Properties

Label 3450.2.a.e.1.1
Level $3450$
Weight $2$
Character 3450.1
Self dual yes
Analytic conductor $27.548$
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] = [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: \(1\)
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -5.00000 q^{11} -1.00000 q^{12} +3.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} -3.00000 q^{19} -1.00000 q^{21} +5.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} -3.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -1.00000 q^{29} -1.00000 q^{32} +5.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} +4.00000 q^{37} +3.00000 q^{38} -3.00000 q^{39} -3.00000 q^{41} +1.00000 q^{42} +5.00000 q^{43} -5.00000 q^{44} -1.00000 q^{46} -6.00000 q^{47} -1.00000 q^{48} -6.00000 q^{49} -2.00000 q^{51} +3.00000 q^{52} +1.00000 q^{54} -1.00000 q^{56} +3.00000 q^{57} +1.00000 q^{58} -6.00000 q^{59} +4.00000 q^{61} +1.00000 q^{63} +1.00000 q^{64} -5.00000 q^{66} -4.00000 q^{67} +2.00000 q^{68} -1.00000 q^{69} -2.00000 q^{71} -1.00000 q^{72} +9.00000 q^{73} -4.00000 q^{74} -3.00000 q^{76} -5.00000 q^{77} +3.00000 q^{78} -7.00000 q^{79} +1.00000 q^{81} +3.00000 q^{82} -1.00000 q^{83} -1.00000 q^{84} -5.00000 q^{86} +1.00000 q^{87} +5.00000 q^{88} -2.00000 q^{89} +3.00000 q^{91} +1.00000 q^{92} +6.00000 q^{94} +1.00000 q^{96} +6.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) −1.00000 −0.288675
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −1.00000 −0.235702
\(19\) −3.00000 −0.688247 −0.344124 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 5.00000 1.06600
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −3.00000 −0.588348
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) 5.00000 0.870388
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 3.00000 0.486664
\(39\) −3.00000 −0.480384
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 1.00000 0.154303
\(43\) 5.00000 0.762493 0.381246 0.924473i \(-0.375495\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) −5.00000 −0.753778
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 3.00000 0.416025
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 3.00000 0.397360
\(58\) 1.00000 0.131306
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −5.00000 −0.615457
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 2.00000 0.242536
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) −1.00000 −0.117851
\(73\) 9.00000 1.05337 0.526685 0.850060i \(-0.323435\pi\)
0.526685 + 0.850060i \(0.323435\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) −3.00000 −0.344124
\(77\) −5.00000 −0.569803
\(78\) 3.00000 0.339683
\(79\) −7.00000 −0.787562 −0.393781 0.919204i \(-0.628833\pi\)
−0.393781 + 0.919204i \(0.628833\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 3.00000 0.331295
\(83\) −1.00000 −0.109764 −0.0548821 0.998493i \(-0.517478\pi\)
−0.0548821 + 0.998493i \(0.517478\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) −5.00000 −0.539164
\(87\) 1.00000 0.107211
\(88\) 5.00000 0.533002
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 3.00000 0.314485
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) 6.00000 0.618853
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 6.00000 0.606092
\(99\) −5.00000 −0.502519
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 2.00000 0.198030
\(103\) 7.00000 0.689730 0.344865 0.938652i \(-0.387925\pi\)
0.344865 + 0.938652i \(0.387925\pi\)
\(104\) −3.00000 −0.294174
\(105\) 0 0
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 1.00000 0.0944911
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −3.00000 −0.280976
\(115\) 0 0
\(116\) −1.00000 −0.0928477
\(117\) 3.00000 0.277350
\(118\) 6.00000 0.552345
