Properties

Label 2175.4.a.a.1.1
Level $2175$
Weight $4$
Character 2175.1
Self dual yes
Analytic conductor $128.329$
Analytic rank $2$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2175,4,Mod(1,2175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2175.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2175.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: \(128.329154262\)
Analytic rank: \(2\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2175.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.00000 q^{2} +3.00000 q^{3} +17.0000 q^{4} -15.0000 q^{6} -16.0000 q^{7} -45.0000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-5.00000 q^{2} +3.00000 q^{3} +17.0000 q^{4} -15.0000 q^{6} -16.0000 q^{7} -45.0000 q^{8} +9.00000 q^{9} -44.0000 q^{11} +51.0000 q^{12} -78.0000 q^{13} +80.0000 q^{14} +89.0000 q^{16} -18.0000 q^{17} -45.0000 q^{18} -28.0000 q^{19} -48.0000 q^{21} +220.000 q^{22} -184.000 q^{23} -135.000 q^{24} +390.000 q^{26} +27.0000 q^{27} -272.000 q^{28} +29.0000 q^{29} -224.000 q^{31} -85.0000 q^{32} -132.000 q^{33} +90.0000 q^{34} +153.000 q^{36} -254.000 q^{37} +140.000 q^{38} -234.000 q^{39} -78.0000 q^{41} +240.000 q^{42} +260.000 q^{43} -748.000 q^{44} +920.000 q^{46} -312.000 q^{47} +267.000 q^{48} -87.0000 q^{49} -54.0000 q^{51} -1326.00 q^{52} -574.000 q^{53} -135.000 q^{54} +720.000 q^{56} -84.0000 q^{57} -145.000 q^{58} +180.000 q^{59} -610.000 q^{61} +1120.00 q^{62} -144.000 q^{63} -287.000 q^{64} +660.000 q^{66} +340.000 q^{67} -306.000 q^{68} -552.000 q^{69} +296.000 q^{71} -405.000 q^{72} -394.000 q^{73} +1270.00 q^{74} -476.000 q^{76} +704.000 q^{77} +1170.00 q^{78} -960.000 q^{79} +81.0000 q^{81} +390.000 q^{82} +908.000 q^{83} -816.000 q^{84} -1300.00 q^{86} +87.0000 q^{87} +1980.00 q^{88} -990.000 q^{89} +1248.00 q^{91} -3128.00 q^{92} -672.000 q^{93} +1560.00 q^{94} -255.000 q^{96} -1234.00 q^{97} +435.000 q^{98} -396.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.00000 −1.76777 −0.883883 0.467707i \(-0.845080\pi\)
−0.883883 + 0.467707i \(0.845080\pi\)
\(3\) 3.00000 0.577350
\(4\) 17.0000 2.12500
\(5\) 0 0
\(6\) −15.0000 −1.02062
\(7\) −16.0000 −0.863919 −0.431959 0.901893i \(-0.642178\pi\)
−0.431959 + 0.901893i \(0.642178\pi\)
\(8\) −45.0000 −1.98874
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −44.0000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 51.0000 1.22687
\(13\) −78.0000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 80.0000 1.52721
\(15\) 0 0
\(16\) 89.0000 1.39062
\(17\) −18.0000 −0.256802 −0.128401 0.991722i \(-0.540985\pi\)
−0.128401 + 0.991722i \(0.540985\pi\)
\(18\) −45.0000 −0.589256
\(19\) −28.0000 −0.338086 −0.169043 0.985609i \(-0.554068\pi\)
−0.169043 + 0.985609i \(0.554068\pi\)
\(20\) 0 0
\(21\) −48.0000 −0.498784
\(22\) 220.000 2.13201
\(23\) −184.000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) −135.000 −1.14820
\(25\) 0 0
\(26\) 390.000 2.94174
\(27\) 27.0000 0.192450
\(28\) −272.000 −1.83583
\(29\) 29.0000 0.185695
\(30\) 0 0
\(31\) −224.000 −1.29779 −0.648897 0.760877i \(-0.724769\pi\)
−0.648897 + 0.760877i \(0.724769\pi\)
\(32\) −85.0000 −0.469563
\(33\) −132.000 −0.696311
\(34\) 90.0000 0.453967
\(35\) 0 0
\(36\) 153.000 0.708333
\(37\) −254.000 −1.12858 −0.564288 0.825578i \(-0.690849\pi\)
−0.564288 + 0.825578i \(0.690849\pi\)
\(38\) 140.000 0.597658
\(39\) −234.000 −0.960769
\(40\) 0 0
\(41\) −78.0000 −0.297111 −0.148556 0.988904i \(-0.547462\pi\)
−0.148556 + 0.988904i \(0.547462\pi\)
\(42\) 240.000 0.881733
\(43\) 260.000 0.922084 0.461042 0.887378i \(-0.347476\pi\)
0.461042 + 0.887378i \(0.347476\pi\)
\(44\) −748.000 −2.56285
\(45\) 0 0
\(46\) 920.000 2.94884
\(47\) −312.000 −0.968295 −0.484148 0.874986i \(-0.660870\pi\)
−0.484148 + 0.874986i \(0.660870\pi\)
\(48\) 267.000 0.802878
\(49\) −87.0000 −0.253644
\(50\) 0 0
\(51\) −54.0000 −0.148265
\(52\) −1326.00 −3.53621
\(53\) −574.000 −1.48764 −0.743820 0.668380i \(-0.766988\pi\)
−0.743820 + 0.668380i \(0.766988\pi\)
\(54\) −135.000 −0.340207
\(55\) 0 0
\(56\) 720.000 1.71811
\(57\) −84.0000 −0.195194
\(58\) −145.000 −0.328266
\(59\) 180.000 0.397187 0.198593 0.980082i \(-0.436363\pi\)
0.198593 + 0.980082i \(0.436363\pi\)
\(60\) 0 0
\(61\) −610.000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 1120.00 2.29420
\(63\) −144.000 −0.287973
\(64\) −287.000 −0.560547
\(65\) 0 0
\(66\) 660.000 1.23091
\(67\) 340.000 0.619964 0.309982 0.950742i \(-0.399677\pi\)
0.309982 + 0.950742i \(0.399677\pi\)
\(68\) −306.000 −0.545705
\(69\) −552.000 −0.963087
\(70\) 0 0
\(71\) 296.000 0.494771 0.247385 0.968917i \(-0.420429\pi\)
0.247385 + 0.968917i \(0.420429\pi\)
\(72\) −405.000 −0.662913
