Properties

Label 280.6.a.i.1.4
Level $280$
Weight $6$
Character 280.1
Self dual yes
Analytic conductor $44.907$
Analytic rank $0$
Dimension $5$
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] = [280,6,Mod(1,280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(280, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("280.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 280.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: \(44.9074695476\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 791x^{3} + 280x^{2} + 24832x + 39040 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(6.54694\) of defining polynomial
Character \(\chi\) \(=\) 280.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.54694 q^{3} -25.0000 q^{5} -49.0000 q^{7} -212.231 q^{9} +O(q^{10})\) \(q+5.54694 q^{3} -25.0000 q^{5} -49.0000 q^{7} -212.231 q^{9} +713.630 q^{11} -157.390 q^{13} -138.674 q^{15} -1669.06 q^{17} +18.8324 q^{19} -271.800 q^{21} +1479.72 q^{23} +625.000 q^{25} -2525.14 q^{27} +7550.14 q^{29} +6293.24 q^{31} +3958.46 q^{33} +1225.00 q^{35} -391.915 q^{37} -873.033 q^{39} -5926.90 q^{41} +20027.4 q^{43} +5305.79 q^{45} -20388.6 q^{47} +2401.00 q^{49} -9258.15 q^{51} +2088.81 q^{53} -17840.8 q^{55} +104.462 q^{57} +45077.4 q^{59} +21786.2 q^{61} +10399.3 q^{63} +3934.75 q^{65} +64262.6 q^{67} +8207.92 q^{69} +16333.8 q^{71} +6540.80 q^{73} +3466.84 q^{75} -34967.9 q^{77} -84587.9 q^{79} +37565.4 q^{81} +28514.9 q^{83} +41726.4 q^{85} +41880.2 q^{87} -13227.8 q^{89} +7712.11 q^{91} +34908.2 q^{93} -470.810 q^{95} +139857. q^{97} -151455. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 3 q^{3} - 125 q^{5} - 245 q^{7} + 372 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 3 q^{3} - 125 q^{5} - 245 q^{7} + 372 q^{9} - 263 q^{11} - 729 q^{13} + 75 q^{15} - 1003 q^{17} - 2506 q^{19} + 147 q^{21} - 1066 q^{23} + 3125 q^{25} + 615 q^{27} + 3489 q^{29} + 7880 q^{31} + 4863 q^{33} + 6125 q^{35} + 13118 q^{37} + 27189 q^{39} + 23972 q^{41} + 3978 q^{43} - 9300 q^{45} + 9057 q^{47} + 12005 q^{49} + 96639 q^{51} + 2128 q^{53} + 6575 q^{55} + 61674 q^{57} - 15512 q^{59} + 4560 q^{61} - 18228 q^{63} + 18225 q^{65} + 7780 q^{67} + 126474 q^{69} + 32752 q^{71} + 189498 q^{73} - 1875 q^{75} + 12887 q^{77} + 42055 q^{79} + 294645 q^{81} + 58420 q^{83} + 25075 q^{85} - 765 q^{87} + 231324 q^{89} + 35721 q^{91} + 395736 q^{93} + 62650 q^{95} + 247569 q^{97} + 30606 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 5.54694 0.355836 0.177918 0.984045i \(-0.443064\pi\)
0.177918 + 0.984045i \(0.443064\pi\)
\(4\) 0 0
\(5\) −25.0000 −0.447214
\(6\) 0 0
\(7\) −49.0000 −0.377964
\(8\) 0 0
\(9\) −212.231 −0.873380
\(10\) 0 0
\(11\) 713.630 1.77824 0.889122 0.457670i \(-0.151316\pi\)
0.889122 + 0.457670i \(0.151316\pi\)
\(12\) 0 0
\(13\) −157.390 −0.258297 −0.129148 0.991625i \(-0.541224\pi\)
−0.129148 + 0.991625i \(0.541224\pi\)
\(14\) 0 0
\(15\) −138.674 −0.159135
\(16\) 0 0
\(17\) −1669.06 −1.40071 −0.700355 0.713794i \(-0.746975\pi\)
−0.700355 + 0.713794i \(0.746975\pi\)
\(18\) 0 0
\(19\) 18.8324 0.0119680 0.00598400 0.999982i \(-0.498095\pi\)
0.00598400 + 0.999982i \(0.498095\pi\)
\(20\) 0 0
\(21\) −271.800 −0.134494
\(22\) 0 0
\(23\) 1479.72 0.583257 0.291628 0.956532i \(-0.405803\pi\)
0.291628 + 0.956532i \(0.405803\pi\)
\(24\) 0 0
\(25\) 625.000 0.200000
\(26\) 0 0
\(27\) −2525.14 −0.666617
\(28\) 0 0
\(29\) 7550.14 1.66709 0.833547 0.552449i \(-0.186307\pi\)
0.833547 + 0.552449i \(0.186307\pi\)
\(30\) 0 0
\(31\) 6293.24 1.17617 0.588085 0.808799i \(-0.299882\pi\)
0.588085 + 0.808799i \(0.299882\pi\)
\(32\) 0 0
\(33\) 3958.46 0.632764
\(34\) 0 0
\(35\) 1225.00 0.169031
\(36\) 0 0
\(37\) −391.915 −0.0470639 −0.0235319 0.999723i \(-0.507491\pi\)
−0.0235319 + 0.999723i \(0.507491\pi\)
\(38\) 0 0
\(39\) −873.033 −0.0919113
\(40\) 0 0
\(41\) −5926.90 −0.550641 −0.275320 0.961353i \(-0.588784\pi\)
−0.275320 + 0.961353i \(0.588784\pi\)
\(42\) 0 0
\(43\) 20027.4 1.65179 0.825893 0.563827i \(-0.190672\pi\)
0.825893 + 0.563827i \(0.190672\pi\)
\(44\) 0 0
\(45\) 5305.79 0.390588
\(46\) 0 0
\(47\) −20388.6 −1.34630 −0.673152 0.739505i \(-0.735060\pi\)
−0.673152 + 0.739505i \(0.735060\pi\)
\(48\) 0 0
\(49\) 2401.00 0.142857
\(50\) 0 0
\(51\) −9258.15 −0.498424
\(52\) 0 0
\(53\) 2088.81 0.102143 0.0510717 0.998695i \(-0.483736\pi\)
