Properties

Label 414.4.a.n.1.1
Level $414$
Weight $4$
Character 414.1
Self dual yes
Analytic conductor $24.427$
Analytic rank $0$
Dimension $4$
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] = [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: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 219x^{2} - 468x + 3240 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.93708\) 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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +4.00000 q^{4} -19.0098 q^{5} -26.4014 q^{7} +8.00000 q^{8} -38.0197 q^{10} +27.2657 q^{11} +69.8938 q^{13} -52.8028 q^{14} +16.0000 q^{16} -15.2407 q^{17} +65.6587 q^{19} -76.0393 q^{20} +54.5314 q^{22} +23.0000 q^{23} +236.374 q^{25} +139.788 q^{26} -105.606 q^{28} +124.619 q^{29} +302.744 q^{31} +32.0000 q^{32} -30.4815 q^{34} +501.886 q^{35} +5.93992 q^{37} +131.317 q^{38} -152.079 q^{40} -166.526 q^{41} -318.295 q^{43} +109.063 q^{44} +46.0000 q^{46} +144.643 q^{47} +354.033 q^{49} +472.747 q^{50} +279.575 q^{52} -502.487 q^{53} -518.317 q^{55} -211.211 q^{56} +249.237 q^{58} -600.540 q^{59} +636.071 q^{61} +605.489 q^{62} +64.0000 q^{64} -1328.67 q^{65} +206.163 q^{67} -60.9630 q^{68} +1003.77 q^{70} +440.948 q^{71} +390.246 q^{73} +11.8798 q^{74} +262.635 q^{76} -719.853 q^{77} +681.317 q^{79} -304.157 q^{80} -333.051 q^{82} -695.269 q^{83} +289.724 q^{85} -636.589 q^{86} +218.126 q^{88} +948.900 q^{89} -1845.29 q^{91} +92.0000 q^{92} +289.286 q^{94} -1248.16 q^{95} +746.861 q^{97} +708.067 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{2} + 16 q^{4} + 18 q^{7} + 32 q^{8} + 42 q^{11} + 108 q^{13} + 36 q^{14} + 64 q^{16} + 26 q^{17} + 132 q^{19} + 84 q^{22} + 92 q^{23} + 260 q^{25} + 216 q^{26} + 72 q^{28} + 252 q^{29} + 428 q^{31}+ \cdots + 2152 q^{98}+O(q^{100}) \) 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
\(5\) −19.0098 −1.70029 −0.850145 0.526548i \(-0.823486\pi\)
−0.850145 + 0.526548i \(0.823486\pi\)
\(6\) 0 0
\(7\) −26.4014 −1.42554 −0.712770 0.701397i \(-0.752560\pi\)
−0.712770 + 0.701397i \(0.752560\pi\)
\(8\) 8.00000 0.353553
\(9\) 0 0
\(10\) −38.0197 −1.20229
\(11\) 27.2657 0.747357 0.373678 0.927558i \(-0.378096\pi\)
0.373678 + 0.927558i \(0.378096\pi\)
\(12\) 0 0
\(13\) 69.8938 1.49116 0.745579 0.666417i \(-0.232173\pi\)
0.745579 + 0.666417i \(0.232173\pi\)
\(14\) −52.8028 −1.00801
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −15.2407 −0.217437 −0.108718 0.994073i \(-0.534675\pi\)
−0.108718 + 0.994073i \(0.534675\pi\)
\(18\) 0 0
\(19\) 65.6587 0.792797 0.396398 0.918079i \(-0.370260\pi\)
0.396398 + 0.918079i \(0.370260\pi\)
\(20\) −76.0393 −0.850145
\(21\) 0 0
\(22\) 54.5314 0.528461
\(23\) 23.0000 0.208514
\(24\) 0 0
\(25\) 236.374 1.89099
\(26\) 139.788 1.05441
\(27\) 0 0
\(28\) −105.606 −0.712770
\(29\) 124.619 0.797968 0.398984 0.916958i \(-0.369363\pi\)
0.398984 + 0.916958i \(0.369363\pi\)
\(30\) 0 0
\(31\) 302.744 1.75402 0.877008 0.480476i \(-0.159536\pi\)
0.877008 + 0.480476i \(0.159536\pi\)
\(32\) 32.0000 0.176777
\(33\) 0 0
\(34\) −30.4815 −0.153751
\(35\) 501.886 2.42383
\(36\) 0 0
\(37\) 5.93992 0.0263924 0.0131962 0.999913i \(-0.495799\pi\)
0.0131962 + 0.999913i \(0.495799\pi\)
\(38\) 131.317 0.560592
\(39\) 0 0
\(40\) −152.079 −0.601144
\(41\) −166.526 −0.634316 −0.317158 0.948373i \(-0.602729\pi\)
−0.317158 + 0.948373i \(0.602729\pi\)
\(42\) 0 0
\(43\) −318.295 −1.12882 −0.564412 0.825493i \(-0.690897\pi\)
−0.564412 + 0.825493i \(0.690897\pi\)
\(44\) 109.063 0.373678
\(45\) 0 0
\(46\) 46.0000 0.147442
\(47\) 144.643 0.448902 0.224451 0.974485i \(-0.427941\pi\)
0.224451 + 0.974485i \(0.427941\pi\)
\(48\) 0 0
\(49\) 354.033 1.03217
\(50\) 472.747 1.33713
\(51\) 0 0
\(52\) 279.575 0.745579
\(53\) −502.487 −1.30230 −0.651150 0.758949i \(-0.725713\pi\)
−0.651150 + 0.758949i \(0.725713\pi\)
\(54\) 0 0
\(55\) −518.317 −1.27072
\(56\) −211.211 −0.504005
\(57\) 0 0
\(58\) 249.237 0.564249
\(59\) −600.540 −1.32515 −0.662573 0.748997i \(-0.730536\pi\)
−0.662573 + 0.748997i \(0.730536\pi\)
\(60\) 0 0
\(61\) 636.071 1.33509 0.667546 0.744569i \(-0.267345\pi\)
0.667546 + 0.744569i \(0.267345\pi\)
\(62\) 605.489 1.24028
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −1328.67 −2.53540
\(66\) 0 0
\(67\) 206.163 0.375922 0.187961 0.982176i \(-0.439812\pi\)
0.187961 + 0.982176i \(0.439812\pi\)
\(68\) −60.9630 −0.108718
\(69\) 0 0
\(70\) 1003.77 1.71391
\(71\) 440.948 0.737055 0.368527 0.929617i \(-0.379862\pi\)