\(73\) −394.000 −0.631702 −0.315851 0.948809i \(-0.602290\pi\)
−0.315851 + 0.948809i \(0.602290\pi\)
\(74\) 1270.00 1.99506
\(75\) 0 0
\(76\) −476.000 −0.718433
\(77\) 704.000 1.04193
\(78\) 1170.00 1.69842
\(79\) −960.000 −1.36720 −0.683598 0.729859i \(-0.739586\pi\)
−0.683598 + 0.729859i \(0.739586\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 390.000 0.525223
\(83\) 908.000 1.20079 0.600397 0.799702i \(-0.295009\pi\)
0.600397 + 0.799702i \(0.295009\pi\)
\(84\) −816.000 −1.05992
\(85\) 0 0
\(86\) −1300.00 −1.63003
\(87\) 87.0000 0.107211
\(88\) 1980.00 2.39851
\(89\) −990.000 −1.17910 −0.589549 0.807732i \(-0.700695\pi\)
−0.589549 + 0.807732i \(0.700695\pi\)
\(90\) 0 0
\(91\) 1248.00 1.43765
\(92\) −3128.00 −3.54475
\(93\) −672.000 −0.749281
\(94\) 1560.00 1.71172
\(95\) 0 0
\(96\) −255.000 −0.271102
\(97\) −1234.00 −1.29169 −0.645844 0.763469i \(-0.723494\pi\)
−0.645844 + 0.763469i \(0.723494\pi\)
\(98\) 435.000 0.448384
\(99\) −396.000 −0.402015
\(100\) 0 0
\(101\) 1022.00 1.00686 0.503430 0.864036i \(-0.332071\pi\)
0.503430 + 0.864036i \(0.332071\pi\)
\(102\) 270.000 0.262098
\(103\) 1248.00 1.19387 0.596937 0.802288i \(-0.296384\pi\)
0.596937 + 0.802288i \(0.296384\pi\)
\(104\) 3510.00 3.30946
\(105\) 0 0
\(106\) 2870.00 2.62980
\(107\) 116.000 0.104805 0.0524025 0.998626i \(-0.483312\pi\)
0.0524025 + 0.998626i \(0.483312\pi\)
\(108\) 459.000 0.408956
\(109\) −826.000 −0.725839 −0.362920 0.931820i \(-0.618220\pi\)
−0.362920 + 0.931820i \(0.618220\pi\)
\(110\) 0 0
\(111\) −762.000 −0.651584
\(112\) −1424.00 −1.20139
\(113\) 2206.00 1.83649 0.918243 0.396016i \(-0.129608\pi\)
0.918243 + 0.396016i \(0.129608\pi\)
\(114\) 420.000 0.345058
\(115\) 0 0
\(116\) 493.000 0.394603
\(117\) −702.000 −0.554700
\(118\) −900.000 −0.702133
\(119\) 288.000 0.221856
\(120\) 0 0
\(121\) 605.000 0.454545
\(122\) 3050.00 2.26339
\(123\) −234.000 −0.171537
\(124\) −3808.00 −2.75781
\(125\) 0 0
\(126\) 720.000 0.509069
\(127\) 2056.00 1.43654 0.718270 0.695765i \(-0.244934\pi\)
0.718270 + 0.695765i \(0.244934\pi\)
\(128\) 2115.00 1.46048
\(129\) 780.000 0.532366
\(130\) 0 0
\(131\) 12.0000 0.00800340 0.00400170 0.999992i \(-0.498726\pi\)
0.00400170 + 0.999992i \(0.498726\pi\)
\(132\) −2244.00 −1.47966
\(133\) 448.000 0.292079
\(134\) −1700.00 −1.09595
\(135\) 0 0
\(136\) 810.000 0.510713
\(137\) 2758.00 1.71994 0.859970 0.510344i \(-0.170482\pi\)
0.859970 + 0.510344i \(0.170482\pi\)
\(138\) 2760.00 1.70251
\(139\) −1436.00 −0.876258 −0.438129 0.898912i \(-0.644359\pi\)
−0.438129 + 0.898912i \(0.644359\pi\)
\(140\) 0 0
\(141\) −936.000 −0.559046
\(142\) −1480.00 −0.874640
\(143\) 3432.00 2.00698
\(144\) 801.000 0.463542
\(145\) 0 0
\(146\) 1970.00 1.11670
\(147\) −261.000 −0.146442
\(148\) −4318.00 −2.39823
\(149\) −498.000 −0.273810 −0.136905 0.990584i \(-0.543716\pi\)
−0.136905 + 0.990584i \(0.543716\pi\)
\(150\) 0 0
\(151\) −2696.00 −1.45296 −0.726481 0.687186i \(-0.758846\pi\)
−0.726481 + 0.687186i \(0.758846\pi\)
\(152\) 1260.00 0.672365
\(153\) −162.000 −0.0856008
\(154\) −3520.00 −1.84188
\(155\) 0 0
\(156\) −3978.00 −2.04163
\(157\) −534.000 −0.271451 −0.135726 0.990746i \(-0.543337\pi\)
−0.135726 + 0.990746i \(0.543337\pi\)
\(158\) 4800.00 2.41688
\(159\) −1722.00 −0.858890
\(160\) 0 0
\(161\) 2944.00 1.44112
\(162\) −405.000 −0.196419
\(163\) −1380.00 −0.663128 −0.331564 0.943433i \(-0.607576\pi\)
−0.331564 + 0.943433i \(0.607576\pi\)
\(164\) −1326.00 −0.631361
\(165\) 0 0
\(166\) −4540.00 −2.12272
\(167\) −2616.00 −1.21217 −0.606084 0.795400i \(-0.707261\pi\)
−0.606084 + 0.795400i \(0.707261\pi\)
\(168\) 2160.00 0.991950
\(169\) 3887.00 1.76923
\(170\) 0 0
\(171\) −252.000 −0.112695
\(172\) 4420.00 1.95943
\(173\) 330.000 0.145026 0.0725128 0.997367i \(-0.476898\pi\)
0.0725128 + 0.997367i \(0.476898\pi\)
\(174\) −435.000 −0.189525
\(175\) 0 0
\(176\) −3916.00 −1.67716
\(177\) 540.000 0.229316
\(178\) 4950.00 2.08437
\(179\) −372.000 −0.155333 −0.0776664 0.996979i \(-0.524747\pi\)
−0.0776664 + 0.996979i \(0.524747\pi\)
\(180\) 0 0
\(181\) −1010.00 −0.414766 −0.207383 0.978260i \(-0.566495\pi\)
−0.207383 + 0.978260i \(0.566495\pi\)
\(182\) −6240.00 −2.54143
\(183\) −1830.00 −0.739221
\(184\) 8280.00 3.31744
\(185\) 0 0
\(186\) 3360.00 1.32455
\(187\) 792.000 0.309715
\(188\) −5304.00 −2.05763
\(189\) −432.000 −0.166261
\(190\) 0 0
\(191\) 2008.00 0.760700 0.380350 0.924843i \(-0.375803\pi\)
0.380350 + 0.924843i \(0.375803\pi\)
\(192\) −861.000 −0.323632
\(193\) −2578.00 −0.961495 −0.480747 0.876859i \(-0.659635\pi\)
−0.480747 + 0.876859i \(0.659635\pi\)
\(194\) 6170.00 2.28340
\(195\) 0 0
\(196\) −1479.00 −0.538994
\(197\) −526.000 −0.190233 −0.0951166 0.995466i \(-0.530322\pi\)
