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Properties

Label 2325.2.a.d.1.1
Level $2325$
Weight $2$
Character 2325.1
Self dual yes
Analytic conductor $18.565$
Analytic rank $1$
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] = [2325,2,Mod(1,2325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2325, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2325.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2325 = 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2325.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: \(18.5652184699\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 465)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2325.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} -4.00000 q^{11} -1.00000 q^{12} -2.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -8.00000 q^{19} +2.00000 q^{21} +4.00000 q^{22} +8.00000 q^{23} +3.00000 q^{24} +1.00000 q^{27} -2.00000 q^{28} +1.00000 q^{31} -5.00000 q^{32} -4.00000 q^{33} +2.00000 q^{34} -1.00000 q^{36} -8.00000 q^{37} +8.00000 q^{38} -6.00000 q^{41} -2.00000 q^{42} +4.00000 q^{44} -8.00000 q^{46} -4.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -2.00000 q^{51} -6.00000 q^{53} -1.00000 q^{54} +6.00000 q^{56} -8.00000 q^{57} +10.0000 q^{59} -14.0000 q^{61} -1.00000 q^{62} +2.00000 q^{63} +7.00000 q^{64} +4.00000 q^{66} -2.00000 q^{67} +2.00000 q^{68} +8.00000 q^{69} +6.00000 q^{71} +3.00000 q^{72} +16.0000 q^{73} +8.00000 q^{74} +8.00000 q^{76} -8.00000 q^{77} +1.00000 q^{81} +6.00000 q^{82} -4.00000 q^{83} -2.00000 q^{84} -12.0000 q^{88} +4.00000 q^{89} -8.00000 q^{92} +1.00000 q^{93} +4.00000 q^{94} -5.00000 q^{96} -6.00000 q^{97} +3.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 − 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −1.00000 −0.235702
\(19\) −8.00000 −1.83533 −0.917663 − 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 4.00000 0.852803
\(23\) 8.00000 1.66812 0.834058 − 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 3.00000 0.612372
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −2.00000 −0.377964
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) −5.00000 −0.883883
\(33\) −4.00000 −0.696311
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) −8.00000 −1.31519 −0.657596 − 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 8.00000 1.29777
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −2.00000 −0.308607
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) −4.00000 −0.583460 −0.291730 − 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 6.00000 0.801784
\(57\) −8.00000 −1.05963
\(58\) 0 0
\(59\) 10.0000 1.30189 0.650945 − 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 − 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) −1.00000 −0.127000
\(63\) 2.00000 0.251976
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) −2.00000 −0.244339 −0.122169 − 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 2.00000 0.242536
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 3.00000 0.353553
\(73\) 16.0000 1.87266 0.936329 − 0.351123i \(-0.114200\pi\)
0.936329 + 0.351123i \(0.114200\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) 8.00000 0.917663
\(77\) −8.00000 −0.911685
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) −12.0000 −1.27920
\(89\) 4.00000 0.423999 0.212000 − 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −8.00000 −0.834058
\(93\) 1.00000 0.103695
\(94\) 4.00000 0.412568
\(95\) 0 0
\(96\) −5.00000 −0.510310
\(97\) −6.00000 −0.609208 −0.304604 − 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 3.00000 0.303046
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) 14.0000 1.39305 0.696526 − 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 2.00000 0.198030
\(103\) −2.00000 −0.197066 −0.0985329 − 0.995134i \(-0.531415\pi\)
−0.0985329 + 0.995134i \(0.531415\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −16.0000 −1.54678 −0.773389 − 0.633932i \(-0.781440\pi\)
