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Properties

Label 3450.2.a.n.1.1
Level $3450$
Weight $2$
Character 3450.1
Self dual yes
Analytic conductor $27.548$
Analytic rank $0$
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] = [3450,2,Mod(1,3450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3450, 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("3450.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.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: \(27.5483886973\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3450.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -5.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -5.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{12} -2.00000 q^{13} -5.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} +1.00000 q^{18} +2.00000 q^{19} +5.00000 q^{21} -1.00000 q^{23} -1.00000 q^{24} -2.00000 q^{26} -1.00000 q^{27} -5.00000 q^{28} +3.00000 q^{29} +2.00000 q^{31} +1.00000 q^{32} -3.00000 q^{34} +1.00000 q^{36} +7.00000 q^{37} +2.00000 q^{38} +2.00000 q^{39} +5.00000 q^{42} -2.00000 q^{43} -1.00000 q^{46} +3.00000 q^{47} -1.00000 q^{48} +18.0000 q^{49} +3.00000 q^{51} -2.00000 q^{52} +12.0000 q^{53} -1.00000 q^{54} -5.00000 q^{56} -2.00000 q^{57} +3.00000 q^{58} +6.00000 q^{59} +2.00000 q^{61} +2.00000 q^{62} -5.00000 q^{63} +1.00000 q^{64} -2.00000 q^{67} -3.00000 q^{68} +1.00000 q^{69} -15.0000 q^{71} +1.00000 q^{72} -11.0000 q^{73} +7.00000 q^{74} +2.00000 q^{76} +2.00000 q^{78} +8.00000 q^{79} +1.00000 q^{81} +9.00000 q^{83} +5.00000 q^{84} -2.00000 q^{86} -3.00000 q^{87} +3.00000 q^{89} +10.0000 q^{91} -1.00000 q^{92} -2.00000 q^{93} +3.00000 q^{94} -1.00000 q^{96} +10.0000 q^{97} +18.0000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) −5.00000 −1.88982 −0.944911 − 0.327327i \(-0.893852\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −5.00000 −1.33631
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 − 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 1.00000 0.235702
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 5.00000 1.09109
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) −5.00000 −0.944911
\(29\) 3.00000 0.557086 0.278543 − 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −3.00000 −0.514496
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 7.00000 1.15079 0.575396 − 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) 2.00000 0.324443
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 5.00000 0.771517
\(43\) −2.00000 −0.304997 −0.152499 − 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 3.00000 0.437595 0.218797 − 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) −1.00000 −0.144338
\(49\) 18.0000 2.57143
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) −2.00000 −0.277350
\(53\) 12.0000 1.64833 0.824163 − 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −5.00000 −0.668153
\(57\) −2.00000 −0.264906
\(58\) 3.00000 0.393919
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 2.00000 0.254000
\(63\) −5.00000 −0.629941
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 − 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) −3.00000 −0.363803
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −15.0000 −1.78017 −0.890086 − 0.455792i \(-0.849356\pi\)
−0.890086 + 0.455792i \(0.849356\pi\)
\(72\) 1.00000 0.117851
\(73\) −11.0000 −1.28745 −0.643726 − 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) 7.00000 0.813733
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 9.00000 0.987878 0.493939 − 0.869496i \(-0.335557\pi\)
0.493939 + 0.869496i \(0.335557\pi\)
\(84\) 5.00000 0.545545
\(85\) 0 0
\(86\) −2.00000 −0.215666
\(87\) −3.00000 −0.321634
\(88\) 0 0
\(89\) 3.00000 0.317999 0.159000 − 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) 10.0000 1.04828
\(92\) −1.00000 −0.104257
\(93\) −2.00000 −0.207390
\(94\) 3.00000 0.309426
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 18.0000 1.81827
\(99\) 0 0
\(100\) 0 0
\(101\) −3.00000 −0.298511 −0.149256 − 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) 3.00000 0.297044