0.0510717 + 0.998695i \(0.483736\pi\)
\(54\) 0 0
\(55\) −17840.8 −0.795255
\(56\) 0 0
\(57\) 104.462 0.00425865
\(58\) 0 0
\(59\) 45077.4 1.68589 0.842943 0.538002i \(-0.180821\pi\)
0.842943 + 0.538002i \(0.180821\pi\)
\(60\) 0 0
\(61\) 21786.2 0.749649 0.374824 0.927096i \(-0.377703\pi\)
0.374824 + 0.927096i \(0.377703\pi\)
\(62\) 0 0
\(63\) 10399.3 0.330107
\(64\) 0 0
\(65\) 3934.75 0.115514
\(66\) 0 0
\(67\) 64262.6 1.74893 0.874463 0.485093i \(-0.161214\pi\)
0.874463 + 0.485093i \(0.161214\pi\)
\(68\) 0 0
\(69\) 8207.92 0.207544
\(70\) 0 0
\(71\) 16333.8 0.384541 0.192270 0.981342i \(-0.438415\pi\)
0.192270 + 0.981342i \(0.438415\pi\)
\(72\) 0 0
\(73\) 6540.80 0.143656 0.0718280 0.997417i \(-0.477117\pi\)
0.0718280 + 0.997417i \(0.477117\pi\)
\(74\) 0 0
\(75\) 3466.84 0.0711673
\(76\) 0 0
\(77\) −34967.9 −0.672113
\(78\) 0 0
\(79\) −84587.9 −1.52490 −0.762448 0.647049i \(-0.776003\pi\)
−0.762448 + 0.647049i \(0.776003\pi\)
\(80\) 0 0
\(81\) 37565.4 0.636174
\(82\) 0 0
\(83\) 28514.9 0.454336 0.227168 0.973856i \(-0.427053\pi\)
0.227168 + 0.973856i \(0.427053\pi\)
\(84\) 0 0
\(85\) 41726.4 0.626417
\(86\) 0 0
\(87\) 41880.2 0.593213
\(88\) 0 0
\(89\) −13227.8 −0.177016 −0.0885080 0.996075i \(-0.528210\pi\)
−0.0885080 + 0.996075i \(0.528210\pi\)
\(90\) 0 0
\(91\) 7712.11 0.0976269
\(92\) 0 0
\(93\) 34908.2 0.418524
\(94\) 0 0
\(95\) −470.810 −0.00535226
\(96\) 0 0
\(97\) 139857. 1.50923 0.754615 0.656167i \(-0.227823\pi\)
0.754615 + 0.656167i \(0.227823\pi\)
\(98\) 0 0
\(99\) −151455. −1.55308
\(100\) 0 0
\(101\) 24058.8 0.234677 0.117339 0.993092i \(-0.462564\pi\)
0.117339 + 0.993092i \(0.462564\pi\)
\(102\) 0 0
\(103\) 7378.93 0.0685331 0.0342666 0.999413i \(-0.489090\pi\)
0.0342666 + 0.999413i \(0.489090\pi\)
\(104\) 0 0
\(105\) 6795.00 0.0601473
\(106\) 0 0
\(107\) 32937.0 0.278115 0.139058 0.990284i \(-0.455593\pi\)
0.139058 + 0.990284i \(0.455593\pi\)
\(108\) 0 0
\(109\) 213922. 1.72460 0.862301 0.506395i \(-0.169022\pi\)
0.862301 + 0.506395i \(0.169022\pi\)
\(110\) 0 0
\(111\) −2173.93 −0.0167470
\(112\) 0 0
\(113\) 12294.4 0.0905753 0.0452876 0.998974i \(-0.485580\pi\)
0.0452876 + 0.998974i \(0.485580\pi\)
\(114\) 0 0
\(115\) −36993.0 −0.260840
\(116\) 0 0
\(117\) 33403.1 0.225591
\(118\) 0 0
\(119\) 81783.7 0.529419
\(120\) 0 0
\(121\) 348217. 2.16215
\(122\) 0 0
\(123\) −32876.2 −0.195938
\(124\) 0 0
\(125\) −15625.0 −0.0894427
\(126\) 0 0
\(127\) −19972.3 −0.109880 −0.0549401 0.998490i \(-0.517497\pi\)
−0.0549401 + 0.998490i \(0.517497\pi\)
\(128\) 0 0
\(129\) 111091. 0.587765
\(130\) 0 0
\(131\) 122192. 0.622108 0.311054 0.950392i \(-0.399318\pi\)
0.311054 + 0.950392i \(0.399318\pi\)
\(132\) 0 0
\(133\) −922.788 −0.00452348
\(134\) 0 0
\(135\) 63128.5 0.298120
\(136\) 0 0
\(137\) −147736. −0.672489 −0.336245 0.941775i \(-0.609157\pi\)
−0.336245 + 0.941775i \(0.609157\pi\)
\(138\) 0 0
\(139\) −97522.3 −0.428121 −0.214061 0.976820i \(-0.568669\pi\)
−0.214061 + 0.976820i \(0.568669\pi\)
\(140\) 0 0
\(141\) −113094. −0.479064
\(142\) 0 0
\(143\) −112318. −0.459314
\(144\) 0 0
\(145\) −188753. −0.745547
\(146\) 0 0
\(147\) 13318.2 0.0508338
\(148\) 0 0
\(149\) 106789. 0.394057 0.197029 0.980398i \(-0.436871\pi\)
0.197029 + 0.980398i \(0.436871\pi\)
\(150\) 0 0
\(151\) −374118. −1.33526 −0.667631 0.744492i \(-0.732692\pi\)
−0.667631 + 0.744492i \(0.732692\pi\)
\(152\) 0 0
\(153\) 354226. 1.22335
\(154\) 0 0
\(155\) −157331. −0.525999
\(156\) 0 0
\(157\) −385106. −1.24690 −0.623449 0.781864i \(-0.714269\pi\)
−0.623449 + 0.781864i \(0.714269\pi\)
\(158\) 0 0
\(159\) 11586.5 0.0363463
\(160\) 0 0
\(161\) −72506.3 −0.220450
\(162\) 0 0
\(163\) −349922. −1.03158 −0.515789 0.856716i \(-0.672501\pi\)
−0.515789 + 0.856716i \(0.672501\pi\)
\(164\) 0 0
\(165\) −98961.6 −0.282981
\(166\) 0 0
\(167\) 63417.5 0.175962 0.0879809 0.996122i \(-0.471959\pi\)
0.0879809 + 0.996122i \(0.471959\pi\)
\(168\) 0 0
\(169\) −346521. −0.933283
\(170\) 0 0
\(171\) −3996.83 −0.0104526
\(172\) 0 0
\(173\) 469308. 1.19218 0.596091 0.802917i \(-0.296720\pi\)
0.596091 + 0.802917i \(0.296720\pi\)
\(174\) 0 0
\(175\) −30625.0 −0.0755929
\(176\) 0 0
\(177\) 250041. 0.599900
\(178\) 0 0
\(179\) 480916. 1.12186 0.560928 0.827865i \(-0.310445\pi\)
0.560928 + 0.827865i \(0.310445\pi\)
\(180\) 0 0
\(181\) −467752. −1.06125 −0.530627 0.847606i \(-0.678043\pi\)
