Properties

Label 414.4.a.g.1.2
Level $414$
Weight $4$
Character 414.1
Self dual yes
Analytic conductor $24.427$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [414,4,Mod(1,414)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(414, 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("414.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 414.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: \(24.4267907424\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 138)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 414.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +4.00000 q^{4} +7.31371 q^{5} +17.3137 q^{7} -8.00000 q^{8} -14.6274 q^{10} -69.2548 q^{11} -3.25483 q^{13} -34.6274 q^{14} +16.0000 q^{16} -85.9411 q^{17} -19.9411 q^{19} +29.2548 q^{20} +138.510 q^{22} -23.0000 q^{23} -71.5097 q^{25} +6.50967 q^{26} +69.2548 q^{28} +47.1371 q^{29} +109.137 q^{31} -32.0000 q^{32} +171.882 q^{34} +126.627 q^{35} +86.9016 q^{37} +39.8823 q^{38} -58.5097 q^{40} -65.7645 q^{41} -398.451 q^{43} -277.019 q^{44} +46.0000 q^{46} +164.118 q^{47} -43.2355 q^{49} +143.019 q^{50} -13.0193 q^{52} -631.862 q^{53} -506.510 q^{55} -138.510 q^{56} -94.2742 q^{58} +665.450 q^{59} -490.431 q^{61} -218.274 q^{62} +64.0000 q^{64} -23.8049 q^{65} +83.4701 q^{67} -343.765 q^{68} -253.255 q^{70} -969.568 q^{71} +462.313 q^{73} -173.803 q^{74} -79.7645 q^{76} -1199.06 q^{77} -857.314 q^{79} +117.019 q^{80} +131.529 q^{82} +1195.61 q^{83} -628.548 q^{85} +796.902 q^{86} +554.039 q^{88} +382.960 q^{89} -56.3532 q^{91} -92.0000 q^{92} -328.235 q^{94} -145.844 q^{95} -817.724 q^{97} +86.4710 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 8 q^{4} - 8 q^{5} + 12 q^{7} - 16 q^{8} + 16 q^{10} - 48 q^{11} + 84 q^{13} - 24 q^{14} + 32 q^{16} - 104 q^{17} + 28 q^{19} - 32 q^{20} + 96 q^{22} - 46 q^{23} + 38 q^{25} - 168 q^{26}+ \cdots + 716 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) 7.31371 0.654158 0.327079 0.944997i \(-0.393936\pi\)
0.327079 + 0.944997i \(0.393936\pi\)
\(6\) 0 0
\(7\) 17.3137 0.934852 0.467426 0.884032i \(-0.345181\pi\)
0.467426 + 0.884032i \(0.345181\pi\)
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) −14.6274 −0.462560
\(11\) −69.2548 −1.89828 −0.949142 0.314849i \(-0.898046\pi\)
−0.949142 + 0.314849i \(0.898046\pi\)
\(12\) 0 0
\(13\) −3.25483 −0.0694407 −0.0347203 0.999397i \(-0.511054\pi\)
−0.0347203 + 0.999397i \(0.511054\pi\)
\(14\) −34.6274 −0.661040
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −85.9411 −1.22610 −0.613052 0.790042i \(-0.710059\pi\)
−0.613052 + 0.790042i \(0.710059\pi\)
\(18\) 0 0
\(19\) −19.9411 −0.240779 −0.120390 0.992727i \(-0.538414\pi\)
−0.120390 + 0.992727i \(0.538414\pi\)
\(20\) 29.2548 0.327079
\(21\) 0 0
\(22\) 138.510 1.34229
\(23\) −23.0000 −0.208514
\(24\) 0 0
\(25\) −71.5097 −0.572077
\(26\) 6.50967 0.0491020
\(27\) 0 0
\(28\) 69.2548 0.467426
\(29\) 47.1371 0.301832 0.150916 0.988547i \(-0.451778\pi\)
0.150916 + 0.988547i \(0.451778\pi\)
\(30\) 0 0
\(31\) 109.137 0.632310 0.316155 0.948708i \(-0.397608\pi\)
0.316155 + 0.948708i \(0.397608\pi\)
\(32\) −32.0000 −0.176777
\(33\) 0 0
\(34\) 171.882 0.866987
\(35\) 126.627 0.611541
\(36\) 0 0
\(37\) 86.9016 0.386123 0.193061 0.981187i \(-0.438158\pi\)
0.193061 + 0.981187i \(0.438158\pi\)
\(38\) 39.8823 0.170257
\(39\) 0 0
\(40\) −58.5097 −0.231280
\(41\) −65.7645 −0.250505 −0.125252 0.992125i \(-0.539974\pi\)
−0.125252 + 0.992125i \(0.539974\pi\)
\(42\) 0 0
\(43\) −398.451 −1.41310 −0.706549 0.707665i \(-0.749749\pi\)
−0.706549 + 0.707665i \(0.749749\pi\)
\(44\) −277.019 −0.949142
\(45\) 0 0
\(46\) 46.0000 0.147442
\(47\) 164.118 0.509341 0.254671 0.967028i \(-0.418033\pi\)
0.254671 + 0.967028i \(0.418033\pi\)
\(48\) 0 0
\(49\) −43.2355 −0.126051
\(50\) 143.019 0.404520
\(51\) 0 0
\(52\) −13.0193 −0.0347203
\(53\) −631.862 −1.63760 −0.818801 0.574077i \(-0.805361\pi\)
−0.818801 + 0.574077i \(0.805361\pi\)
\(54\) 0 0
\(55\) −506.510 −1.24178
\(56\) −138.510 −0.330520
\(57\) 0 0
\(58\) −94.2742 −0.213428
\(59\) 665.450 1.46838 0.734188 0.678946i \(-0.237563\pi\)
0.734188 + 0.678946i \(0.237563\pi\)
\(60\) 0 0
\(61\) −490.431 −1.02940 −0.514698 0.857371i \(-0.672096\pi\)
−0.514698 + 0.857371i \(0.672096\pi\)
\(62\) −218.274 −0.447110
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −23.8049 −0.0454252
\(66\) 0 0
\(67\) 83.4701 0.152201 0.0761007 0.997100i \(-0.475753\pi\)
0.0761007 + 0.997100i \(0.475753\pi\)
\(68\) −343.765 −0.613052
\(69\) 0 0