0.368527 + 0.929617i \(0.379862\pi\)
\(72\) 0 0
\(73\) 390.246 0.625683 0.312841 0.949805i \(-0.398719\pi\)
0.312841 + 0.949805i \(0.398719\pi\)
\(74\) 11.8798 0.0186622
\(75\) 0 0
\(76\) 262.635 0.396398
\(77\) −719.853 −1.06539
\(78\) 0 0
\(79\) 681.317 0.970306 0.485153 0.874429i \(-0.338764\pi\)
0.485153 + 0.874429i \(0.338764\pi\)
\(80\) −304.157 −0.425073
\(81\) 0 0
\(82\) −333.051 −0.448529
\(83\) −695.269 −0.919466 −0.459733 0.888057i \(-0.652055\pi\)
−0.459733 + 0.888057i \(0.652055\pi\)
\(84\) 0 0
\(85\) 289.724 0.369705
\(86\) −636.589 −0.798200
\(87\) 0 0
\(88\) 218.126 0.264230
\(89\) 948.900 1.13015 0.565074 0.825040i \(-0.308848\pi\)
0.565074 + 0.825040i \(0.308848\pi\)
\(90\) 0 0
\(91\) −1845.29 −2.12571
\(92\) 92.0000 0.104257
\(93\) 0 0
\(94\) 289.286 0.317421
\(95\) −1248.16 −1.34798
\(96\) 0 0
\(97\) 746.861 0.781775 0.390888 0.920438i \(-0.372168\pi\)
0.390888 + 0.920438i \(0.372168\pi\)
\(98\) 708.067 0.729852
\(99\) 0 0
\(100\) 945.494 0.945494
\(101\) 1168.00 1.15070 0.575349 0.817908i \(-0.304866\pi\)
0.575349 + 0.817908i \(0.304866\pi\)
\(102\) 0 0
\(103\) 608.376 0.581991 0.290995 0.956724i \(-0.406014\pi\)
0.290995 + 0.956724i \(0.406014\pi\)
\(104\) 559.150 0.527204
\(105\) 0 0
\(106\) −1004.97 −0.920866
\(107\) 1134.79 1.02528 0.512638 0.858605i \(-0.328668\pi\)
0.512638 + 0.858605i \(0.328668\pi\)
\(108\) 0 0
\(109\) 1602.61 1.40828 0.704140 0.710061i \(-0.251333\pi\)
0.704140 + 0.710061i \(0.251333\pi\)
\(110\) −1036.63 −0.898537
\(111\) 0 0
\(112\) −422.422 −0.356385
\(113\) −2049.37 −1.70609 −0.853046 0.521836i \(-0.825247\pi\)
−0.853046 + 0.521836i \(0.825247\pi\)
\(114\) 0 0
\(115\) −437.226 −0.354535
\(116\) 498.474 0.398984
\(117\) 0 0
\(118\) −1201.08 −0.937020
\(119\) 402.377 0.309965
\(120\) 0 0
\(121\) −587.581 −0.441458
\(122\) 1272.14 0.944052
\(123\) 0 0
\(124\) 1210.98 0.877008
\(125\) −2117.19 −1.51494
\(126\) 0 0
\(127\) 1322.84 0.924278 0.462139 0.886807i \(-0.347082\pi\)
0.462139 + 0.886807i \(0.347082\pi\)
\(128\) 128.000 0.0883883
\(129\) 0 0
\(130\) −2657.34 −1.79280
\(131\) 353.591 0.235827 0.117914 0.993024i \(-0.462379\pi\)
0.117914 + 0.993024i \(0.462379\pi\)
\(132\) 0 0
\(133\) −1733.48 −1.13016
\(134\) 412.326 0.265817
\(135\) 0 0
\(136\) −121.926 −0.0768755
\(137\) 3171.79 1.97799 0.988995 0.147948i \(-0.0472670\pi\)
0.988995 + 0.147948i \(0.0472670\pi\)
\(138\) 0 0
\(139\) 508.055 0.310019 0.155010 0.987913i \(-0.450459\pi\)
0.155010 + 0.987913i \(0.450459\pi\)
\(140\) 2007.54 1.21192
\(141\) 0 0
\(142\) 881.896 0.521176
\(143\) 1905.71 1.11443
\(144\) 0 0
\(145\) −2368.98 −1.35678
\(146\) 780.492 0.442424
\(147\) 0 0
\(148\) 23.7597 0.0131962
\(149\) −2954.62 −1.62451 −0.812255 0.583303i \(-0.801760\pi\)
−0.812255 + 0.583303i \(0.801760\pi\)
\(150\) 0 0
\(151\) −3383.54 −1.82350 −0.911749 0.410747i \(-0.865268\pi\)
−0.911749 + 0.410747i \(0.865268\pi\)
\(152\) 525.270 0.280296
\(153\) 0 0
\(154\) −1439.71 −0.753343
\(155\) −5755.12 −2.98234
\(156\) 0 0
\(157\) −293.879 −0.149389 −0.0746945 0.997206i \(-0.523798\pi\)
−0.0746945 + 0.997206i \(0.523798\pi\)
\(158\) 1362.63 0.686110
\(159\) 0 0
\(160\) −608.314 −0.300572
\(161\) −607.232 −0.297246
\(162\) 0 0
\(163\) 1277.17 0.613717 0.306859 0.951755i \(-0.400722\pi\)
0.306859 + 0.951755i \(0.400722\pi\)
\(164\) −666.103 −0.317158
\(165\) 0 0
\(166\) −1390.54 −0.650161
\(167\) 1509.84 0.699609 0.349805 0.936823i \(-0.386248\pi\)
0.349805 + 0.936823i \(0.386248\pi\)
\(168\) 0 0
\(169\) 2688.15 1.22355
\(170\) 579.448 0.261421
\(171\) 0 0
\(172\) −1273.18 −0.564412
\(173\) 137.777 0.0605490 0.0302745 0.999542i \(-0.490362\pi\)
0.0302745 + 0.999542i \(0.490362\pi\)
\(174\) 0 0
\(175\) −6240.59 −2.69568
\(176\) 436.252 0.186839
\(177\) 0 0
\(178\) 1897.80 0.799135
\(179\) −3043.97 −1.27105 −0.635523 0.772082i \(-0.719216\pi\)
−0.635523 + 0.772082i \(0.719216\pi\)
\(180\) 0 0
\(181\) 3906.90 1.60441 0.802204 0.597051i \(-0.203661\pi\)
0.802204 + 0.597051i \(0.203661\pi\)
\(182\) −3690.59 −1.50310
\(183\) 0 0
\(184\) 184.000 0.0737210
\(185\) −112.917 −0.0448747
\(186\) 0 0
\(187\) −415.550 −0.162503
\(188\) 578.573 0.224451
\(189\) 0 0
\(190\) −2496.32 −0.953169
\(191\) 1018.93 0.386007 0.193004 0.981198i \(-0.438177\pi\)
0.193004 + 0.981198i \(0.438177\pi\)
\(192\) 0 0
\(193\) −4302.92 −1.60482 −0.802412 0.596771i \(-0.796450\pi\)
−0.802412 + 0.596771i \(0.796450\pi\)
\(194\) 1493.72 0.552799
\(195\) 0 0