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) −4.00000 −0.362143
\(123\) 3.00000 0.270501
\(124\) 0 0
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) −18.0000 −1.59724 −0.798621 0.601834i \(-0.794437\pi\)
−0.798621 + 0.601834i \(0.794437\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −5.00000 −0.440225
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 5.00000 0.435194
\(133\) −3.00000 −0.260133
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) 16.0000 1.36697 0.683486 0.729964i \(-0.260463\pi\)
0.683486 + 0.729964i \(0.260463\pi\)
\(138\) 1.00000 0.0851257
\(139\) −10.0000 −0.848189 −0.424094 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 2.00000 0.167836
\(143\) −15.0000 −1.25436
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −9.00000 −0.744845
\(147\) 6.00000 0.494872
\(148\) 4.00000 0.328798
\(149\) −20.0000 −1.63846 −0.819232 0.573462i \(-0.805600\pi\)
−0.819232 + 0.573462i \(0.805600\pi\)
\(150\) 0 0
\(151\) 18.0000 1.46482 0.732410 0.680864i \(-0.238396\pi\)
0.732410 + 0.680864i \(0.238396\pi\)
\(152\) 3.00000 0.243332
\(153\) 2.00000 0.161690
\(154\) 5.00000 0.402911
\(155\) 0 0
\(156\) −3.00000 −0.240192
\(157\) −4.00000 −0.319235 −0.159617 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) 7.00000 0.556890
\(159\) 0 0
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) −1.00000 −0.0785674
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −3.00000 −0.234261
\(165\) 0 0
\(166\) 1.00000 0.0776151
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 1.00000 0.0771517
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) −3.00000 −0.229416
\(172\) 5.00000 0.381246
\(173\) −9.00000 −0.684257 −0.342129 0.939653i \(-0.611148\pi\)
−0.342129 + 0.939653i \(0.611148\pi\)
\(174\) −1.00000 −0.0758098
\(175\) 0 0
\(176\) −5.00000 −0.376889
\(177\) 6.00000 0.450988
\(178\) 2.00000 0.149906
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) −3.00000 −0.222375
\(183\) −4.00000 −0.295689
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 0 0
\(187\) −10.0000 −0.731272
\(188\) −6.00000 −0.437595
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −3.00000 −0.217072 −0.108536 0.994092i \(-0.534616\pi\)
−0.108536 + 0.994092i \(0.534616\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 15.0000 1.06871 0.534353 0.845262i \(-0.320555\pi\)
0.534353 + 0.845262i \(0.320555\pi\)
\(198\) 5.00000 0.355335
\(199\) −11.0000 −0.779769 −0.389885 0.920864i \(-0.627485\pi\)
−0.389885 + 0.920864i \(0.627485\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) −6.00000 −0.422159
\(203\) −1.00000 −0.0701862
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) −7.00000 −0.487713
\(207\) 1.00000 0.0695048
\(208\) 3.00000 0.208013
\(209\) 15.0000 1.03757
\(210\) 0 0
\(211\) −2.00000 −0.137686 −0.0688428 0.997628i \(-0.521931\pi\)
−0.0688428 + 0.997628i \(0.521931\pi\)
\(212\) 0 0
\(213\) 2.00000 0.137038
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 4.00000 0.270914
\(219\) −9.00000 −0.608164
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 4.00000 0.268462
\(223\) 2.00000 0.133930 0.0669650 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) −24.0000 −1.59294 −0.796468 0.604681i \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(228\) 3.00000 0.198680
\(229\) −8.00000 −0.528655 −0.264327 0.964433i \(-0.585150\pi\)
−0.264327 + 0.964433i \(0.585150\pi\)
\(230\) 0 0
\(231\) 5.00000 0.328976
\(232\) 1.00000 0.0656532
\(233\) −11.0000 −0.720634 −0.360317 0.932830i \(-0.617331\pi\)
−0.360317 + 0.932830i \(0.617331\pi\)
\(234\) −3.00000 −0.196116
\(235\) 0 0
\(236\) −6.00000 −0.390567
\(237\) 7.00000 0.454699
\(238\) −2.00000 −0.129641
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) −14.0000 −0.899954