−0.0951166 + 0.995466i \(0.530322\pi\)
\(198\) 1980.00 0.710669
\(199\) 4440.00 1.58162 0.790812 0.612059i \(-0.209658\pi\)
0.790812 + 0.612059i \(0.209658\pi\)
\(200\) 0 0
\(201\) 1020.00 0.357937
\(202\) −5110.00 −1.77989
\(203\) −464.000 −0.160426
\(204\) −918.000 −0.315063
\(205\) 0 0
\(206\) −6240.00 −2.11049
\(207\) −1656.00 −0.556038
\(208\) −6942.00 −2.31414
\(209\) 1232.00 0.407747
\(210\) 0 0
\(211\) 308.000 0.100491 0.0502455 0.998737i \(-0.484000\pi\)
0.0502455 + 0.998737i \(0.484000\pi\)
\(212\) −9758.00 −3.16124
\(213\) 888.000 0.285656
\(214\) −580.000 −0.185271
\(215\) 0 0
\(216\) −1215.00 −0.382733
\(217\) 3584.00 1.12119
\(218\) 4130.00 1.28311
\(219\) −1182.00 −0.364713
\(220\) 0 0
\(221\) 1404.00 0.427345
\(222\) 3810.00 1.15185
\(223\) −4120.00 −1.23720 −0.618600 0.785706i \(-0.712300\pi\)
−0.618600 + 0.785706i \(0.712300\pi\)
\(224\) 1360.00 0.405664
\(225\) 0 0
\(226\) −11030.0 −3.24648
\(227\) −4932.00 −1.44206 −0.721032 0.692902i \(-0.756332\pi\)
−0.721032 + 0.692902i \(0.756332\pi\)
\(228\) −1428.00 −0.414788
\(229\) −3050.00 −0.880130 −0.440065 0.897966i \(-0.645045\pi\)
−0.440065 + 0.897966i \(0.645045\pi\)
\(230\) 0 0
\(231\) 2112.00 0.601556
\(232\) −1305.00 −0.369299
\(233\) −82.0000 −0.0230558 −0.0115279 0.999934i \(-0.503670\pi\)
−0.0115279 + 0.999934i \(0.503670\pi\)
\(234\) 3510.00 0.980581
\(235\) 0 0
\(236\) 3060.00 0.844021
\(237\) −2880.00 −0.789351
\(238\) −1440.00 −0.392190
\(239\) −5104.00 −1.38138 −0.690691 0.723150i \(-0.742694\pi\)
−0.690691 + 0.723150i \(0.742694\pi\)
\(240\) 0 0
\(241\) −2158.00 −0.576801 −0.288400 0.957510i \(-0.593123\pi\)
−0.288400 + 0.957510i \(0.593123\pi\)
\(242\) −3025.00 −0.803530
\(243\) 243.000 0.0641500
\(244\) −10370.0 −2.72078
\(245\) 0 0
\(246\) 1170.00 0.303238
\(247\) 2184.00 0.562610
\(248\) 10080.0 2.58097
\(249\) 2724.00 0.693279
\(250\) 0 0
\(251\) 6116.00 1.53800 0.769001 0.639248i \(-0.220754\pi\)
0.769001 + 0.639248i \(0.220754\pi\)
\(252\) −2448.00 −0.611942
\(253\) 8096.00 2.01182
\(254\) −10280.0 −2.53947
\(255\) 0 0
\(256\) −8279.00 −2.02124
\(257\) −3418.00 −0.829607 −0.414803 0.909911i \(-0.636150\pi\)
−0.414803 + 0.909911i \(0.636150\pi\)
\(258\) −3900.00 −0.941098
\(259\) 4064.00 0.974999
\(260\) 0 0
\(261\) 261.000 0.0618984
\(262\) −60.0000 −0.0141481
\(263\) −7440.00 −1.74437 −0.872186 0.489174i \(-0.837298\pi\)
−0.872186 + 0.489174i \(0.837298\pi\)
\(264\) 5940.00 1.38478
\(265\) 0 0
\(266\) −2240.00 −0.516328
\(267\) −2970.00 −0.680753
\(268\) 5780.00 1.31742
\(269\) 6582.00 1.49186 0.745932 0.666022i \(-0.232004\pi\)
0.745932 + 0.666022i \(0.232004\pi\)
\(270\) 0 0
\(271\) −5504.00 −1.23374 −0.616871 0.787064i \(-0.711600\pi\)
−0.616871 + 0.787064i \(0.711600\pi\)
\(272\) −1602.00 −0.357116
\(273\) 3744.00 0.830026
\(274\) −13790.0 −3.04045
\(275\) 0 0
\(276\) −9384.00 −2.04656
\(277\) −3718.00 −0.806473 −0.403236 0.915096i \(-0.632115\pi\)
−0.403236 + 0.915096i \(0.632115\pi\)
\(278\) 7180.00 1.54902
\(279\) −2016.00 −0.432598
\(280\) 0 0
\(281\) 1754.00 0.372366 0.186183 0.982515i \(-0.440388\pi\)
0.186183 + 0.982515i \(0.440388\pi\)
\(282\) 4680.00 0.988262
\(283\) −3572.00 −0.750295 −0.375147 0.926965i \(-0.622408\pi\)
−0.375147 + 0.926965i \(0.622408\pi\)
\(284\) 5032.00 1.05139
\(285\) 0 0
\(286\) −17160.0 −3.54787
\(287\) 1248.00 0.256680
\(288\) −765.000 −0.156521
\(289\) −4589.00 −0.934053
\(290\) 0 0
\(291\) −3702.00 −0.745756
\(292\) −6698.00 −1.34237
\(293\) −126.000 −0.0251229 −0.0125614 0.999921i \(-0.503999\pi\)
−0.0125614 + 0.999921i \(0.503999\pi\)
\(294\) 1305.00 0.258875
\(295\) 0 0
\(296\) 11430.0 2.24444
\(297\) −1188.00 −0.232104
\(298\) 2490.00 0.484033
\(299\) 14352.0 2.77591
\(300\) 0 0
\(301\) −4160.00 −0.796606
\(302\) 13480.0 2.56850
\(303\) 3066.00 0.581311
\(304\) −2492.00 −0.470151
\(305\) 0 0
\(306\) 810.000 0.151322
\(307\) 2412.00 0.448404 0.224202 0.974543i \(-0.428022\pi\)
0.224202 + 0.974543i \(0.428022\pi\)
\(308\) 11968.0 2.21409
\(309\) 3744.00 0.689284
\(310\) 0 0
\(311\) −2928.00 −0.533864 −0.266932 0.963715i \(-0.586010\pi\)
−0.266932 + 0.963715i \(0.586010\pi\)
\(312\) 10530.0 1.91072
\(313\) −2874.00 −0.519003 −0.259502 0.965743i \(-0.583558\pi\)
−0.259502 + 0.965743i \(0.583558\pi\)
\(314\) 2670.00 0.479862
\(315\) 0 0
\(316\) −16320.0 −2.90529
\(317\) 26.0000 0.00460664 0.00230332 0.999997i \(-0.499267\pi\)
0.00230332 + 0.999997i \(0.499267\pi\)
\(318\) 8610.00 1.51832
\(319\) −1276.00 −0.223957
\(320\) 0 0
\(321\) 348.000 0.0605092
\(322\) −14720.0 −2.54756
\(323\) 504.000 0.0868214
\(324\) 1377.00 0.236111
\(325\) 0 0
\(326\) 6900.00 1.17226
\(327\) −2478.00 −0.419063
\(328\) 3510.00 0.590876