−0.530627 + 0.847606i \(0.678043\pi\)
\(182\) 0 0
\(183\) 120847. 0.266752
\(184\) 0 0
\(185\) 9797.88 0.0210476
\(186\) 0 0
\(187\) −1.19109e6 −2.49081
\(188\) 0 0
\(189\) 123732. 0.251958
\(190\) 0 0
\(191\) −523673. −1.03867 −0.519334 0.854572i \(-0.673820\pi\)
−0.519334 + 0.854572i \(0.673820\pi\)
\(192\) 0 0
\(193\) 663432. 1.28204 0.641022 0.767522i \(-0.278511\pi\)
0.641022 + 0.767522i \(0.278511\pi\)
\(194\) 0 0
\(195\) 21825.8 0.0411040
\(196\) 0 0
\(197\) 102398. 0.187987 0.0939933 0.995573i \(-0.470037\pi\)
0.0939933 + 0.995573i \(0.470037\pi\)
\(198\) 0 0
\(199\) −208690. −0.373567 −0.186784 0.982401i \(-0.559806\pi\)
−0.186784 + 0.982401i \(0.559806\pi\)
\(200\) 0 0
\(201\) 356461. 0.622331
\(202\) 0 0
\(203\) −369957. −0.630102
\(204\) 0 0
\(205\) 148173. 0.246254
\(206\) 0 0
\(207\) −314043. −0.509405
\(208\) 0 0
\(209\) 13439.4 0.0212820
\(210\) 0 0
\(211\) −238134. −0.368227 −0.184114 0.982905i \(-0.558941\pi\)
−0.184114 + 0.982905i \(0.558941\pi\)
\(212\) 0 0
\(213\) 90602.8 0.136834
\(214\) 0 0
\(215\) −500685. −0.738701
\(216\) 0 0
\(217\) −308369. −0.444551
\(218\) 0 0
\(219\) 36281.4 0.0511180
\(220\) 0 0
\(221\) 262692. 0.361799
\(222\) 0 0
\(223\) −577920. −0.778225 −0.389113 0.921190i \(-0.627218\pi\)
−0.389113 + 0.921190i \(0.627218\pi\)
\(224\) 0 0
\(225\) −132645. −0.174676
\(226\) 0 0
\(227\) −1.32560e6 −1.70745 −0.853723 0.520727i \(-0.825661\pi\)
−0.853723 + 0.520727i \(0.825661\pi\)
\(228\) 0 0
\(229\) −1.56401e6 −1.97084 −0.985420 0.170137i \(-0.945579\pi\)
−0.985420 + 0.170137i \(0.945579\pi\)
\(230\) 0 0
\(231\) −193965. −0.239162
\(232\) 0 0
\(233\) −1.35581e6 −1.63610 −0.818048 0.575150i \(-0.804944\pi\)
−0.818048 + 0.575150i \(0.804944\pi\)
\(234\) 0 0
\(235\) 509715. 0.602085
\(236\) 0 0
\(237\) −469204. −0.542614
\(238\) 0 0
\(239\) 389979. 0.441618 0.220809 0.975317i \(-0.429130\pi\)
0.220809 + 0.975317i \(0.429130\pi\)
\(240\) 0 0
\(241\) 952265. 1.05612 0.528062 0.849206i \(-0.322919\pi\)
0.528062 + 0.849206i \(0.322919\pi\)
\(242\) 0 0
\(243\) 821983. 0.892991
\(244\) 0 0
\(245\) −60025.0 −0.0638877
\(246\) 0 0
\(247\) −2964.03 −0.00309129
\(248\) 0 0
\(249\) 158171. 0.161669
\(250\) 0 0
\(251\) 1.58021e6 1.58318 0.791591 0.611052i \(-0.209253\pi\)
0.791591 + 0.611052i \(0.209253\pi\)
\(252\) 0 0
\(253\) 1.05597e6 1.03717
\(254\) 0 0
\(255\) 231454. 0.222902
\(256\) 0 0
\(257\) 1.52332e6 1.43866 0.719329 0.694669i \(-0.244449\pi\)
0.719329 + 0.694669i \(0.244449\pi\)
\(258\) 0 0
\(259\) 19203.8 0.0177885
\(260\) 0 0
\(261\) −1.60238e6 −1.45601
\(262\) 0 0
\(263\) −999652. −0.891168 −0.445584 0.895240i \(-0.647004\pi\)
−0.445584 + 0.895240i \(0.647004\pi\)
\(264\) 0 0
\(265\) −52220.3 −0.0456799
\(266\) 0 0
\(267\) −73373.8 −0.0629887
\(268\) 0 0
\(269\) 1.67825e6 1.41408 0.707041 0.707172i \(-0.250030\pi\)
0.707041 + 0.707172i \(0.250030\pi\)
\(270\) 0 0
\(271\) −89342.2 −0.0738981 −0.0369491 0.999317i \(-0.511764\pi\)
−0.0369491 + 0.999317i \(0.511764\pi\)
\(272\) 0 0
\(273\) 42778.6 0.0347392
\(274\) 0 0
\(275\) 446019. 0.355649
\(276\) 0 0
\(277\) 15650.8 0.0122556 0.00612782 0.999981i \(-0.498049\pi\)
0.00612782 + 0.999981i \(0.498049\pi\)
\(278\) 0 0
\(279\) −1.33562e6 −1.02724
\(280\) 0 0
\(281\) −1.03527e6 −0.782145 −0.391072 0.920360i \(-0.627896\pi\)
−0.391072 + 0.920360i \(0.627896\pi\)
\(282\) 0 0
\(283\) 21123.9 0.0156786 0.00783932 0.999969i \(-0.497505\pi\)
0.00783932 + 0.999969i \(0.497505\pi\)
\(284\) 0 0
\(285\) −2611.56 −0.00190453
\(286\) 0 0
\(287\) 290418. 0.208123
\(288\) 0 0
\(289\) 1.36589e6 0.961991
\(290\) 0 0
\(291\) 775780. 0.537039
\(292\) 0 0
\(293\) −1.05832e6 −0.720193 −0.360097 0.932915i \(-0.617256\pi\)
−0.360097 + 0.932915i \(0.617256\pi\)
\(294\) 0 0
\(295\) −1.12693e6 −0.753952
\(296\) 0 0
\(297\) −1.80202e6 −1.18541
\(298\) 0 0
\(299\) −232893. −0.150653
\(300\) 0 0
\(301\) −981343. −0.624316
\(302\) 0 0
\(303\) 133453. 0.0835068
\(304\) 0 0
\(305\) −544656. −0.335253
\(306\) 0 0
\(307\) −921205. −0.557841 −0.278920 0.960314i \(-0.589977\pi\)
−0.278920 + 0.960314i \(0.589977\pi\)
\(308\) 0 0
\(309\) 40930.5 0.0243866
\(310\) 0 0
\(311\) −436095. −0.255670 −0.127835 0.991795i \(-0.540803\pi\)
−0.127835 + 0.991795i \(0.540803\pi\)
\(312\) 0 0
\(313\) 3.12004e6 1.80011 0.900057 0.435773i \(-0.143525\pi\)
0.900057 + 0.435773i \(0.143525\pi\)