\(196\) 1416.13 0.516083
\(197\) 1439.36 0.520558 0.260279 0.965533i \(-0.416185\pi\)
0.260279 + 0.965533i \(0.416185\pi\)
\(198\) 0 0
\(199\) 5262.93 1.87477 0.937385 0.348295i \(-0.113239\pi\)
0.937385 + 0.348295i \(0.113239\pi\)
\(200\) 1890.99 0.668565
\(201\) 0 0
\(202\) 2336.00 0.813666
\(203\) −3290.10 −1.13754
\(204\) 0 0
\(205\) 3165.63 1.07852
\(206\) 1216.75 0.411530
\(207\) 0 0
\(208\) 1118.30 0.372790
\(209\) 1790.23 0.592502
\(210\) 0 0
\(211\) 353.694 0.115399 0.0576997 0.998334i \(-0.481623\pi\)
0.0576997 + 0.998334i \(0.481623\pi\)
\(212\) −2009.95 −0.651150
\(213\) 0 0
\(214\) 2269.58 0.724979
\(215\) 6050.73 1.91933
\(216\) 0 0
\(217\) −7992.87 −2.50042
\(218\) 3205.23 0.995804
\(219\) 0 0
\(220\) −2073.27 −0.635362
\(221\) −1065.23 −0.324232
\(222\) 0 0
\(223\) −5336.31 −1.60245 −0.801223 0.598366i \(-0.795817\pi\)
−0.801223 + 0.598366i \(0.795817\pi\)
\(224\) −844.844 −0.252002
\(225\) 0 0
\(226\) −4098.74 −1.20639
\(227\) 2213.30 0.647144 0.323572 0.946204i \(-0.395116\pi\)
0.323572 + 0.946204i \(0.395116\pi\)
\(228\) 0 0
\(229\) −5141.09 −1.48355 −0.741774 0.670650i \(-0.766015\pi\)
−0.741774 + 0.670650i \(0.766015\pi\)
\(230\) −874.452 −0.250694
\(231\) 0 0
\(232\) 996.948 0.282124
\(233\) 6582.34 1.85074 0.925372 0.379061i \(-0.123753\pi\)
0.925372 + 0.379061i \(0.123753\pi\)
\(234\) 0 0
\(235\) −2749.64 −0.763263
\(236\) −2402.16 −0.662573
\(237\) 0 0
\(238\) 804.753 0.219178
\(239\) 6205.96 1.67962 0.839812 0.542877i \(-0.182665\pi\)
0.839812 + 0.542877i \(0.182665\pi\)
\(240\) 0 0
\(241\) −6363.48 −1.70086 −0.850432 0.526086i \(-0.823659\pi\)
−0.850432 + 0.526086i \(0.823659\pi\)
\(242\) −1175.16 −0.312158
\(243\) 0 0
\(244\) 2544.28 0.667546
\(245\) −6730.11 −1.75498
\(246\) 0 0
\(247\) 4589.14 1.18219
\(248\) 2421.95 0.620138
\(249\) 0 0
\(250\) −4234.38 −1.07122
\(251\) 1761.97 0.443087 0.221544 0.975150i \(-0.428891\pi\)
0.221544 + 0.975150i \(0.428891\pi\)
\(252\) 0 0
\(253\) 627.112 0.155835
\(254\) 2645.69 0.653563
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) −446.547 −0.108385 −0.0541923 0.998531i \(-0.517258\pi\)
−0.0541923 + 0.998531i \(0.517258\pi\)
\(258\) 0 0
\(259\) −156.822 −0.0376234
\(260\) −5314.68 −1.26770
\(261\) 0 0
\(262\) 707.182 0.166755
\(263\) −8390.91 −1.96732 −0.983661 0.180033i \(-0.942379\pi\)
−0.983661 + 0.180033i \(0.942379\pi\)
\(264\) 0 0
\(265\) 9552.20 2.21429
\(266\) −3466.96 −0.799147
\(267\) 0 0
\(268\) 824.651 0.187961
\(269\) −4219.17 −0.956310 −0.478155 0.878275i \(-0.658694\pi\)
−0.478155 + 0.878275i \(0.658694\pi\)
\(270\) 0 0
\(271\) 4711.39 1.05608 0.528038 0.849221i \(-0.322928\pi\)
0.528038 + 0.849221i \(0.322928\pi\)
\(272\) −243.852 −0.0543592
\(273\) 0 0
\(274\) 6343.59 1.39865
\(275\) 6444.89 1.41324
\(276\) 0 0
\(277\) −6017.91 −1.30535 −0.652674 0.757639i \(-0.726353\pi\)
−0.652674 + 0.757639i \(0.726353\pi\)
\(278\) 1016.11 0.219217
\(279\) 0 0
\(280\) 4015.09 0.856955
\(281\) −5210.98 −1.10627 −0.553134 0.833092i \(-0.686568\pi\)
−0.553134 + 0.833092i \(0.686568\pi\)
\(282\) 0 0
\(283\) −5156.61 −1.08314 −0.541570 0.840655i \(-0.682170\pi\)
−0.541570 + 0.840655i \(0.682170\pi\)
\(284\) 1763.79 0.368527
\(285\) 0 0
\(286\) 3811.41 0.788019
\(287\) 4396.51 0.904243
\(288\) 0 0
\(289\) −4680.72 −0.952721
\(290\) −4737.95 −0.959387
\(291\) 0 0
\(292\) 1560.98 0.312841
\(293\) 4978.19 0.992590 0.496295 0.868154i \(-0.334693\pi\)
0.496295 + 0.868154i \(0.334693\pi\)
\(294\) 0 0
\(295\) 11416.2 2.25313
\(296\) 47.5194 0.00933111
\(297\) 0 0
\(298\) −5909.24 −1.14870
\(299\) 1607.56 0.310928
\(300\) 0 0
\(301\) 8403.42 1.60919
\(302\) −6767.07 −1.28941
\(303\) 0 0
\(304\) 1050.54 0.198199
\(305\) −12091.6 −2.27004
\(306\) 0 0
\(307\) 3035.93 0.564396 0.282198 0.959356i \(-0.408937\pi\)
0.282198 + 0.959356i \(0.408937\pi\)
\(308\) −2879.41 −0.532694
\(309\) 0 0
\(310\) −11510.2 −2.10883
\(311\) 4811.62 0.877305 0.438652 0.898657i \(-0.355456\pi\)
0.438652 + 0.898657i \(0.355456\pi\)
\(312\) 0 0
\(313\) −6959.90 −1.25686 −0.628429 0.777867i \(-0.716302\pi\)
−0.628429 + 0.777867i \(0.716302\pi\)
\(314\) −587.758 −0.105634
\(315\) 0 0
\(316\) 2725.27 0.485153
\(317\) −2817.16 −0.499140 −0.249570 0.968357i \(-0.580289\pi\)
−0.249570 + 0.968357i \(0.580289\pi\)
\(318\) 0 0
\(319\) 3397.81 0.596367
\(320\) −1216.63 −0.212536
\(321\) 0 0
\(322\) −1214.46 −0.210185
\(323\) −1000.69 −0.172383
\(324\) 0 0
\(325\) 16521.0 2.81976
\(326\) 2554.35 0.433964