\(243\) −1.00000 −0.0641500
\(244\) 4.00000 0.256074
\(245\) 0 0
\(246\) −3.00000 −0.191273
\(247\) −9.00000 −0.572656
\(248\) 0 0
\(249\) 1.00000 0.0633724
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 1.00000 0.0629941
\(253\) −5.00000 −0.314347
\(254\) 18.0000 1.12942
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −22.0000 −1.37232 −0.686161 0.727450i \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) 5.00000 0.311286
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) −12.0000 −0.741362
\(263\) 28.0000 1.72655 0.863277 0.504730i \(-0.168408\pi\)
0.863277 + 0.504730i \(0.168408\pi\)
\(264\) −5.00000 −0.307729
\(265\) 0 0
\(266\) 3.00000 0.183942
\(267\) 2.00000 0.122398
\(268\) −4.00000 −0.244339
\(269\) −9.00000 −0.548740 −0.274370 0.961624i \(-0.588469\pi\)
−0.274370 + 0.961624i \(0.588469\pi\)
\(270\) 0 0
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) 2.00000 0.121268
\(273\) −3.00000 −0.181568
\(274\) −16.0000 −0.966595
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) −31.0000 −1.86261 −0.931305 0.364241i \(-0.881328\pi\)
−0.931305 + 0.364241i \(0.881328\pi\)
\(278\) 10.0000 0.599760
\(279\) 0 0
\(280\) 0 0
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) −6.00000 −0.357295
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −2.00000 −0.118678
\(285\) 0 0
\(286\) 15.0000 0.886969
\(287\) −3.00000 −0.177084
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 9.00000 0.526685
\(293\) −28.0000 −1.63578 −0.817889 0.575376i \(-0.804856\pi\)
−0.817889 + 0.575376i \(0.804856\pi\)
\(294\) −6.00000 −0.349927
\(295\) 0 0
\(296\) −4.00000 −0.232495
\(297\) 5.00000 0.290129
\(298\) 20.0000 1.15857
\(299\) 3.00000 0.173494
\(300\) 0 0
\(301\) 5.00000 0.288195
\(302\) −18.0000 −1.03578
\(303\) −6.00000 −0.344691
\(304\) −3.00000 −0.172062
\(305\) 0 0
\(306\) −2.00000 −0.114332
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) −5.00000 −0.284901
\(309\) −7.00000 −0.398216
\(310\) 0 0
\(311\) −20.0000 −1.13410 −0.567048 0.823685i \(-0.691915\pi\)
−0.567048 + 0.823685i \(0.691915\pi\)
\(312\) 3.00000 0.169842
\(313\) 32.0000 1.80875 0.904373 0.426742i \(-0.140339\pi\)
0.904373 + 0.426742i \(0.140339\pi\)
\(314\) 4.00000 0.225733
\(315\) 0 0
\(316\) −7.00000 −0.393781
\(317\) −13.0000 −0.730153 −0.365076 0.930978i \(-0.618957\pi\)
−0.365076 + 0.930978i \(0.618957\pi\)
\(318\) 0 0
\(319\) 5.00000 0.279946
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) −1.00000 −0.0557278
\(323\) −6.00000 −0.333849
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 4.00000 0.221540
\(327\) 4.00000 0.221201
\(328\) 3.00000 0.165647
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −1.00000 −0.0548821
\(333\) 4.00000 0.219199
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 4.00000 0.217571
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) 3.00000 0.162221
\(343\) −13.0000 −0.701934
\(344\) −5.00000 −0.269582
\(345\) 0 0
\(346\) 9.00000 0.483843
\(347\) −20.0000 −1.07366 −0.536828 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) 1.00000 0.0536056
\(349\) −15.0000 −0.802932 −0.401466 0.915874i \(-0.631499\pi\)
−0.401466 + 0.915874i \(0.631499\pi\)
\(350\) 0 0
\(351\) −3.00000 −0.160128
\(352\) 5.00000 0.266501
\(353\) 11.0000 0.585471 0.292735 0.956193i \(-0.405434\pi\)
0.292735 + 0.956193i \(0.405434\pi\)
\(354\) −6.00000 −0.318896
\(355\) 0 0
\(356\) −2.00000 −0.106000
\(357\) −2.00000 −0.105851
\(358\) −10.0000 −0.528516
\(359\) −9.00000 −0.475002 −0.237501 0.971387i \(-0.576328\pi\)
−0.237501 + 0.971387i \(0.576328\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 6.00000 0.315353
\(363\) −14.0000 −0.734809
\(364\) 3.00000 0.157243