\(314\) 0 0
\(315\) −259984. −0.147628
\(316\) 0 0
\(317\) −2.40017e6 −1.34151 −0.670756 0.741678i \(-0.734030\pi\)
−0.670756 + 0.741678i \(0.734030\pi\)
\(318\) 0 0
\(319\) 5.38801e6 2.96450
\(320\) 0 0
\(321\) 182700. 0.0989636
\(322\) 0 0
\(323\) −31432.3 −0.0167637
\(324\) 0 0
\(325\) −98368.7 −0.0516593
\(326\) 0 0
\(327\) 1.18661e6 0.613676
\(328\) 0 0
\(329\) 999042. 0.508855
\(330\) 0 0
\(331\) 2.50079e6 1.25461 0.627303 0.778776i \(-0.284159\pi\)
0.627303 + 0.778776i \(0.284159\pi\)
\(332\) 0 0
\(333\) 83176.7 0.0411047
\(334\) 0 0
\(335\) −1.60656e6 −0.782143
\(336\) 0 0
\(337\) −2.20634e6 −1.05828 −0.529138 0.848536i \(-0.677484\pi\)
−0.529138 + 0.848536i \(0.677484\pi\)
\(338\) 0 0
\(339\) 68196.1 0.0322300
\(340\) 0 0
\(341\) 4.49105e6 2.09152
\(342\) 0 0
\(343\) −117649. −0.0539949
\(344\) 0 0
\(345\) −205198. −0.0928165
\(346\) 0 0
\(347\) 2.07193e6 0.923745 0.461872 0.886946i \(-0.347178\pi\)
0.461872 + 0.886946i \(0.347178\pi\)
\(348\) 0 0
\(349\) 2.88623e6 1.26843 0.634215 0.773157i \(-0.281323\pi\)
0.634215 + 0.773157i \(0.281323\pi\)
\(350\) 0 0
\(351\) 397432. 0.172185
\(352\) 0 0
\(353\) −2.32386e6 −0.992598 −0.496299 0.868152i \(-0.665308\pi\)
−0.496299 + 0.868152i \(0.665308\pi\)
\(354\) 0 0
\(355\) −408346. −0.171972
\(356\) 0 0
\(357\) 453649. 0.188387
\(358\) 0 0
\(359\) 4.55224e6 1.86419 0.932093 0.362219i \(-0.117981\pi\)
0.932093 + 0.362219i \(0.117981\pi\)
\(360\) 0 0
\(361\) −2.47574e6 −0.999857
\(362\) 0 0
\(363\) 1.93154e6 0.769373
\(364\) 0 0
\(365\) −163520. −0.0642449
\(366\) 0 0
\(367\) −2.51918e6 −0.976324 −0.488162 0.872753i \(-0.662332\pi\)
−0.488162 + 0.872753i \(0.662332\pi\)
\(368\) 0 0
\(369\) 1.25788e6 0.480919
\(370\) 0 0
\(371\) −102352. −0.0386065
\(372\) 0 0
\(373\) 2.35441e6 0.876213 0.438107 0.898923i \(-0.355649\pi\)
0.438107 + 0.898923i \(0.355649\pi\)
\(374\) 0 0
\(375\) −86671.0 −0.0318270
\(376\) 0 0
\(377\) −1.18832e6 −0.430604
\(378\) 0 0
\(379\) 1.74826e6 0.625185 0.312593 0.949887i \(-0.398802\pi\)
0.312593 + 0.949887i \(0.398802\pi\)
\(380\) 0 0
\(381\) −110785. −0.0390994
\(382\) 0 0
\(383\) 2.57306e6 0.896300 0.448150 0.893958i \(-0.352083\pi\)
0.448150 + 0.893958i \(0.352083\pi\)
\(384\) 0 0
\(385\) 874197. 0.300578
\(386\) 0 0
\(387\) −4.25045e6 −1.44264
\(388\) 0 0
\(389\) 2.22324e6 0.744924 0.372462 0.928047i \(-0.378514\pi\)
0.372462 + 0.928047i \(0.378514\pi\)
\(390\) 0 0
\(391\) −2.46973e6 −0.816974
\(392\) 0 0
\(393\) 677794. 0.221369
\(394\) 0 0
\(395\) 2.11470e6 0.681955
\(396\) 0 0
\(397\) −4.95025e6 −1.57635 −0.788173 0.615454i \(-0.788973\pi\)
−0.788173 + 0.615454i \(0.788973\pi\)
\(398\) 0 0
\(399\) −5118.65 −0.00160962
\(400\) 0 0
\(401\) 5.91241e6 1.83613 0.918066 0.396428i \(-0.129750\pi\)
0.918066 + 0.396428i \(0.129750\pi\)
\(402\) 0 0
\(403\) −990493. −0.303801
\(404\) 0 0
\(405\) −939136. −0.284506
\(406\) 0 0
\(407\) −279683. −0.0836911
\(408\) 0 0
\(409\) −3.28867e6 −0.972101 −0.486050 0.873931i \(-0.661563\pi\)
−0.486050 + 0.873931i \(0.661563\pi\)
\(410\) 0 0
\(411\) −819484. −0.239296
\(412\) 0 0
\(413\) −2.20879e6 −0.637205
\(414\) 0 0
\(415\) −712873. −0.203185
\(416\) 0 0
\(417\) −540950. −0.152341
\(418\) 0 0
\(419\) 917263. 0.255246 0.127623 0.991823i \(-0.459265\pi\)
0.127623 + 0.991823i \(0.459265\pi\)
\(420\) 0 0
\(421\) −2.00150e6 −0.550365 −0.275183 0.961392i \(-0.588738\pi\)
−0.275183 + 0.961392i \(0.588738\pi\)
\(422\) 0 0
\(423\) 4.32710e6 1.17583
\(424\) 0 0
\(425\) −1.04316e6 −0.280142
\(426\) 0 0
\(427\) −1.06753e6 −0.283341
\(428\) 0 0
\(429\) −623022. −0.163441
\(430\) 0 0
\(431\) 4.22623e6 1.09587 0.547936 0.836520i \(-0.315414\pi\)
0.547936 + 0.836520i \(0.315414\pi\)
\(432\) 0 0
\(433\) 5.45557e6 1.39836 0.699182 0.714943i \(-0.253548\pi\)
0.699182 + 0.714943i \(0.253548\pi\)
\(434\) 0 0
\(435\) −1.04700e6 −0.265293
\(436\) 0 0
\(437\) 27866.7 0.00698042
\(438\) 0 0
\(439\) 4.43517e6 1.09837 0.549185 0.835701i \(-0.314938\pi\)
0.549185 + 0.835701i \(0.314938\pi\)
\(440\) 0 0
\(441\) −509568. −0.124769
\(442\) 0 0
\(443\) 2.18645e6 0.529335 0.264667 0.964340i \(-0.414738\pi\)
0.264667 + 0.964340i \(0.414738\pi\)
\(444\) 0 0
\(445\) 330695. 0.0791640
\(446\) 0 0
\(447\) 592350. 0.140220
\(448\) 0 0
\(449\) −1.47302e6 −0.344820 −0.172410 0.985025i \(-0.555155\pi\)
−0.172410 + 0.985025i \(0.555155\pi\)