−0.773389 + 0.633932i \(0.781440\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −14.0000 −1.34096 −0.670478 − 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) −2.00000 −0.188982
\(113\) 6.00000 0.564433 0.282216 − 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 8.00000 0.749269
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) −10.0000 −0.920575
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 14.0000 1.26750
\(123\) −6.00000 −0.541002
\(124\) −1.00000 −0.0898027
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) 4.00000 0.354943 0.177471 − 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 3.00000 0.265165
\(129\) 0 0
\(130\) 0 0
\(131\) −14.0000 −1.22319 −0.611593 − 0.791173i \(-0.709471\pi\)
−0.611593 + 0.791173i \(0.709471\pi\)
\(132\) 4.00000 0.348155
\(133\) −16.0000 −1.38738
\(134\) 2.00000 0.172774
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) 18.0000 1.53784 0.768922 − 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) −8.00000 −0.681005
\(139\) −12.0000 −1.01783 −0.508913 − 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) −6.00000 −0.503509
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −16.0000 −1.32417
\(147\) −3.00000 −0.247436
\(148\) 8.00000 0.657596
\(149\) −18.0000 −1.47462 −0.737309 − 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) −24.0000 −1.95309 −0.976546 − 0.215308i \(-0.930924\pi\)
−0.976546 + 0.215308i \(0.930924\pi\)
\(152\) −24.0000 −1.94666
\(153\) −2.00000 −0.161690
\(154\) 8.00000 0.644658
\(155\) 0 0
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 − 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 16.0000 1.26098
\(162\) −1.00000 −0.0785674
\(163\) −2.00000 −0.156652 −0.0783260 − 0.996928i \(-0.524958\pi\)
−0.0783260 + 0.996928i \(0.524958\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) −16.0000 −1.23812 −0.619059 − 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 6.00000 0.462910
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) −8.00000 −0.611775
\(172\) 0 0
\(173\) −6.00000 −0.456172 −0.228086 − 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 10.0000 0.751646
\(178\) −4.00000 −0.299813
\(179\) −4.00000 −0.298974 −0.149487 − 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 − 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) −14.0000 −1.03491
\(184\) 24.0000 1.76930
\(185\) 0 0
\(186\) −1.00000 −0.0733236
\(187\) 8.00000 0.585018
\(188\) 4.00000 0.291730
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) −14.0000 −1.01300 −0.506502 − 0.862239i \(-0.669062\pi\)
−0.506502 + 0.862239i \(0.669062\pi\)
\(192\) 7.00000 0.505181
\(193\) −6.00000 −0.431889 −0.215945 − 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) −2.00000 −0.142494 −0.0712470 − 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 4.00000 0.284268
\(199\) −8.00000 −0.567105 −0.283552 − 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) −2.00000 −0.141069
\(202\) −14.0000 −0.985037
\(203\) 0 0
\(204\) 2.00000 0.140028
\(205\) 0 0
\(206\) 2.00000 0.139347
\(207\) 8.00000 0.556038
\(208\) 0 0
\(209\) 32.0000 2.21349
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 − 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 6.00000 0.412082
\(213\) 6.00000 0.411113
\(214\) 16.0000 1.09374
\(215\) 0 0
\(216\) 3.00000 0.204124
\(217\) 2.00000 0.135769
\(218\) 14.0000 0.948200
\(219\) 16.0000 1.08118
\(220\) 0 0
\(221\) 0 0
\(222\) 8.00000 0.536925
\(223\) −16.0000 −1.07144 −0.535720 − 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) −10.0000 −0.668153
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 20.0000 1.32745 0.663723 − 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 8.00000 0.529813
\(229\) 6.00000 0.396491 0.198246 − 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) −8.00000 −0.526361
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 − 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −10.0000 −0.650945
\(237\) 0 0
\(238\) 4.00000 0.259281