\(365\) 0 0
\(366\) 4.00000 0.209083
\(367\) 37.0000 1.93138 0.965692 0.259690i \(-0.0836203\pi\)
0.965692 + 0.259690i \(0.0836203\pi\)
\(368\) 1.00000 0.0521286
\(369\) −3.00000 −0.156174
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 10.0000 0.517088
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) −3.00000 −0.154508
\(378\) 1.00000 0.0514344
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 18.0000 0.922168
\(382\) 3.00000 0.153493
\(383\) −9.00000 −0.459879 −0.229939 0.973205i \(-0.573853\pi\)
−0.229939 + 0.973205i \(0.573853\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) 5.00000 0.254164
\(388\) 0 0
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) 2.00000 0.101144
\(392\) 6.00000 0.303046
\(393\) −12.0000 −0.605320
\(394\) −15.0000 −0.755689
\(395\) 0 0
\(396\) −5.00000 −0.251259
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 11.0000 0.551380
\(399\) 3.00000 0.150188
\(400\) 0 0
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) −4.00000 −0.199502
\(403\) 0 0
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 1.00000 0.0496292
\(407\) −20.0000 −0.991363
\(408\) 2.00000 0.0990148
\(409\) 17.0000 0.840596 0.420298 0.907386i \(-0.361926\pi\)
0.420298 + 0.907386i \(0.361926\pi\)
\(410\) 0 0
\(411\) −16.0000 −0.789222
\(412\) 7.00000 0.344865
\(413\) −6.00000 −0.295241
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) −3.00000 −0.147087
\(417\) 10.0000 0.489702
\(418\) −15.0000 −0.733674
\(419\) 25.0000 1.22133 0.610665 0.791889i \(-0.290902\pi\)
0.610665 + 0.791889i \(0.290902\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) 2.00000 0.0973585
\(423\) −6.00000 −0.291730
\(424\) 0 0
\(425\) 0 0
\(426\) −2.00000 −0.0969003
\(427\) 4.00000 0.193574
\(428\) −12.0000 −0.580042
\(429\) 15.0000 0.724207
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −12.0000 −0.576683 −0.288342 0.957528i \(-0.593104\pi\)
−0.288342 + 0.957528i \(0.593104\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) −3.00000 −0.143509
\(438\) 9.00000 0.430037
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) −6.00000 −0.285391
\(443\) −20.0000 −0.950229 −0.475114 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) −4.00000 −0.189832
\(445\) 0 0
\(446\) −2.00000 −0.0947027
\(447\) 20.0000 0.945968
\(448\) 1.00000 0.0472456
\(449\) 14.0000 0.660701 0.330350 0.943858i \(-0.392833\pi\)
0.330350 + 0.943858i \(0.392833\pi\)
\(450\) 0 0
\(451\) 15.0000 0.706322
\(452\) −6.00000 −0.282216
\(453\) −18.0000 −0.845714
\(454\) 24.0000 1.12638
\(455\) 0 0
\(456\) −3.00000 −0.140488
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) 8.00000 0.373815
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) −37.0000 −1.72326 −0.861631 0.507535i \(-0.830557\pi\)
−0.861631 + 0.507535i \(0.830557\pi\)
\(462\) −5.00000 −0.232621
\(463\) 12.0000 0.557687 0.278844 0.960337i \(-0.410049\pi\)
0.278844 + 0.960337i \(0.410049\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 0 0
\(466\) 11.0000 0.509565
\(467\) −27.0000 −1.24941 −0.624705 0.780860i \(-0.714781\pi\)
−0.624705 + 0.780860i \(0.714781\pi\)
\(468\) 3.00000 0.138675
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) 4.00000 0.184310
\(472\) 6.00000 0.276172
\(473\) −25.0000 −1.14950
\(474\) −7.00000 −0.321521
\(475\) 0 0
\(476\) 2.00000 0.0916698
\(477\) 0 0
\(478\) 8.00000 0.365911
\(479\) 17.0000 0.776750 0.388375 0.921501i \(-0.373037\pi\)
0.388375 + 0.921501i \(0.373037\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) 2.00000 0.0910975
\(483\) −1.00000 −0.0455016
\(484\) 14.0000 0.636364
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 18.0000 0.815658 0.407829 0.913058i \(-0.366286\pi\)
0.407829 + 0.913058i \(0.366286\pi\)