\(103\) 13.0000 1.28093 0.640464 − 0.767988i \(-0.278742\pi\)
0.640464 + 0.767988i \(0.278742\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 12.0000 1.16554
\(107\) −12.0000 −1.16008 −0.580042 − 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 17.0000 1.62830 0.814152 − 0.580651i \(-0.197202\pi\)
0.814152 + 0.580651i \(0.197202\pi\)
\(110\) 0 0
\(111\) −7.00000 −0.664411
\(112\) −5.00000 −0.472456
\(113\) 9.00000 0.846649 0.423324 − 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(114\) −2.00000 −0.187317
\(115\) 0 0
\(116\) 3.00000 0.278543
\(117\) −2.00000 −0.184900
\(118\) 6.00000 0.552345
\(119\) 15.0000 1.37505
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) 2.00000 0.179605
\(125\) 0 0
\(126\) −5.00000 −0.445435
\(127\) 16.0000 1.41977 0.709885 − 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 − 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) −10.0000 −0.867110
\(134\) −2.00000 −0.172774
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) 15.0000 1.28154 0.640768 − 0.767734i \(-0.278616\pi\)
0.640768 + 0.767734i \(0.278616\pi\)
\(138\) 1.00000 0.0851257
\(139\) 5.00000 0.424094 0.212047 − 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) 0 0
\(141\) −3.00000 −0.252646
\(142\) −15.0000 −1.25877
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −11.0000 −0.910366
\(147\) −18.0000 −1.48461
\(148\) 7.00000 0.575396
\(149\) 12.0000 0.983078 0.491539 − 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 − 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 2.00000 0.162221
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) −2.00000 −0.159617 −0.0798087 − 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 8.00000 0.636446
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) 5.00000 0.394055
\(162\) 1.00000 0.0785674
\(163\) 16.0000 1.25322 0.626608 − 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 9.00000 0.698535
\(167\) −15.0000 −1.16073 −0.580367 − 0.814355i \(-0.697091\pi\)
−0.580367 + 0.814355i \(0.697091\pi\)
\(168\) 5.00000 0.385758
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) −2.00000 −0.152499
\(173\) −6.00000 −0.456172 −0.228086 − 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) −3.00000 −0.227429
\(175\) 0 0
\(176\) 0 0
\(177\) −6.00000 −0.450988
\(178\) 3.00000 0.224860
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 5.00000 0.371647 0.185824 − 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) 10.0000 0.741249
\(183\) −2.00000 −0.147844
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) −2.00000 −0.146647
\(187\) 0 0
\(188\) 3.00000 0.218797
\(189\) 5.00000 0.363696
\(190\) 0 0
\(191\) −6.00000 −0.434145 −0.217072 − 0.976156i \(-0.569651\pi\)
−0.217072 + 0.976156i \(0.569651\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 1.00000 0.0719816 0.0359908 − 0.999352i \(-0.488541\pi\)
0.0359908 + 0.999352i \(0.488541\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 18.0000 1.28571
\(197\) −3.00000 −0.213741 −0.106871 − 0.994273i \(-0.534083\pi\)
−0.106871 + 0.994273i \(0.534083\pi\)
\(198\) 0 0
\(199\) −19.0000 −1.34687 −0.673437 − 0.739244i \(-0.735183\pi\)
−0.673437 + 0.739244i \(0.735183\pi\)
\(200\) 0 0
\(201\) 2.00000 0.141069
\(202\) −3.00000 −0.211079
\(203\) −15.0000 −1.05279
\(204\) 3.00000 0.210042
\(205\) 0 0
\(206\) 13.0000 0.905753
\(207\) −1.00000 −0.0695048
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) 23.0000 1.58339 0.791693 − 0.610920i \(-0.209200\pi\)
0.791693 + 0.610920i \(0.209200\pi\)
\(212\) 12.0000 0.824163
\(213\) 15.0000 1.02778
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) −10.0000 −0.678844
\(218\) 17.0000 1.15139
\(219\) 11.0000 0.743311
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) −7.00000 −0.469809
\(223\) −26.0000 −1.74109 −0.870544 − 0.492090i \(-0.836233\pi\)
−0.870544 + 0.492090i \(0.836233\pi\)
\(224\) −5.00000 −0.334077
\(225\) 0 0
\(226\) 9.00000 0.598671
\(227\) −3.00000 −0.199117 −0.0995585 − 0.995032i \(-0.531743\pi\)
−0.0995585 + 0.995032i \(0.531743\pi\)
\(228\) −2.00000 −0.132453
\(229\) 14.0000 0.925146 0.462573 − 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 3.00000 0.196960