\(450\) 0 0
\(451\) −4.22962e6 −0.979174
\(452\) 0 0
\(453\) −2.07521e6 −0.475135
\(454\) 0 0
\(455\) −192803. −0.0436601
\(456\) 0 0
\(457\) 3.58032e6 0.801920 0.400960 0.916096i \(-0.368677\pi\)
0.400960 + 0.916096i \(0.368677\pi\)
\(458\) 0 0
\(459\) 4.21460e6 0.933738
\(460\) 0 0
\(461\) −7.45341e6 −1.63344 −0.816719 0.577035i \(-0.804210\pi\)
−0.816719 + 0.577035i \(0.804210\pi\)
\(462\) 0 0
\(463\) −819161. −0.177589 −0.0887947 0.996050i \(-0.528301\pi\)
−0.0887947 + 0.996050i \(0.528301\pi\)
\(464\) 0 0
\(465\) −872706. −0.187170
\(466\) 0 0
\(467\) 571988. 0.121365 0.0606827 0.998157i \(-0.480672\pi\)
0.0606827 + 0.998157i \(0.480672\pi\)
\(468\) 0 0
\(469\) −3.14887e6 −0.661032
\(470\) 0 0
\(471\) −2.13616e6 −0.443692
\(472\) 0 0
\(473\) 1.42922e7 2.93728
\(474\) 0 0
\(475\) 11770.3 0.00239360
\(476\) 0 0
\(477\) −443312. −0.0892100
\(478\) 0 0
\(479\) 1.73825e6 0.346157 0.173079 0.984908i \(-0.444629\pi\)
0.173079 + 0.984908i \(0.444629\pi\)
\(480\) 0 0
\(481\) 61683.5 0.0121564
\(482\) 0 0
\(483\) −402188. −0.0784443
\(484\) 0 0
\(485\) −3.49643e6 −0.674948
\(486\) 0 0
\(487\) 5.79360e6 1.10694 0.553472 0.832867i \(-0.313302\pi\)
0.553472 + 0.832867i \(0.313302\pi\)
\(488\) 0 0
\(489\) −1.94099e6 −0.367073
\(490\) 0 0
\(491\) 6.43388e6 1.20440 0.602198 0.798347i \(-0.294292\pi\)
0.602198 + 0.798347i \(0.294292\pi\)
\(492\) 0 0
\(493\) −1.26016e7 −2.33512
\(494\) 0 0
\(495\) 3.78637e6 0.694560
\(496\) 0 0
\(497\) −800358. −0.145343
\(498\) 0 0
\(499\) 8.28825e6 1.49009 0.745043 0.667017i \(-0.232429\pi\)
0.745043 + 0.667017i \(0.232429\pi\)
\(500\) 0 0
\(501\) 351773. 0.0626136
\(502\) 0 0
\(503\) −1.00504e7 −1.77118 −0.885590 0.464468i \(-0.846245\pi\)
−0.885590 + 0.464468i \(0.846245\pi\)
\(504\) 0 0
\(505\) −601471. −0.104951
\(506\) 0 0
\(507\) −1.92213e6 −0.332096
\(508\) 0 0
\(509\) −2.73931e6 −0.468648 −0.234324 0.972159i \(-0.575288\pi\)
−0.234324 + 0.972159i \(0.575288\pi\)
\(510\) 0 0
\(511\) −320499. −0.0542969
\(512\) 0 0
\(513\) −47554.5 −0.00797808
\(514\) 0 0
\(515\) −184473. −0.0306489
\(516\) 0 0
\(517\) −1.45499e7 −2.39406
\(518\) 0 0
\(519\) 2.60322e6 0.424222
\(520\) 0 0
\(521\) 4.69300e6 0.757454 0.378727 0.925508i \(-0.376362\pi\)
0.378727 + 0.925508i \(0.376362\pi\)
\(522\) 0 0
\(523\) −7.92425e6 −1.26679 −0.633394 0.773829i \(-0.718339\pi\)
−0.633394 + 0.773829i \(0.718339\pi\)
\(524\) 0 0
\(525\) −169875. −0.0268987
\(526\) 0 0
\(527\) −1.05038e7 −1.64747
\(528\) 0 0
\(529\) −4.24677e6 −0.659812
\(530\) 0 0
\(531\) −9.56683e6 −1.47242
\(532\) 0 0
\(533\) 932835. 0.142229
\(534\) 0 0
\(535\) −823426. −0.124377
\(536\) 0 0
\(537\) 2.66761e6 0.399197
\(538\) 0 0
\(539\) 1.71343e6 0.254035
\(540\) 0 0
\(541\) 6.60043e6 0.969569 0.484785 0.874634i \(-0.338898\pi\)
0.484785 + 0.874634i \(0.338898\pi\)
\(542\) 0 0
\(543\) −2.59459e6 −0.377633
\(544\) 0 0
\(545\) −5.34805e6 −0.771266
\(546\) 0 0
\(547\) −3.74178e6 −0.534700 −0.267350 0.963600i \(-0.586148\pi\)
−0.267350 + 0.963600i \(0.586148\pi\)
\(548\) 0 0
\(549\) −4.62373e6 −0.654729
\(550\) 0 0
\(551\) 142187. 0.0199518
\(552\) 0 0
\(553\) 4.14481e6 0.576357
\(554\) 0 0
\(555\) 54348.3 0.00748951
\(556\) 0 0
\(557\) −1.38624e6 −0.189322 −0.0946608 0.995510i \(-0.530177\pi\)
−0.0946608 + 0.995510i \(0.530177\pi\)
\(558\) 0 0
\(559\) −3.15211e6 −0.426650
\(560\) 0 0
\(561\) −6.60690e6 −0.886320
\(562\) 0 0
\(563\) 1.05387e7 1.40125 0.700624 0.713531i \(-0.252905\pi\)
0.700624 + 0.713531i \(0.252905\pi\)
\(564\) 0 0
\(565\) −307359. −0.0405065
\(566\) 0 0
\(567\) −1.84071e6 −0.240451
\(568\) 0 0
\(569\) −1.89568e6 −0.245462 −0.122731 0.992440i \(-0.539165\pi\)
−0.122731 + 0.992440i \(0.539165\pi\)
\(570\) 0 0
\(571\) 1.77283e6 0.227550 0.113775 0.993507i \(-0.463706\pi\)
0.113775 + 0.993507i \(0.463706\pi\)
\(572\) 0 0
\(573\) −2.90478e6 −0.369596
\(574\) 0 0
\(575\) 924825. 0.116651
\(576\) 0 0
\(577\) −9.91545e6 −1.23986 −0.619930 0.784657i \(-0.712839\pi\)
−0.619930 + 0.784657i \(0.712839\pi\)
\(578\) 0 0
\(579\) 3.68002e6 0.456198
\(580\) 0 0
\(581\) −1.39723e6 −0.171723
\(582\) 0 0
\(583\) 1.49064e6 0.181636
\(584\) 0 0
\(585\) −835077. −0.100887
\(586\) 0 0
\(587\) −7.50312e6 −0.898766 −0.449383 0.893339i \(-0.648356\pi\)
−0.449383 + 0.893339i \(0.648356\pi\)
\(588\) 0 0
\(589\) 118517. 0.0140764
\(590\) 0 0
\(591\) 567997. 0.0668924
