709\) 35234.1 1.86635 0.933177 0.359418i \(-0.117025\pi\)
0.933177 + 0.359418i \(0.117025\pi\)
\(710\) 14862.5 0.785606
\(711\) 0 0
\(712\) −6042.34 −0.318043
\(713\) 58671.3 3.08171
\(714\) 0 0
\(715\) −93.2470 −0.00487726
\(716\) −11357.3 −0.592797
\(717\) 0 0
\(718\) −3465.18 −0.180111
\(719\) −22089.7 −1.14577 −0.572885 0.819636i \(-0.694176\pi\)
−0.572885 + 0.819636i \(0.694176\pi\)
\(720\) 0 0
\(721\) −21153.9 −1.09267
\(722\) −17681.5 −0.911409
\(723\) 0 0
\(724\) 9669.60 0.496364
\(725\) 1922.72 0.0984939
\(726\) 0 0
\(727\) −19435.2 −0.991488 −0.495744 0.868469i \(-0.665105\pi\)
−0.495744 + 0.868469i \(0.665105\pi\)
\(728\) −872.596 −0.0444238
\(729\) 0 0
\(730\) −18840.4 −0.955227
\(731\) −2653.54 −0.134261
\(732\) 0 0
\(733\) 14974.2 0.754550 0.377275 0.926101i \(-0.376861\pi\)
0.377275 + 0.926101i \(0.376861\pi\)
\(734\) −23116.5 −1.16246
\(735\) 0 0
\(736\) 55719.3 2.79055
\(737\) 1501.36 0.0750384
\(738\) 0 0
\(739\) −30663.4 −1.52635 −0.763174 0.646193i \(-0.776360\pi\)
−0.763174 + 0.646193i \(0.776360\pi\)
\(740\) 6145.13 0.305270
\(741\) 0 0
\(742\) −37112.8 −1.83619
\(743\) 3414.68 0.168604 0.0843018 0.996440i \(-0.473134\pi\)
0.0843018 + 0.996440i \(0.473134\pi\)
\(744\) 0 0
\(745\) 12911.5 0.634952
\(746\) 5431.49 0.266570
\(747\) 0 0
\(748\) −135.024 −0.00660023
\(749\) 29434.3 1.43592
\(750\) 0 0
\(751\) −746.259 −0.0362601 −0.0181301 0.999836i \(-0.505771\pi\)
−0.0181301 + 0.999836i \(0.505771\pi\)
\(752\) −10816.2 −0.524505
\(753\) 0 0
\(754\) 3518.82 0.169957
\(755\) −12689.6 −0.611685
\(756\) 0 0
\(757\) −22929.5 −1.10091 −0.550454 0.834865i \(-0.685546\pi\)
−0.550454 + 0.834865i \(0.685546\pi\)
\(758\) −20595.2 −0.986876
\(759\) 0 0
\(760\) 993.029 0.0473960
\(761\) −21176.6 −1.00874 −0.504370 0.863487i \(-0.668275\pi\)
−0.504370 + 0.863487i \(0.668275\pi\)
\(762\) 0 0
\(763\) −18398.8 −0.872978
\(764\) −2566.81 −0.121550
\(765\) 0 0
\(766\) −9232.92 −0.435508
\(767\) −5346.63 −0.251702
\(768\) 0 0
\(769\) 2774.18 0.130091 0.0650453 0.997882i \(-0.479281\pi\)
0.0650453 + 0.997882i \(0.479281\pi\)
\(770\) 691.792 0.0323772
\(771\) 0 0
\(772\) −41635.4 −1.94105
\(773\) 28891.7 1.34433 0.672163 0.740403i \(-0.265365\pi\)
0.672163 + 0.740403i \(0.265365\pi\)
\(774\) 0 0
\(775\) −6838.43 −0.316960
\(776\) −4398.52 −0.203476
\(777\) 0 0
\(778\) −46912.1 −2.16180
\(779\) −2701.53 −0.124252
\(780\) 0 0
\(781\) −1211.62 −0.0555124
\(782\) 7936.56 0.362930
\(783\) 0 0
\(784\) −3212.83 −0.146357
\(785\) −525.175 −0.0238781