\(70\) −253.255 −0.432425
\(71\) −969.568 −1.62066 −0.810328 0.585977i \(-0.800711\pi\)
−0.810328 + 0.585977i \(0.800711\pi\)
\(72\) 0 0
\(73\) 462.313 0.741228 0.370614 0.928787i \(-0.379147\pi\)
0.370614 + 0.928787i \(0.379147\pi\)
\(74\) −173.803 −0.273030
\(75\) 0 0
\(76\) −79.7645 −0.120390
\(77\) −1199.06 −1.77461
\(78\) 0 0
\(79\) −857.314 −1.22095 −0.610477 0.792034i \(-0.709022\pi\)
−0.610477 + 0.792034i \(0.709022\pi\)
\(80\) 117.019 0.163539
\(81\) 0 0
\(82\) 131.529 0.177134
\(83\) 1195.61 1.58114 0.790571 0.612370i \(-0.209784\pi\)
0.790571 + 0.612370i \(0.209784\pi\)
\(84\) 0 0
\(85\) −628.548 −0.802066
\(86\) 796.902 0.999211
\(87\) 0 0
\(88\) 554.039 0.671145
\(89\) 382.960 0.456109 0.228055 0.973648i \(-0.426764\pi\)
0.228055 + 0.973648i \(0.426764\pi\)
\(90\) 0 0
\(91\) −56.3532 −0.0649168
\(92\) −92.0000 −0.104257
\(93\) 0 0
\(94\) −328.235 −0.360159
\(95\) −145.844 −0.157508
\(96\) 0 0
\(97\) −817.724 −0.855952 −0.427976 0.903790i \(-0.640773\pi\)
−0.427976 + 0.903790i \(0.640773\pi\)
\(98\) 86.4710 0.0891315
\(99\) 0 0
\(100\) −286.039 −0.286039
\(101\) −530.548 −0.522688 −0.261344 0.965246i \(-0.584166\pi\)
−0.261344 + 0.965246i \(0.584166\pi\)
\(102\) 0 0
\(103\) −1427.12 −1.36522 −0.682612 0.730781i \(-0.739156\pi\)
−0.682612 + 0.730781i \(0.739156\pi\)
\(104\) 26.0387 0.0245510
\(105\) 0 0
\(106\) 1263.72 1.15796
\(107\) −1085.49 −0.980732 −0.490366 0.871517i \(-0.663137\pi\)
−0.490366 + 0.871517i \(0.663137\pi\)
\(108\) 0 0
\(109\) −100.432 −0.0882539 −0.0441269 0.999026i \(-0.514051\pi\)
−0.0441269 + 0.999026i \(0.514051\pi\)
\(110\) 1013.02 0.878069
\(111\) 0 0
\(112\) 277.019 0.233713
\(113\) −2289.43 −1.90594 −0.952971 0.303062i \(-0.901991\pi\)
−0.952971 + 0.303062i \(0.901991\pi\)
\(114\) 0 0
\(115\) −168.215 −0.136401
\(116\) 188.548 0.150916
\(117\) 0 0
\(118\) −1330.90 −1.03830
\(119\) −1487.96 −1.14623
\(120\) 0 0
\(121\) 3465.23 2.60348
\(122\) 980.861 0.727893
\(123\) 0 0
\(124\) 436.548 0.316155
\(125\) −1437.21 −1.02839
\(126\) 0 0
\(127\) 313.531 0.219066 0.109533 0.993983i \(-0.465065\pi\)
0.109533 + 0.993983i \(0.465065\pi\)
\(128\) −128.000 −0.0883883
\(129\) 0 0
\(130\) 47.6098 0.0321204
\(131\) 517.568 0.345192 0.172596 0.984993i \(-0.444785\pi\)
0.172596 + 0.984993i \(0.444785\pi\)
\(132\) 0 0
\(133\) −345.255 −0.225093
\(134\) −166.940 −0.107623
\(135\) 0 0
\(136\) 687.529 0.433494
\(137\) 435.038 0.271298 0.135649 0.990757i \(-0.456688\pi\)
0.135649 + 0.990757i \(0.456688\pi\)
\(138\) 0 0
\(139\) 124.626 0.0760476 0.0380238 0.999277i \(-0.487894\pi\)
0.0380238 + 0.999277i \(0.487894\pi\)
\(140\) 506.510 0.305771
\(141\) 0 0
\(142\) 1939.14 1.14598
\(143\) 225.413 0.131818
\(144\) 0 0
\(145\) 344.747 0.197446
\(146\) −924.626 −0.524127
\(147\) 0 0
\(148\) 347.606 0.193061
\(149\) 2790.29 1.53416 0.767079 0.641553i \(-0.221709\pi\)
0.767079 + 0.641553i \(0.221709\pi\)
\(150\) 0 0
\(151\) 2883.14 1.55382 0.776908 0.629614i \(-0.216787\pi\)
0.776908 + 0.629614i \(0.216787\pi\)
\(152\) 159.529 0.0851284
\(153\) 0 0
\(154\) 2398.12 1.25484
\(155\) 798.197 0.413630
\(156\) 0 0
\(157\) 1524.86 0.775142 0.387571 0.921840i \(-0.373314\pi\)
0.387571 + 0.921840i \(0.373314\pi\)
\(158\) 1714.63 0.863345
\(159\) 0 0
\(160\) −234.039 −0.115640
\(161\) −398.215 −0.194930
\(162\) 0 0
\(163\) −1796.74 −0.863385 −0.431693 0.902021i \(-0.642083\pi\)
−0.431693 + 0.902021i \(0.642083\pi\)
\(164\) −263.058 −0.125252
\(165\) 0 0
\(166\) −2391.21 −1.11804
\(167\) 1339.61 0.620730 0.310365 0.950618i \(-0.399549\pi\)
0.310365 + 0.950618i \(0.399549\pi\)
\(168\) 0 0
\(169\) −2186.41 −0.995178
\(170\) 1257.10 0.567147
\(171\) 0 0
\(172\) −1593.80 −0.706549
\(173\) 1753.65 0.770678 0.385339 0.922775i \(-0.374085\pi\)
0.385339 + 0.922775i \(0.374085\pi\)
\(174\) 0 0
\(175\) −1238.10 −0.534808
\(176\) −1108.08 −0.474571
\(177\) 0 0
\(178\) −765.921 −0.322518
\(179\) 4375.18 1.82690 0.913452 0.406945i \(-0.133406\pi\)
0.913452 + 0.406945i \(0.133406\pi\)
\(180\) 0 0
\(181\) −3619.02 −1.48619 −0.743093 0.669188i \(-0.766642\pi\)
−0.743093 + 0.669188i \(0.766642\pi\)
\(182\) 112.706 0.0459031
\(183\) 0 0
\(184\) 184.000 0.0737210
\(185\) 635.573 0.252585
\(186\) 0 0
\(187\) 5951.84 2.32749
\(188\) 656.471 0.254671
\(189\) 0 0
\(190\) 291.687 0.111375
\(191\) −537.369 −0.203574 −0.101787 0.994806i \(-0.532456\pi\)
−0.101787 + 0.994806i \(0.532456\pi\)
\(192\) 0 0
\(193\) −334.784 −0.124861 −0.0624307 0.998049i \(-0.519885\pi\)
−0.0624307 + 0.998049i \(0.519885\pi\)