\(239\) −20.0000 −1.29369 −0.646846 − 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 − 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) −5.00000 −0.321412
\(243\) 1.00000 0.0641500
\(244\) 14.0000 0.896258
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) 0 0
\(248\) 3.00000 0.190500
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 20.0000 1.26239 0.631194 − 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) −2.00000 −0.125988
\(253\) −32.0000 −2.01182
\(254\) −4.00000 −0.250982
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 18.0000 1.12281 0.561405 − 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) −16.0000 −0.994192
\(260\) 0 0
\(261\) 0 0
\(262\) 14.0000 0.864923
\(263\) 24.0000 1.47990 0.739952 − 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) −12.0000 −0.738549
\(265\) 0 0
\(266\) 16.0000 0.981023
\(267\) 4.00000 0.244796
\(268\) 2.00000 0.122169
\(269\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 − 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) 0 0
\(276\) −8.00000 −0.481543
\(277\) 28.0000 1.68236 0.841178 − 0.540758i \(-0.181862\pi\)
0.841178 + 0.540758i \(0.181862\pi\)
\(278\) 12.0000 0.719712
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 − 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 4.00000 0.238197
\(283\) −10.0000 −0.594438 −0.297219 − 0.954809i \(-0.596059\pi\)
−0.297219 + 0.954809i \(0.596059\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) −5.00000 −0.294628
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −6.00000 −0.351726
\(292\) −16.0000 −0.936329
\(293\) −26.0000 −1.51894 −0.759468 − 0.650545i \(-0.774541\pi\)
−0.759468 + 0.650545i \(0.774541\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) −24.0000 −1.39497
\(297\) −4.00000 −0.232104
\(298\) 18.0000 1.04271
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 24.0000 1.38104
\(303\) 14.0000 0.804279
\(304\) 8.00000 0.458831
\(305\) 0 0
\(306\) 2.00000 0.114332
\(307\) −22.0000 −1.25561 −0.627803 − 0.778372i \(-0.716046\pi\)
−0.627803 + 0.778372i \(0.716046\pi\)
\(308\) 8.00000 0.455842
\(309\) −2.00000 −0.113776
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 − 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) 8.00000 0.452187 0.226093 − 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) 0 0
\(317\) 6.00000 0.336994 0.168497 − 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 6.00000 0.336463
\(319\) 0 0
\(320\) 0 0
\(321\) −16.0000 −0.893033
\(322\) −16.0000 −0.891645
\(323\) 16.0000 0.890264
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 2.00000 0.110770
\(327\) −14.0000 −0.774202
\(328\) −18.0000 −0.993884
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 − 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 4.00000 0.219529
\(333\) −8.00000 −0.438397
\(334\) 16.0000 0.875481
\(335\) 0 0
\(336\) −2.00000 −0.109109
\(337\) −28.0000 −1.52526 −0.762629 − 0.646837i \(-0.776092\pi\)
−0.762629 + 0.646837i \(0.776092\pi\)
\(338\) 13.0000 0.707107
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) −4.00000 −0.216612
\(342\) 8.00000 0.432590
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 − 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 20.0000 1.06600
\(353\) 14.0000 0.745145 0.372572 − 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) −10.0000 −0.531494
\(355\) 0 0
\(356\) −4.00000 −0.212000
\(357\) −4.00000 −0.211702
\(358\) 4.00000 0.211407
\(359\) 26.0000 1.37223 0.686114 − 0.727494i \(-0.259315\pi\)
0.686114 + 0.727494i \(0.259315\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 6.00000 0.315353
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) 0 0
\(366\) 14.0000 0.731792
\(367\) 32.0000 1.67039 0.835193 − 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) −8.00000 −0.417029
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) −1.00000 −0.0518476
\(373\) 22.0000 1.13912 0.569558 − 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 0 0
\(378\) −2.00000 −0.102869
\(379\) 16.0000 0.821865 0.410932 − 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) 4.00000 0.204926