\(329\) 4992.00 0.836528
\(330\) 0 0
\(331\) 9340.00 1.55098 0.775488 0.631363i \(-0.217504\pi\)
0.775488 + 0.631363i \(0.217504\pi\)
\(332\) 15436.0 2.55169
\(333\) −2286.00 −0.376192
\(334\) 13080.0 2.14283
\(335\) 0 0
\(336\) −4272.00 −0.693621
\(337\) 10302.0 1.66524 0.832620 0.553845i \(-0.186840\pi\)
0.832620 + 0.553845i \(0.186840\pi\)
\(338\) −19435.0 −3.12759
\(339\) 6618.00 1.06030
\(340\) 0 0
\(341\) 9856.00 1.56520
\(342\) 1260.00 0.199219
\(343\) 6880.00 1.08305
\(344\) −11700.0 −1.83378
\(345\) 0 0
\(346\) −1650.00 −0.256372
\(347\) 8020.00 1.24074 0.620369 0.784310i \(-0.286983\pi\)
0.620369 + 0.784310i \(0.286983\pi\)
\(348\) 1479.00 0.227824
\(349\) −1306.00 −0.200311 −0.100156 0.994972i \(-0.531934\pi\)
−0.100156 + 0.994972i \(0.531934\pi\)
\(350\) 0 0
\(351\) −2106.00 −0.320256
\(352\) 3740.00 0.566314
\(353\) −5658.00 −0.853102 −0.426551 0.904464i \(-0.640272\pi\)
−0.426551 + 0.904464i \(0.640272\pi\)
\(354\) −2700.00 −0.405377
\(355\) 0 0
\(356\) −16830.0 −2.50558
\(357\) 864.000 0.128089
\(358\) 1860.00 0.274592
\(359\) −12240.0 −1.79945 −0.899725 0.436457i \(-0.856233\pi\)
−0.899725 + 0.436457i \(0.856233\pi\)
\(360\) 0 0
\(361\) −6075.00 −0.885698
\(362\) 5050.00 0.733210
\(363\) 1815.00 0.262432
\(364\) 21216.0 3.05500
\(365\) 0 0
\(366\) 9150.00 1.30677
\(367\) −8984.00 −1.27782 −0.638911 0.769280i \(-0.720615\pi\)
−0.638911 + 0.769280i \(0.720615\pi\)
\(368\) −16376.0 −2.31972
\(369\) −702.000 −0.0990370
\(370\) 0 0
\(371\) 9184.00 1.28520
\(372\) −11424.0 −1.59222
\(373\) 938.000 0.130209 0.0651043 0.997878i \(-0.479262\pi\)
0.0651043 + 0.997878i \(0.479262\pi\)
\(374\) −3960.00 −0.547505
\(375\) 0 0
\(376\) 14040.0 1.92569
\(377\) −2262.00 −0.309016
\(378\) 2160.00 0.293911
\(379\) −7796.00 −1.05661 −0.528303 0.849056i \(-0.677171\pi\)
−0.528303 + 0.849056i \(0.677171\pi\)
\(380\) 0 0
\(381\) 6168.00 0.829386
\(382\) −10040.0 −1.34474
\(383\) 6944.00 0.926428 0.463214 0.886247i \(-0.346696\pi\)
0.463214 + 0.886247i \(0.346696\pi\)
\(384\) 6345.00 0.843208
\(385\) 0 0
\(386\) 12890.0 1.69970
\(387\) 2340.00 0.307361
\(388\) −20978.0 −2.74484
\(389\) 2126.00 0.277101 0.138551 0.990355i \(-0.455756\pi\)
0.138551 + 0.990355i \(0.455756\pi\)
\(390\) 0 0
\(391\) 3312.00 0.428376
\(392\) 3915.00 0.504432
\(393\) 36.0000 0.00462076
\(394\) 2630.00 0.336288
\(395\) 0 0
\(396\) −6732.00 −0.854282
\(397\) 3346.00 0.423000 0.211500 0.977378i \(-0.432165\pi\)
0.211500 + 0.977378i \(0.432165\pi\)
\(398\) −22200.0 −2.79594
\(399\) 1344.00 0.168632
\(400\) 0 0
\(401\) 12850.0 1.60025 0.800123 0.599836i \(-0.204768\pi\)
0.800123 + 0.599836i \(0.204768\pi\)
\(402\) −5100.00 −0.632748
\(403\) 17472.0 2.15966
\(404\) 17374.0 2.13958
\(405\) 0 0
\(406\) 2320.00 0.283595
\(407\) 11176.0 1.36111
\(408\) 2430.00 0.294860
\(409\) 6122.00 0.740131 0.370065 0.929006i \(-0.379335\pi\)
0.370065 + 0.929006i \(0.379335\pi\)
\(410\) 0 0
\(411\) 8274.00 0.993008
\(412\) 21216.0 2.53698
\(413\) −2880.00 −0.343137
\(414\) 8280.00 0.982946
\(415\) 0 0
\(416\) 6630.00 0.781400
\(417\) −4308.00 −0.505908
\(418\) −6160.00 −0.720803
\(419\) 1372.00 0.159968 0.0799840 0.996796i \(-0.474513\pi\)
0.0799840 + 0.996796i \(0.474513\pi\)
\(420\) 0 0
\(421\) 12150.0 1.40654 0.703272 0.710921i \(-0.251722\pi\)
0.703272 + 0.710921i \(0.251722\pi\)
\(422\) −1540.00 −0.177645
\(423\) −2808.00 −0.322765
\(424\) 25830.0 2.95853
\(425\) 0 0
\(426\) −4440.00 −0.504973
\(427\) 9760.00 1.10613
\(428\) 1972.00 0.222711
\(429\) 10296.0 1.15873
\(430\) 0 0
\(431\) 6288.00 0.702743 0.351372 0.936236i \(-0.385715\pi\)
0.351372 + 0.936236i \(0.385715\pi\)
\(432\) 2403.00 0.267626
\(433\) −15650.0 −1.73693 −0.868465 0.495750i \(-0.834893\pi\)
−0.868465 + 0.495750i \(0.834893\pi\)
\(434\) −17920.0 −1.98200
\(435\) 0 0
\(436\) −14042.0 −1.54241
\(437\) 5152.00 0.563967
\(438\) 5910.00 0.644728
\(439\) −14520.0 −1.57859 −0.789296 0.614013i \(-0.789554\pi\)
−0.789296 + 0.614013i \(0.789554\pi\)
\(440\) 0 0
\(441\) −783.000 −0.0845481
\(442\) −7020.00 −0.755446
\(443\) −7372.00 −0.790642 −0.395321 0.918543i \(-0.629367\pi\)
−0.395321 + 0.918543i \(0.629367\pi\)
\(444\) −12954.0 −1.38462
\(445\) 0 0
\(446\) 20600.0 2.18708
\(447\) −1494.00 −0.158085
\(448\) 4592.00 0.484267
\(449\) 10666.0 1.12107 0.560534 0.828131i \(-0.310596\pi\)
0.560534 + 0.828131i \(0.310596\pi\)
\(450\) 0 0
\(451\) 3432.00 0.358329
\(452\) 37502.0 3.90253
\(453\) −8088.00 −0.838868
\(454\) 24660.0 2.54923
\(455\) 0 0
\(456\) 3780.00 0.388190
\(457\) 8006.00 0.819486 0.409743 0.912201i \(-0.365618\pi\)
0.409743 + 0.912201i \(0.365618\pi\)
\(458\) 15250.0 1.55586
\(459\) −486.000 −0.0494217
\(460\) 0 0
\(461\) 1254.00 0.126691 0.0633456 0.997992i \(-0.479823\pi\)