\(592\) 0 0
\(593\) 1.33626e7 1.56047 0.780235 0.625487i \(-0.215100\pi\)
0.780235 + 0.625487i \(0.215100\pi\)
\(594\) 0 0
\(595\) −2.04459e6 −0.236763
\(596\) 0 0
\(597\) −1.15759e6 −0.132929
\(598\) 0 0
\(599\) 4.67515e6 0.532388 0.266194 0.963919i \(-0.414234\pi\)
0.266194 + 0.963919i \(0.414234\pi\)
\(600\) 0 0
\(601\) 1.29686e7 1.46456 0.732279 0.681005i \(-0.238457\pi\)
0.732279 + 0.681005i \(0.238457\pi\)
\(602\) 0 0
\(603\) −1.36385e7 −1.52748
\(604\) 0 0
\(605\) −8.70542e6 −0.966944
\(606\) 0 0
\(607\) −1.12171e7 −1.23569 −0.617845 0.786300i \(-0.711994\pi\)
−0.617845 + 0.786300i \(0.711994\pi\)
\(608\) 0 0
\(609\) −2.05213e6 −0.224213
\(610\) 0 0
\(611\) 3.20896e6 0.347745
\(612\) 0 0
\(613\) 1.09615e7 1.17820 0.589102 0.808059i \(-0.299482\pi\)
0.589102 + 0.808059i \(0.299482\pi\)
\(614\) 0 0
\(615\) 821905. 0.0876261
\(616\) 0 0
\(617\) −9.20607e6 −0.973557 −0.486779 0.873525i \(-0.661828\pi\)
−0.486779 + 0.873525i \(0.661828\pi\)
\(618\) 0 0
\(619\) −7.81664e6 −0.819962 −0.409981 0.912094i \(-0.634465\pi\)
−0.409981 + 0.912094i \(0.634465\pi\)
\(620\) 0 0
\(621\) −3.73650e6 −0.388809
\(622\) 0 0
\(623\) 648162. 0.0669058
\(624\) 0 0
\(625\) 390625. 0.0400000
\(626\) 0 0
\(627\) 74547.4 0.00757293
\(628\) 0 0
\(629\) 654128. 0.0659229
\(630\) 0 0
\(631\) −6.26789e6 −0.626683 −0.313341 0.949641i \(-0.601448\pi\)
−0.313341 + 0.949641i \(0.601448\pi\)
\(632\) 0 0
\(633\) −1.32092e6 −0.131029
\(634\) 0 0
\(635\) 499309. 0.0491400
\(636\) 0 0
\(637\) −377893. −0.0368995
\(638\) 0 0
\(639\) −3.46655e6 −0.335850
\(640\) 0 0
\(641\) 1.50806e7 1.44968 0.724840 0.688917i \(-0.241914\pi\)
0.724840 + 0.688917i \(0.241914\pi\)
\(642\) 0 0
\(643\) −1.33403e7 −1.27245 −0.636223 0.771506i \(-0.719504\pi\)
−0.636223 + 0.771506i \(0.719504\pi\)
\(644\) 0 0
\(645\) −2.77727e6 −0.262857
\(646\) 0 0
\(647\) −1.94521e6 −0.182686 −0.0913431 0.995819i \(-0.529116\pi\)
−0.0913431 + 0.995819i \(0.529116\pi\)
\(648\) 0 0
\(649\) 3.21686e7 2.99792
\(650\) 0 0
\(651\) −1.71050e6 −0.158187
\(652\) 0 0
\(653\) −2.04880e6 −0.188025 −0.0940125 0.995571i \(-0.529969\pi\)
−0.0940125 + 0.995571i \(0.529969\pi\)
\(654\) 0 0
\(655\) −3.05481e6 −0.278215
\(656\) 0 0
\(657\) −1.38816e6 −0.125466
\(658\) 0 0
\(659\) 7.45814e6 0.668986 0.334493 0.942398i \(-0.391435\pi\)
0.334493 + 0.942398i \(0.391435\pi\)
\(660\) 0 0
\(661\) 1.86599e6 0.166113 0.0830567 0.996545i \(-0.473532\pi\)
0.0830567 + 0.996545i \(0.473532\pi\)
\(662\) 0 0
\(663\) 1.45714e6 0.128741
\(664\) 0 0
\(665\) 23069.7 0.00202296
\(666\) 0 0
\(667\) 1.11721e7 0.972344
\(668\) 0 0
\(669\) −3.20569e6 −0.276921
\(670\) 0 0
\(671\) 1.55473e7 1.33306
\(672\) 0 0
\(673\) −7.06697e6 −0.601445 −0.300722 0.953712i \(-0.597228\pi\)
−0.300722 + 0.953712i \(0.597228\pi\)
\(674\) 0 0
\(675\) −1.57821e6 −0.133323
\(676\) 0 0
\(677\) 1.92223e7 1.61189 0.805943 0.591992i \(-0.201658\pi\)
0.805943 + 0.591992i \(0.201658\pi\)
\(678\) 0 0
\(679\) −6.85300e6 −0.570436
\(680\) 0 0
\(681\) −7.35301e6 −0.607572
\(682\) 0 0
\(683\) 2.16710e7 1.77757 0.888786 0.458322i \(-0.151549\pi\)
0.888786 + 0.458322i \(0.151549\pi\)
\(684\) 0 0
\(685\) 3.69340e6 0.300746
\(686\) 0 0
\(687\) −8.67549e6 −0.701297
\(688\) 0 0
\(689\) −328758. −0.0263833
\(690\) 0 0
\(691\) −1.29271e7 −1.02993 −0.514964 0.857212i \(-0.672195\pi\)
−0.514964 + 0.857212i \(0.672195\pi\)
\(692\) 0 0
\(693\) 7.42128e6 0.587011
\(694\) 0 0
\(695\) 2.43806e6 0.191462
\(696\) 0 0
\(697\) 9.89233e6 0.771288
\(698\) 0 0
\(699\) −7.52059e6 −0.582183
\(700\) 0 0
\(701\) −1.46000e7 −1.12217 −0.561083 0.827759i \(-0.689615\pi\)
−0.561083 + 0.827759i \(0.689615\pi\)
\(702\) 0 0
\(703\) −7380.71 −0.000563261 0
\(704\) 0 0
\(705\) 2.82736e6 0.214244
\(706\) 0 0
\(707\) −1.17888e6 −0.0886997
\(708\) 0 0
\(709\) −1.34336e7 −1.00364 −0.501818 0.864973i \(-0.667335\pi\)
−0.501818 + 0.864973i \(0.667335\pi\)
\(710\) 0 0
\(711\) 1.79522e7 1.33182
\(712\) 0 0
\(713\) 9.31223e6 0.686009
\(714\) 0 0
\(715\) 2.80795e6 0.205412
\(716\) 0 0
\(717\) 2.16319e6 0.157144
\(718\) 0 0
\(719\) 1.10819e7 0.799454 0.399727 0.916634i \(-0.369105\pi\)
0.399727 + 0.916634i \(0.369105\pi\)
\(720\) 0 0
\(721\) −361568. −0.0259031
\(722\) 0 0
\(723\) 5.28216e6 0.375808
\(724\) 0 0
\(725\) 4.71884e6 0.333419
\(726\) 0 0
\(727\) 9.30383e6 0.652869 0.326434 0.945220i \(-0.394153\pi\)