\(382\) 14.0000 0.716302
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) 6.00000 0.305392
\(387\) 0 0
\(388\) 6.00000 0.304604
\(389\) 16.0000 0.811232 0.405616 − 0.914044i \(-0.367057\pi\)
0.405616 + 0.914044i \(0.367057\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) −9.00000 −0.454569
\(393\) −14.0000 −0.706207
\(394\) 2.00000 0.100759
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) −14.0000 −0.702640 −0.351320 − 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 8.00000 0.401004
\(399\) −16.0000 −0.801002
\(400\) 0 0
\(401\) 16.0000 0.799002 0.399501 − 0.916733i \(-0.369183\pi\)
0.399501 + 0.916733i \(0.369183\pi\)
\(402\) 2.00000 0.0997509
\(403\) 0 0
\(404\) −14.0000 −0.696526
\(405\) 0 0
\(406\) 0 0
\(407\) 32.0000 1.58618
\(408\) −6.00000 −0.297044
\(409\) −18.0000 −0.890043 −0.445021 − 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 0 0
\(411\) 18.0000 0.887875
\(412\) 2.00000 0.0985329
\(413\) 20.0000 0.984136
\(414\) −8.00000 −0.393179
\(415\) 0 0
\(416\) 0 0
\(417\) −12.0000 −0.587643
\(418\) −32.0000 −1.56517
\(419\) 14.0000 0.683945 0.341972 − 0.939710i \(-0.388905\pi\)
0.341972 + 0.939710i \(0.388905\pi\)
\(420\) 0 0
\(421\) 18.0000 0.877266 0.438633 − 0.898666i \(-0.355463\pi\)
0.438633 + 0.898666i \(0.355463\pi\)
\(422\) −8.00000 −0.389434
\(423\) −4.00000 −0.194487
\(424\) −18.0000 −0.874157
\(425\) 0 0
\(426\) −6.00000 −0.290701
\(427\) −28.0000 −1.35501
\(428\) 16.0000 0.773389
\(429\) 0 0
\(430\) 0 0
\(431\) 2.00000 0.0963366 0.0481683 − 0.998839i \(-0.484662\pi\)
0.0481683 + 0.998839i \(0.484662\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −24.0000 −1.15337 −0.576683 − 0.816968i \(-0.695653\pi\)
−0.576683 + 0.816968i \(0.695653\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) −64.0000 −3.06154
\(438\) −16.0000 −0.764510
\(439\) 24.0000 1.14546 0.572729 − 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 20.0000 0.950229 0.475114 − 0.879924i \(-0.342407\pi\)
0.475114 + 0.879924i \(0.342407\pi\)
\(444\) 8.00000 0.379663
\(445\) 0 0
\(446\) 16.0000 0.757622
\(447\) −18.0000 −0.851371
\(448\) 14.0000 0.661438
\(449\) −8.00000 −0.377543 −0.188772 − 0.982021i \(-0.560451\pi\)
−0.188772 + 0.982021i \(0.560451\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) −6.00000 −0.282216
\(453\) −24.0000 −1.12762
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) −24.0000 −1.12390
\(457\) 8.00000 0.374224 0.187112 − 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) −6.00000 −0.280362
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 − 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 8.00000 0.372194
\(463\) −20.0000 −0.929479 −0.464739 − 0.885448i \(-0.653852\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) −40.0000 −1.85098 −0.925490 − 0.378773i \(-0.876346\pi\)
−0.925490 + 0.378773i \(0.876346\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) 30.0000 1.38086
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 4.00000 0.183340
\(477\) −6.00000 −0.274721
\(478\) 20.0000 0.914779
\(479\) −10.0000 −0.456912 −0.228456 − 0.973554i \(-0.573368\pi\)
−0.228456 + 0.973554i \(0.573368\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 22.0000 1.00207
\(483\) 16.0000 0.728025
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 4.00000 0.181257 0.0906287 − 0.995885i \(-0.471112\pi\)
0.0906287 + 0.995885i \(0.471112\pi\)
\(488\) −42.0000 −1.90125
\(489\) −2.00000 −0.0904431
\(490\) 0 0
\(491\) −24.0000 −1.08310 −0.541552 − 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) 6.00000 0.270501
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −1.00000 −0.0449013
\(497\) 12.0000 0.538274
\(498\) 4.00000 0.179244
\(499\) −20.0000 −0.895323 −0.447661 − 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 0 0
\(501\) −16.0000 −0.714827
\(502\) −20.0000 −0.892644
\(503\) 12.0000 0.535054 0.267527 − 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 6.00000 0.267261
\(505\) 0 0
\(506\) 32.0000 1.42257