0.0633456 + 0.997992i \(0.479823\pi\)
\(462\) −10560.0 −1.06341
\(463\) 4584.00 0.460122 0.230061 0.973176i \(-0.426107\pi\)
0.230061 + 0.973176i \(0.426107\pi\)
\(464\) 2581.00 0.258233
\(465\) 0 0
\(466\) 410.000 0.0407573
\(467\) −11588.0 −1.14824 −0.574121 0.818771i \(-0.694656\pi\)
−0.574121 + 0.818771i \(0.694656\pi\)
\(468\) −11934.0 −1.17874
\(469\) −5440.00 −0.535599
\(470\) 0 0
\(471\) −1602.00 −0.156722
\(472\) −8100.00 −0.789900
\(473\) −11440.0 −1.11208
\(474\) 14400.0 1.39539
\(475\) 0 0
\(476\) 4896.00 0.471445
\(477\) −5166.00 −0.495880
\(478\) 25520.0 2.44196
\(479\) −18200.0 −1.73607 −0.868037 0.496500i \(-0.834618\pi\)
−0.868037 + 0.496500i \(0.834618\pi\)
\(480\) 0 0
\(481\) 19812.0 1.87807
\(482\) 10790.0 1.01965
\(483\) 8832.00 0.832029
\(484\) 10285.0 0.965909
\(485\) 0 0
\(486\) −1215.00 −0.113402
\(487\) −3824.00 −0.355815 −0.177908 0.984047i \(-0.556933\pi\)
−0.177908 + 0.984047i \(0.556933\pi\)
\(488\) 27450.0 2.54632
\(489\) −4140.00 −0.382857
\(490\) 0 0
\(491\) −3100.00 −0.284931 −0.142465 0.989800i \(-0.545503\pi\)
−0.142465 + 0.989800i \(0.545503\pi\)
\(492\) −3978.00 −0.364516
\(493\) −522.000 −0.0476870
\(494\) −10920.0 −0.994563
\(495\) 0 0
\(496\) −19936.0 −1.80474
\(497\) −4736.00 −0.427442
\(498\) −13620.0 −1.22556
\(499\) 19740.0 1.77091 0.885455 0.464726i \(-0.153847\pi\)
0.885455 + 0.464726i \(0.153847\pi\)
\(500\) 0 0
\(501\) −7848.00 −0.699846
\(502\) −30580.0 −2.71883
\(503\) 6720.00 0.595686 0.297843 0.954615i \(-0.403733\pi\)
0.297843 + 0.954615i \(0.403733\pi\)
\(504\) 6480.00 0.572703
\(505\) 0 0
\(506\) −40480.0 −3.55643
\(507\) 11661.0 1.02147
\(508\) 34952.0 3.05265
\(509\) 10886.0 0.947964 0.473982 0.880535i \(-0.342816\pi\)
0.473982 + 0.880535i \(0.342816\pi\)
\(510\) 0 0
\(511\) 6304.00 0.545739
\(512\) 24475.0 2.11260
\(513\) −756.000 −0.0650647
\(514\) 17090.0 1.46655
\(515\) 0 0
\(516\) 13260.0 1.13128
\(517\) 13728.0 1.16781
\(518\) −20320.0 −1.72357
\(519\) 990.000 0.0837306
\(520\) 0 0
\(521\) 22522.0 1.89387 0.946935 0.321424i \(-0.104161\pi\)
0.946935 + 0.321424i \(0.104161\pi\)
\(522\) −1305.00 −0.109422
\(523\) 13212.0 1.10463 0.552314 0.833636i \(-0.313745\pi\)
0.552314 + 0.833636i \(0.313745\pi\)
\(524\) 204.000 0.0170072
\(525\) 0 0
\(526\) 37200.0 3.08364
\(527\) 4032.00 0.333276
\(528\) −11748.0 −0.968307
\(529\) 21689.0 1.78261
\(530\) 0 0
\(531\) 1620.00 0.132396
\(532\) 7616.00 0.620668
\(533\) 6084.00 0.494423
\(534\) 14850.0 1.20341
\(535\) 0 0
\(536\) −15300.0 −1.23295
\(537\) −1116.00 −0.0896815
\(538\) −32910.0 −2.63727
\(539\) 3828.00 0.305907
\(540\) 0 0
\(541\) −4642.00 −0.368900 −0.184450 0.982842i \(-0.559050\pi\)
−0.184450 + 0.982842i \(0.559050\pi\)
\(542\) 27520.0 2.18097
\(543\) −3030.00 −0.239465
\(544\) 1530.00 0.120585
\(545\) 0 0
\(546\) −18720.0 −1.46729
\(547\) −4060.00 −0.317355 −0.158677 0.987330i \(-0.550723\pi\)
−0.158677 + 0.987330i \(0.550723\pi\)
\(548\) 46886.0 3.65487
\(549\) −5490.00 −0.426790
\(550\) 0 0
\(551\) −812.000 −0.0627811
\(552\) 24840.0 1.91533
\(553\) 15360.0 1.18115
\(554\) 18590.0 1.42566
\(555\) 0 0
\(556\) −24412.0 −1.86205
\(557\) 1386.00 0.105434 0.0527170 0.998609i \(-0.483212\pi\)
0.0527170 + 0.998609i \(0.483212\pi\)
\(558\) 10080.0 0.764732
\(559\) −20280.0 −1.53444
\(560\) 0 0
\(561\) 2376.00 0.178814
\(562\) −8770.00 −0.658256
\(563\) −2452.00 −0.183551 −0.0917757 0.995780i \(-0.529254\pi\)
−0.0917757 + 0.995780i \(0.529254\pi\)
\(564\) −15912.0 −1.18797
\(565\) 0 0
\(566\) 17860.0 1.32635
\(567\) −1296.00 −0.0959910
\(568\) −13320.0 −0.983970
\(569\) −20862.0 −1.53705 −0.768524 0.639821i \(-0.779009\pi\)
−0.768524 + 0.639821i \(0.779009\pi\)
\(570\) 0 0
\(571\) −9420.00 −0.690394 −0.345197 0.938530i \(-0.612188\pi\)
−0.345197 + 0.938530i \(0.612188\pi\)
\(572\) 58344.0 4.26483
\(573\) 6024.00 0.439191
\(574\) −6240.00 −0.453750
\(575\) 0 0
\(576\) −2583.00 −0.186849
\(577\) −13202.0 −0.952524 −0.476262 0.879303i \(-0.658009\pi\)
−0.476262 + 0.879303i \(0.658009\pi\)
\(578\) 22945.0 1.65119
\(579\) −7734.00 −0.555119
\(580\) 0 0
\(581\) −14528.0 −1.03739
\(582\) 18510.0 1.31832
\(583\) 25256.0 1.79416
\(584\) 17730.0 1.25629
\(585\) 0 0
\(586\) 630.000 0.0444114
\(587\) 8708.00 0.612296 0.306148 0.951984i \(-0.400960\pi\)
0.306148 + 0.951984i \(0.400960\pi\)
\(588\) −4437.00 −0.311188
\(589\) 6272.00 0.438766
\(590\) 0 0
\(591\) −1578.00 −0.109831
\(592\) −22606.0 −1.56943
\(593\) 4390.00 0.304006 0.152003 0.988380i \(-0.451428\pi\)
0.152003 + 0.988380i \(0.451428\pi\)
\(594\) 5940.00 0.410305
\(595\) 0 0
\(596\) −8466.00 −0.581847
\(597\) 13320.0 0.913151
\(598\) −71760.0 −4.90716