0.326434 + 0.945220i \(0.394153\pi\)
\(728\) 0 0
\(729\) −4.56891e6 −0.318415
\(730\) 0 0
\(731\) −3.34269e7 −2.31367
\(732\) 0 0
\(733\) −4.49677e6 −0.309130 −0.154565 0.987983i \(-0.549398\pi\)
−0.154565 + 0.987983i \(0.549398\pi\)
\(734\) 0 0
\(735\) −332955. −0.0227336
\(736\) 0 0
\(737\) 4.58597e7 3.11002
\(738\) 0 0
\(739\) −1.04581e7 −0.704434 −0.352217 0.935918i \(-0.614572\pi\)
−0.352217 + 0.935918i \(0.614572\pi\)
\(740\) 0 0
\(741\) −16441.3 −0.00110000
\(742\) 0 0
\(743\) −285721. −0.0189876 −0.00949379 0.999955i \(-0.503022\pi\)
−0.00949379 + 0.999955i \(0.503022\pi\)
\(744\) 0 0
\(745\) −2.66971e6 −0.176228
\(746\) 0 0
\(747\) −6.05176e6 −0.396808
\(748\) 0 0
\(749\) −1.61391e6 −0.105118
\(750\) 0 0
\(751\) −1.76200e7 −1.14000 −0.570002 0.821643i \(-0.693058\pi\)
−0.570002 + 0.821643i \(0.693058\pi\)
\(752\) 0 0
\(753\) 8.76533e6 0.563354
\(754\) 0 0
\(755\) 9.35296e6 0.597147
\(756\) 0 0
\(757\) 2.63145e6 0.166899 0.0834497 0.996512i \(-0.473406\pi\)
0.0834497 + 0.996512i \(0.473406\pi\)
\(758\) 0 0
\(759\) 5.85742e6 0.369064
\(760\) 0 0
\(761\) 1.25303e7 0.784332 0.392166 0.919894i \(-0.371726\pi\)
0.392166 + 0.919894i \(0.371726\pi\)
\(762\) 0 0
\(763\) −1.04822e7 −0.651839
\(764\) 0 0
\(765\) −8.85565e6 −0.547100
\(766\) 0 0
\(767\) −7.09472e6 −0.435459
\(768\) 0 0
\(769\) −1.44887e7 −0.883515 −0.441757 0.897135i \(-0.645645\pi\)
−0.441757 + 0.897135i \(0.645645\pi\)
\(770\) 0 0
\(771\) 8.44976e6 0.511927
\(772\) 0 0
\(773\) 2.08149e7 1.25292 0.626462 0.779452i \(-0.284502\pi\)
0.626462 + 0.779452i \(0.284502\pi\)
\(774\) 0 0
\(775\) 3.93328e6 0.235234
\(776\) 0 0
\(777\) 106523. 0.00632979
\(778\) 0 0
\(779\) −111618. −0.00659007
\(780\) 0 0
\(781\) 1.16563e7 0.683807
\(782\) 0 0
\(783\) −1.90652e7 −1.11131
\(784\) 0 0
\(785\) 9.62765e6 0.557630
\(786\) 0 0
\(787\) −1.92751e6 −0.110933 −0.0554665 0.998461i \(-0.517665\pi\)
−0.0554665 + 0.998461i \(0.517665\pi\)
\(788\) 0 0
\(789\) −5.54501e6 −0.317110
\(790\) 0 0
\(791\) −602423. −0.0342342
\(792\) 0 0
\(793\) −3.42894e6 −0.193632
\(794\) 0 0
\(795\) −289663. −0.0162546
\(796\) 0 0
\(797\) 1.00107e7 0.558239 0.279120 0.960256i \(-0.409957\pi\)
0.279120 + 0.960256i \(0.409957\pi\)
\(798\) 0 0
\(799\) 3.40297e7 1.88578
\(800\) 0 0
\(801\) 2.80735e6 0.154602
\(802\) 0 0
\(803\) 4.66771e6 0.255455
\(804\) 0 0
\(805\) 1.81266e6 0.0985884
\(806\) 0 0
\(807\) 9.30913e6 0.503182
\(808\) 0 0
\(809\) 2.60552e7 1.39966 0.699832 0.714308i \(-0.253258\pi\)
0.699832 + 0.714308i \(0.253258\pi\)
\(810\) 0 0
\(811\) −2.84980e7 −1.52147 −0.760733 0.649064i \(-0.775161\pi\)
−0.760733 + 0.649064i \(0.775161\pi\)
\(812\) 0 0
\(813\) −495576. −0.0262956
\(814\) 0 0
\(815\) 8.74804e6 0.461335
\(816\) 0 0
\(817\) 377164. 0.0197686
\(818\) 0 0
\(819\) −1.63675e6 −0.0852654
\(820\) 0 0
\(821\) 2.20586e7 1.14214 0.571071 0.820901i \(-0.306528\pi\)
0.571071 + 0.820901i \(0.306528\pi\)
\(822\) 0 0
\(823\) 2.88032e6 0.148232 0.0741158 0.997250i \(-0.476387\pi\)
0.0741158 + 0.997250i \(0.476387\pi\)
\(824\) 0 0
\(825\) 2.47404e6 0.126553
\(826\) 0 0
\(827\) 2.90804e7 1.47855 0.739275 0.673403i \(-0.235168\pi\)
0.739275 + 0.673403i \(0.235168\pi\)
\(828\) 0 0
\(829\) −3.12006e7 −1.57680 −0.788399 0.615164i \(-0.789090\pi\)
−0.788399 + 0.615164i \(0.789090\pi\)
\(830\) 0 0
\(831\) 86813.8 0.00436100
\(832\) 0 0
\(833\) −4.00740e6 −0.200102
\(834\) 0 0
\(835\) −1.58544e6 −0.0786925
\(836\) 0 0
\(837\) −1.58913e7 −0.784055
\(838\) 0 0
\(839\) −3.36267e6 −0.164922 −0.0824612 0.996594i \(-0.526278\pi\)
−0.0824612 + 0.996594i \(0.526278\pi\)
\(840\) 0 0
\(841\) 3.64934e7 1.77920
\(842\) 0 0
\(843\) −5.74257e6 −0.278316
\(844\) 0 0
\(845\) 8.66304e6 0.417377
\(846\) 0 0
\(847\) −1.70626e7 −0.817217
\(848\) 0 0
\(849\) 117173. 0.00557903
\(850\) 0 0
\(851\) −579925. −0.0274503
\(852\) 0 0
\(853\) 1.95962e7 0.922144 0.461072 0.887363i \(-0.347465\pi\)
0.461072 + 0.887363i \(0.347465\pi\)
\(854\) 0 0
\(855\) 99920.7 0.00467456
\(856\) 0 0
\(857\) 9.56996e6 0.445101 0.222550 0.974921i \(-0.428562\pi\)
0.222550 + 0.974921i \(0.428562\pi\)
\(858\) 0 0
\(859\) 2.49323e7 1.15287 0.576435 0.817143i \(-0.304443\pi\)
0.576435 + 0.817143i \(0.304443\pi\)
\(860\) 0 0
\(861\) 1.61093e6 0.0740576
\(862\) 0 0
\(863\) −1.31911e7 −0.602913 −0.301457 0.953480i \(-0.597473\pi\)