\(786\) 0 0
\(787\) 8808.75 0.398981 0.199490 0.979900i \(-0.436071\pi\)
0.199490 + 0.979900i \(0.436071\pi\)
\(788\) −15172.8 −0.685926
\(789\) 0 0
\(790\) −1228.29 −0.0553172
\(791\) 19767.0 0.888537
\(792\) 0 0
\(793\) 7510.29 0.336316
\(794\) −61455.6 −2.74682
\(795\) 0 0
\(796\) −29093.5 −1.29547
\(797\) −780.503 −0.0346886 −0.0173443 0.999850i \(-0.505521\pi\)
−0.0173443 + 0.999850i \(0.505521\pi\)
\(798\) 0 0
\(799\) −1752.28 −0.0775859
\(800\) −6494.36 −0.287013
\(801\) 0 0
\(802\) −29503.4 −1.29900
\(803\) 1535.91 0.0674981
\(804\) 0 0
\(805\) −21473.7 −0.940185
\(806\) −12515.2 −0.546934
\(807\) 0 0
\(808\) 6896.32 0.300262
\(809\) 15234.0 0.662051 0.331025 0.943622i \(-0.392605\pi\)
0.331025 + 0.943622i \(0.392605\pi\)
\(810\) 0 0
\(811\) 10825.2 0.468711 0.234356 0.972151i \(-0.424702\pi\)
0.234356 + 0.972151i \(0.424702\pi\)
\(812\) −13786.3 −0.595818
\(813\) 0 0
\(814\) −948.625 −0.0408468
\(815\) −11144.4 −0.478983
\(816\) 0 0
\(817\) −14953.0 −0.640319
\(818\) −46625.2 −1.99292
\(819\) 0 0
\(820\) −2387.89 −0.101693
\(821\) 39971.2 1.69915 0.849577 0.527464i \(-0.176857\pi\)
0.849577 + 0.527464i \(0.176857\pi\)
\(822\) 0 0
\(823\) 28620.6 1.21221 0.606107 0.795383i \(-0.292730\pi\)
0.606107 + 0.795383i \(0.292730\pi\)
\(824\) 4143.28 0.175168
\(825\) 0 0
\(826\) 39666.2 1.67090
\(827\) 2265.62 0.0952639 0.0476320 0.998865i \(-0.484833\pi\)
0.0476320 + 0.998865i \(0.484833\pi\)
\(828\) 0 0
\(829\) −14651.6 −0.613839 −0.306919 0.951736i \(-0.599298\pi\)
−0.306919 + 0.951736i \(0.599298\pi\)
\(830\) 15293.4 0.639566
\(831\) 0 0
\(832\) −6954.05 −0.289770
\(833\) −520.493 −0.0216495
\(834\) 0 0
\(835\) 14069.0 0.583087
\(836\) −760.879 −0.0314779
\(837\) 0 0
\(838\) −52289.4 −2.15550
\(839\) −22466.4 −0.924466 −0.462233 0.886758i \(-0.652952\pi\)
−0.462233 + 0.886758i \(0.652952\pi\)
\(840\) 0 0
\(841\) −18474.0 −0.757474
\(842\) −70665.0 −2.89225
\(843\) 0 0
\(844\) −9655.79 −0.393799
\(845\) 10367.6 0.422078
\(846\) 0 0
\(847\) 26594.1 1.07885
\(848\) −24972.4 −1.01127
\(849\) 0 0
\(850\) −925.045 −0.0373280
\(851\) 29446.0 1.18613
\(852\) 0 0
\(853\) 44471.0 1.78506 0.892531 0.450986i \(-0.148927\pi\)
0.892531 + 0.450986i \(0.148927\pi\)
\(854\) −55718.2 −2.23260
\(855\) 0 0
\(856\) −5765.12 −0.230196
\(857\) −31480.6 −1.25479 −0.627396 0.778700i \(-0.715879\pi\)
−0.627396 + 0.778700i \(0.715879\pi\)
\(858\) 0 0
\(859\) 42983.1 1.70729 0.853646 0.520853i \(-0.174386\pi\)
0.853646 + 0.520853i \(0.174386\pi\)