\(233\) −12.0000 −0.786146 −0.393073 − 0.919507i \(-0.628588\pi\)
−0.393073 + 0.919507i \(0.628588\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) −8.00000 −0.519656
\(238\) 15.0000 0.972306
\(239\) −15.0000 −0.970269 −0.485135 − 0.874439i \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 − 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) −11.0000 −0.707107
\(243\) −1.00000 −0.0641500
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) 2.00000 0.127000
\(249\) −9.00000 −0.570352
\(250\) 0 0
\(251\) 3.00000 0.189358 0.0946792 − 0.995508i \(-0.469817\pi\)
0.0946792 + 0.995508i \(0.469817\pi\)
\(252\) −5.00000 −0.314970
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 − 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 2.00000 0.124515
\(259\) −35.0000 −2.17479
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) −12.0000 −0.741362
\(263\) 18.0000 1.10993 0.554964 − 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −10.0000 −0.613139
\(267\) −3.00000 −0.183597
\(268\) −2.00000 −0.122169
\(269\) 18.0000 1.09748 0.548740 − 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 − 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −3.00000 −0.181902
\(273\) −10.0000 −0.605228
\(274\) 15.0000 0.906183
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) −2.00000 −0.120168 −0.0600842 − 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 5.00000 0.299880
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) 27.0000 1.61068 0.805342 − 0.592810i \(-0.201981\pi\)
0.805342 + 0.592810i \(0.201981\pi\)
\(282\) −3.00000 −0.178647
\(283\) 4.00000 0.237775 0.118888 − 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −15.0000 −0.890086
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) −11.0000 −0.643726
\(293\) 30.0000 1.75262 0.876309 − 0.481749i \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) −18.0000 −1.04978
\(295\) 0 0
\(296\) 7.00000 0.406867
\(297\) 0 0
\(298\) 12.0000 0.695141
\(299\) 2.00000 0.115663
\(300\) 0 0
\(301\) 10.0000 0.576390
\(302\) −4.00000 −0.230174
\(303\) 3.00000 0.172345
\(304\) 2.00000 0.114708
\(305\) 0 0
\(306\) −3.00000 −0.171499
\(307\) −29.0000 −1.65512 −0.827559 − 0.561379i \(-0.810271\pi\)
−0.827559 + 0.561379i \(0.810271\pi\)
\(308\) 0 0
\(309\) −13.0000 −0.739544
\(310\) 0 0
\(311\) 27.0000 1.53103 0.765515 − 0.643418i \(-0.222484\pi\)
0.765515 + 0.643418i \(0.222484\pi\)
\(312\) 2.00000 0.113228
\(313\) −8.00000 −0.452187 −0.226093 − 0.974106i \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 9.00000 0.505490 0.252745 − 0.967533i \(-0.418667\pi\)
0.252745 + 0.967533i \(0.418667\pi\)
\(318\) −12.0000 −0.672927
\(319\) 0 0
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 5.00000 0.278639
\(323\) −6.00000 −0.333849
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 16.0000 0.886158
\(327\) −17.0000 −0.940102
\(328\) 0 0
\(329\) −15.0000 −0.826977
\(330\) 0 0
\(331\) −19.0000 −1.04433 −0.522167 − 0.852843i \(-0.674876\pi\)
−0.522167 + 0.852843i \(0.674876\pi\)
\(332\) 9.00000 0.493939
\(333\) 7.00000 0.383598
\(334\) −15.0000 −0.820763
\(335\) 0 0
\(336\) 5.00000 0.272772
\(337\) −20.0000 −1.08947 −0.544735 − 0.838608i \(-0.683370\pi\)
−0.544735 + 0.838608i \(0.683370\pi\)
\(338\) −9.00000 −0.489535
\(339\) −9.00000 −0.488813
\(340\) 0 0
\(341\) 0 0
\(342\) 2.00000 0.108148
\(343\) −55.0000 −2.96972
\(344\) −2.00000 −0.107833
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 18.0000 0.966291 0.483145 − 0.875540i \(-0.339494\pi\)
0.483145 + 0.875540i \(0.339494\pi\)
\(348\) −3.00000 −0.160817
\(349\) 26.0000 1.39175 0.695874 − 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) 6.00000 0.319348 0.159674 − 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) −6.00000 −0.318896
\(355\) 0 0
\(356\) 3.00000 0.159000
\(357\) −15.0000 −0.793884
\(358\) 12.0000 0.634220
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 5.00000 0.262794
\(363\) 11.0000 0.577350
\(364\) 10.0000 0.524142
\(365\) 0 0
\(366\) −2.00000 −0.104542
\(367\) 16.0000 0.835193 0.417597 − 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) 0 0
\(371\) −60.0000 −3.11504