\(327\) 0 0
\(328\) −1332.21 −0.224265
\(329\) −3818.78 −0.639928
\(330\) 0 0
\(331\) 8262.04 1.37197 0.685986 0.727614i \(-0.259371\pi\)
0.685986 + 0.727614i \(0.259371\pi\)
\(332\) −2781.08 −0.459733
\(333\) 0 0
\(334\) 3019.68 0.494699
\(335\) −3919.12 −0.639177
\(336\) 0 0
\(337\) −8303.24 −1.34215 −0.671077 0.741387i \(-0.734168\pi\)
−0.671077 + 0.741387i \(0.734168\pi\)
\(338\) 5376.29 0.865182
\(339\) 0 0
\(340\) 1158.90 0.184853
\(341\) 8254.54 1.31088
\(342\) 0 0
\(343\) −291.293 −0.0458553
\(344\) −2546.36 −0.399100
\(345\) 0 0
\(346\) 275.554 0.0428146
\(347\) −307.537 −0.0475776 −0.0237888 0.999717i \(-0.507573\pi\)
−0.0237888 + 0.999717i \(0.507573\pi\)
\(348\) 0 0
\(349\) −10.9250 −0.00167564 −0.000837822 1.00000i \(-0.500267\pi\)
−0.000837822 1.00000i \(0.500267\pi\)
\(350\) −12481.2 −1.90613
\(351\) 0 0
\(352\) 872.503 0.132115
\(353\) −525.894 −0.0792932 −0.0396466 0.999214i \(-0.512623\pi\)
−0.0396466 + 0.999214i \(0.512623\pi\)
\(354\) 0 0
\(355\) −8382.34 −1.25321
\(356\) 3795.60 0.565074
\(357\) 0 0
\(358\) −6087.95 −0.898766
\(359\) −1726.99 −0.253891 −0.126946 0.991910i \(-0.540517\pi\)
−0.126946 + 0.991910i \(0.540517\pi\)
\(360\) 0 0
\(361\) −2547.94 −0.371473
\(362\) 7813.80 1.13449
\(363\) 0 0
\(364\) −7381.17 −1.06285
\(365\) −7418.51 −1.06384
\(366\) 0 0
\(367\) −537.816 −0.0764953 −0.0382477 0.999268i \(-0.512178\pi\)
−0.0382477 + 0.999268i \(0.512178\pi\)
\(368\) 368.000 0.0521286
\(369\) 0 0
\(370\) −225.834 −0.0317312
\(371\) 13266.4 1.85648
\(372\) 0 0
\(373\) 3074.20 0.426745 0.213373 0.976971i \(-0.431555\pi\)
0.213373 + 0.976971i \(0.431555\pi\)
\(374\) −831.100 −0.114907
\(375\) 0 0
\(376\) 1157.15 0.158711
\(377\) 8710.07 1.18990
\(378\) 0 0
\(379\) 9245.15 1.25301 0.626506 0.779417i \(-0.284485\pi\)
0.626506 + 0.779417i \(0.284485\pi\)
\(380\) −4992.64 −0.673992
\(381\) 0 0
\(382\) 2037.87 0.272948
\(383\) 2029.45 0.270758 0.135379 0.990794i \(-0.456775\pi\)
0.135379 + 0.990794i \(0.456775\pi\)
\(384\) 0 0
\(385\) 13684.3 1.81147
\(386\) −8605.84 −1.13478
\(387\) 0 0
\(388\) 2987.44 0.390888
\(389\) 7081.67 0.923020 0.461510 0.887135i \(-0.347308\pi\)
0.461510 + 0.887135i \(0.347308\pi\)
\(390\) 0 0
\(391\) −350.537 −0.0453387
\(392\) 2832.27 0.364926
\(393\) 0 0
\(394\) 2878.71 0.368090
\(395\) −12951.7 −1.64980
\(396\) 0 0
\(397\) 3974.11 0.502405 0.251203 0.967935i \(-0.419174\pi\)
0.251203 + 0.967935i \(0.419174\pi\)
\(398\) 10525.9 1.32566
\(399\) 0 0
\(400\) 3781.98 0.472747
\(401\) −1228.32 −0.152966 −0.0764830 0.997071i \(-0.524369\pi\)
−0.0764830 + 0.997071i \(0.524369\pi\)
\(402\) 0 0
\(403\) 21160.0 2.61551
\(404\) 4672.01 0.575349
\(405\) 0 0
\(406\) −6580.21 −0.804360
\(407\) 161.956 0.0197245
\(408\) 0 0
\(409\) −4559.87 −0.551274 −0.275637 0.961262i \(-0.588889\pi\)
−0.275637 + 0.961262i \(0.588889\pi\)
\(410\) 6331.25 0.762630
\(411\) 0 0
\(412\) 2433.50 0.290995
\(413\) 15855.1 1.88905
\(414\) 0 0
\(415\) 13216.9 1.56336
\(416\) 2236.60 0.263602
\(417\) 0 0
\(418\) 3580.46 0.418962
\(419\) 16364.6 1.90803 0.954013 0.299766i \(-0.0969084\pi\)
0.954013 + 0.299766i \(0.0969084\pi\)
\(420\) 0 0
\(421\) 8467.50 0.980239 0.490120 0.871655i \(-0.336953\pi\)
0.490120 + 0.871655i \(0.336953\pi\)
\(422\) 707.387 0.0815997
\(423\) 0 0
\(424\) −4019.90 −0.460433
\(425\) −3602.51 −0.411170
\(426\) 0 0
\(427\) −16793.2 −1.90323
\(428\) 4539.17 0.512638
\(429\) 0 0
\(430\) 12101.5 1.35717
\(431\) 12440.5 1.39034 0.695172 0.718843i \(-0.255328\pi\)
0.695172 + 0.718843i \(0.255328\pi\)
\(432\) 0 0
\(433\) 4079.23 0.452737 0.226369 0.974042i \(-0.427315\pi\)
0.226369 + 0.974042i \(0.427315\pi\)
\(434\) −15985.7 −1.76806
\(435\) 0 0
\(436\) 6410.45 0.704140
\(437\) 1510.15 0.165310
\(438\) 0 0
\(439\) 3486.22 0.379017 0.189508 0.981879i \(-0.439311\pi\)
0.189508 + 0.981879i \(0.439311\pi\)
\(440\) −4146.53 −0.449269
\(441\) 0 0
\(442\) −2130.47 −0.229267
\(443\) 11320.3 1.21409 0.607047 0.794666i \(-0.292354\pi\)
0.607047 + 0.794666i \(0.292354\pi\)
\(444\) 0 0
\(445\) −18038.4 −1.92158
\(446\) −10672.6 −1.13310
\(447\) 0 0
\(448\) −1689.69 −0.178193
\(449\) −16854.6 −1.77154 −0.885768 0.464129i \(-0.846367\pi\)
−0.885768 + 0.464129i \(0.846367\pi\)
\(450\) 0 0
\(451\) −4540.44 −0.474060
\(452\) −8197.47 −0.853046
\(453\) 0 0
\(454\) 4426.59 0.457600
\(455\) 35078.7 3.61432
\(456\) 0 0
\(457\) −5329.53 −0.545525 −0.272762 0.962081i \(-0.587937\pi\)
−0.272762 + 0.962081i \(0.587937\pi\)