\(507\) −13.0000 −0.577350
\(508\) −4.00000 −0.177471
\(509\) −28.0000 −1.24108 −0.620539 − 0.784176i \(-0.713086\pi\)
−0.620539 + 0.784176i \(0.713086\pi\)
\(510\) 0 0
\(511\) 32.0000 1.41560
\(512\) 11.0000 0.486136
\(513\) −8.00000 −0.353209
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) 0 0
\(517\) 16.0000 0.703679
\(518\) 16.0000 0.703000
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 − 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) 8.00000 0.349816 0.174908 − 0.984585i \(-0.444037\pi\)
0.174908 + 0.984585i \(0.444037\pi\)
\(524\) 14.0000 0.611593
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) −2.00000 −0.0871214
\(528\) 4.00000 0.174078
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 10.0000 0.433963
\(532\) 16.0000 0.693688
\(533\) 0 0
\(534\) −4.00000 −0.173097
\(535\) 0 0
\(536\) −6.00000 −0.259161
\(537\) −4.00000 −0.172613
\(538\) 0 0
\(539\) 12.0000 0.516877
\(540\) 0 0
\(541\) 34.0000 1.46177 0.730887 − 0.682498i \(-0.239107\pi\)
0.730887 + 0.682498i \(0.239107\pi\)
\(542\) −8.00000 −0.343629
\(543\) −6.00000 −0.257485
\(544\) 10.0000 0.428746
\(545\) 0 0
\(546\) 0 0
\(547\) 22.0000 0.940652 0.470326 − 0.882493i \(-0.344136\pi\)
0.470326 + 0.882493i \(0.344136\pi\)
\(548\) −18.0000 −0.768922
\(549\) −14.0000 −0.597505
\(550\) 0 0
\(551\) 0 0
\(552\) 24.0000 1.02151
\(553\) 0 0
\(554\) −28.0000 −1.18961
\(555\) 0 0
\(556\) 12.0000 0.508913
\(557\) −22.0000 −0.932170 −0.466085 − 0.884740i \(-0.654336\pi\)
−0.466085 + 0.884740i \(0.654336\pi\)
\(558\) −1.00000 −0.0423334
\(559\) 0 0
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) −30.0000 −1.26547
\(563\) −24.0000 −1.01148 −0.505740 − 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 4.00000 0.168430
\(565\) 0 0
\(566\) 10.0000 0.420331
\(567\) 2.00000 0.0839921
\(568\) 18.0000 0.755263
\(569\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 − 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) −14.0000 −0.584858
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) 38.0000 1.58196 0.790980 − 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) 13.0000 0.540729
\(579\) −6.00000 −0.249351
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) 6.00000 0.248708
\(583\) 24.0000 0.993978
\(584\) 48.0000 1.98625
\(585\) 0 0
\(586\) 26.0000 1.07405
\(587\) 12.0000 0.495293 0.247647 − 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 3.00000 0.123718
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) −2.00000 −0.0822690
\(592\) 8.00000 0.328798
\(593\) −18.0000 −0.739171 −0.369586 − 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) −8.00000 −0.327418
\(598\) 0 0
\(599\) −14.0000 −0.572024 −0.286012 − 0.958226i \(-0.592330\pi\)
−0.286012 + 0.958226i \(0.592330\pi\)
\(600\) 0 0
\(601\) 14.0000 0.571072 0.285536 − 0.958368i \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) 0 0
\(603\) −2.00000 −0.0814463
\(604\) 24.0000 0.976546
\(605\) 0 0
\(606\) −14.0000 −0.568711
\(607\) 46.0000 1.86708 0.933541 − 0.358470i \(-0.116702\pi\)
0.933541 + 0.358470i \(0.116702\pi\)
\(608\) 40.0000 1.62221
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 2.00000 0.0808452
\(613\) 44.0000 1.77714 0.888572 − 0.458738i \(-0.151698\pi\)
0.888572 + 0.458738i \(0.151698\pi\)
\(614\) 22.0000 0.887848
\(615\) 0 0
\(616\) −24.0000 −0.966988
\(617\) 38.0000 1.52982 0.764911 − 0.644136i \(-0.222783\pi\)
0.764911 + 0.644136i \(0.222783\pi\)
\(618\) 2.00000 0.0804518
\(619\) 20.0000 0.803868 0.401934 − 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) 8.00000 0.321029
\(622\) −18.0000 −0.721734
\(623\) 8.00000 0.320513
\(624\) 0 0
\(625\) 0 0
\(626\) −8.00000 −0.319744
\(627\) 32.0000 1.27796
\(628\) 2.00000 0.0798087
\(629\) 16.0000 0.637962
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 − 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 0 0
\(633\) 8.00000 0.317971
\(634\) −6.00000 −0.238290
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) 0 0
\(639\) 6.00000 0.237356
\(640\) 0 0