\(194\) 1635.45 0.605249
\(195\) 0 0
\(196\) −172.942 −0.0630255
\(197\) 1674.19 0.605488 0.302744 0.953072i \(-0.402097\pi\)
0.302744 + 0.953072i \(0.402097\pi\)
\(198\) 0 0
\(199\) 767.231 0.273304 0.136652 0.990619i \(-0.456366\pi\)
0.136652 + 0.990619i \(0.456366\pi\)
\(200\) 572.077 0.202260
\(201\) 0 0
\(202\) 1061.10 0.369597
\(203\) 816.118 0.282169
\(204\) 0 0
\(205\) −480.982 −0.163870
\(206\) 2854.23 0.965359
\(207\) 0 0
\(208\) −52.0773 −0.0173602
\(209\) 1381.02 0.457067
\(210\) 0 0
\(211\) −4779.40 −1.55937 −0.779687 0.626170i \(-0.784622\pi\)
−0.779687 + 0.626170i \(0.784622\pi\)
\(212\) −2527.45 −0.818801
\(213\) 0 0
\(214\) 2170.98 0.693482
\(215\) −2914.15 −0.924389
\(216\) 0 0
\(217\) 1889.57 0.591116
\(218\) 200.865 0.0624049
\(219\) 0 0
\(220\) −2026.04 −0.620889
\(221\) 279.724 0.0851415
\(222\) 0 0
\(223\) −797.410 −0.239455 −0.119728 0.992807i \(-0.538202\pi\)
−0.119728 + 0.992807i \(0.538202\pi\)
\(224\) −554.039 −0.165260
\(225\) 0 0
\(226\) 4578.86 1.34770
\(227\) 5898.63 1.72469 0.862347 0.506317i \(-0.168994\pi\)
0.862347 + 0.506317i \(0.168994\pi\)
\(228\) 0 0
\(229\) −168.826 −0.0487176 −0.0243588 0.999703i \(-0.507754\pi\)
−0.0243588 + 0.999703i \(0.507754\pi\)
\(230\) 336.431 0.0964503
\(231\) 0 0
\(232\) −377.097 −0.106714
\(233\) −4184.67 −1.17660 −0.588298 0.808644i \(-0.700202\pi\)
−0.588298 + 0.808644i \(0.700202\pi\)
\(234\) 0 0
\(235\) 1200.31 0.333190
\(236\) 2661.80 0.734188
\(237\) 0 0
\(238\) 2975.92 0.810505
\(239\) −2573.96 −0.696635 −0.348317 0.937377i \(-0.613247\pi\)
−0.348317 + 0.937377i \(0.613247\pi\)
\(240\) 0 0
\(241\) −1092.40 −0.291981 −0.145990 0.989286i \(-0.546637\pi\)
−0.145990 + 0.989286i \(0.546637\pi\)
\(242\) −6930.46 −1.84094
\(243\) 0 0
\(244\) −1961.72 −0.514698
\(245\) −316.212 −0.0824573
\(246\) 0 0
\(247\) 64.9051 0.0167199
\(248\) −873.097 −0.223555
\(249\) 0 0
\(250\) 2874.43 0.727179
\(251\) 5504.11 1.38413 0.692065 0.721836i \(-0.256701\pi\)
0.692065 + 0.721836i \(0.256701\pi\)
\(252\) 0 0
\(253\) 1592.86 0.395819
\(254\) −627.061 −0.154903
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) −3654.39 −0.886984 −0.443492 0.896278i \(-0.646260\pi\)
−0.443492 + 0.896278i \(0.646260\pi\)
\(258\) 0 0
\(259\) 1504.59 0.360968
\(260\) −95.2196 −0.0227126
\(261\) 0 0
\(262\) −1035.14 −0.244087
\(263\) 5352.43 1.25492 0.627462 0.778648i \(-0.284094\pi\)
0.627462 + 0.778648i \(0.284094\pi\)
\(264\) 0 0
\(265\) −4621.25 −1.07125
\(266\) 690.510 0.159165
\(267\) 0 0
\(268\) 333.881 0.0761007
\(269\) −4423.21 −1.00256 −0.501279 0.865286i \(-0.667137\pi\)
−0.501279 + 0.865286i \(0.667137\pi\)
\(270\) 0 0
\(271\) −3179.72 −0.712747 −0.356374 0.934344i \(-0.615987\pi\)
−0.356374 + 0.934344i \(0.615987\pi\)
\(272\) −1375.06 −0.306526
\(273\) 0 0
\(274\) −870.076 −0.191836
\(275\) 4952.39 1.08596
\(276\) 0 0
\(277\) 1135.26 0.246249 0.123125 0.992391i \(-0.460708\pi\)
0.123125 + 0.992391i \(0.460708\pi\)
\(278\) −249.251 −0.0537738
\(279\) 0 0
\(280\) −1013.02 −0.216212
\(281\) 2869.86 0.609257 0.304629 0.952471i \(-0.401468\pi\)
0.304629 + 0.952471i \(0.401468\pi\)
\(282\) 0 0
\(283\) 857.505 0.180118 0.0900590 0.995936i \(-0.471294\pi\)
0.0900590 + 0.995936i \(0.471294\pi\)
\(284\) −3878.27 −0.810328
\(285\) 0 0
\(286\) −450.826 −0.0932094
\(287\) −1138.63 −0.234185
\(288\) 0 0
\(289\) 2472.88 0.503333
\(290\) −689.494 −0.139615
\(291\) 0 0
\(292\) 1849.25 0.370614
\(293\) −8823.46 −1.75929 −0.879645 0.475631i \(-0.842220\pi\)
−0.879645 + 0.475631i \(0.842220\pi\)
\(294\) 0 0
\(295\) 4866.91 0.960550
\(296\) −695.213 −0.136515
\(297\) 0 0
\(298\) −5580.58 −1.08481
\(299\) 74.8612 0.0144794
\(300\) 0 0
\(301\) −6898.66 −1.32104
\(302\) −5766.27 −1.09871
\(303\) 0 0
\(304\) −319.058 −0.0601948
\(305\) −3586.87 −0.673388
\(306\) 0 0
\(307\) 8123.41 1.51019 0.755094 0.655617i \(-0.227591\pi\)
0.755094 + 0.655617i \(0.227591\pi\)
\(308\) −4796.23 −0.887307
\(309\) 0 0
\(310\) −1596.39 −0.292481
\(311\) −3416.66 −0.622961 −0.311481 0.950252i \(-0.600825\pi\)
−0.311481 + 0.950252i \(0.600825\pi\)
\(312\) 0 0
\(313\) 5325.14 0.961644 0.480822 0.876818i \(-0.340338\pi\)
0.480822 + 0.876818i \(0.340338\pi\)
\(314\) −3049.73 −0.548108
\(315\) 0 0
\(316\) −3429.25 −0.610477
\(317\) 952.315 0.168730 0.0843649 0.996435i \(-0.473114\pi\)
0.0843649 + 0.996435i \(0.473114\pi\)
\(318\) 0 0
\(319\) −3264.47 −0.572963
\(320\) 468.077 0.0817697
\(321\) 0 0
\(322\) 796.431 0.137836
\(323\) 1713.76 0.295221
\(324\) 0 0