\(458\) −10282.2 −1.04903
\(459\) 0 0
\(460\) −1748.90 −0.177268
\(461\) 8612.31 0.870098 0.435049 0.900407i \(-0.356731\pi\)
0.435049 + 0.900407i \(0.356731\pi\)
\(462\) 0 0
\(463\) −11553.9 −1.15973 −0.579867 0.814711i \(-0.696896\pi\)
−0.579867 + 0.814711i \(0.696896\pi\)
\(464\) 1993.90 0.199492
\(465\) 0 0
\(466\) 13164.7 1.30867
\(467\) 6425.53 0.636698 0.318349 0.947974i \(-0.396872\pi\)
0.318349 + 0.947974i \(0.396872\pi\)
\(468\) 0 0
\(469\) −5442.98 −0.535893
\(470\) −5499.28 −0.539709
\(471\) 0 0
\(472\) −4804.32 −0.468510
\(473\) −8678.53 −0.843635
\(474\) 0 0
\(475\) 15520.0 1.49917
\(476\) 1609.51 0.154982
\(477\) 0 0
\(478\) 12411.9 1.18767
\(479\) 8666.10 0.826647 0.413324 0.910584i \(-0.364368\pi\)
0.413324 + 0.910584i \(0.364368\pi\)
\(480\) 0 0
\(481\) 415.164 0.0393552
\(482\) −12727.0 −1.20269
\(483\) 0 0
\(484\) −2350.32 −0.220729
\(485\) −14197.7 −1.32925
\(486\) 0 0
\(487\) −18563.6 −1.72730 −0.863652 0.504088i \(-0.831829\pi\)
−0.863652 + 0.504088i \(0.831829\pi\)
\(488\) 5088.57 0.472026
\(489\) 0 0
\(490\) −13460.2 −1.24096
\(491\) −11797.2 −1.08432 −0.542161 0.840274i \(-0.682394\pi\)
−0.542161 + 0.840274i \(0.682394\pi\)
\(492\) 0 0
\(493\) −1899.28 −0.173508
\(494\) 9178.27 0.835931
\(495\) 0 0
\(496\) 4843.91 0.438504
\(497\) −11641.6 −1.05070
\(498\) 0 0
\(499\) −3453.29 −0.309801 −0.154900 0.987930i \(-0.549506\pi\)
−0.154900 + 0.987930i \(0.549506\pi\)
\(500\) −8468.77 −0.757469
\(501\) 0 0
\(502\) 3523.95 0.313310
\(503\) 9326.34 0.826721 0.413361 0.910567i \(-0.364355\pi\)
0.413361 + 0.910567i \(0.364355\pi\)
\(504\) 0 0
\(505\) −22203.5 −1.95652
\(506\) 1254.22 0.110192
\(507\) 0 0
\(508\) 5291.37 0.462139
\(509\) −12603.6 −1.09754 −0.548768 0.835975i \(-0.684903\pi\)
−0.548768 + 0.835975i \(0.684903\pi\)
\(510\) 0 0
\(511\) −10303.0 −0.891936
\(512\) 512.000 0.0441942
\(513\) 0 0
\(514\) −893.094 −0.0766394
\(515\) −11565.1 −0.989554
\(516\) 0 0
\(517\) 3943.80 0.335490
\(518\) −313.645 −0.0266038
\(519\) 0 0
\(520\) −10629.4 −0.896400
\(521\) −17821.9 −1.49864 −0.749322 0.662206i \(-0.769620\pi\)
−0.749322 + 0.662206i \(0.769620\pi\)
\(522\) 0 0
\(523\) −3816.68 −0.319104 −0.159552 0.987189i \(-0.551005\pi\)
−0.159552 + 0.987189i \(0.551005\pi\)
\(524\) 1414.36 0.117914
\(525\) 0 0
\(526\) −16781.8 −1.39111
\(527\) −4614.05 −0.381387
\(528\) 0 0
\(529\) 529.000 0.0434783
\(530\) 19104.4 1.56574
\(531\) 0 0
\(532\) −6933.92 −0.565082
\(533\) −11639.1 −0.945865
\(534\) 0 0
\(535\) −21572.2 −1.74327
\(536\) 1649.30 0.132909
\(537\) 0 0
\(538\) −8438.34 −0.676213
\(539\) 9652.97 0.771397
\(540\) 0 0
\(541\) 12730.5 1.01169 0.505846 0.862624i \(-0.331180\pi\)
0.505846 + 0.862624i \(0.331180\pi\)
\(542\) 9422.78 0.746759
\(543\) 0 0
\(544\) −487.704 −0.0384377
\(545\) −30465.4 −2.39449
\(546\) 0 0
\(547\) 175.554 0.0137224 0.00686119 0.999976i \(-0.497816\pi\)
0.00686119 + 0.999976i \(0.497816\pi\)
\(548\) 12687.2 0.988995
\(549\) 0 0
\(550\) 12889.8 0.999313
\(551\) 8182.29 0.632627
\(552\) 0 0
\(553\) −17987.7 −1.38321
\(554\) −12035.8 −0.923020
\(555\) 0 0
\(556\) 2032.22 0.155010
\(557\) 4603.34 0.350179 0.175089 0.984553i \(-0.443979\pi\)
0.175089 + 0.984553i \(0.443979\pi\)
\(558\) 0 0
\(559\) −22246.8 −1.68326
\(560\) 8030.17 0.605958
\(561\) 0 0
\(562\) −10422.0 −0.782249
\(563\) 3425.06 0.256393 0.128196 0.991749i \(-0.459081\pi\)
0.128196 + 0.991749i \(0.459081\pi\)
\(564\) 0 0
\(565\) 38958.1 2.90085
\(566\) −10313.2 −0.765896
\(567\) 0 0
\(568\) 3527.58 0.260588
\(569\) −23899.8 −1.76086 −0.880431 0.474175i \(-0.842746\pi\)
−0.880431 + 0.474175i \(0.842746\pi\)
\(570\) 0 0
\(571\) −12192.1 −0.893563 −0.446781 0.894643i \(-0.647430\pi\)
−0.446781 + 0.894643i \(0.647430\pi\)
\(572\) 7622.82 0.557214
\(573\) 0 0
\(574\) 8793.02 0.639397
\(575\) 5436.59 0.394298
\(576\) 0 0
\(577\) 16729.8 1.20706 0.603529 0.797341i \(-0.293761\pi\)
0.603529 + 0.797341i \(0.293761\pi\)
\(578\) −9361.44 −0.673676
\(579\) 0 0
\(580\) −9475.91 −0.678389
\(581\) 18356.1 1.31074
\(582\) 0 0
\(583\) −13700.7 −0.973283
\(584\) 3121.97 0.221212
\(585\) 0 0
\(586\) 9956.38 0.701867
\(587\) 10502.3 0.738463 0.369231 0.929338i \(-0.379621\pi\)
0.369231 + 0.929338i \(0.379621\pi\)
\(588\) 0 0
\(589\) 19877.8 1.39058
\(590\) 22832.3 1.59321
\(591\) 0 0
\(592\) 95.0388 0.00659809
\(593\) 19802.9 1.37134 0.685671 0.727912i \(-0.259509\pi\)
0.685671 + 0.727912i \(0.259509\pi\)
\(594\) 0 0
\(595\) −7649.11 −0.527030