\(860\) −13217.0 −0.524066
\(861\) 0 0
\(862\) −45829.9 −1.81087
\(863\) −25203.4 −0.994131 −0.497065 0.867713i \(-0.665589\pi\)
−0.497065 + 0.867713i \(0.665589\pi\)
\(864\) 0 0
\(865\) −16422.2 −0.645518
\(866\) 31602.8 1.24008
\(867\) 0 0
\(868\) 49033.0 1.91738
\(869\) 100.132 0.00390882
\(870\) 0 0
\(871\) 9940.95 0.386723
\(872\) 3603.67 0.139949
\(873\) 0 0
\(874\) 44723.6 1.73089
\(875\) 2502.86 0.0966997
\(876\) 0 0
\(877\) 3472.71 0.133712 0.0668558 0.997763i \(-0.478703\pi\)
0.0668558 + 0.997763i \(0.478703\pi\)
\(878\) 13849.9 0.532359
\(879\) 0 0
\(880\) 465.490 0.0178314
\(881\) 26066.1 0.996811 0.498405 0.866944i \(-0.333919\pi\)
0.498405 + 0.866944i \(0.333919\pi\)
\(882\) 0 0
\(883\) −3032.93 −0.115590 −0.0577952 0.998328i \(-0.518407\pi\)
−0.0577952 + 0.998328i \(0.518407\pi\)
\(884\) −894.035 −0.0340154
\(885\) 0 0
\(886\) 32502.5 1.23244
\(887\) −42339.4 −1.60273 −0.801363 0.598179i \(-0.795891\pi\)
−0.801363 + 0.598179i \(0.795891\pi\)
\(888\) 0 0
\(889\) −29635.9 −1.11806
\(890\) 31718.3 1.19461
\(891\) 0 0
\(892\) −46045.2 −1.72837
\(893\) −9874.32 −0.370024
\(894\) 0 0
\(895\) 6343.09 0.236901
\(896\) 9979.99 0.372107
\(897\) 0 0
\(898\) 9241.44 0.343419
\(899\) −21037.4 −0.780464
\(900\) 0 0
\(901\) −4045.63 −0.149589
\(902\) 368.619 0.0136072
\(903\) 0 0
\(904\) −3871.64 −0.142443
\(905\) −5400.50 −0.198363
\(906\) 0 0
\(907\) −20209.8 −0.739861 −0.369931 0.929059i \(-0.620619\pi\)
−0.369931 + 0.929059i \(0.620619\pi\)
\(908\) −26048.8 −0.952047
\(909\) 0 0
\(910\) 4580.56 0.166861
\(911\) −11254.6 −0.409310 −0.204655 0.978834i \(-0.565607\pi\)
−0.204655 + 0.978834i \(0.565607\pi\)
\(912\) 0 0
\(913\) −1246.74 −0.0451929
\(914\) 60727.9 2.19770
\(915\) 0 0
\(916\) 15450.4 0.557309
\(917\) −10143.5 −0.365287
\(918\) 0 0
\(919\) −17825.9 −0.639851 −0.319925 0.947443i \(-0.603658\pi\)
−0.319925 + 0.947443i \(0.603658\pi\)
\(920\) 4205.92 0.150723
\(921\) 0 0
\(922\) 51382.4 1.83535
\(923\) −8022.49 −0.286093
\(924\) 0 0
\(925\) −3432.08 −0.121996
\(926\) −39852.3 −1.41428
\(927\) 0 0
\(928\) −19978.9 −0.706725
\(929\) 2084.24 0.0736077 0.0368039 0.999323i \(-0.488282\pi\)
0.0368039 + 0.999323i \(0.488282\pi\)
\(930\) 0 0
\(931\) −2933.04 −0.103251
\(932\) −9735.67 −0.342170
\(933\) 0 0
\(934\) 27645.2 0.968499
\(935\) 75.4114 0.00263767
\(936\) 0 0
\(937\) −5419.51 −0.188952 −0.0944758 0.995527i \(-0.530117\pi\)
−0.0944758 + 0.995527i \(0.530117\pi\)
\(938\) −73751.0 −2.56722
\(939\) 0 0
\(940\) −8727.93 −0.302844