\(325\) 232.752 0.0397254
\(326\) 3593.49 0.610506
\(327\) 0 0
\(328\) 526.116 0.0885668
\(329\) 2841.49 0.476159
\(330\) 0 0
\(331\) 5728.43 0.951248 0.475624 0.879649i \(-0.342222\pi\)
0.475624 + 0.879649i \(0.342222\pi\)
\(332\) 4782.43 0.790571
\(333\) 0 0
\(334\) −2679.21 −0.438922
\(335\) 610.476 0.0995638
\(336\) 0 0
\(337\) −6304.39 −1.01906 −0.509528 0.860454i \(-0.670180\pi\)
−0.509528 + 0.860454i \(0.670180\pi\)
\(338\) 4372.81 0.703697
\(339\) 0 0
\(340\) −2514.19 −0.401033
\(341\) −7558.27 −1.20030
\(342\) 0 0
\(343\) −6687.17 −1.05269
\(344\) 3187.61 0.499605
\(345\) 0 0
\(346\) −3507.29 −0.544952
\(347\) −9338.30 −1.44469 −0.722344 0.691534i \(-0.756935\pi\)
−0.722344 + 0.691534i \(0.756935\pi\)
\(348\) 0 0
\(349\) 4224.27 0.647908 0.323954 0.946073i \(-0.394988\pi\)
0.323954 + 0.946073i \(0.394988\pi\)
\(350\) 2476.20 0.378166
\(351\) 0 0
\(352\) 2216.15 0.335572
\(353\) 6291.13 0.948564 0.474282 0.880373i \(-0.342708\pi\)
0.474282 + 0.880373i \(0.342708\pi\)
\(354\) 0 0
\(355\) −7091.14 −1.06016
\(356\) 1531.84 0.228055
\(357\) 0 0
\(358\) −8750.35 −1.29182
\(359\) −12068.0 −1.77416 −0.887079 0.461618i \(-0.847269\pi\)
−0.887079 + 0.461618i \(0.847269\pi\)
\(360\) 0 0
\(361\) −6461.35 −0.942025
\(362\) 7238.04 1.05089
\(363\) 0 0
\(364\) −225.413 −0.0324584
\(365\) 3381.22 0.484880
\(366\) 0 0
\(367\) 8119.00 1.15479 0.577395 0.816465i \(-0.304069\pi\)
0.577395 + 0.816465i \(0.304069\pi\)
\(368\) −368.000 −0.0521286
\(369\) 0 0
\(370\) −1271.15 −0.178605
\(371\) −10939.9 −1.53092
\(372\) 0 0
\(373\) 5256.30 0.729654 0.364827 0.931075i \(-0.381128\pi\)
0.364827 + 0.931075i \(0.381128\pi\)
\(374\) −11903.7 −1.64579
\(375\) 0 0
\(376\) −1312.94 −0.180079
\(377\) −153.423 −0.0209594
\(378\) 0 0
\(379\) 3375.35 0.457467 0.228734 0.973489i \(-0.426542\pi\)
0.228734 + 0.973489i \(0.426542\pi\)
\(380\) −583.374 −0.0787539
\(381\) 0 0
\(382\) 1074.74 0.143949
\(383\) 8034.61 1.07193 0.535966 0.844240i \(-0.319948\pi\)
0.535966 + 0.844240i \(0.319948\pi\)
\(384\) 0 0
\(385\) −8769.56 −1.16088
\(386\) 669.568 0.0882904
\(387\) 0 0
\(388\) −3270.90 −0.427976
\(389\) −26.5264 −0.00345743 −0.00172872 0.999999i \(-0.500550\pi\)
−0.00172872 + 0.999999i \(0.500550\pi\)
\(390\) 0 0
\(391\) 1976.65 0.255661
\(392\) 345.884 0.0445658
\(393\) 0 0
\(394\) −3348.38 −0.428145
\(395\) −6270.14 −0.798696
\(396\) 0 0
\(397\) 9160.19 1.15803 0.579014 0.815318i \(-0.303438\pi\)
0.579014 + 0.815318i \(0.303438\pi\)
\(398\) −1534.46 −0.193255
\(399\) 0 0
\(400\) −1144.15 −0.143019
\(401\) −4494.80 −0.559750 −0.279875 0.960036i \(-0.590293\pi\)
−0.279875 + 0.960036i \(0.590293\pi\)
\(402\) 0 0
\(403\) −355.223 −0.0439080
\(404\) −2122.19 −0.261344
\(405\) 0 0
\(406\) −1632.24 −0.199523
\(407\) −6018.35 −0.732970
\(408\) 0 0
\(409\) −7988.04 −0.965729 −0.482864 0.875695i \(-0.660404\pi\)
−0.482864 + 0.875695i \(0.660404\pi\)
\(410\) 961.965 0.115873
\(411\) 0 0
\(412\) −5708.47 −0.682612
\(413\) 11521.4 1.37272
\(414\) 0 0
\(415\) 8744.32 1.03432
\(416\) 104.155 0.0122755
\(417\) 0 0
\(418\) −2762.04 −0.323196
\(419\) 12558.9 1.46430 0.732150 0.681143i \(-0.238517\pi\)
0.732150 + 0.681143i \(0.238517\pi\)
\(420\) 0 0
\(421\) −12609.7 −1.45976 −0.729879 0.683577i \(-0.760423\pi\)
−0.729879 + 0.683577i \(0.760423\pi\)
\(422\) 9558.81 1.10264
\(423\) 0 0
\(424\) 5054.90 0.578980
\(425\) 6145.62 0.701427
\(426\) 0 0
\(427\) −8491.17 −0.962334
\(428\) −4341.96 −0.490366
\(429\) 0 0
\(430\) 5828.31 0.653642
\(431\) 1715.05 0.191673 0.0958364 0.995397i \(-0.469447\pi\)
0.0958364 + 0.995397i \(0.469447\pi\)
\(432\) 0 0
\(433\) 16686.2 1.85194 0.925969 0.377601i \(-0.123251\pi\)
0.925969 + 0.377601i \(0.123251\pi\)
\(434\) −3779.14 −0.417982
\(435\) 0 0
\(436\) −401.729 −0.0441269
\(437\) 458.646 0.0502060
\(438\) 0 0
\(439\) 11447.0 1.24450 0.622249 0.782819i \(-0.286219\pi\)
0.622249 + 0.782819i \(0.286219\pi\)
\(440\) 4052.08 0.439035
\(441\) 0 0
\(442\) −559.448 −0.0602042
\(443\) −12008.1 −1.28786 −0.643932 0.765082i \(-0.722698\pi\)
−0.643932 + 0.765082i \(0.722698\pi\)
\(444\) 0 0
\(445\) 2800.86 0.298368
\(446\) 1594.82 0.169320
\(447\) 0 0
\(448\) 1108.08 0.116857
\(449\) −3225.83 −0.339057 −0.169528 0.985525i \(-0.554224\pi\)
−0.169528 + 0.985525i \(0.554224\pi\)
\(450\) 0 0
\(451\) 4554.51 0.475529
\(452\) −9157.72 −0.952971
\(453\) 0 0
\(454\) −11797.3 −1.21954
\(455\) −412.151 −0.0424658
\(456\) 0 0
\(457\) 1318.20 0.134929 0.0674645 0.997722i \(-0.478509\pi\)
0.0674645 + 0.997722i \(0.478509\pi\)