\(488\) −4.00000 −0.181071
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 3.00000 0.135250
\(493\) −2.00000 −0.0900755
\(494\) 9.00000 0.404929
\(495\) 0 0
\(496\) 0 0
\(497\) −2.00000 −0.0897123
\(498\) −1.00000 −0.0448111
\(499\) −34.0000 −1.52205 −0.761025 0.648723i \(-0.775303\pi\)
−0.761025 + 0.648723i \(0.775303\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 12.0000 0.535586
\(503\) −11.0000 −0.490466 −0.245233 0.969464i \(-0.578864\pi\)
−0.245233 + 0.969464i \(0.578864\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) 5.00000 0.222277
\(507\) 4.00000 0.177646
\(508\) −18.0000 −0.798621
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) 9.00000 0.398137
\(512\) −1.00000 −0.0441942
\(513\) 3.00000 0.132453
\(514\) 22.0000 0.970378
\(515\) 0 0
\(516\) −5.00000 −0.220113
\(517\) 30.0000 1.31940
\(518\) −4.00000 −0.175750
\(519\) 9.00000 0.395056
\(520\) 0 0
\(521\) −4.00000 −0.175243 −0.0876216 0.996154i \(-0.527927\pi\)
−0.0876216 + 0.996154i \(0.527927\pi\)
\(522\) 1.00000 0.0437688
\(523\) 19.0000 0.830812 0.415406 0.909636i \(-0.363640\pi\)
0.415406 + 0.909636i \(0.363640\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) −28.0000 −1.22086
\(527\) 0 0
\(528\) 5.00000 0.217597
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) −3.00000 −0.130066
\(533\) −9.00000 −0.389833
\(534\) −2.00000 −0.0865485
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) −10.0000 −0.431532
\(538\) 9.00000 0.388018
\(539\) 30.0000 1.29219
\(540\) 0 0
\(541\) −31.0000 −1.33279 −0.666397 0.745597i \(-0.732164\pi\)
−0.666397 + 0.745597i \(0.732164\pi\)
\(542\) 28.0000 1.20270
\(543\) 6.00000 0.257485
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) 3.00000 0.128388
\(547\) 6.00000 0.256541 0.128271 0.991739i \(-0.459057\pi\)
0.128271 + 0.991739i \(0.459057\pi\)
\(548\) 16.0000 0.683486
\(549\) 4.00000 0.170716
\(550\) 0 0
\(551\) 3.00000 0.127804
\(552\) 1.00000 0.0425628
\(553\) −7.00000 −0.297670
\(554\) 31.0000 1.31706
\(555\) 0 0
\(556\) −10.0000 −0.424094
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 0 0
\(559\) 15.0000 0.634432
\(560\) 0 0
\(561\) 10.0000 0.422200
\(562\) 22.0000 0.928014
\(563\) 11.0000 0.463595 0.231797 0.972764i \(-0.425539\pi\)
0.231797 + 0.972764i \(0.425539\pi\)
\(564\) 6.00000 0.252646
\(565\) 0 0
\(566\) 4.00000 0.168133
\(567\) 1.00000 0.0419961
\(568\) 2.00000 0.0839181
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) 36.0000 1.50655 0.753277 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(572\) −15.0000 −0.627182
\(573\) 3.00000 0.125327
\(574\) 3.00000 0.125218
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 17.0000 0.707719 0.353860 0.935299i \(-0.384869\pi\)
0.353860 + 0.935299i \(0.384869\pi\)
\(578\) 13.0000 0.540729
\(579\) −10.0000 −0.415586
\(580\) 0 0
\(581\) −1.00000 −0.0414870
\(582\) 0 0
\(583\) 0 0
\(584\) −9.00000 −0.372423
\(585\) 0 0
\(586\) 28.0000 1.15667
\(587\) 2.00000 0.0825488 0.0412744 0.999148i \(-0.486858\pi\)
0.0412744 + 0.999148i \(0.486858\pi\)
\(588\) 6.00000 0.247436
\(589\) 0 0
\(590\) 0 0
\(591\) −15.0000 −0.617018
\(592\) 4.00000 0.164399
\(593\) 9.00000 0.369586 0.184793 0.982777i \(-0.440839\pi\)
0.184793 + 0.982777i \(0.440839\pi\)
\(594\) −5.00000 −0.205152
\(595\) 0 0
\(596\) −20.0000 −0.819232
\(597\) 11.0000 0.450200
\(598\) −3.00000 −0.122679
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) −5.00000 −0.203785
\(603\) −4.00000 −0.162893
\(604\) 18.0000 0.732410
\(605\) 0 0
\(606\) 6.00000 0.243733
\(607\) −14.0000 −0.568242 −0.284121 0.958788i \(-0.591702\pi\)
−0.284121 + 0.958788i \(0.591702\pi\)
\(608\) 3.00000 0.121666
\(609\) 1.00000 0.0405220
\(610\) 0 0
\(611\) −18.0000 −0.728202
\(612\) 2.00000 0.0808452