\(596\) −11818.5 −0.812255
\(597\) 0 0
\(598\) 3215.12 0.219859
\(599\) −26702.7 −1.82144 −0.910720 0.413024i \(-0.864472\pi\)
−0.910720 + 0.413024i \(0.864472\pi\)
\(600\) 0 0
\(601\) −18863.6 −1.28030 −0.640150 0.768250i \(-0.721128\pi\)
−0.640150 + 0.768250i \(0.721128\pi\)
\(602\) 16806.8 1.13787
\(603\) 0 0
\(604\) −13534.1 −0.911749
\(605\) 11169.8 0.750607
\(606\) 0 0
\(607\) 10095.7 0.675079 0.337539 0.941311i \(-0.390405\pi\)
0.337539 + 0.941311i \(0.390405\pi\)
\(608\) 2101.08 0.140148
\(609\) 0 0
\(610\) −24183.2 −1.60516
\(611\) 10109.7 0.669383
\(612\) 0 0
\(613\) 770.460 0.0507644 0.0253822 0.999678i \(-0.491920\pi\)
0.0253822 + 0.999678i \(0.491920\pi\)
\(614\) 6071.85 0.399088
\(615\) 0 0
\(616\) −5758.82 −0.376671
\(617\) −6297.91 −0.410931 −0.205465 0.978664i \(-0.565871\pi\)
−0.205465 + 0.978664i \(0.565871\pi\)
\(618\) 0 0
\(619\) 893.359 0.0580083 0.0290041 0.999579i \(-0.490766\pi\)
0.0290041 + 0.999579i \(0.490766\pi\)
\(620\) −23020.5 −1.49117
\(621\) 0 0
\(622\) 9623.23 0.620348
\(623\) −25052.3 −1.61107
\(624\) 0 0
\(625\) 10700.8 0.684848
\(626\) −13919.8 −0.888733
\(627\) 0 0
\(628\) −1175.52 −0.0746945
\(629\) −90.5289 −0.00573867
\(630\) 0 0
\(631\) 16031.2 1.01140 0.505698 0.862711i \(-0.331235\pi\)
0.505698 + 0.862711i \(0.331235\pi\)
\(632\) 5450.54 0.343055
\(633\) 0 0
\(634\) −5634.31 −0.352945
\(635\) −25147.0 −1.57154
\(636\) 0 0
\(637\) 24744.7 1.53912
\(638\) 6795.63 0.421695
\(639\) 0 0
\(640\) −2433.26 −0.150286
\(641\) 11366.2 0.700370 0.350185 0.936681i \(-0.386119\pi\)
0.350185 + 0.936681i \(0.386119\pi\)
\(642\) 0 0
\(643\) −267.056 −0.0163789 −0.00818946 0.999966i \(-0.502607\pi\)
−0.00818946 + 0.999966i \(0.502607\pi\)
\(644\) −2428.93 −0.148623
\(645\) 0 0
\(646\) −2001.37 −0.121893
\(647\) −10111.1 −0.614389 −0.307194 0.951647i \(-0.599390\pi\)
−0.307194 + 0.951647i \(0.599390\pi\)
\(648\) 0 0
\(649\) −16374.1 −0.990357
\(650\) 33042.1 1.99387
\(651\) 0 0
\(652\) 5108.69 0.306859
\(653\) 12953.9 0.776299 0.388150 0.921596i \(-0.373114\pi\)
0.388150 + 0.921596i \(0.373114\pi\)
\(654\) 0 0
\(655\) −6721.70 −0.400975
\(656\) −2664.41 −0.158579
\(657\) 0 0
\(658\) −7637.56 −0.452497
\(659\) −920.513 −0.0544129 −0.0272064 0.999630i \(-0.508661\pi\)
−0.0272064 + 0.999630i \(0.508661\pi\)
\(660\) 0 0
\(661\) 6224.81 0.366289 0.183144 0.983086i \(-0.441372\pi\)
0.183144 + 0.983086i \(0.441372\pi\)
\(662\) 16524.1 0.970131
\(663\) 0 0
\(664\) −5562.15 −0.325080
\(665\) 32953.2 1.92161
\(666\) 0 0
\(667\) 2866.23 0.166388
\(668\) 6039.35 0.349805
\(669\) 0 0
\(670\) −7838.24 −0.451966
\(671\) 17342.9 0.997789
\(672\) 0 0
\(673\) 7195.10 0.412111 0.206056 0.978540i \(-0.433937\pi\)
0.206056 + 0.978540i \(0.433937\pi\)
\(674\) −16606.5 −0.949047
\(675\) 0 0
\(676\) 10752.6 0.611776
\(677\) −2584.01 −0.146694 −0.0733468 0.997306i \(-0.523368\pi\)
−0.0733468 + 0.997306i \(0.523368\pi\)
\(678\) 0 0
\(679\) −19718.2 −1.11445
\(680\) 2317.79 0.130711
\(681\) 0 0
\(682\) 16509.1 0.926929
\(683\) −15445.0 −0.865280 −0.432640 0.901567i \(-0.642418\pi\)
−0.432640 + 0.901567i \(0.642418\pi\)
\(684\) 0 0
\(685\) −60295.3 −3.36316
\(686\) −582.587 −0.0324246
\(687\) 0 0
\(688\) −5092.71 −0.282206
\(689\) −35120.7 −1.94194
\(690\) 0 0
\(691\) 28002.8 1.54164 0.770822 0.637050i \(-0.219846\pi\)
0.770822 + 0.637050i \(0.219846\pi\)
\(692\) 551.107 0.0302745
\(693\) 0 0
\(694\) −615.074 −0.0336425
\(695\) −9658.05 −0.527123
\(696\) 0 0
\(697\) 2537.98 0.137924
\(698\) −21.8499 −0.00118486
\(699\) 0 0
\(700\) −24962.4 −1.34784
\(701\) −31379.7 −1.69072 −0.845360 0.534198i \(-0.820614\pi\)
−0.845360 + 0.534198i \(0.820614\pi\)
\(702\) 0 0
\(703\) 390.008 0.0209238
\(704\) 1745.01 0.0934196
\(705\) 0 0
\(706\) −1051.79 −0.0560688
\(707\) −30836.9 −1.64037
\(708\) 0 0
\(709\) −12612.6 −0.668092 −0.334046 0.942557i \(-0.608414\pi\)
−0.334046 + 0.942557i \(0.608414\pi\)
\(710\) −16764.7 −0.886151
\(711\) 0 0
\(712\) 7591.20 0.399568
\(713\) 6963.12 0.365738
\(714\) 0 0
\(715\) −36227.1 −1.89485
\(716\) −12175.9 −0.635523
\(717\) 0 0
\(718\) −3453.98 −0.179528
\(719\) −8522.72 −0.442064 −0.221032 0.975267i \(-0.570942\pi\)
−0.221032 + 0.975267i \(0.570942\pi\)
\(720\) 0 0
\(721\) −16062.0 −0.829652
\(722\) −5095.87 −0.262671
\(723\) 0 0
\(724\) 15627.6 0.802204
\(725\) 29456.5 1.50895
\(726\) 0 0
\(727\) 9853.76 0.502690 0.251345 0.967898i \(-0.419127\pi\)
0.251345 + 0.967898i \(0.419127\pi\)
\(728\) −14762.3 −0.751551
\(729\) 0 0