−0.301457 + 0.953480i \(0.597473\pi\)
\(864\) 0 0
\(865\) −1.17327e7 −0.533160
\(866\) 0 0
\(867\) 7.57651e6 0.342311
\(868\) 0 0
\(869\) −6.03645e7 −2.71164
\(870\) 0 0
\(871\) −1.01143e7 −0.451741
\(872\) 0 0
\(873\) −2.96821e7 −1.31813
\(874\) 0 0
\(875\) 765625. 0.0338062
\(876\) 0 0
\(877\) −8.92706e6 −0.391931 −0.195965 0.980611i \(-0.562784\pi\)
−0.195965 + 0.980611i \(0.562784\pi\)
\(878\) 0 0
\(879\) −5.87045e6 −0.256271
\(880\) 0 0
\(881\) 6.59013e6 0.286058 0.143029 0.989718i \(-0.454316\pi\)
0.143029 + 0.989718i \(0.454316\pi\)
\(882\) 0 0
\(883\) −2.43558e7 −1.05124 −0.525618 0.850721i \(-0.676166\pi\)
−0.525618 + 0.850721i \(0.676166\pi\)
\(884\) 0 0
\(885\) −6.25104e6 −0.268283
\(886\) 0 0
\(887\) −6.49008e6 −0.276975 −0.138488 0.990364i \(-0.544224\pi\)
−0.138488 + 0.990364i \(0.544224\pi\)
\(888\) 0 0
\(889\) 978645. 0.0415308
\(890\) 0 0
\(891\) 2.68078e7 1.13127
\(892\) 0 0
\(893\) −383967. −0.0161126
\(894\) 0 0
\(895\) −1.20229e7 −0.501709
\(896\) 0 0
\(897\) −1.29184e6 −0.0536079
\(898\) 0 0
\(899\) 4.75149e7 1.96079
\(900\) 0 0
\(901\) −3.48635e6 −0.143073
\(902\) 0 0
\(903\) −5.44345e6 −0.222154
\(904\) 0 0
\(905\) 1.16938e7 0.474607
\(906\) 0 0
\(907\) 4.63222e7 1.86970 0.934848 0.355049i \(-0.115536\pi\)
0.934848 + 0.355049i \(0.115536\pi\)
\(908\) 0 0
\(909\) −5.10604e6 −0.204963
\(910\) 0 0
\(911\) 1.91803e7 0.765701 0.382850 0.923810i \(-0.374942\pi\)
0.382850 + 0.923810i \(0.374942\pi\)
\(912\) 0 0
\(913\) 2.03491e7 0.807920
\(914\) 0 0
\(915\) −3.02118e6 −0.119295
\(916\) 0 0
\(917\) −5.98742e6 −0.235135
\(918\) 0 0
\(919\) −1.93274e7 −0.754892 −0.377446 0.926032i \(-0.623198\pi\)
−0.377446 + 0.926032i \(0.623198\pi\)
\(920\) 0 0
\(921\) −5.10987e6 −0.198500
\(922\) 0 0
\(923\) −2.57078e6 −0.0993255
\(924\) 0 0
\(925\) −244947. −0.00941278
\(926\) 0 0
\(927\) −1.56604e6 −0.0598555
\(928\) 0 0
\(929\) −2.56658e7 −0.975697 −0.487848 0.872928i \(-0.662218\pi\)
−0.487848 + 0.872928i \(0.662218\pi\)
\(930\) 0 0
\(931\) 45216.6 0.00170972
\(932\) 0 0
\(933\) −2.41899e6 −0.0909767
\(934\) 0 0
\(935\) 2.97772e7 1.11392
\(936\) 0 0
\(937\) 6.72369e6 0.250184 0.125092 0.992145i \(-0.460077\pi\)
0.125092 + 0.992145i \(0.460077\pi\)
\(938\) 0 0
\(939\) 1.73067e7 0.640546
\(940\) 0 0
\(941\) −2.99755e7 −1.10355 −0.551775 0.833993i \(-0.686049\pi\)
−0.551775 + 0.833993i \(0.686049\pi\)
\(942\) 0 0
\(943\) −8.77016e6 −0.321165
\(944\) 0 0
\(945\) −3.09330e6 −0.112679
\(946\) 0 0
\(947\) −2.10799e7 −0.763823 −0.381912 0.924199i \(-0.624734\pi\)
−0.381912 + 0.924199i \(0.624734\pi\)
\(948\) 0 0
\(949\) −1.02946e6 −0.0371058
\(950\) 0 0
\(951\) −1.33136e7 −0.477359
\(952\) 0 0
\(953\) 231114. 0.00824318 0.00412159 0.999992i \(-0.498688\pi\)
0.00412159 + 0.999992i \(0.498688\pi\)
\(954\) 0 0
\(955\) 1.30918e7 0.464506
\(956\) 0 0
\(957\) 2.98870e7 1.05488
\(958\) 0 0
\(959\) 7.23907e6 0.254177
\(960\) 0 0
\(961\) 1.09757e7 0.383377
\(962\) 0 0
\(963\) −6.99027e6 −0.242901
\(964\) 0 0
\(965\) −1.65858e7 −0.573348
\(966\) 0 0
\(967\) −1.67022e7 −0.574391 −0.287195 0.957872i \(-0.592723\pi\)
−0.287195 + 0.957872i \(0.592723\pi\)
\(968\) 0 0
\(969\) −174353. −0.00596514
\(970\) 0 0
\(971\) −2.46767e7 −0.839924 −0.419962 0.907542i \(-0.637957\pi\)
−0.419962 + 0.907542i \(0.637957\pi\)
\(972\) 0 0
\(973\) 4.77859e6 0.161815
\(974\) 0 0
\(975\) −545645. −0.0183823
\(976\) 0 0
\(977\) 4.73438e6 0.158682 0.0793409 0.996848i \(-0.474718\pi\)
0.0793409 + 0.996848i \(0.474718\pi\)
\(978\) 0 0
\(979\) −9.43975e6 −0.314778
\(980\) 0 0
\(981\) −4.54010e7 −1.50623
\(982\) 0 0
\(983\) −3.25544e7 −1.07455 −0.537273 0.843408i \(-0.680546\pi\)
−0.537273 + 0.843408i \(0.680546\pi\)
\(984\) 0 0
\(985\) −2.55995e6 −0.0840701
\(986\) 0 0
\(987\) 5.54162e6 0.181069
\(988\) 0 0
\(989\) 2.96349e7 0.963415
\(990\) 0 0
\(991\) 4.80110e6 0.155295 0.0776473 0.996981i \(-0.475259\pi\)
0.0776473 + 0.996981i \(0.475259\pi\)
\(992\) 0 0
\(993\) 1.38717e7 0.446434
\(994\) 0 0
\(995\) 5.21725e6 0.167064
\(996\) 0 0
\(997\) −4.58078e6 −0.145949 −0.0729745 0.997334i \(-0.523249\pi\)
−0.0729745 + 0.997334i \(0.523249\pi\)
\(998\) 0 0
\(999\) 989642. 0.0313736
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 280.6.a.i.1.4 5
4.3 odd 2 560.6.a.z.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.6.a.i.1.4 5 1.1 even 1 trivial
560.6.a.z.1.2 5 4.3 odd 2