\(372\) −2.00000 −0.103695
\(373\) −17.0000 −0.880227 −0.440113 − 0.897942i \(-0.645062\pi\)
−0.440113 + 0.897942i \(0.645062\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 3.00000 0.154713
\(377\) −6.00000 −0.309016
\(378\) 5.00000 0.257172
\(379\) −16.0000 −0.821865 −0.410932 − 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) −6.00000 −0.306987
\(383\) −24.0000 −1.22634 −0.613171 − 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 1.00000 0.0508987
\(387\) −2.00000 −0.101666
\(388\) 10.0000 0.507673
\(389\) 18.0000 0.912636 0.456318 − 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) 3.00000 0.151717
\(392\) 18.0000 0.909137
\(393\) 12.0000 0.605320
\(394\) −3.00000 −0.151138
\(395\) 0 0
\(396\) 0 0
\(397\) −20.0000 −1.00377 −0.501886 − 0.864934i \(-0.667360\pi\)
−0.501886 + 0.864934i \(0.667360\pi\)
\(398\) −19.0000 −0.952384
\(399\) 10.0000 0.500626
\(400\) 0 0
\(401\) 6.00000 0.299626 0.149813 − 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 2.00000 0.0997509
\(403\) −4.00000 −0.199254
\(404\) −3.00000 −0.149256
\(405\) 0 0
\(406\) −15.0000 −0.744438
\(407\) 0 0
\(408\) 3.00000 0.148522
\(409\) −7.00000 −0.346128 −0.173064 − 0.984911i \(-0.555367\pi\)
−0.173064 + 0.984911i \(0.555367\pi\)
\(410\) 0 0
\(411\) −15.0000 −0.739895
\(412\) 13.0000 0.640464
\(413\) −30.0000 −1.47620
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) −5.00000 −0.244851
\(418\) 0 0
\(419\) 15.0000 0.732798 0.366399 − 0.930458i \(-0.380591\pi\)
0.366399 + 0.930458i \(0.380591\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 − 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 23.0000 1.11962
\(423\) 3.00000 0.145865
\(424\) 12.0000 0.582772
\(425\) 0 0
\(426\) 15.0000 0.726752
\(427\) −10.0000 −0.483934
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) 6.00000 0.289010 0.144505 − 0.989504i \(-0.453841\pi\)
0.144505 + 0.989504i \(0.453841\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −2.00000 −0.0961139 −0.0480569 − 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) −10.0000 −0.480015
\(435\) 0 0
\(436\) 17.0000 0.814152
\(437\) −2.00000 −0.0956730
\(438\) 11.0000 0.525600
\(439\) −28.0000 −1.33637 −0.668184 − 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) 18.0000 0.857143
\(442\) 6.00000 0.285391
\(443\) 24.0000 1.14027 0.570137 − 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) −7.00000 −0.332205
\(445\) 0 0
\(446\) −26.0000 −1.23114
\(447\) −12.0000 −0.567581
\(448\) −5.00000 −0.236228
\(449\) 30.0000 1.41579 0.707894 − 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 9.00000 0.423324
\(453\) 4.00000 0.187936
\(454\) −3.00000 −0.140797
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) 28.0000 1.30978 0.654892 − 0.755722i \(-0.272714\pi\)
0.654892 + 0.755722i \(0.272714\pi\)
\(458\) 14.0000 0.654177
\(459\) 3.00000 0.140028
\(460\) 0 0
\(461\) −15.0000 −0.698620 −0.349310 − 0.937007i \(-0.613584\pi\)
−0.349310 + 0.937007i \(0.613584\pi\)
\(462\) 0 0
\(463\) 40.0000 1.85896 0.929479 − 0.368875i \(-0.120257\pi\)
0.929479 + 0.368875i \(0.120257\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) −12.0000 −0.555889
\(467\) −15.0000 −0.694117 −0.347059 − 0.937843i \(-0.612820\pi\)
−0.347059 + 0.937843i \(0.612820\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 10.0000 0.461757
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 6.00000 0.276172
\(473\) 0 0
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) 15.0000 0.687524
\(477\) 12.0000 0.549442
\(478\) −15.0000 −0.686084
\(479\) 30.0000 1.37073 0.685367 − 0.728197i \(-0.259642\pi\)
0.685367 + 0.728197i \(0.259642\pi\)
\(480\) 0 0
\(481\) −14.0000 −0.638345
\(482\) 8.00000 0.364390
\(483\) −5.00000 −0.227508
\(484\) −11.0000 −0.500000
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −8.00000 −0.362515 −0.181257 − 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 2.00000 0.0905357
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) −6.00000 −0.270776 −0.135388 − 0.990793i \(-0.543228\pi\)
−0.135388 + 0.990793i \(0.543228\pi\)
\(492\) 0 0
\(493\) −9.00000 −0.405340
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) 75.0000 3.36421