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) 20.0000 0.807134
\(615\) 0 0
\(616\) 5.00000 0.201456
\(617\) 10.0000 0.402585 0.201292 0.979531i \(-0.435486\pi\)
0.201292 + 0.979531i \(0.435486\pi\)
\(618\) 7.00000 0.281581
\(619\) −36.0000 −1.44696 −0.723481 0.690344i \(-0.757459\pi\)
−0.723481 + 0.690344i \(0.757459\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 20.0000 0.801927
\(623\) −2.00000 −0.0801283
\(624\) −3.00000 −0.120096
\(625\) 0 0
\(626\) −32.0000 −1.27898
\(627\) −15.0000 −0.599042
\(628\) −4.00000 −0.159617
\(629\) 8.00000 0.318981
\(630\) 0 0
\(631\) −25.0000 −0.995234 −0.497617 0.867397i \(-0.665792\pi\)
−0.497617 + 0.867397i \(0.665792\pi\)
\(632\) 7.00000 0.278445
\(633\) 2.00000 0.0794929
\(634\) 13.0000 0.516296
\(635\) 0 0
\(636\) 0 0
\(637\) −18.0000 −0.713186
\(638\) −5.00000 −0.197952
\(639\) −2.00000 −0.0791188
\(640\) 0 0
\(641\) 4.00000 0.157991 0.0789953 0.996875i \(-0.474829\pi\)
0.0789953 + 0.996875i \(0.474829\pi\)
\(642\) −12.0000 −0.473602
\(643\) −19.0000 −0.749287 −0.374643 0.927169i \(-0.622235\pi\)
−0.374643 + 0.927169i \(0.622235\pi\)
\(644\) 1.00000 0.0394055
\(645\) 0 0
\(646\) 6.00000 0.236067
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 30.0000 1.17760
\(650\) 0 0
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 1.00000 0.0391330 0.0195665 0.999809i \(-0.493771\pi\)
0.0195665 + 0.999809i \(0.493771\pi\)
\(654\) −4.00000 −0.156412
\(655\) 0 0
\(656\) −3.00000 −0.117130
\(657\) 9.00000 0.351123
\(658\) 6.00000 0.233904
\(659\) 45.0000 1.75295 0.876476 0.481446i \(-0.159888\pi\)
0.876476 + 0.481446i \(0.159888\pi\)
\(660\) 0 0
\(661\) 24.0000 0.933492 0.466746 0.884391i \(-0.345426\pi\)
0.466746 + 0.884391i \(0.345426\pi\)
\(662\) −20.0000 −0.777322
\(663\) −6.00000 −0.233021
\(664\) 1.00000 0.0388075
\(665\) 0 0
\(666\) −4.00000 −0.154997
\(667\) −1.00000 −0.0387202
\(668\) −12.0000 −0.464294
\(669\) −2.00000 −0.0773245
\(670\) 0 0
\(671\) −20.0000 −0.772091
\(672\) 1.00000 0.0385758
\(673\) 27.0000 1.04077 0.520387 0.853931i \(-0.325788\pi\)
0.520387 + 0.853931i \(0.325788\pi\)
\(674\) −8.00000 −0.308148
\(675\) 0 0
\(676\) −4.00000 −0.153846
\(677\) 8.00000 0.307465 0.153732 0.988113i \(-0.450871\pi\)
0.153732 + 0.988113i \(0.450871\pi\)
\(678\) −6.00000 −0.230429
\(679\) 0 0
\(680\) 0 0
\(681\) 24.0000 0.919682
\(682\) 0 0
\(683\) −18.0000 −0.688751 −0.344375 0.938832i \(-0.611909\pi\)
−0.344375 + 0.938832i \(0.611909\pi\)
\(684\) −3.00000 −0.114708
\(685\) 0 0
\(686\) 13.0000 0.496342
\(687\) 8.00000 0.305219
\(688\) 5.00000 0.190623
\(689\) 0 0
\(690\) 0 0
\(691\) −14.0000 −0.532585 −0.266293 0.963892i \(-0.585799\pi\)
−0.266293 + 0.963892i \(0.585799\pi\)
\(692\) −9.00000 −0.342129
\(693\) −5.00000 −0.189934
\(694\) 20.0000 0.759190
\(695\) 0 0
\(696\) −1.00000 −0.0379049
\(697\) −6.00000 −0.227266
\(698\) 15.0000 0.567758
\(699\) 11.0000 0.416058
\(700\) 0 0
\(701\) −36.0000 −1.35970 −0.679851 0.733351i \(-0.737955\pi\)
−0.679851 + 0.733351i \(0.737955\pi\)
\(702\) 3.00000 0.113228
\(703\) −12.0000 −0.452589
\(704\) −5.00000 −0.188445
\(705\) 0 0
\(706\) −11.0000 −0.413990
\(707\) 6.00000 0.225653
\(708\) 6.00000 0.225494
\(709\) −44.0000 −1.65245 −0.826227 0.563337i \(-0.809517\pi\)
−0.826227 + 0.563337i \(0.809517\pi\)
\(710\) 0 0
\(711\) −7.00000 −0.262521
\(712\) 2.00000 0.0749532
\(713\) 0 0
\(714\) 2.00000 0.0748481
\(715\) 0 0
\(716\) 10.0000 0.373718
\(717\) 8.00000 0.298765
\(718\) 9.00000 0.335877
\(719\) 30.0000 1.11881 0.559406 0.828894i \(-0.311029\pi\)
0.559406 + 0.828894i \(0.311029\pi\)
\(720\) 0 0
\(721\) 7.00000 0.260694
\(722\) 10.0000 0.372161
\(723\) 2.00000 0.0743808
\(724\) −6.00000 −0.222988
\(725\) 0 0
\(726\) 14.0000 0.519589