\(599\) −20256.0 −1.38170 −0.690850 0.722999i \(-0.742763\pi\)
−0.690850 + 0.722999i \(0.742763\pi\)
\(600\) 0 0
\(601\) 9610.00 0.652246 0.326123 0.945327i \(-0.394258\pi\)
0.326123 + 0.945327i \(0.394258\pi\)
\(602\) 20800.0 1.40821
\(603\) 3060.00 0.206655
\(604\) −45832.0 −3.08755
\(605\) 0 0
\(606\) −15330.0 −1.02762
\(607\) −10376.0 −0.693820 −0.346910 0.937898i \(-0.612769\pi\)
−0.346910 + 0.937898i \(0.612769\pi\)
\(608\) 2380.00 0.158753
\(609\) −1392.00 −0.0926218
\(610\) 0 0
\(611\) 24336.0 1.61134
\(612\) −2754.00 −0.181902
\(613\) −6822.00 −0.449491 −0.224746 0.974417i \(-0.572155\pi\)
−0.224746 + 0.974417i \(0.572155\pi\)
\(614\) −12060.0 −0.792674
\(615\) 0 0
\(616\) −31680.0 −2.07212
\(617\) 20070.0 1.30954 0.654771 0.755827i \(-0.272765\pi\)
0.654771 + 0.755827i \(0.272765\pi\)
\(618\) −18720.0 −1.21849
\(619\) 19228.0 1.24853 0.624264 0.781214i \(-0.285399\pi\)
0.624264 + 0.781214i \(0.285399\pi\)
\(620\) 0 0
\(621\) −4968.00 −0.321029
\(622\) 14640.0 0.943747
\(623\) 15840.0 1.01865
\(624\) −20826.0 −1.33607
\(625\) 0 0
\(626\) 14370.0 0.917477
\(627\) 3696.00 0.235413
\(628\) −9078.00 −0.576834
\(629\) 4572.00 0.289821
\(630\) 0 0
\(631\) 6552.00 0.413361 0.206681 0.978408i \(-0.433734\pi\)
0.206681 + 0.978408i \(0.433734\pi\)
\(632\) 43200.0 2.71899
\(633\) 924.000 0.0580185
\(634\) −130.000 −0.00814347
\(635\) 0 0
\(636\) −29274.0 −1.82514
\(637\) 6786.00 0.422090
\(638\) 6380.00 0.395904
\(639\) 2664.00 0.164924
\(640\) 0 0
\(641\) −14422.0 −0.888666 −0.444333 0.895862i \(-0.646559\pi\)
−0.444333 + 0.895862i \(0.646559\pi\)
\(642\) −1740.00 −0.106966
\(643\) 6212.00 0.380991 0.190496 0.981688i \(-0.438991\pi\)
0.190496 + 0.981688i \(0.438991\pi\)
\(644\) 50048.0 3.06237
\(645\) 0 0
\(646\) −2520.00 −0.153480
\(647\) −22024.0 −1.33826 −0.669129 0.743146i \(-0.733333\pi\)
−0.669129 + 0.743146i \(0.733333\pi\)
\(648\) −3645.00 −0.220971
\(649\) −7920.00 −0.479025
\(650\) 0 0
\(651\) 10752.0 0.647318
\(652\) −23460.0 −1.40915
\(653\) −16630.0 −0.996604 −0.498302 0.867004i \(-0.666043\pi\)
−0.498302 + 0.867004i \(0.666043\pi\)
\(654\) 12390.0 0.740806
\(655\) 0 0
\(656\) −6942.00 −0.413170
\(657\) −3546.00 −0.210567
\(658\) −24960.0 −1.47879
\(659\) −24468.0 −1.44634 −0.723170 0.690670i \(-0.757316\pi\)
−0.723170 + 0.690670i \(0.757316\pi\)
\(660\) 0 0
\(661\) −10226.0 −0.601733 −0.300866 0.953666i \(-0.597276\pi\)
−0.300866 + 0.953666i \(0.597276\pi\)
\(662\) −46700.0 −2.74176
\(663\) 4212.00 0.246728
\(664\) −40860.0 −2.38807
\(665\) 0 0
\(666\) 11430.0 0.665020
\(667\) −5336.00 −0.309761
\(668\) −44472.0 −2.57586
\(669\) −12360.0 −0.714298
\(670\) 0 0
\(671\) 26840.0 1.54418
\(672\) 4080.00 0.234210
\(673\) −13458.0 −0.770829 −0.385414 0.922744i \(-0.625941\pi\)
−0.385414 + 0.922744i \(0.625941\pi\)
\(674\) −51510.0 −2.94376
\(675\) 0 0
\(676\) 66079.0 3.75962
\(677\) −22174.0 −1.25881 −0.629406 0.777076i \(-0.716702\pi\)
−0.629406 + 0.777076i \(0.716702\pi\)
\(678\) −33090.0 −1.87436
\(679\) 19744.0 1.11591
\(680\) 0 0
\(681\) −14796.0 −0.832576
\(682\) −49280.0 −2.76690
\(683\) 2404.00 0.134680 0.0673400 0.997730i \(-0.478549\pi\)
0.0673400 + 0.997730i \(0.478549\pi\)
\(684\) −4284.00 −0.239478
\(685\) 0 0
\(686\) −34400.0 −1.91457
\(687\) −9150.00 −0.508143
\(688\) 23140.0 1.28227
\(689\) 44772.0 2.47558
\(690\) 0 0
\(691\) −5956.00 −0.327897 −0.163949 0.986469i \(-0.552423\pi\)
−0.163949 + 0.986469i \(0.552423\pi\)
\(692\) 5610.00 0.308179
\(693\) 6336.00 0.347308
\(694\) −40100.0 −2.19334
\(695\) 0 0
\(696\) −3915.00 −0.213215
\(697\) 1404.00 0.0762988
\(698\) 6530.00 0.354103
\(699\) −246.000 −0.0133113
\(700\) 0 0
\(701\) −14586.0 −0.785885 −0.392943 0.919563i \(-0.628543\pi\)
−0.392943 + 0.919563i \(0.628543\pi\)
\(702\) 10530.0 0.566139
\(703\) 7112.00 0.381556
\(704\) 12628.0 0.676045
\(705\) 0 0
\(706\) 28290.0 1.50809
\(707\) −16352.0 −0.869845
\(708\) 9180.00 0.487296
\(709\) −4370.00 −0.231479 −0.115740 0.993280i \(-0.536924\pi\)
−0.115740 + 0.993280i \(0.536924\pi\)
\(710\) 0 0
\(711\) −8640.00 −0.455732
\(712\) 44550.0 2.34492
\(713\) 41216.0 2.16487
\(714\) −4320.00 −0.226431
\(715\) 0 0
\(716\) −6324.00 −0.330082
\(717\) −15312.0 −0.797541
\(718\) 61200.0 3.18101
\(719\) 144.000 0.00746912 0.00373456 0.999993i \(-0.498811\pi\)
0.00373456 + 0.999993i \(0.498811\pi\)
\(720\) 0 0
\(721\) −19968.0 −1.03141
\(722\) 30375.0 1.56571
\(723\) −6474.00 −0.333016
\(724\) −17170.0 −0.881378
\(725\) 0 0
\(726\) −9075.00 −0.463919
\(727\) 9632.00 0.491377 0.245689 0.969349i \(-0.420986\pi\)
0.245689 + 0.969349i \(0.420986\pi\)
\(728\) −56160.0 −2.85910
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −4680.00 −0.236794
\(732\) −31110.0 −1.57085