\(641\) 40.0000 1.57991 0.789953 − 0.613168i \(-0.210105\pi\)
0.789953 + 0.613168i \(0.210105\pi\)
\(642\) 16.0000 0.631470
\(643\) −16.0000 −0.630978 −0.315489 − 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) −16.0000 −0.630488
\(645\) 0 0
\(646\) −16.0000 −0.629512
\(647\) 16.0000 0.629025 0.314512 − 0.949253i \(-0.398159\pi\)
0.314512 + 0.949253i \(0.398159\pi\)
\(648\) 3.00000 0.117851
\(649\) −40.0000 −1.57014
\(650\) 0 0
\(651\) 2.00000 0.0783862
\(652\) 2.00000 0.0783260
\(653\) −10.0000 −0.391330 −0.195665 − 0.980671i \(-0.562687\pi\)
−0.195665 + 0.980671i \(0.562687\pi\)
\(654\) 14.0000 0.547443
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) 16.0000 0.624219
\(658\) 8.00000 0.311872
\(659\) −42.0000 −1.63609 −0.818044 − 0.575156i \(-0.804941\pi\)
−0.818044 + 0.575156i \(0.804941\pi\)
\(660\) 0 0
\(661\) −2.00000 −0.0777910 −0.0388955 − 0.999243i \(-0.512384\pi\)
−0.0388955 + 0.999243i \(0.512384\pi\)
\(662\) 4.00000 0.155464
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) 8.00000 0.309994
\(667\) 0 0
\(668\) 16.0000 0.619059
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) 56.0000 2.16186
\(672\) −10.0000 −0.385758
\(673\) 44.0000 1.69608 0.848038 − 0.529936i \(-0.177784\pi\)
0.848038 + 0.529936i \(0.177784\pi\)
\(674\) 28.0000 1.07852
\(675\) 0 0
\(676\) 13.0000 0.500000
\(677\) 26.0000 0.999261 0.499631 − 0.866239i \(-0.333469\pi\)
0.499631 + 0.866239i \(0.333469\pi\)
\(678\) −6.00000 −0.230429
\(679\) −12.0000 −0.460518
\(680\) 0 0
\(681\) 20.0000 0.766402
\(682\) 4.00000 0.153168
\(683\) 4.00000 0.153056 0.0765279 − 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 8.00000 0.305888
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) 6.00000 0.228914
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 12.0000 0.456502 0.228251 − 0.973602i \(-0.426699\pi\)
0.228251 + 0.973602i \(0.426699\pi\)
\(692\) 6.00000 0.228086
\(693\) −8.00000 −0.303895
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) 0 0
\(697\) 12.0000 0.454532
\(698\) 14.0000 0.529908
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) −6.00000 −0.226617 −0.113308 − 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 0 0
\(703\) 64.0000 2.41381
\(704\) −28.0000 −1.05529
\(705\) 0 0
\(706\) −14.0000 −0.526897
\(707\) 28.0000 1.05305
\(708\) −10.0000 −0.375823
\(709\) −42.0000 −1.57734 −0.788672 − 0.614815i \(-0.789231\pi\)
−0.788672 + 0.614815i \(0.789231\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 12.0000 0.449719
\(713\) 8.00000 0.299602
\(714\) 4.00000 0.149696
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) −20.0000 −0.746914
\(718\) −26.0000 −0.970311
\(719\) −8.00000 −0.298350 −0.149175 − 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) −45.0000 −1.67473
\(723\) −22.0000 −0.818189
\(724\) 6.00000 0.222988
\(725\) 0 0
\(726\) −5.00000 −0.185567
\(727\) −30.0000 −1.11264 −0.556319 − 0.830969i \(-0.687787\pi\)
−0.556319 + 0.830969i \(0.687787\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 14.0000 0.517455
\(733\) 26.0000 0.960332 0.480166 − 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) −32.0000 −1.18114
\(735\) 0 0
\(736\) −40.0000 −1.47442
\(737\) 8.00000 0.294684
\(738\) 6.00000 0.220863
\(739\) 28.0000 1.03000 0.514998 − 0.857191i \(-0.327793\pi\)
0.514998 + 0.857191i \(0.327793\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 12.0000 0.440534
\(743\) −24.0000 −0.880475 −0.440237 − 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 3.00000 0.109985
\(745\) 0 0
\(746\) −22.0000 −0.805477
\(747\) −4.00000 −0.146352
\(748\) −8.00000 −0.292509
\(749\) −32.0000 −1.16925
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 − 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 4.00000 0.145865
\(753\) 20.0000 0.728841
\(754\) 0 0
\(755\) 0 0
\(756\) −2.00000 −0.0727393
\(757\) 8.00000 0.290765 0.145382 − 0.989376i \(-0.453559\pi\)
0.145382 + 0.989376i \(0.453559\pi\)
\(758\) −16.0000 −0.581146
\(759\) −32.0000 −1.16153
\(760\) 0 0