\(498\) −9.00000 −0.403300
\(499\) 5.00000 0.223831 0.111915 − 0.993718i \(-0.464301\pi\)
0.111915 + 0.993718i \(0.464301\pi\)
\(500\) 0 0
\(501\) 15.0000 0.670151
\(502\) 3.00000 0.133897
\(503\) −12.0000 −0.535054 −0.267527 − 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) −5.00000 −0.222718
\(505\) 0 0
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 16.0000 0.709885
\(509\) 15.0000 0.664863 0.332432 − 0.943127i \(-0.392131\pi\)
0.332432 + 0.943127i \(0.392131\pi\)
\(510\) 0 0
\(511\) 55.0000 2.43306
\(512\) 1.00000 0.0441942
\(513\) −2.00000 −0.0883022
\(514\) 18.0000 0.793946
\(515\) 0 0
\(516\) 2.00000 0.0880451
\(517\) 0 0
\(518\) −35.0000 −1.53781
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −39.0000 −1.70862 −0.854311 − 0.519763i \(-0.826020\pi\)
−0.854311 + 0.519763i \(0.826020\pi\)
\(522\) 3.00000 0.131306
\(523\) −20.0000 −0.874539 −0.437269 − 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 18.0000 0.784837
\(527\) −6.00000 −0.261364
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) −10.0000 −0.433555
\(533\) 0 0
\(534\) −3.00000 −0.129823
\(535\) 0 0
\(536\) −2.00000 −0.0863868
\(537\) −12.0000 −0.517838
\(538\) 18.0000 0.776035
\(539\) 0 0
\(540\) 0 0
\(541\) 20.0000 0.859867 0.429934 − 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) −16.0000 −0.687259
\(543\) −5.00000 −0.214571
\(544\) −3.00000 −0.128624
\(545\) 0 0
\(546\) −10.0000 −0.427960
\(547\) 1.00000 0.0427569 0.0213785 − 0.999771i \(-0.493195\pi\)
0.0213785 + 0.999771i \(0.493195\pi\)
\(548\) 15.0000 0.640768
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 6.00000 0.255609
\(552\) 1.00000 0.0425628
\(553\) −40.0000 −1.70097
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) 5.00000 0.212047
\(557\) −18.0000 −0.762684 −0.381342 − 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 2.00000 0.0846668
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) 27.0000 1.13893
\(563\) −24.0000 −1.01148 −0.505740 − 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) −3.00000 −0.126323
\(565\) 0 0
\(566\) 4.00000 0.168133
\(567\) −5.00000 −0.209980
\(568\) −15.0000 −0.629386
\(569\) 30.0000 1.25767 0.628833 − 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) 26.0000 1.08807 0.544033 − 0.839064i \(-0.316897\pi\)
0.544033 + 0.839064i \(0.316897\pi\)
\(572\) 0 0
\(573\) 6.00000 0.250654
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −2.00000 −0.0832611 −0.0416305 − 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) −8.00000 −0.332756
\(579\) −1.00000 −0.0415586
\(580\) 0 0
\(581\) −45.0000 −1.86691
\(582\) −10.0000 −0.414513
\(583\) 0 0
\(584\) −11.0000 −0.455183
\(585\) 0 0
\(586\) 30.0000 1.23929
\(587\) 12.0000 0.495293 0.247647 − 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) −18.0000 −0.742307
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) 3.00000 0.123404
\(592\) 7.00000 0.287698
\(593\) 24.0000 0.985562 0.492781 − 0.870153i \(-0.335980\pi\)
0.492781 + 0.870153i \(0.335980\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 12.0000 0.491539
\(597\) 19.0000 0.777618
\(598\) 2.00000 0.0817861
\(599\) −24.0000 −0.980613 −0.490307 − 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −43.0000 −1.75401 −0.877003 − 0.480484i \(-0.840461\pi\)
−0.877003 + 0.480484i \(0.840461\pi\)
\(602\) 10.0000 0.407570
\(603\) −2.00000 −0.0814463
\(604\) −4.00000 −0.162758
\(605\) 0 0
\(606\) 3.00000 0.121867
\(607\) −14.0000 −0.568242 −0.284121 − 0.958788i \(-0.591702\pi\)
−0.284121 + 0.958788i \(0.591702\pi\)
\(608\) 2.00000 0.0811107
\(609\) 15.0000 0.607831
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) −3.00000 −0.121268
\(613\) 25.0000 1.00974 0.504870 − 0.863195i \(-0.331540\pi\)
0.504870 + 0.863195i \(0.331540\pi\)
\(614\) −29.0000 −1.17034
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 − 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) −13.0000 −0.522937
\(619\) −40.0000 −1.60774 −0.803868 − 0.594808i \(-0.797228\pi\)
−0.803868 + 0.594808i \(0.797228\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 27.0000 1.08260
\(623\) −15.0000 −0.600962