\(733\) 19306.0 0.972829 0.486414 0.873728i \(-0.338305\pi\)
0.486414 + 0.873728i \(0.338305\pi\)
\(734\) 44920.0 2.25889
\(735\) 0 0
\(736\) 15640.0 0.783285
\(737\) −14960.0 −0.747705
\(738\) 3510.00 0.175074
\(739\) −36540.0 −1.81887 −0.909435 0.415845i \(-0.863486\pi\)
−0.909435 + 0.415845i \(0.863486\pi\)
\(740\) 0 0
\(741\) 6552.00 0.324823
\(742\) −45920.0 −2.27194
\(743\) −5408.00 −0.267026 −0.133513 0.991047i \(-0.542626\pi\)
−0.133513 + 0.991047i \(0.542626\pi\)
\(744\) 30240.0 1.49012
\(745\) 0 0
\(746\) −4690.00 −0.230178
\(747\) 8172.00 0.400265
\(748\) 13464.0 0.658145
\(749\) −1856.00 −0.0905431
\(750\) 0 0
\(751\) −13952.0 −0.677917 −0.338959 0.940801i \(-0.610075\pi\)
−0.338959 + 0.940801i \(0.610075\pi\)
\(752\) −27768.0 −1.34654
\(753\) 18348.0 0.887966
\(754\) 11310.0 0.546268
\(755\) 0 0
\(756\) −7344.00 −0.353305
\(757\) 4274.00 0.205206 0.102603 0.994722i \(-0.467283\pi\)
0.102603 + 0.994722i \(0.467283\pi\)
\(758\) 38980.0 1.86783
\(759\) 24288.0 1.16153
\(760\) 0 0
\(761\) −230.000 −0.0109560 −0.00547799 0.999985i \(-0.501744\pi\)
−0.00547799 + 0.999985i \(0.501744\pi\)
\(762\) −30840.0 −1.46616
\(763\) 13216.0 0.627066
\(764\) 34136.0 1.61649
\(765\) 0 0
\(766\) −34720.0 −1.63771
\(767\) −14040.0 −0.660958
\(768\) −24837.0 −1.16696
\(769\) −7854.00 −0.368300 −0.184150 0.982898i \(-0.558953\pi\)
−0.184150 + 0.982898i \(0.558953\pi\)
\(770\) 0 0
\(771\) −10254.0 −0.478974
\(772\) −43826.0 −2.04318
\(773\) −19550.0 −0.909657 −0.454828 0.890579i \(-0.650299\pi\)
−0.454828 + 0.890579i \(0.650299\pi\)
\(774\) −11700.0 −0.543343
\(775\) 0 0
\(776\) 55530.0 2.56883
\(777\) 12192.0 0.562916
\(778\) −10630.0 −0.489851
\(779\) 2184.00 0.100449
\(780\) 0 0
\(781\) −13024.0 −0.596716
\(782\) −16560.0 −0.757269
\(783\) 783.000 0.0357371
\(784\) −7743.00 −0.352724
\(785\) 0 0
\(786\) −180.000 −0.00816843
\(787\) −27228.0 −1.23326 −0.616629 0.787254i \(-0.711502\pi\)
−0.616629 + 0.787254i \(0.711502\pi\)
\(788\) −8942.00 −0.404246
\(789\) −22320.0 −1.00711
\(790\) 0 0
\(791\) −35296.0 −1.58658
\(792\) 17820.0 0.799503
\(793\) 47580.0 2.13066
\(794\) −16730.0 −0.747765
\(795\) 0 0
\(796\) 75480.0 3.36095
\(797\) 3386.00 0.150487 0.0752436 0.997165i \(-0.476027\pi\)
0.0752436 + 0.997165i \(0.476027\pi\)
\(798\) −6720.00 −0.298102
\(799\) 5616.00 0.248661
\(800\) 0 0
\(801\) −8910.00 −0.393033
\(802\) −64250.0 −2.82886
\(803\) 17336.0 0.761861
\(804\) 17340.0 0.760615
\(805\) 0 0
\(806\) −87360.0 −3.81777
\(807\) 19746.0 0.861329
\(808\) −45990.0 −2.00238
\(809\) 26994.0 1.17313 0.586563 0.809904i \(-0.300481\pi\)
0.586563 + 0.809904i \(0.300481\pi\)
\(810\) 0 0
\(811\) 8356.00 0.361799 0.180899 0.983502i \(-0.442099\pi\)
0.180899 + 0.983502i \(0.442099\pi\)
\(812\) −7888.00 −0.340905
\(813\) −16512.0 −0.712302
\(814\) −55880.0 −2.40613
\(815\) 0 0
\(816\) −4806.00 −0.206181
\(817\) −7280.00 −0.311744
\(818\) −30610.0 −1.30838
\(819\) 11232.0 0.479216
\(820\) 0 0
\(821\) 9838.00 0.418208 0.209104 0.977893i \(-0.432945\pi\)
0.209104 + 0.977893i \(0.432945\pi\)
\(822\) −41370.0 −1.75541
\(823\) 29552.0 1.25166 0.625831 0.779959i \(-0.284760\pi\)
0.625831 + 0.779959i \(0.284760\pi\)
\(824\) −56160.0 −2.37430
\(825\) 0 0
\(826\) 14400.0 0.606586
\(827\) −18556.0 −0.780236 −0.390118 0.920765i \(-0.627566\pi\)
−0.390118 + 0.920765i \(0.627566\pi\)
\(828\) −28152.0 −1.18158
\(829\) 7966.00 0.333740 0.166870 0.985979i \(-0.446634\pi\)
0.166870 + 0.985979i \(0.446634\pi\)
\(830\) 0 0
\(831\) −11154.0 −0.465617
\(832\) 22386.0 0.932806
\(833\) 1566.00 0.0651365
\(834\) 21540.0 0.894328
\(835\) 0 0
\(836\) 20944.0 0.866463
\(837\) −6048.00 −0.249760
\(838\) −6860.00 −0.282786
\(839\) −42048.0 −1.73022 −0.865112 0.501578i \(-0.832753\pi\)
−0.865112 + 0.501578i \(0.832753\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) −60750.0 −2.48644
\(843\) 5262.00 0.214986
\(844\) 5236.00 0.213543
\(845\) 0 0
\(846\) 14040.0 0.570573
\(847\) −9680.00 −0.392690
\(848\) −51086.0 −2.06875
\(849\) −10716.0 −0.433183
\(850\) 0 0
\(851\) 46736.0 1.88260
\(852\) 15096.0 0.607019
\(853\) −29950.0 −1.20219 −0.601095 0.799177i \(-0.705269\pi\)
−0.601095 + 0.799177i \(0.705269\pi\)
\(854\) −48800.0 −1.95539
\(855\) 0 0
\(856\) −5220.00 −0.208430
\(857\) 31454.0 1.25373 0.626866 0.779127i \(-0.284337\pi\)
0.626866 + 0.779127i \(0.284337\pi\)
\(858\) −51480.0 −2.04837
\(859\) −22036.0 −0.875272 −0.437636 0.899152i \(-0.644184\pi\)
−0.437636 + 0.899152i \(0.644184\pi\)
\(860\) 0 0
\(861\) 3744.00 0.148194
\(862\) −31440.0 −1.24229
\(863\) −14048.0 −0.554113 −0.277056 0.960854i \(-0.589359\pi\)
−0.277056 + 0.960854i \(0.589359\pi\)
\(864\) −2295.00 −0.0903675
\(865\) 0 0
\(866\) 78250.0 3.07049