\(458\) 337.652 0.0344486
\(459\) 0 0
\(460\) −672.861 −0.0682007
\(461\) 7837.18 0.791787 0.395894 0.918296i \(-0.370435\pi\)
0.395894 + 0.918296i \(0.370435\pi\)
\(462\) 0 0
\(463\) 15812.5 1.58719 0.793594 0.608448i \(-0.208208\pi\)
0.793594 + 0.608448i \(0.208208\pi\)
\(464\) 754.193 0.0754581
\(465\) 0 0
\(466\) 8369.34 0.831979
\(467\) 8142.16 0.806797 0.403399 0.915024i \(-0.367829\pi\)
0.403399 + 0.915024i \(0.367829\pi\)
\(468\) 0 0
\(469\) 1445.18 0.142286
\(470\) −2400.62 −0.235601
\(471\) 0 0
\(472\) −5323.60 −0.519149
\(473\) 27594.6 2.68246
\(474\) 0 0
\(475\) 1425.98 0.137744
\(476\) −5951.84 −0.573114
\(477\) 0 0
\(478\) 5147.92 0.492595
\(479\) 2216.85 0.211463 0.105731 0.994395i \(-0.466282\pi\)
0.105731 + 0.994395i \(0.466282\pi\)
\(480\) 0 0
\(481\) −282.850 −0.0268126
\(482\) 2184.79 0.206462
\(483\) 0 0
\(484\) 13860.9 1.30174
\(485\) −5980.60 −0.559928
\(486\) 0 0
\(487\) −85.0055 −0.00790958 −0.00395479 0.999992i \(-0.501259\pi\)
−0.00395479 + 0.999992i \(0.501259\pi\)
\(488\) 3923.44 0.363947
\(489\) 0 0
\(490\) 632.424 0.0583061
\(491\) 1200.29 0.110322 0.0551611 0.998477i \(-0.482433\pi\)
0.0551611 + 0.998477i \(0.482433\pi\)
\(492\) 0 0
\(493\) −4051.01 −0.370078
\(494\) −129.810 −0.0118227
\(495\) 0 0
\(496\) 1746.19 0.158077
\(497\) −16786.8 −1.51507
\(498\) 0 0
\(499\) −18846.0 −1.69071 −0.845353 0.534208i \(-0.820610\pi\)
−0.845353 + 0.534208i \(0.820610\pi\)
\(500\) −5748.86 −0.514193
\(501\) 0 0
\(502\) −11008.2 −0.978727
\(503\) 8489.56 0.752546 0.376273 0.926509i \(-0.377205\pi\)
0.376273 + 0.926509i \(0.377205\pi\)
\(504\) 0 0
\(505\) −3880.28 −0.341921
\(506\) −3185.72 −0.279887
\(507\) 0 0
\(508\) 1254.12 0.109533
\(509\) −7158.36 −0.623357 −0.311679 0.950188i \(-0.600891\pi\)
−0.311679 + 0.950188i \(0.600891\pi\)
\(510\) 0 0
\(511\) 8004.35 0.692939
\(512\) −512.000 −0.0441942
\(513\) 0 0
\(514\) 7308.79 0.627192
\(515\) −10437.5 −0.893072
\(516\) 0 0
\(517\) −11365.9 −0.966874
\(518\) −3009.18 −0.255243
\(519\) 0 0
\(520\) 190.439 0.0160602
\(521\) 4283.35 0.360186 0.180093 0.983650i \(-0.442360\pi\)
0.180093 + 0.983650i \(0.442360\pi\)
\(522\) 0 0
\(523\) 19831.0 1.65803 0.829014 0.559228i \(-0.188902\pi\)
0.829014 + 0.559228i \(0.188902\pi\)
\(524\) 2070.27 0.172596
\(525\) 0 0
\(526\) −10704.9 −0.887365
\(527\) −9379.36 −0.775278
\(528\) 0 0
\(529\) 529.000 0.0434783
\(530\) 9242.51 0.757489
\(531\) 0 0
\(532\) −1381.02 −0.112547
\(533\) 214.053 0.0173952
\(534\) 0 0
\(535\) −7938.96 −0.641554
\(536\) −667.761 −0.0538114
\(537\) 0 0
\(538\) 8846.43 0.708916
\(539\) 2994.27 0.239281
\(540\) 0 0
\(541\) −15713.7 −1.24877 −0.624384 0.781117i \(-0.714650\pi\)
−0.624384 + 0.781117i \(0.714650\pi\)
\(542\) 6359.45 0.503988
\(543\) 0 0
\(544\) 2750.12 0.216747
\(545\) −734.533 −0.0577320
\(546\) 0 0
\(547\) 21143.1 1.65267 0.826337 0.563176i \(-0.190421\pi\)
0.826337 + 0.563176i \(0.190421\pi\)
\(548\) 1740.15 0.135649
\(549\) 0 0
\(550\) −9904.78 −0.767893
\(551\) −939.967 −0.0726750
\(552\) 0 0
\(553\) −14843.3 −1.14141
\(554\) −2270.52 −0.174125
\(555\) 0 0
\(556\) 498.503 0.0380238
\(557\) −3746.75 −0.285018 −0.142509 0.989794i \(-0.545517\pi\)
−0.142509 + 0.989794i \(0.545517\pi\)
\(558\) 0 0
\(559\) 1296.89 0.0981264
\(560\) 2026.04 0.152885
\(561\) 0 0
\(562\) −5739.71 −0.430810
\(563\) 16060.2 1.20223 0.601116 0.799162i \(-0.294723\pi\)
0.601116 + 0.799162i \(0.294723\pi\)
\(564\) 0 0
\(565\) −16744.2 −1.24679
\(566\) −1715.01 −0.127363
\(567\) 0 0
\(568\) 7756.54 0.572988
\(569\) −17547.1 −1.29282 −0.646409 0.762991i \(-0.723730\pi\)
−0.646409 + 0.762991i \(0.723730\pi\)
\(570\) 0 0
\(571\) −5551.81 −0.406893 −0.203446 0.979086i \(-0.565214\pi\)
−0.203446 + 0.979086i \(0.565214\pi\)
\(572\) 901.652 0.0659090
\(573\) 0 0
\(574\) 2277.25 0.165594
\(575\) 1644.72 0.119286
\(576\) 0 0
\(577\) 1394.14 0.100587 0.0502937 0.998734i \(-0.483984\pi\)
0.0502937 + 0.998734i \(0.483984\pi\)
\(578\) −4945.75 −0.355910
\(579\) 0 0
\(580\) 1378.99 0.0987230
\(581\) 20700.4 1.47814
\(582\) 0 0
\(583\) 43759.5 3.10863
\(584\) −3698.50 −0.262064
\(585\) 0 0
\(586\) 17646.9 1.24401
\(587\) 26307.6 1.84980 0.924900 0.380210i \(-0.124148\pi\)
0.924900 + 0.380210i \(0.124148\pi\)
\(588\) 0 0
\(589\) −2176.32 −0.152247
\(590\) −9733.81 −0.679211
\(591\) 0 0
\(592\) 1390.43 0.0965306
\(593\) −10657.9 −0.738056 −0.369028 0.929418i \(-0.620309\pi\)
−0.369028 + 0.929418i \(0.620309\pi\)
\(594\) 0 0
\(595\) −10882.5 −0.749814
\(596\) 11161.2 0.767079