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) −8.00000 −0.319744
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) −21.0000 −0.837325
\(630\) 0 0
\(631\) −1.00000 −0.0398094 −0.0199047 − 0.999802i \(-0.506336\pi\)
−0.0199047 + 0.999802i \(0.506336\pi\)
\(632\) 8.00000 0.318223
\(633\) −23.0000 −0.914168
\(634\) 9.00000 0.357436
\(635\) 0 0
\(636\) −12.0000 −0.475831
\(637\) −36.0000 −1.42637
\(638\) 0 0
\(639\) −15.0000 −0.593391
\(640\) 0 0
\(641\) 3.00000 0.118493 0.0592464 − 0.998243i \(-0.481130\pi\)
0.0592464 + 0.998243i \(0.481130\pi\)
\(642\) 12.0000 0.473602
\(643\) −14.0000 −0.552106 −0.276053 − 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) 5.00000 0.197028
\(645\) 0 0
\(646\) −6.00000 −0.236067
\(647\) 9.00000 0.353827 0.176913 − 0.984226i \(-0.443389\pi\)
0.176913 + 0.984226i \(0.443389\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) 10.0000 0.391931
\(652\) 16.0000 0.626608
\(653\) 45.0000 1.76099 0.880493 − 0.474059i \(-0.157212\pi\)
0.880493 + 0.474059i \(0.157212\pi\)
\(654\) −17.0000 −0.664753
\(655\) 0 0
\(656\) 0 0
\(657\) −11.0000 −0.429151
\(658\) −15.0000 −0.584761
\(659\) −33.0000 −1.28550 −0.642749 − 0.766077i \(-0.722206\pi\)
−0.642749 + 0.766077i \(0.722206\pi\)
\(660\) 0 0
\(661\) −13.0000 −0.505641 −0.252821 − 0.967513i \(-0.581358\pi\)
−0.252821 + 0.967513i \(0.581358\pi\)
\(662\) −19.0000 −0.738456
\(663\) −6.00000 −0.233021
\(664\) 9.00000 0.349268
\(665\) 0 0
\(666\) 7.00000 0.271244
\(667\) −3.00000 −0.116160
\(668\) −15.0000 −0.580367
\(669\) 26.0000 1.00522
\(670\) 0 0
\(671\) 0 0
\(672\) 5.00000 0.192879
\(673\) −17.0000 −0.655302 −0.327651 − 0.944799i \(-0.606257\pi\)
−0.327651 + 0.944799i \(0.606257\pi\)
\(674\) −20.0000 −0.770371
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −48.0000 −1.84479 −0.922395 − 0.386248i \(-0.873771\pi\)
−0.922395 + 0.386248i \(0.873771\pi\)
\(678\) −9.00000 −0.345643
\(679\) −50.0000 −1.91882
\(680\) 0 0
\(681\) 3.00000 0.114960
\(682\) 0 0
\(683\) −24.0000 −0.918334 −0.459167 − 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) 2.00000 0.0764719
\(685\) 0 0
\(686\) −55.0000 −2.09991
\(687\) −14.0000 −0.534133
\(688\) −2.00000 −0.0762493
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) 35.0000 1.33146 0.665731 − 0.746191i \(-0.268120\pi\)
0.665731 + 0.746191i \(0.268120\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) 18.0000 0.683271
\(695\) 0 0
\(696\) −3.00000 −0.113715
\(697\) 0 0
\(698\) 26.0000 0.984115
\(699\) 12.0000 0.453882
\(700\) 0 0
\(701\) −42.0000 −1.58632 −0.793159 − 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 2.00000 0.0754851
\(703\) 14.0000 0.528020
\(704\) 0 0
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 15.0000 0.564133
\(708\) −6.00000 −0.225494
\(709\) −49.0000 −1.84023 −0.920117 − 0.391644i \(-0.871906\pi\)
−0.920117 + 0.391644i \(0.871906\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 3.00000 0.112430
\(713\) −2.00000 −0.0749006
\(714\) −15.0000 −0.561361
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 15.0000 0.560185
\(718\) 0 0
\(719\) 3.00000 0.111881 0.0559406 − 0.998434i \(-0.482184\pi\)
0.0559406 + 0.998434i \(0.482184\pi\)
\(720\) 0 0
\(721\) −65.0000 −2.42073
\(722\) −15.0000 −0.558242
\(723\) −8.00000 −0.297523
\(724\) 5.00000 0.185824
\(725\) 0 0
\(726\) 11.0000 0.408248
\(727\) 40.0000 1.48352 0.741759 − 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) 10.0000 0.370625
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 6.00000 0.221918
\(732\) −2.00000 −0.0739221
\(733\) 49.0000 1.80986 0.904928 − 0.425564i \(-0.139924\pi\)
0.904928 + 0.425564i \(0.139924\pi\)
\(734\) 16.0000 0.590571
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 0 0
\(738\) 0 0
\(739\) −25.0000 −0.919640 −0.459820 − 0.888012i \(-0.652086\pi\)
−0.459820 + 0.888012i \(0.652086\pi\)
\(740\) 0 0
\(741\) 4.00000 0.146944
\(742\) −60.0000 −2.20267
\(743\) 6.00000 0.220119 0.110059 − 0.993925i \(-0.464896\pi\)
0.110059 + 0.993925i \(0.464896\pi\)
\(744\) −2.00000 −0.0733236
\(745\) 0 0
\(746\) −17.0000 −0.622414
\(747\) 9.00000 0.329293
\(748\) 0 0
\(749\) 60.0000 2.19235