\(867\) −13767.0 −0.539275
\(868\) 60928.0 2.38252
\(869\) 42240.0 1.64890
\(870\) 0 0
\(871\) −26520.0 −1.03168
\(872\) 37170.0 1.44350
\(873\) −11106.0 −0.430563
\(874\) −25760.0 −0.996962
\(875\) 0 0
\(876\) −20094.0 −0.775015
\(877\) 39922.0 1.53714 0.768569 0.639767i \(-0.220969\pi\)
0.768569 + 0.639767i \(0.220969\pi\)
\(878\) 72600.0 2.79058
\(879\) −378.000 −0.0145047
\(880\) 0 0
\(881\) 38730.0 1.48110 0.740549 0.672003i \(-0.234566\pi\)
0.740549 + 0.672003i \(0.234566\pi\)
\(882\) 3915.00 0.149461
\(883\) 32948.0 1.25571 0.627853 0.778332i \(-0.283934\pi\)
0.627853 + 0.778332i \(0.283934\pi\)
\(884\) 23868.0 0.908108
\(885\) 0 0
\(886\) 36860.0 1.39767
\(887\) 21680.0 0.820680 0.410340 0.911933i \(-0.365410\pi\)
0.410340 + 0.911933i \(0.365410\pi\)
\(888\) 34290.0 1.29583
\(889\) −32896.0 −1.24105
\(890\) 0 0
\(891\) −3564.00 −0.134005
\(892\) −70040.0 −2.62905
\(893\) 8736.00 0.327367
\(894\) 7470.00 0.279457
\(895\) 0 0
\(896\) −33840.0 −1.26174
\(897\) 43056.0 1.60267
\(898\) −53330.0 −1.98179
\(899\) −6496.00 −0.240994
\(900\) 0 0
\(901\) 10332.0 0.382030
\(902\) −17160.0 −0.633443
\(903\) −12480.0 −0.459921
\(904\) −99270.0 −3.65229
\(905\) 0 0
\(906\) 40440.0 1.48292
\(907\) −2236.00 −0.0818580 −0.0409290 0.999162i \(-0.513032\pi\)
−0.0409290 + 0.999162i \(0.513032\pi\)
\(908\) −83844.0 −3.06438
\(909\) 9198.00 0.335620
\(910\) 0 0
\(911\) 35816.0 1.30257 0.651283 0.758835i \(-0.274231\pi\)
0.651283 + 0.758835i \(0.274231\pi\)
\(912\) −7476.00 −0.271442
\(913\) −39952.0 −1.44821
\(914\) −40030.0 −1.44866
\(915\) 0 0
\(916\) −51850.0 −1.87028
\(917\) −192.000 −0.00691428
\(918\) 2430.00 0.0873660
\(919\) 39704.0 1.42515 0.712576 0.701595i \(-0.247529\pi\)
0.712576 + 0.701595i \(0.247529\pi\)
\(920\) 0 0
\(921\) 7236.00 0.258886
\(922\) −6270.00 −0.223960
\(923\) −23088.0 −0.823349
\(924\) 35904.0 1.27831
\(925\) 0 0
\(926\) −22920.0 −0.813389
\(927\) 11232.0 0.397958
\(928\) −2465.00 −0.0871957
\(929\) −19534.0 −0.689871 −0.344935 0.938626i \(-0.612099\pi\)
−0.344935 + 0.938626i \(0.612099\pi\)
\(930\) 0 0
\(931\) 2436.00 0.0857537
\(932\) −1394.00 −0.0489935
\(933\) −8784.00 −0.308226
\(934\) 57940.0 2.02982
\(935\) 0 0
\(936\) 31590.0 1.10315
\(937\) 8678.00 0.302559 0.151280 0.988491i \(-0.451661\pi\)
0.151280 + 0.988491i \(0.451661\pi\)
\(938\) 27200.0 0.946814
\(939\) −8622.00 −0.299647
\(940\) 0 0
\(941\) −7050.00 −0.244233 −0.122117 0.992516i \(-0.538968\pi\)
−0.122117 + 0.992516i \(0.538968\pi\)
\(942\) 8010.00 0.277049
\(943\) 14352.0 0.495616
\(944\) 16020.0 0.552337
\(945\) 0 0
\(946\) 57200.0 1.96589
\(947\) −23396.0 −0.802817 −0.401409 0.915899i \(-0.631479\pi\)
−0.401409 + 0.915899i \(0.631479\pi\)
\(948\) −48960.0 −1.67737
\(949\) 30732.0 1.05121
\(950\) 0 0
\(951\) 78.0000 0.00265965
\(952\) −12960.0 −0.441214
\(953\) 36126.0 1.22795 0.613975 0.789326i \(-0.289570\pi\)
0.613975 + 0.789326i \(0.289570\pi\)
\(954\) 25830.0 0.876601
\(955\) 0 0
\(956\) −86768.0 −2.93544
\(957\) −3828.00 −0.129302
\(958\) 91000.0 3.06897
\(959\) −44128.0 −1.48589
\(960\) 0 0
\(961\) 20385.0 0.684267
\(962\) −99060.0 −3.31998
\(963\) 1044.00 0.0349350
\(964\) −36686.0 −1.22570
\(965\) 0 0
\(966\) −44160.0 −1.47083
\(967\) −38624.0 −1.28445 −0.642225 0.766516i \(-0.721989\pi\)
−0.642225 + 0.766516i \(0.721989\pi\)
\(968\) −27225.0 −0.903972
\(969\) 1512.00 0.0501264
\(970\) 0 0
\(971\) −7292.00 −0.241000 −0.120500 0.992713i \(-0.538450\pi\)
−0.120500 + 0.992713i \(0.538450\pi\)
\(972\) 4131.00 0.136319
\(973\) 22976.0 0.757016
\(974\) 19120.0 0.628998
\(975\) 0 0
\(976\) −54290.0 −1.78051
\(977\) 26838.0 0.878837 0.439418 0.898282i \(-0.355185\pi\)
0.439418 + 0.898282i \(0.355185\pi\)
\(978\) 20700.0 0.676803
\(979\) 43560.0 1.42205
\(980\) 0 0
\(981\) −7434.00 −0.241946
\(982\) 15500.0 0.503691
\(983\) −20192.0 −0.655163 −0.327581 0.944823i \(-0.606234\pi\)
−0.327581 + 0.944823i \(0.606234\pi\)
\(984\) 10530.0 0.341142
\(985\) 0 0
\(986\) 2610.00 0.0842995
\(987\) 14976.0 0.482970
\(988\) 37128.0 1.19555
\(989\) −47840.0 −1.53814
\(990\) 0 0
\(991\) 34400.0 1.10268 0.551338 0.834282i \(-0.314117\pi\)
0.551338 + 0.834282i \(0.314117\pi\)
\(992\) 19040.0 0.609396
\(993\) 28020.0 0.895456
\(994\) 23680.0 0.755618
\(995\) 0 0
\(996\) 46308.0 1.47322
\(997\) −58430.0 −1.85606 −0.928032 0.372499i \(-0.878501\pi\)
−0.928032 + 0.372499i \(0.878501\pi\)
\(998\) −98700.0 −3.13056
\(999\) −6858.00 −0.217195
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 2175.4.a.a.1.1 1
5.4 even 2 435.4.a.c.1.1 1
15.14 odd 2 1305.4.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.4.a.c.1.1 1 5.4 even 2
1305.4.a.a.1.1 1 15.14 odd 2
2175.4.a.a.1.1 1 1.1 even 1 trivial