\(597\) 0 0
\(598\) −149.722 −0.0102385
\(599\) −5018.67 −0.342333 −0.171166 0.985242i \(-0.554754\pi\)
−0.171166 + 0.985242i \(0.554754\pi\)
\(600\) 0 0
\(601\) 7715.55 0.523667 0.261833 0.965113i \(-0.415673\pi\)
0.261833 + 0.965113i \(0.415673\pi\)
\(602\) 13797.3 0.934114
\(603\) 0 0
\(604\) 11532.5 0.776908
\(605\) 25343.7 1.70309
\(606\) 0 0
\(607\) 26998.5 1.80533 0.902664 0.430346i \(-0.141608\pi\)
0.902664 + 0.430346i \(0.141608\pi\)
\(608\) 638.116 0.0425642
\(609\) 0 0
\(610\) 7173.73 0.476157
\(611\) −534.176 −0.0353690
\(612\) 0 0
\(613\) 3756.96 0.247540 0.123770 0.992311i \(-0.460501\pi\)
0.123770 + 0.992311i \(0.460501\pi\)
\(614\) −16246.8 −1.06786
\(615\) 0 0
\(616\) 9592.46 0.627421
\(617\) −14186.2 −0.925632 −0.462816 0.886454i \(-0.653161\pi\)
−0.462816 + 0.886454i \(0.653161\pi\)
\(618\) 0 0
\(619\) −25670.2 −1.66684 −0.833418 0.552643i \(-0.813619\pi\)
−0.833418 + 0.552643i \(0.813619\pi\)
\(620\) 3192.79 0.206815
\(621\) 0 0
\(622\) 6833.32 0.440500
\(623\) 6630.47 0.426395
\(624\) 0 0
\(625\) −1572.66 −0.100650
\(626\) −10650.3 −0.679985
\(627\) 0 0
\(628\) 6099.45 0.387571
\(629\) −7468.42 −0.473427
\(630\) 0 0
\(631\) −948.530 −0.0598421 −0.0299211 0.999552i \(-0.509526\pi\)
−0.0299211 + 0.999552i \(0.509526\pi\)
\(632\) 6858.51 0.431672
\(633\) 0 0
\(634\) −1904.63 −0.119310
\(635\) 2293.07 0.143304
\(636\) 0 0
\(637\) 140.724 0.00875307
\(638\) 6528.94 0.405146
\(639\) 0 0
\(640\) −936.155 −0.0578199
\(641\) 28678.3 1.76712 0.883560 0.468318i \(-0.155140\pi\)
0.883560 + 0.468318i \(0.155140\pi\)
\(642\) 0 0
\(643\) 7606.41 0.466513 0.233256 0.972415i \(-0.425062\pi\)
0.233256 + 0.972415i \(0.425062\pi\)
\(644\) −1592.86 −0.0974651
\(645\) 0 0
\(646\) −3427.53 −0.208753
\(647\) −546.880 −0.0332304 −0.0166152 0.999862i \(-0.505289\pi\)
−0.0166152 + 0.999862i \(0.505289\pi\)
\(648\) 0 0
\(649\) −46085.6 −2.78739
\(650\) −465.504 −0.0280901
\(651\) 0 0
\(652\) −7186.97 −0.431693
\(653\) −2615.66 −0.156751 −0.0783756 0.996924i \(-0.524973\pi\)
−0.0783756 + 0.996924i \(0.524973\pi\)
\(654\) 0 0
\(655\) 3785.34 0.225810
\(656\) −1052.23 −0.0626262
\(657\) 0 0
\(658\) −5682.97 −0.336695
\(659\) −5101.12 −0.301535 −0.150767 0.988569i \(-0.548174\pi\)
−0.150767 + 0.988569i \(0.548174\pi\)
\(660\) 0 0
\(661\) −33184.1 −1.95267 −0.976334 0.216268i \(-0.930611\pi\)
−0.976334 + 0.216268i \(0.930611\pi\)
\(662\) −11456.9 −0.672634
\(663\) 0 0
\(664\) −9564.85 −0.559018
\(665\) −2525.09 −0.147246
\(666\) 0 0
\(667\) −1084.15 −0.0629364
\(668\) 5358.43 0.310365
\(669\) 0 0
\(670\) −1220.95 −0.0704022
\(671\) 33964.7 1.95409
\(672\) 0 0
\(673\) −22540.9 −1.29107 −0.645533 0.763733i \(-0.723365\pi\)
−0.645533 + 0.763733i \(0.723365\pi\)
\(674\) 12608.8 0.720582
\(675\) 0 0
\(676\) −8745.62 −0.497589
\(677\) −30165.0 −1.71246 −0.856229 0.516596i \(-0.827199\pi\)
−0.856229 + 0.516596i \(0.827199\pi\)
\(678\) 0 0
\(679\) −14157.8 −0.800188
\(680\) 5028.39 0.283573
\(681\) 0 0
\(682\) 15116.5 0.848742
\(683\) −417.726 −0.0234024 −0.0117012 0.999932i \(-0.503725\pi\)
−0.0117012 + 0.999932i \(0.503725\pi\)
\(684\) 0 0
\(685\) 3181.74 0.177472
\(686\) 13374.3 0.744365
\(687\) 0 0
\(688\) −6375.21 −0.353274
\(689\) 2056.61 0.113716
\(690\) 0 0
\(691\) −5421.98 −0.298498 −0.149249 0.988800i \(-0.547686\pi\)
−0.149249 + 0.988800i \(0.547686\pi\)
\(692\) 7014.59 0.385339
\(693\) 0 0
\(694\) 18676.6 1.02155
\(695\) 911.476 0.0497471
\(696\) 0 0
\(697\) 5651.88 0.307145
\(698\) −8448.54 −0.458140
\(699\) 0 0
\(700\) −4952.39 −0.267404
\(701\) −14840.0 −0.799572 −0.399786 0.916608i \(-0.630916\pi\)
−0.399786 + 0.916608i \(0.630916\pi\)
\(702\) 0 0
\(703\) −1732.92 −0.0929703
\(704\) −4432.31 −0.237285
\(705\) 0 0
\(706\) −12582.3 −0.670736
\(707\) −9185.76 −0.488637
\(708\) 0 0
\(709\) 15302.4 0.810569 0.405284 0.914191i \(-0.367173\pi\)
0.405284 + 0.914191i \(0.367173\pi\)
\(710\) 14182.3 0.749649
\(711\) 0 0
\(712\) −3063.68 −0.161259
\(713\) −2510.15 −0.131846
\(714\) 0 0
\(715\) 1648.60 0.0862298
\(716\) 17500.7 0.913452
\(717\) 0 0
\(718\) 24135.9 1.25452
\(719\) −26125.2 −1.35508 −0.677542 0.735484i \(-0.736955\pi\)
−0.677542 + 0.735484i \(0.736955\pi\)
\(720\) 0 0
\(721\) −24708.7 −1.27628
\(722\) 12922.7 0.666112
\(723\) 0 0
\(724\) −14476.1 −0.743093
\(725\) −3370.76 −0.172671
\(726\) 0 0
\(727\) −4928.32 −0.251419 −0.125709 0.992067i \(-0.540121\pi\)
−0.125709 + 0.992067i \(0.540121\pi\)
\(728\) 450.826 0.0229515
\(729\) 0 0
\(730\) −6762.44 −0.342862