\(750\) 0 0
\(751\) 41.0000 1.49611 0.748056 − 0.663636i \(-0.230988\pi\)
0.748056 + 0.663636i \(0.230988\pi\)
\(752\) 3.00000 0.109399
\(753\) −3.00000 −0.109326
\(754\) −6.00000 −0.218507
\(755\) 0 0
\(756\) 5.00000 0.181848
\(757\) 7.00000 0.254419 0.127210 − 0.991876i \(-0.459398\pi\)
0.127210 + 0.991876i \(0.459398\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 − 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) −16.0000 −0.579619
\(763\) −85.0000 −3.07721
\(764\) −6.00000 −0.217072
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) −12.0000 −0.433295
\(768\) −1.00000 −0.0360844
\(769\) −34.0000 −1.22607 −0.613036 − 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 1.00000 0.0359908
\(773\) 42.0000 1.51064 0.755318 − 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) −2.00000 −0.0718885
\(775\) 0 0
\(776\) 10.0000 0.358979
\(777\) 35.0000 1.25562
\(778\) 18.0000 0.645331
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 3.00000 0.107280
\(783\) −3.00000 −0.107211
\(784\) 18.0000 0.642857
\(785\) 0 0
\(786\) 12.0000 0.428026
\(787\) −32.0000 −1.14068 −0.570338 − 0.821410i \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(788\) −3.00000 −0.106871
\(789\) −18.0000 −0.640817
\(790\) 0 0
\(791\) −45.0000 −1.60002
\(792\) 0 0
\(793\) −4.00000 −0.142044
\(794\) −20.0000 −0.709773
\(795\) 0 0
\(796\) −19.0000 −0.673437
\(797\) 42.0000 1.48772 0.743858 − 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 10.0000 0.353996
\(799\) −9.00000 −0.318397
\(800\) 0 0
\(801\) 3.00000 0.106000
\(802\) 6.00000 0.211867
\(803\) 0 0
\(804\) 2.00000 0.0705346
\(805\) 0 0
\(806\) −4.00000 −0.140894
\(807\) −18.0000 −0.633630
\(808\) −3.00000 −0.105540
\(809\) −18.0000 −0.632846 −0.316423 − 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) −4.00000 −0.140459 −0.0702295 − 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) −15.0000 −0.526397
\(813\) 16.0000 0.561144
\(814\) 0 0
\(815\) 0 0
\(816\) 3.00000 0.105021
\(817\) −4.00000 −0.139942
\(818\) −7.00000 −0.244749
\(819\) 10.0000 0.349428
\(820\) 0 0
\(821\) 30.0000 1.04701 0.523504 − 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) −15.0000 −0.523185
\(823\) 28.0000 0.976019 0.488009 − 0.872838i \(-0.337723\pi\)
0.488009 + 0.872838i \(0.337723\pi\)
\(824\) 13.0000 0.452876
\(825\) 0 0
\(826\) −30.0000 −1.04383
\(827\) 9.00000 0.312961 0.156480 − 0.987681i \(-0.449985\pi\)
0.156480 + 0.987681i \(0.449985\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 2.00000 0.0694629 0.0347314 − 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) −2.00000 −0.0693375
\(833\) −54.0000 −1.87099
\(834\) −5.00000 −0.173136
\(835\) 0 0
\(836\) 0 0
\(837\) −2.00000 −0.0691301
\(838\) 15.0000 0.518166
\(839\) 24.0000 0.828572 0.414286 − 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −10.0000 −0.344623
\(843\) −27.0000 −0.929929
\(844\) 23.0000 0.791693
\(845\) 0 0
\(846\) 3.00000 0.103142
\(847\) 55.0000 1.88982
\(848\) 12.0000 0.412082
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) −7.00000 −0.239957
\(852\) 15.0000 0.513892
\(853\) 4.00000 0.136957 0.0684787 − 0.997653i \(-0.478185\pi\)
0.0684787 + 0.997653i \(0.478185\pi\)
\(854\) −10.0000 −0.342193
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −54.0000 −1.84460 −0.922302 − 0.386469i \(-0.873695\pi\)
−0.922302 + 0.386469i \(0.873695\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 − 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 6.00000 0.204361
\(863\) −33.0000 −1.12333 −0.561667 − 0.827364i \(-0.689840\pi\)
−0.561667 + 0.827364i \(0.689840\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −2.00000 −0.0679628
\(867\) 8.00000 0.271694
\(868\) −10.0000 −0.339422
\(869\) 0 0
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) 17.0000 0.575693
\(873\) 10.0000 0.338449
\(874\) −2.00000 −0.0676510
\(875\) 0 0
\(876\) 11.0000 0.371656
\(877\) 22.0000 0.742887 0.371444 − 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) −28.0000 −0.944954
\(879\) −30.0000 −1.01187
\(880\) 0 0
\(881\) −42.0000 −1.41502 −0.707508 − 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 18.0000 0.606092