\(730\) −14837.0 −0.752250
\(731\) 4851.05 0.245448
\(732\) 0 0
\(733\) −21851.1 −1.10107 −0.550537 0.834810i \(-0.685577\pi\)
−0.550537 + 0.834810i \(0.685577\pi\)
\(734\) −1075.63 −0.0540903
\(735\) 0 0
\(736\) 736.000 0.0368605
\(737\) 5621.18 0.280948
\(738\) 0 0
\(739\) 9049.47 0.450460 0.225230 0.974306i \(-0.427687\pi\)
0.225230 + 0.974306i \(0.427687\pi\)
\(740\) −451.668 −0.0224373
\(741\) 0 0
\(742\) 26532.7 1.31273
\(743\) −10650.0 −0.525855 −0.262927 0.964816i \(-0.584688\pi\)
−0.262927 + 0.964816i \(0.584688\pi\)
\(744\) 0 0
\(745\) 56166.8 2.76214
\(746\) 6148.39 0.301754
\(747\) 0 0
\(748\) −1662.20 −0.0812514
\(749\) −29960.1 −1.46157
\(750\) 0 0
\(751\) −9340.20 −0.453833 −0.226917 0.973914i \(-0.572864\pi\)
−0.226917 + 0.973914i \(0.572864\pi\)
\(752\) 2314.29 0.112225
\(753\) 0 0
\(754\) 17420.1 0.841384
\(755\) 64320.4 3.10048
\(756\) 0 0
\(757\) −12383.4 −0.594562 −0.297281 0.954790i \(-0.596080\pi\)
−0.297281 + 0.954790i \(0.596080\pi\)
\(758\) 18490.3 0.886013
\(759\) 0 0
\(760\) −9985.28 −0.476585
\(761\) 5906.69 0.281363 0.140682 0.990055i \(-0.455071\pi\)
0.140682 + 0.990055i \(0.455071\pi\)
\(762\) 0 0
\(763\) −42311.2 −2.00756
\(764\) 4075.73 0.193004
\(765\) 0 0
\(766\) 4058.91 0.191455
\(767\) −41974.0 −1.97600
\(768\) 0 0
\(769\) 17056.5 0.799834 0.399917 0.916551i \(-0.369039\pi\)
0.399917 + 0.916551i \(0.369039\pi\)
\(770\) 27368.6 1.28090
\(771\) 0 0
\(772\) −17211.7 −0.802412
\(773\) −28498.6 −1.32603 −0.663017 0.748604i \(-0.730724\pi\)
−0.663017 + 0.748604i \(0.730724\pi\)
\(774\) 0 0
\(775\) 71560.7 3.31682
\(776\) 5974.89 0.276399
\(777\) 0 0
\(778\) 14163.3 0.652674
\(779\) −10933.9 −0.502884
\(780\) 0 0
\(781\) 12022.8 0.550843
\(782\) −701.074 −0.0320593
\(783\) 0 0
\(784\) 5664.53 0.258042
\(785\) 5586.59 0.254005
\(786\) 0 0
\(787\) 30401.9 1.37701 0.688507 0.725230i \(-0.258266\pi\)
0.688507 + 0.725230i \(0.258266\pi\)
\(788\) 5757.43 0.260279
\(789\) 0 0
\(790\) −25903.4 −1.16659
\(791\) 54106.2 2.43210
\(792\) 0 0
\(793\) 44457.4 1.99083
\(794\) 7948.22 0.355254
\(795\) 0 0
\(796\) 21051.7 0.937385
\(797\) −3453.59 −0.153491 −0.0767456 0.997051i \(-0.524453\pi\)
−0.0767456 + 0.997051i \(0.524453\pi\)
\(798\) 0 0
\(799\) −2204.47 −0.0976077
\(800\) 7563.95 0.334283
\(801\) 0 0
\(802\) −2456.64 −0.108163
\(803\) 10640.3 0.467608
\(804\) 0 0
\(805\) 11543.4 0.505404
\(806\) 42319.9 1.84945
\(807\) 0 0
\(808\) 9344.01 0.406833
\(809\) 31655.0 1.37569 0.687844 0.725859i \(-0.258557\pi\)
0.687844 + 0.725859i \(0.258557\pi\)
\(810\) 0 0
\(811\) −2071.19 −0.0896786 −0.0448393 0.998994i \(-0.514278\pi\)
−0.0448393 + 0.998994i \(0.514278\pi\)
\(812\) −13160.4 −0.568768
\(813\) 0 0
\(814\) 323.913 0.0139473
\(815\) −24278.8 −1.04350
\(816\) 0 0
\(817\) −20898.8 −0.894929
\(818\) −9119.74 −0.389809
\(819\) 0 0
\(820\) 12662.5 0.539261
\(821\) 3082.16 0.131021 0.0655105 0.997852i \(-0.479132\pi\)
0.0655105 + 0.997852i \(0.479132\pi\)
\(822\) 0 0
\(823\) −16572.0 −0.701901 −0.350951 0.936394i \(-0.614142\pi\)
−0.350951 + 0.936394i \(0.614142\pi\)
\(824\) 4867.01 0.205765
\(825\) 0 0
\(826\) 31710.2 1.33576
\(827\) 5374.02 0.225965 0.112983 0.993597i \(-0.463960\pi\)
0.112983 + 0.993597i \(0.463960\pi\)
\(828\) 0 0
\(829\) −19312.7 −0.809117 −0.404559 0.914512i \(-0.632575\pi\)
−0.404559 + 0.914512i \(0.632575\pi\)
\(830\) 26433.9 1.10546
\(831\) 0 0
\(832\) 4473.20 0.186395
\(833\) −5395.73 −0.224431
\(834\) 0 0
\(835\) −28701.8 −1.18954
\(836\) 7160.93 0.296251
\(837\) 0 0
\(838\) 32729.2 1.34918
\(839\) 8072.94 0.332192 0.166096 0.986110i \(-0.446884\pi\)
0.166096 + 0.986110i \(0.446884\pi\)
\(840\) 0 0
\(841\) −8859.21 −0.363246
\(842\) 16935.0 0.693134
\(843\) 0 0
\(844\) 1414.77 0.0576997
\(845\) −51101.2 −2.08039
\(846\) 0 0
\(847\) 15512.9 0.629316
\(848\) −8039.80 −0.325575
\(849\) 0 0
\(850\) −7205.02 −0.290741
\(851\) 136.618 0.00550319
\(852\) 0 0
\(853\) −3676.62 −0.147579 −0.0737896 0.997274i \(-0.523509\pi\)
−0.0737896 + 0.997274i \(0.523509\pi\)
\(854\) −33586.3 −1.34578
\(855\) 0 0
\(856\) 9078.34 0.362490
\(857\) 33806.9 1.34752 0.673758 0.738952i \(-0.264679\pi\)
0.673758 + 0.738952i \(0.264679\pi\)
\(858\) 0 0
\(859\) −16752.0 −0.665391 −0.332696 0.943034i \(-0.607958\pi\)
−0.332696 + 0.943034i \(0.607958\pi\)
\(860\) 24202.9 0.959665
\(861\) 0 0
\(862\) 24881.0 0.983122
\(863\) −26052.3 −1.02761 −0.513806 0.857906i \(-0.671765\pi\)
−0.513806 + 0.857906i \(0.671765\pi\)