\(731\) 34243.3 1.73261
\(732\) 0 0
\(733\) 16142.2 0.813405 0.406703 0.913561i \(-0.366679\pi\)
0.406703 + 0.913561i \(0.366679\pi\)
\(734\) −16238.0 −0.816560
\(735\) 0 0
\(736\) 736.000 0.0368605
\(737\) −5780.71 −0.288922
\(738\) 0 0
\(739\) −1612.09 −0.0802457 −0.0401229 0.999195i \(-0.512775\pi\)
−0.0401229 + 0.999195i \(0.512775\pi\)
\(740\) 2542.29 0.126293
\(741\) 0 0
\(742\) 21879.8 1.08252
\(743\) 33768.1 1.66734 0.833668 0.552265i \(-0.186236\pi\)
0.833668 + 0.552265i \(0.186236\pi\)
\(744\) 0 0
\(745\) 20407.4 1.00358
\(746\) −10512.6 −0.515943
\(747\) 0 0
\(748\) 23807.4 1.16375
\(749\) −18793.9 −0.916840
\(750\) 0 0
\(751\) −12656.6 −0.614977 −0.307488 0.951552i \(-0.599489\pi\)
−0.307488 + 0.951552i \(0.599489\pi\)
\(752\) 2625.88 0.127335
\(753\) 0 0
\(754\) 306.847 0.0148206
\(755\) 21086.4 1.01644
\(756\) 0 0
\(757\) −35451.3 −1.70211 −0.851056 0.525075i \(-0.824037\pi\)
−0.851056 + 0.525075i \(0.824037\pi\)
\(758\) −6750.70 −0.323478
\(759\) 0 0
\(760\) 1166.75 0.0556874
\(761\) −4234.99 −0.201732 −0.100866 0.994900i \(-0.532161\pi\)
−0.100866 + 0.994900i \(0.532161\pi\)
\(762\) 0 0
\(763\) −1738.86 −0.0825043
\(764\) −2149.48 −0.101787
\(765\) 0 0
\(766\) −16069.2 −0.757970
\(767\) −2165.93 −0.101965
\(768\) 0 0
\(769\) 23381.4 1.09643 0.548216 0.836337i \(-0.315307\pi\)
0.548216 + 0.836337i \(0.315307\pi\)
\(770\) 17539.1 0.820865
\(771\) 0 0
\(772\) −1339.14 −0.0624307
\(773\) −8976.44 −0.417672 −0.208836 0.977951i \(-0.566967\pi\)
−0.208836 + 0.977951i \(0.566967\pi\)
\(774\) 0 0
\(775\) −7804.36 −0.361730
\(776\) 6541.79 0.302625
\(777\) 0 0
\(778\) 53.0528 0.00244478
\(779\) 1311.42 0.0603163
\(780\) 0 0
\(781\) 67147.2 3.07646
\(782\) −3953.29 −0.180779
\(783\) 0 0
\(784\) −691.768 −0.0315128
\(785\) 11152.4 0.507065
\(786\) 0 0
\(787\) −33632.8 −1.52335 −0.761677 0.647957i \(-0.775624\pi\)
−0.761677 + 0.647957i \(0.775624\pi\)
\(788\) 6696.77 0.302744
\(789\) 0 0
\(790\) 12540.3 0.564764
\(791\) −39638.5 −1.78177
\(792\) 0 0
\(793\) 1596.27 0.0714820
\(794\) −18320.4 −0.818849
\(795\) 0 0
\(796\) 3068.92 0.136652
\(797\) −38371.3 −1.70537 −0.852686 0.522424i \(-0.825028\pi\)
−0.852686 + 0.522424i \(0.825028\pi\)
\(798\) 0 0
\(799\) −14104.5 −0.624506
\(800\) 2288.31 0.101130
\(801\) 0 0
\(802\) 8989.60 0.395803
\(803\) −32017.4 −1.40706
\(804\) 0 0
\(805\) −2912.43 −0.127515
\(806\) 710.446 0.0310476
\(807\) 0 0
\(808\) 4244.39 0.184798
\(809\) 15249.5 0.662722 0.331361 0.943504i \(-0.392492\pi\)
0.331361 + 0.943504i \(0.392492\pi\)
\(810\) 0 0
\(811\) 9708.02 0.420339 0.210169 0.977665i \(-0.432598\pi\)
0.210169 + 0.977665i \(0.432598\pi\)
\(812\) 3264.47 0.141084
\(813\) 0 0
\(814\) 12036.7 0.518288
\(815\) −13140.9 −0.564790
\(816\) 0 0
\(817\) 7945.56 0.340245
\(818\) 15976.1 0.682873
\(819\) 0 0
\(820\) −1923.93 −0.0819348
\(821\) −9028.52 −0.383797 −0.191899 0.981415i \(-0.561464\pi\)
−0.191899 + 0.981415i \(0.561464\pi\)
\(822\) 0 0
\(823\) 42539.1 1.80172 0.900862 0.434105i \(-0.142935\pi\)
0.900862 + 0.434105i \(0.142935\pi\)
\(824\) 11416.9 0.482679
\(825\) 0 0
\(826\) −23042.8 −0.970656
\(827\) −21526.4 −0.905134 −0.452567 0.891730i \(-0.649492\pi\)
−0.452567 + 0.891730i \(0.649492\pi\)
\(828\) 0 0
\(829\) 38550.4 1.61509 0.807546 0.589804i \(-0.200795\pi\)
0.807546 + 0.589804i \(0.200795\pi\)
\(830\) −17488.6 −0.731373
\(831\) 0 0
\(832\) −208.309 −0.00868008
\(833\) 3715.71 0.154552
\(834\) 0 0
\(835\) 9797.49 0.406055
\(836\) 5524.08 0.228534
\(837\) 0 0
\(838\) −25117.8 −1.03542
\(839\) −30566.4 −1.25777 −0.628884 0.777499i \(-0.716488\pi\)
−0.628884 + 0.777499i \(0.716488\pi\)
\(840\) 0 0
\(841\) −22167.1 −0.908897
\(842\) 25219.3 1.03220
\(843\) 0 0
\(844\) −19117.6 −0.779687
\(845\) −15990.7 −0.651004
\(846\) 0 0
\(847\) 59996.0 2.43387
\(848\) −10109.8 −0.409401
\(849\) 0 0
\(850\) −12291.2 −0.495984
\(851\) −1998.74 −0.0805121
\(852\) 0 0
\(853\) 20617.3 0.827576 0.413788 0.910373i \(-0.364205\pi\)
0.413788 + 0.910373i \(0.364205\pi\)
\(854\) 16982.3 0.680473
\(855\) 0 0
\(856\) 8683.92 0.346741
\(857\) −1553.09 −0.0619048 −0.0309524 0.999521i \(-0.509854\pi\)
−0.0309524 + 0.999521i \(0.509854\pi\)
\(858\) 0 0
\(859\) −34732.9 −1.37959 −0.689797 0.724003i \(-0.742300\pi\)
−0.689797 + 0.724003i \(0.742300\pi\)
\(860\) −11656.6 −0.462194
\(861\) 0 0
\(862\) −3430.10 −0.135533
\(863\) −9162.01 −0.361389 −0.180694 0.983539i \(-0.557834\pi\)
−0.180694 + 0.983539i \(0.557834\pi\)
\(864\) 0 0
\(865\) 12825.7 0.504145
\(866\) −33372.5 −1.30952