\(761\) −24.0000 −0.869999 −0.435000 − 0.900431i \(-0.643252\pi\)
−0.435000 + 0.900431i \(0.643252\pi\)
\(762\) −4.00000 −0.144905
\(763\) −28.0000 −1.01367
\(764\) 14.0000 0.506502
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −17.0000 −0.613435
\(769\) 46.0000 1.65880 0.829401 − 0.558653i \(-0.188682\pi\)
0.829401 + 0.558653i \(0.188682\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 6.00000 0.215945
\(773\) −30.0000 −1.07903 −0.539513 − 0.841978i \(-0.681391\pi\)
−0.539513 + 0.841978i \(0.681391\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −18.0000 −0.646162
\(777\) −16.0000 −0.573997
\(778\) −16.0000 −0.573628
\(779\) 48.0000 1.71978
\(780\) 0 0
\(781\) −24.0000 −0.858788
\(782\) 16.0000 0.572159
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) 14.0000 0.499363
\(787\) −52.0000 −1.85360 −0.926800 − 0.375555i \(-0.877452\pi\)
−0.926800 + 0.375555i \(0.877452\pi\)
\(788\) 2.00000 0.0712470
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) −12.0000 −0.426401
\(793\) 0 0
\(794\) 14.0000 0.496841
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) −42.0000 −1.48772 −0.743858 − 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 16.0000 0.566394
\(799\) 8.00000 0.283020
\(800\) 0 0
\(801\) 4.00000 0.141333
\(802\) −16.0000 −0.564980
\(803\) −64.0000 −2.25851
\(804\) 2.00000 0.0705346
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 42.0000 1.47755
\(809\) 24.0000 0.843795 0.421898 − 0.906644i \(-0.361364\pi\)
0.421898 + 0.906644i \(0.361364\pi\)
\(810\) 0 0
\(811\) 32.0000 1.12367 0.561836 − 0.827249i \(-0.310095\pi\)
0.561836 + 0.827249i \(0.310095\pi\)
\(812\) 0 0
\(813\) 8.00000 0.280572
\(814\) −32.0000 −1.12160
\(815\) 0 0
\(816\) 2.00000 0.0700140
\(817\) 0 0
\(818\) 18.0000 0.629355
\(819\) 0 0
\(820\) 0 0
\(821\) 24.0000 0.837606 0.418803 − 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) −18.0000 −0.627822
\(823\) 44.0000 1.53374 0.766872 − 0.641800i \(-0.221812\pi\)
0.766872 + 0.641800i \(0.221812\pi\)
\(824\) −6.00000 −0.209020
\(825\) 0 0
\(826\) −20.0000 −0.695889
\(827\) 12.0000 0.417281 0.208640 − 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) −8.00000 −0.278019
\(829\) −22.0000 −0.764092 −0.382046 − 0.924143i \(-0.624780\pi\)
−0.382046 + 0.924143i \(0.624780\pi\)
\(830\) 0 0
\(831\) 28.0000 0.971309
\(832\) 0 0
\(833\) 6.00000 0.207888
\(834\) 12.0000 0.415526
\(835\) 0 0
\(836\) −32.0000 −1.10674
\(837\) 1.00000 0.0345651
\(838\) −14.0000 −0.483622
\(839\) −6.00000 −0.207143 −0.103572 − 0.994622i \(-0.533027\pi\)
−0.103572 + 0.994622i \(0.533027\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) −18.0000 −0.620321
\(843\) 30.0000 1.03325
\(844\) −8.00000 −0.275371
\(845\) 0 0
\(846\) 4.00000 0.137523
\(847\) 10.0000 0.343604
\(848\) 6.00000 0.206041
\(849\) −10.0000 −0.343199
\(850\) 0 0
\(851\) −64.0000 −2.19389
\(852\) −6.00000 −0.205557
\(853\) 2.00000 0.0684787 0.0342393 − 0.999414i \(-0.489099\pi\)
0.0342393 + 0.999414i \(0.489099\pi\)
\(854\) 28.0000 0.958140
\(855\) 0 0
\(856\) −48.0000 −1.64061
\(857\) 6.00000 0.204956 0.102478 − 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) 0 0
\(859\) 28.0000 0.955348 0.477674 − 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 0 0
\(861\) −12.0000 −0.408959
\(862\) −2.00000 −0.0681203
\(863\) −8.00000 −0.272323 −0.136162 − 0.990687i \(-0.543477\pi\)
−0.136162 + 0.990687i \(0.543477\pi\)
\(864\) −5.00000 −0.170103
\(865\) 0 0
\(866\) 24.0000 0.815553
\(867\) −13.0000 −0.441503
\(868\) −2.00000 −0.0678844
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −42.0000 −1.42230
\(873\) −6.00000 −0.203069
\(874\) 64.0000 2.16483
\(875\) 0 0
\(876\) −16.0000 −0.540590
\(877\) 18.0000 0.607817 0.303908 − 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) −24.0000 −0.809961
\(879\) −26.0000 −0.876958
\(880\) 0 0
\(881\) −28.0000 −0.943344 −0.471672 − 0.881774i \(-0.656349\pi\)
−0.471672 + 0.881774i \(0.656349\pi\)
\(882\) 3.00000 0.101015
\(883\) −28.0000 −0.942275 −0.471138 − 0.882060i \(-0.656156\pi\)