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) −3.00000 −0.111187
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 10.0000 0.369863
\(732\) −4.00000 −0.147844
\(733\) −24.0000 −0.886460 −0.443230 0.896408i \(-0.646168\pi\)
−0.443230 + 0.896408i \(0.646168\pi\)
\(734\) −37.0000 −1.36569
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 20.0000 0.736709
\(738\) 3.00000 0.110432
\(739\) 24.0000 0.882854 0.441427 0.897297i \(-0.354472\pi\)
0.441427 + 0.897297i \(0.354472\pi\)
\(740\) 0 0
\(741\) 9.00000 0.330623
\(742\) 0 0
\(743\) −5.00000 −0.183432 −0.0917161 0.995785i \(-0.529235\pi\)
−0.0917161 + 0.995785i \(0.529235\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 22.0000 0.805477
\(747\) −1.00000 −0.0365881
\(748\) −10.0000 −0.365636
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) 7.00000 0.255434 0.127717 0.991811i \(-0.459235\pi\)
0.127717 + 0.991811i \(0.459235\pi\)
\(752\) −6.00000 −0.218797
\(753\) 12.0000 0.437304
\(754\) 3.00000 0.109254
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) 28.0000 1.01768 0.508839 0.860862i \(-0.330075\pi\)
0.508839 + 0.860862i \(0.330075\pi\)
\(758\) 20.0000 0.726433
\(759\) 5.00000 0.181489
\(760\) 0 0
\(761\) 23.0000 0.833749 0.416875 0.908964i \(-0.363125\pi\)
0.416875 + 0.908964i \(0.363125\pi\)
\(762\) −18.0000 −0.652071
\(763\) −4.00000 −0.144810
\(764\) −3.00000 −0.108536
\(765\) 0 0
\(766\) 9.00000 0.325183
\(767\) −18.0000 −0.649942
\(768\) −1.00000 −0.0360844
\(769\) 6.00000 0.216366 0.108183 0.994131i \(-0.465497\pi\)
0.108183 + 0.994131i \(0.465497\pi\)
\(770\) 0 0
\(771\) 22.0000 0.792311
\(772\) 10.0000 0.359908
\(773\) 12.0000 0.431610 0.215805 0.976436i \(-0.430762\pi\)
0.215805 + 0.976436i \(0.430762\pi\)
\(774\) −5.00000 −0.179721
\(775\) 0 0
\(776\) 0 0
\(777\) −4.00000 −0.143499
\(778\) −10.0000 −0.358517
\(779\) 9.00000 0.322458
\(780\) 0 0
\(781\) 10.0000 0.357828
\(782\) −2.00000 −0.0715199
\(783\) 1.00000 0.0357371
\(784\) −6.00000 −0.214286
\(785\) 0 0
\(786\) 12.0000 0.428026
\(787\) −37.0000 −1.31891 −0.659454 0.751745i \(-0.729212\pi\)
−0.659454 + 0.751745i \(0.729212\pi\)
\(788\) 15.0000 0.534353
\(789\) −28.0000 −0.996826
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 5.00000 0.177667
\(793\) 12.0000 0.426132
\(794\) −2.00000 −0.0709773
\(795\) 0 0
\(796\) −11.0000 −0.389885
\(797\) −10.0000 −0.354218 −0.177109 0.984191i \(-0.556675\pi\)
−0.177109 + 0.984191i \(0.556675\pi\)
\(798\) −3.00000 −0.106199
\(799\) −12.0000 −0.424529
\(800\) 0 0
\(801\) −2.00000 −0.0706665
\(802\) −12.0000 −0.423735
\(803\) −45.0000 −1.58802
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 0 0
\(807\) 9.00000 0.316815
\(808\) −6.00000 −0.211079
\(809\) 39.0000 1.37117 0.685583 0.727994i \(-0.259547\pi\)
0.685583 + 0.727994i \(0.259547\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) −1.00000 −0.0350931
\(813\) 28.0000 0.982003
\(814\) 20.0000 0.701000
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) −15.0000 −0.524784
\(818\) −17.0000 −0.594391
\(819\) 3.00000 0.104828
\(820\) 0 0
\(821\) 53.0000 1.84971 0.924856 0.380317i \(-0.124185\pi\)
0.924856 + 0.380317i \(0.124185\pi\)
\(822\) 16.0000 0.558064
\(823\) 2.00000 0.0697156 0.0348578 0.999392i \(-0.488902\pi\)
0.0348578 + 0.999392i \(0.488902\pi\)
\(824\) −7.00000 −0.243857
\(825\) 0 0
\(826\) 6.00000 0.208767
\(827\) 53.0000 1.84299 0.921495 0.388390i \(-0.126968\pi\)
0.921495 + 0.388390i \(0.126968\pi\)
\(828\) 1.00000 0.0347524
\(829\) −23.0000 −0.798823 −0.399412 0.916772i \(-0.630786\pi\)
−0.399412 + 0.916772i \(0.630786\pi\)
\(830\) 0 0
\(831\) 31.0000 1.07538
\(832\) 3.00000 0.104006
\(833\) −12.0000 −0.415775
\(834\) −10.0000 −0.346272
\(835\) 0 0
\(836\) 15.0000 0.518786