\(883\) −11.0000 −0.370179 −0.185090 − 0.982722i \(-0.559258\pi\)
−0.185090 + 0.982722i \(0.559258\pi\)
\(884\) 6.00000 0.201802
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) 27.0000 0.906571 0.453286 − 0.891365i \(-0.350252\pi\)
0.453286 + 0.891365i \(0.350252\pi\)
\(888\) −7.00000 −0.234905
\(889\) −80.0000 −2.68311
\(890\) 0 0
\(891\) 0 0
\(892\) −26.0000 −0.870544
\(893\) 6.00000 0.200782
\(894\) −12.0000 −0.401340
\(895\) 0 0
\(896\) −5.00000 −0.167038
\(897\) −2.00000 −0.0667781
\(898\) 30.0000 1.00111
\(899\) 6.00000 0.200111
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) −10.0000 −0.332779
\(904\) 9.00000 0.299336
\(905\) 0 0
\(906\) 4.00000 0.132891
\(907\) −26.0000 −0.863316 −0.431658 − 0.902037i \(-0.642071\pi\)
−0.431658 + 0.902037i \(0.642071\pi\)
\(908\) −3.00000 −0.0995585
\(909\) −3.00000 −0.0995037
\(910\) 0 0
\(911\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) −2.00000 −0.0662266
\(913\) 0 0
\(914\) 28.0000 0.926158
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) 60.0000 1.98137
\(918\) 3.00000 0.0990148
\(919\) 11.0000 0.362857 0.181428 − 0.983404i \(-0.441928\pi\)
0.181428 + 0.983404i \(0.441928\pi\)
\(920\) 0 0
\(921\) 29.0000 0.955582
\(922\) −15.0000 −0.493999
\(923\) 30.0000 0.987462
\(924\) 0 0
\(925\) 0 0
\(926\) 40.0000 1.31448
\(927\) 13.0000 0.426976
\(928\) 3.00000 0.0984798
\(929\) −6.00000 −0.196854 −0.0984268 − 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) 36.0000 1.17985
\(932\) −12.0000 −0.393073
\(933\) −27.0000 −0.883940
\(934\) −15.0000 −0.490815
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) 58.0000 1.89478 0.947389 − 0.320085i \(-0.103712\pi\)
0.947389 + 0.320085i \(0.103712\pi\)
\(938\) 10.0000 0.326512
\(939\) 8.00000 0.261070
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 − 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 2.00000 0.0651635
\(943\) 0 0
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) 0 0
\(947\) 24.0000 0.779895 0.389948 − 0.920837i \(-0.372493\pi\)
0.389948 + 0.920837i \(0.372493\pi\)
\(948\) −8.00000 −0.259828
\(949\) 22.0000 0.714150
\(950\) 0 0
\(951\) −9.00000 −0.291845
\(952\) 15.0000 0.486153
\(953\) 33.0000 1.06897 0.534487 − 0.845176i \(-0.320505\pi\)
0.534487 + 0.845176i \(0.320505\pi\)
\(954\) 12.0000 0.388514
\(955\) 0 0
\(956\) −15.0000 −0.485135
\(957\) 0 0
\(958\) 30.0000 0.969256
\(959\) −75.0000 −2.42188
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −14.0000 −0.451378
\(963\) −12.0000 −0.386695
\(964\) 8.00000 0.257663
\(965\) 0 0
\(966\) −5.00000 −0.160872
\(967\) 58.0000 1.86515 0.932577 − 0.360971i \(-0.117555\pi\)
0.932577 + 0.360971i \(0.117555\pi\)
\(968\) −11.0000 −0.353553
\(969\) 6.00000 0.192748
\(970\) 0 0
\(971\) 27.0000 0.866471 0.433236 − 0.901281i \(-0.357372\pi\)
0.433236 + 0.901281i \(0.357372\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −25.0000 −0.801463
\(974\) −8.00000 −0.256337
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 51.0000 1.63163 0.815817 − 0.578310i \(-0.196287\pi\)
0.815817 + 0.578310i \(0.196287\pi\)
\(978\) −16.0000 −0.511624
\(979\) 0 0
\(980\) 0 0
\(981\) 17.0000 0.542768
\(982\) −6.00000 −0.191468
\(983\) 48.0000 1.53096 0.765481 − 0.643458i \(-0.222501\pi\)
0.765481 + 0.643458i \(0.222501\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −9.00000 −0.286618
\(987\) 15.0000 0.477455
\(988\) −4.00000 −0.127257
\(989\) 2.00000 0.0635963
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 − 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 2.00000 0.0635001
\(993\) 19.0000 0.602947
\(994\) 75.0000 2.37886
\(995\) 0 0
\(996\) −9.00000 −0.285176
\(997\) −26.0000 −0.823428 −0.411714 − 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) 5.00000 0.158272
\(999\) −7.00000 −0.221470
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 3450.2.a.n.1.1 yes 1
5.2 odd 4 3450.2.d.h.2899.2 2
5.3 odd 4 3450.2.d.h.2899.1 2
5.4 even 2 3450.2.a.m.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3450.2.a.m.1.1 ✓ 1 5.4 even 2
3450.2.a.n.1.1 yes 1 1.1 even 1 trivial
3450.2.d.h.2899.1 2 5.3 odd 4
3450.2.d.h.2899.2 2 5.2 odd 4