\(941\) 46413.8 1.60792 0.803958 0.594687i \(-0.202724\pi\)
0.803958 + 0.594687i \(0.202724\pi\)
\(942\) 0 0
\(943\) −11442.2 −0.395131
\(944\) 26690.4 0.920232
\(945\) 0 0
\(946\) 2040.31 0.0701230
\(947\) −18783.1 −0.644527 −0.322264 0.946650i \(-0.604444\pi\)
−0.322264 + 0.946650i \(0.604444\pi\)
\(948\) 0 0
\(949\) 10169.7 0.347863
\(950\) −5212.75 −0.178025
\(951\) 0 0
\(952\) 705.692 0.0240248
\(953\) 11796.7 0.400979 0.200489 0.979696i \(-0.435747\pi\)
0.200489 + 0.979696i \(0.435747\pi\)
\(954\) 0 0
\(955\) 1433.57 0.0485752
\(956\) 2858.94 0.0967203
\(957\) 0 0
\(958\) 8759.55 0.295416
\(959\) 19536.8 0.657849
\(960\) 0 0
\(961\) 45031.6 1.51158
\(962\) −6281.13 −0.210511
\(963\) 0 0
\(964\) 32639.3 1.09050
\(965\) 23253.5 0.775707
\(966\) 0 0
\(967\) −10161.7 −0.337932 −0.168966 0.985622i \(-0.554043\pi\)
−0.168966 + 0.985622i \(0.554043\pi\)
\(968\) −5208.83 −0.172953
\(969\) 0 0
\(970\) 23089.3 0.764283
\(971\) −36614.9 −1.21012 −0.605060 0.796180i \(-0.706851\pi\)
−0.605060 + 0.796180i \(0.706851\pi\)
\(972\) 0 0
\(973\) 37819.8 1.24609
\(974\) −26964.6 −0.887067
\(975\) 0 0
\(976\) −37491.5 −1.22958
\(977\) 50990.9 1.66975 0.834874 0.550441i \(-0.185541\pi\)
0.834874 + 0.550441i \(0.185541\pi\)
\(978\) 0 0
\(979\) −2585.74 −0.0844132
\(980\) −2592.52 −0.0845052
\(981\) 0 0
\(982\) 53252.5 1.73051
\(983\) 21783.8 0.706812 0.353406 0.935470i \(-0.385023\pi\)
0.353406 + 0.935470i \(0.385023\pi\)
\(984\) 0 0
\(985\) 8474.08 0.274118
\(986\) −2845.77 −0.0919145
\(987\) 0 0
\(988\) −5038.00 −0.162227
\(989\) −63332.8 −2.03626
\(990\) 0 0
\(991\) −50955.6 −1.63336 −0.816679 0.577092i \(-0.804187\pi\)
−0.816679 + 0.577092i \(0.804187\pi\)
\(992\) 71057.9 2.27429
\(993\) 0 0
\(994\) 59518.2 1.89920
\(995\) 16248.8 0.517711
\(996\) 0 0
\(997\) −19162.8 −0.608719 −0.304360 0.952557i \(-0.598442\pi\)
−0.304360 + 0.952557i \(0.598442\pi\)
\(998\) 21083.5 0.668724
\(999\) 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 405.4.a.k.1.6 6
3.2 odd 2 405.4.a.l.1.1 yes 6
5.4 even 2 2025.4.a.z.1.1 6
9.2 odd 6 405.4.e.w.271.6 12
9.4 even 3 405.4.e.x.136.1 12
9.5 odd 6 405.4.e.w.136.6 12
9.7 even 3 405.4.e.x.271.1 12
15.14 odd 2 2025.4.a.y.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
405.4.a.k.1.6 6 1.1 even 1 trivial
405.4.a.l.1.1 yes 6 3.2 odd 2
405.4.e.w.136.6 12 9.5 odd 6
405.4.e.w.271.6 12 9.2 odd 6
405.4.e.x.136.1 12 9.4 even 3
405.4.e.x.271.1 12 9.7 even 3
2025.4.a.y.1.6 6 15.14 odd 2
2025.4.a.z.1.1 6 5.4 even 2