\(864\) 0 0
\(865\) −2619.11 −0.102951
\(866\) 8158.46 0.320134
\(867\) 0 0
\(868\) −31971.5 −1.25021
\(869\) 18576.6 0.725165
\(870\) 0 0
\(871\) 14409.5 0.560560
\(872\) 12820.9 0.497902
\(873\) 0 0
\(874\) 3020.30 0.116891
\(875\) 55896.8 2.15961
\(876\) 0 0
\(877\) −7214.34 −0.277778 −0.138889 0.990308i \(-0.544353\pi\)
−0.138889 + 0.990308i \(0.544353\pi\)
\(878\) 6972.45 0.268005
\(879\) 0 0
\(880\) −8293.07 −0.317681
\(881\) −30172.3 −1.15384 −0.576919 0.816801i \(-0.695745\pi\)
−0.576919 + 0.816801i \(0.695745\pi\)
\(882\) 0 0
\(883\) 37498.4 1.42913 0.714564 0.699570i \(-0.246625\pi\)
0.714564 + 0.699570i \(0.246625\pi\)
\(884\) −4260.93 −0.162116
\(885\) 0 0
\(886\) 22640.6 0.858494
\(887\) 17991.8 0.681065 0.340533 0.940233i \(-0.389393\pi\)
0.340533 + 0.940233i \(0.389393\pi\)
\(888\) 0 0
\(889\) −34924.9 −1.31760
\(890\) −36076.8 −1.35876
\(891\) 0 0
\(892\) −21345.2 −0.801223
\(893\) 9497.08 0.355888
\(894\) 0 0
\(895\) 57865.4 2.16115
\(896\) −3379.38 −0.126001
\(897\) 0 0
\(898\) −33709.3 −1.25266
\(899\) 37727.6 1.39965
\(900\) 0 0
\(901\) 7658.28 0.283168
\(902\) −9080.89 −0.335211
\(903\) 0 0
\(904\) −16394.9 −0.603194
\(905\) −74269.5 −2.72796
\(906\) 0 0
\(907\) 44123.2 1.61531 0.807655 0.589655i \(-0.200736\pi\)
0.807655 + 0.589655i \(0.200736\pi\)
\(908\) 8853.19 0.323572
\(909\) 0 0
\(910\) 70157.4 2.55571
\(911\) −10096.6 −0.367196 −0.183598 0.983001i \(-0.558774\pi\)
−0.183598 + 0.983001i \(0.558774\pi\)
\(912\) 0 0
\(913\) −18957.0 −0.687169
\(914\) −10659.1 −0.385744
\(915\) 0 0
\(916\) −20564.4 −0.741774
\(917\) −9335.29 −0.336181
\(918\) 0 0
\(919\) 40900.3 1.46809 0.734045 0.679100i \(-0.237630\pi\)
0.734045 + 0.679100i \(0.237630\pi\)
\(920\) −3497.81 −0.125347
\(921\) 0 0
\(922\) 17224.6 0.615252
\(923\) 30819.5 1.09907
\(924\) 0 0
\(925\) 1404.04 0.0499077
\(926\) −23107.9 −0.820056
\(927\) 0 0
\(928\) 3987.79 0.141062
\(929\) 22417.0 0.791688 0.395844 0.918318i \(-0.370452\pi\)
0.395844 + 0.918318i \(0.370452\pi\)
\(930\) 0 0
\(931\) 23245.4 0.818298
\(932\) 26329.3 0.925372
\(933\) 0 0
\(934\) 12851.1 0.450214
\(935\) 7899.53 0.276302
\(936\) 0 0
\(937\) 3106.56 0.108310 0.0541552 0.998533i \(-0.482753\pi\)
0.0541552 + 0.998533i \(0.482753\pi\)
\(938\) −10886.0 −0.378933
\(939\) 0 0
\(940\) −10998.6 −0.381632
\(941\) −19428.6 −0.673066 −0.336533 0.941672i \(-0.609254\pi\)
−0.336533 + 0.941672i \(0.609254\pi\)
\(942\) 0 0
\(943\) −3830.09 −0.132264
\(944\) −9608.63 −0.331286
\(945\) 0 0
\(946\) −17357.1 −0.596540
\(947\) −26080.2 −0.894924 −0.447462 0.894303i \(-0.647672\pi\)
−0.447462 + 0.894303i \(0.647672\pi\)
\(948\) 0 0
\(949\) 27275.8 0.932992
\(950\) 31040.0 1.06007
\(951\) 0 0
\(952\) 3219.01 0.109589
\(953\) −12821.6 −0.435815 −0.217907 0.975969i \(-0.569923\pi\)
−0.217907 + 0.975969i \(0.569923\pi\)
\(954\) 0 0
\(955\) −19369.7 −0.656325
\(956\) 24823.8 0.839812
\(957\) 0 0
\(958\) 17332.2 0.584528
\(959\) −83739.8 −2.81971
\(960\) 0 0
\(961\) 61863.1 2.07657
\(962\) 830.328 0.0278283
\(963\) 0 0
\(964\) −25453.9 −0.850432
\(965\) 81797.8 2.72867
\(966\) 0 0
\(967\) −4778.68 −0.158916 −0.0794582 0.996838i \(-0.525319\pi\)
−0.0794582 + 0.996838i \(0.525319\pi\)
\(968\) −4700.64 −0.156079
\(969\) 0 0
\(970\) −28395.4 −0.939918
\(971\) −7439.58 −0.245878 −0.122939 0.992414i \(-0.539232\pi\)
−0.122939 + 0.992414i \(0.539232\pi\)
\(972\) 0 0
\(973\) −13413.4 −0.441945
\(974\) −37127.2 −1.22139
\(975\) 0 0
\(976\) 10177.1 0.333773
\(977\) 9975.97 0.326673 0.163336 0.986570i \(-0.447774\pi\)
0.163336 + 0.986570i \(0.447774\pi\)
\(978\) 0 0
\(979\) 25872.4 0.844624
\(980\) −26920.4 −0.877492
\(981\) 0 0
\(982\) −23594.5 −0.766732
\(983\) 21419.6 0.694995 0.347497 0.937681i \(-0.387032\pi\)
0.347497 + 0.937681i \(0.387032\pi\)
\(984\) 0 0
\(985\) −27361.9 −0.885100
\(986\) −3798.56 −0.122688
\(987\) 0 0
\(988\) 18356.5 0.591093
\(989\) −7320.78 −0.235376
\(990\) 0 0
\(991\) −43927.4 −1.40807 −0.704037 0.710164i \(-0.748621\pi\)
−0.704037 + 0.710164i \(0.748621\pi\)
\(992\) 9687.82 0.310069
\(993\) 0 0
\(994\) −23283.3 −0.742958
\(995\) −100047. −3.18765
\(996\) 0 0
\(997\) −39862.3 −1.26625 −0.633125 0.774049i \(-0.718228\pi\)
−0.633125 + 0.774049i \(0.718228\pi\)
\(998\) −6906.58 −0.219062
\(999\) 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.n.1.1 yes 4
3.2 odd 2 414.4.a.m.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
414.4.a.m.1.4 4 3.2 odd 2
414.4.a.n.1.1 yes 4 1.1 even 1 trivial