−0.471138 + 0.882060i \(0.656156\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −20.0000 −0.671913
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) −24.0000 −0.805387
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 16.0000 0.535720
\(893\) 32.0000 1.07084
\(894\) 18.0000 0.602010
\(895\) 0 0
\(896\) 6.00000 0.200446
\(897\) 0 0
\(898\) 8.00000 0.266963
\(899\) 0 0
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) −24.0000 −0.799113
\(903\) 0 0
\(904\) 18.0000 0.598671
\(905\) 0 0
\(906\) 24.0000 0.797347
\(907\) 18.0000 0.597680 0.298840 − 0.954303i \(-0.403400\pi\)
0.298840 + 0.954303i \(0.403400\pi\)
\(908\) −20.0000 −0.663723
\(909\) 14.0000 0.464351
\(910\) 0 0
\(911\) −8.00000 −0.265052 −0.132526 − 0.991180i \(-0.542309\pi\)
−0.132526 + 0.991180i \(0.542309\pi\)
\(912\) 8.00000 0.264906
\(913\) 16.0000 0.529523
\(914\) −8.00000 −0.264616
\(915\) 0 0
\(916\) −6.00000 −0.198246
\(917\) −28.0000 −0.924641
\(918\) 2.00000 0.0660098
\(919\) 20.0000 0.659739 0.329870 − 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) 0 0
\(921\) −22.0000 −0.724925
\(922\) −12.0000 −0.395199
\(923\) 0 0
\(924\) 8.00000 0.263181
\(925\) 0 0
\(926\) 20.0000 0.657241
\(927\) −2.00000 −0.0656886
\(928\) 0 0
\(929\) −44.0000 −1.44359 −0.721797 − 0.692105i \(-0.756683\pi\)
−0.721797 + 0.692105i \(0.756683\pi\)
\(930\) 0 0
\(931\) 24.0000 0.786568
\(932\) 6.00000 0.196537
\(933\) 18.0000 0.589294
\(934\) 40.0000 1.30884
\(935\) 0 0
\(936\) 0 0
\(937\) −6.00000 −0.196011 −0.0980057 − 0.995186i \(-0.531246\pi\)
−0.0980057 + 0.995186i \(0.531246\pi\)
\(938\) 4.00000 0.130605
\(939\) 8.00000 0.261070
\(940\) 0 0
\(941\) 24.0000 0.782378 0.391189 − 0.920310i \(-0.372064\pi\)
0.391189 + 0.920310i \(0.372064\pi\)
\(942\) 2.00000 0.0651635
\(943\) −48.0000 −1.56310
\(944\) −10.0000 −0.325472
\(945\) 0 0
\(946\) 0 0
\(947\) −12.0000 −0.389948 −0.194974 − 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 6.00000 0.194563
\(952\) −12.0000 −0.388922
\(953\) −30.0000 −0.971795 −0.485898 − 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 20.0000 0.646846
\(957\) 0 0
\(958\) 10.0000 0.323085
\(959\) 36.0000 1.16250
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) −16.0000 −0.515593
\(964\) 22.0000 0.708572
\(965\) 0 0
\(966\) −16.0000 −0.514792
\(967\) −24.0000 −0.771788 −0.385894 − 0.922543i \(-0.626107\pi\)
−0.385894 + 0.922543i \(0.626107\pi\)
\(968\) 15.0000 0.482118
\(969\) 16.0000 0.513994
\(970\) 0 0
\(971\) −42.0000 −1.34784 −0.673922 − 0.738802i \(-0.735392\pi\)
−0.673922 + 0.738802i \(0.735392\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −24.0000 −0.769405
\(974\) −4.00000 −0.128168
\(975\) 0 0
\(976\) 14.0000 0.448129
\(977\) 30.0000 0.959785 0.479893 − 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 2.00000 0.0639529
\(979\) −16.0000 −0.511362
\(980\) 0 0
\(981\) −14.0000 −0.446986
\(982\) 24.0000 0.765871
\(983\) 8.00000 0.255160 0.127580 − 0.991828i \(-0.459279\pi\)
0.127580 + 0.991828i \(0.459279\pi\)
\(984\) −18.0000 −0.573819
\(985\) 0 0
\(986\) 0 0
\(987\) −8.00000 −0.254643
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −56.0000 −1.77890 −0.889449 − 0.457034i \(-0.848912\pi\)
−0.889449 + 0.457034i \(0.848912\pi\)
\(992\) −5.00000 −0.158750
\(993\) −4.00000 −0.126936
\(994\) −12.0000 −0.380617
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) −42.0000 −1.33015 −0.665077 − 0.746775i \(-0.731601\pi\)
−0.665077 + 0.746775i \(0.731601\pi\)
\(998\) 20.0000 0.633089
\(999\) −8.00000 −0.253109
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 2325.2.a.d.1.1 1
3.2 odd 2 6975.2.a.q.1.1 1
5.2 odd 4 2325.2.c.d.1024.1 2
5.3 odd 4 2325.2.c.d.1024.2 2
5.4 even 2 465.2.a.b.1.1 ✓ 1
15.14 odd 2 1395.2.a.a.1.1 1
20.19 odd 2 7440.2.a.ba.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.a.b.1.1 ✓ 1 5.4 even 2
1395.2.a.a.1.1 1 15.14 odd 2
2325.2.a.d.1.1 1 1.1 even 1 trivial
2325.2.c.d.1024.1 2 5.2 odd 4
2325.2.c.d.1024.2 2 5.3 odd 4
6975.2.a.q.1.1 1 3.2 odd 2
7440.2.a.ba.1.1 1 20.19 odd 2