\(867\) 0 0
\(868\) 7558.27 0.295558
\(869\) 59373.1 2.31772
\(870\) 0 0
\(871\) −271.681 −0.0105690
\(872\) 803.459 0.0312025
\(873\) 0 0
\(874\) −917.292 −0.0355010
\(875\) −24883.5 −0.961390
\(876\) 0 0
\(877\) −5772.11 −0.222247 −0.111123 0.993807i \(-0.535445\pi\)
−0.111123 + 0.993807i \(0.535445\pi\)
\(878\) −22894.0 −0.879993
\(879\) 0 0
\(880\) −8104.15 −0.310444
\(881\) −9412.65 −0.359955 −0.179977 0.983671i \(-0.557602\pi\)
−0.179977 + 0.983671i \(0.557602\pi\)
\(882\) 0 0
\(883\) −45232.2 −1.72388 −0.861940 0.507011i \(-0.830750\pi\)
−0.861940 + 0.507011i \(0.830750\pi\)
\(884\) 1118.90 0.0425708
\(885\) 0 0
\(886\) 24016.3 0.910658
\(887\) −31353.8 −1.18687 −0.593437 0.804880i \(-0.702229\pi\)
−0.593437 + 0.804880i \(0.702229\pi\)
\(888\) 0 0
\(889\) 5428.38 0.204794
\(890\) −5601.72 −0.210978
\(891\) 0 0
\(892\) −3189.64 −0.119728
\(893\) −3272.69 −0.122639
\(894\) 0 0
\(895\) 31998.8 1.19508
\(896\) −2216.15 −0.0826301
\(897\) 0 0
\(898\) 6451.66 0.239749
\(899\) 5144.40 0.190851
\(900\) 0 0
\(901\) 54302.9 2.00787
\(902\) −9109.02 −0.336250
\(903\) 0 0
\(904\) 18315.4 0.673852
\(905\) −26468.5 −0.972200
\(906\) 0 0
\(907\) −8652.83 −0.316772 −0.158386 0.987377i \(-0.550629\pi\)
−0.158386 + 0.987377i \(0.550629\pi\)
\(908\) 23594.5 0.862347
\(909\) 0 0
\(910\) 824.302 0.0300279
\(911\) −23107.3 −0.840370 −0.420185 0.907438i \(-0.638035\pi\)
−0.420185 + 0.907438i \(0.638035\pi\)
\(912\) 0 0
\(913\) −82801.5 −3.00146
\(914\) −2636.39 −0.0954092
\(915\) 0 0
\(916\) −675.304 −0.0243588
\(917\) 8961.02 0.322703
\(918\) 0 0
\(919\) −42056.1 −1.50958 −0.754790 0.655966i \(-0.772261\pi\)
−0.754790 + 0.655966i \(0.772261\pi\)
\(920\) 1345.72 0.0482252
\(921\) 0 0
\(922\) −15674.4 −0.559878
\(923\) 3155.78 0.112539
\(924\) 0 0
\(925\) −6214.30 −0.220892
\(926\) −31624.9 −1.12231
\(927\) 0 0
\(928\) −1508.39 −0.0533569
\(929\) −1263.92 −0.0446369 −0.0223185 0.999751i \(-0.507105\pi\)
−0.0223185 + 0.999751i \(0.507105\pi\)
\(930\) 0 0
\(931\) 862.164 0.0303505
\(932\) −16738.7 −0.588298
\(933\) 0 0
\(934\) −16284.3 −0.570492
\(935\) 43530.0 1.52255
\(936\) 0 0
\(937\) −27756.8 −0.967743 −0.483871 0.875139i \(-0.660770\pi\)
−0.483871 + 0.875139i \(0.660770\pi\)
\(938\) −2890.35 −0.100611
\(939\) 0 0
\(940\) 4801.24 0.166595
\(941\) 7081.47 0.245323 0.122662 0.992449i \(-0.460857\pi\)
0.122662 + 0.992449i \(0.460857\pi\)
\(942\) 0 0
\(943\) 1512.58 0.0522338
\(944\) 10647.2 0.367094
\(945\) 0 0
\(946\) −55189.3 −1.89678
\(947\) 34048.7 1.16836 0.584179 0.811625i \(-0.301417\pi\)
0.584179 + 0.811625i \(0.301417\pi\)
\(948\) 0 0
\(949\) −1504.75 −0.0514713
\(950\) −2851.97 −0.0974000
\(951\) 0 0
\(952\) 11903.7 0.405252
\(953\) 10453.3 0.355314 0.177657 0.984092i \(-0.443148\pi\)
0.177657 + 0.984092i \(0.443148\pi\)
\(954\) 0 0
\(955\) −3930.16 −0.133170
\(956\) −10295.8 −0.348317
\(957\) 0 0
\(958\) −4433.71 −0.149527
\(959\) 7532.12 0.253623
\(960\) 0 0
\(961\) −17880.1 −0.600185
\(962\) 565.700 0.0189594
\(963\) 0 0
\(964\) −4369.58 −0.145990
\(965\) −2448.51 −0.0816791
\(966\) 0 0
\(967\) −22280.5 −0.740945 −0.370473 0.928843i \(-0.620804\pi\)
−0.370473 + 0.928843i \(0.620804\pi\)
\(968\) −27721.9 −0.920469
\(969\) 0 0
\(970\) 11961.2 0.395929
\(971\) 39430.9 1.30319 0.651595 0.758567i \(-0.274100\pi\)
0.651595 + 0.758567i \(0.274100\pi\)
\(972\) 0 0
\(973\) 2157.73 0.0710933
\(974\) 170.011 0.00559292
\(975\) 0 0
\(976\) −7846.89 −0.257349
\(977\) −56586.0 −1.85297 −0.926483 0.376337i \(-0.877183\pi\)
−0.926483 + 0.376337i \(0.877183\pi\)
\(978\) 0 0
\(979\) −26521.9 −0.865825
\(980\) −1264.85 −0.0412286
\(981\) 0 0
\(982\) −2400.57 −0.0780095
\(983\) −42422.5 −1.37647 −0.688235 0.725488i \(-0.741614\pi\)
−0.688235 + 0.725488i \(0.741614\pi\)
\(984\) 0 0
\(985\) 12244.5 0.396085
\(986\) 8102.03 0.261685
\(987\) 0 0
\(988\) 259.620 0.00835994
\(989\) 9164.37 0.294651
\(990\) 0 0
\(991\) −23798.6 −0.762853 −0.381426 0.924399i \(-0.624567\pi\)
−0.381426 + 0.924399i \(0.624567\pi\)
\(992\) −3492.39 −0.111778
\(993\) 0 0
\(994\) 33573.6 1.07132
\(995\) 5611.31 0.178784
\(996\) 0 0
\(997\) −35708.9 −1.13431 −0.567157 0.823610i \(-0.691957\pi\)
−0.567157 + 0.823610i \(0.691957\pi\)
\(998\) 37692.0 1.19551
\(999\) 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 414.4.a.g.1.2 2
3.2 odd 2 138.4.a.e.1.1 2
12.11 even 2 1104.4.a.p.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.4.a.e.1.1 2 3.2 odd 2
414.4.a.g.1.2 2 1.1 even 1 trivial
1104.4.a.p.1.1 2 12.11 even 2