\(837\) 0 0
\(838\) −25.0000 −0.863611
\(839\) 25.0000 0.863096 0.431548 0.902090i \(-0.357968\pi\)
0.431548 + 0.902090i \(0.357968\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) −6.00000 −0.206774
\(843\) 22.0000 0.757720
\(844\) −2.00000 −0.0688428
\(845\) 0 0
\(846\) 6.00000 0.206284
\(847\) 14.0000 0.481046
\(848\) 0 0
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) 4.00000 0.137118
\(852\) 2.00000 0.0685189
\(853\) −49.0000 −1.67773 −0.838864 0.544341i \(-0.816780\pi\)
−0.838864 + 0.544341i \(0.816780\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 6.00000 0.204956 0.102478 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) −15.0000 −0.512092
\(859\) 6.00000 0.204717 0.102359 0.994748i \(-0.467361\pi\)
0.102359 + 0.994748i \(0.467361\pi\)
\(860\) 0 0
\(861\) 3.00000 0.102240
\(862\) 0 0
\(863\) 18.0000 0.612727 0.306364 0.951915i \(-0.400888\pi\)
0.306364 + 0.951915i \(0.400888\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 12.0000 0.407777
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) 35.0000 1.18729
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) 4.00000 0.135457
\(873\) 0 0
\(874\) 3.00000 0.101477
\(875\) 0 0
\(876\) −9.00000 −0.304082
\(877\) 22.0000 0.742887 0.371444 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) −20.0000 −0.674967
\(879\) 28.0000 0.944417
\(880\) 0 0
\(881\) 40.0000 1.34763 0.673817 0.738898i \(-0.264654\pi\)
0.673817 + 0.738898i \(0.264654\pi\)
\(882\) 6.00000 0.202031
\(883\) 24.0000 0.807664 0.403832 0.914833i \(-0.367678\pi\)
0.403832 + 0.914833i \(0.367678\pi\)
\(884\) 6.00000 0.201802
\(885\) 0 0
\(886\) 20.0000 0.671913
\(887\) 6.00000 0.201460 0.100730 0.994914i \(-0.467882\pi\)
0.100730 + 0.994914i \(0.467882\pi\)
\(888\) 4.00000 0.134231
\(889\) −18.0000 −0.603701
\(890\) 0 0
\(891\) −5.00000 −0.167506
\(892\) 2.00000 0.0669650
\(893\) 18.0000 0.602347
\(894\) −20.0000 −0.668900
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) −3.00000 −0.100167
\(898\) −14.0000 −0.467186
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) −15.0000 −0.499445
\(903\) −5.00000 −0.166390
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) 18.0000 0.598010
\(907\) −33.0000 −1.09575 −0.547874 0.836561i \(-0.684562\pi\)
−0.547874 + 0.836561i \(0.684562\pi\)
\(908\) −24.0000 −0.796468
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) −9.00000 −0.298183 −0.149092 0.988823i \(-0.547635\pi\)
−0.149092 + 0.988823i \(0.547635\pi\)
\(912\) 3.00000 0.0993399
\(913\) 5.00000 0.165476
\(914\) −18.0000 −0.595387
\(915\) 0 0
\(916\) −8.00000 −0.264327
\(917\) 12.0000 0.396275
\(918\) 2.00000 0.0660098
\(919\) 4.00000 0.131948 0.0659739 0.997821i \(-0.478985\pi\)
0.0659739 + 0.997821i \(0.478985\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) 37.0000 1.21853
\(923\) −6.00000 −0.197492
\(924\) 5.00000 0.164488
\(925\) 0 0
\(926\) −12.0000 −0.394344
\(927\) 7.00000 0.229910
\(928\) 1.00000 0.0328266
\(929\) 33.0000 1.08269 0.541347 0.840799i \(-0.317914\pi\)
0.541347 + 0.840799i \(0.317914\pi\)
\(930\) 0 0
\(931\) 18.0000 0.589926
\(932\) −11.0000 −0.360317
\(933\) 20.0000 0.654771
\(934\) 27.0000 0.883467
\(935\) 0 0
\(936\) −3.00000 −0.0980581
\(937\) 60.0000 1.96011 0.980057 0.198715i \(-0.0636769\pi\)
0.980057 + 0.198715i \(0.0636769\pi\)
\(938\) 4.00000 0.130605
\(939\) −32.0000 −1.04428
\(940\) 0 0
\(941\) 32.0000 1.04317 0.521585 0.853199i \(-0.325341\pi\)
0.521585 + 0.853199i \(0.325341\pi\)
\(942\) −4.00000 −0.130327
\(943\) −3.00000 −0.0976934
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) 25.0000 0.812820
\(947\) 32.0000 1.03986 0.519930 0.854209i \(-0.325958\pi\)
0.519930 + 0.854209i \(0.325958\pi\)
\(948\) 7.00000 0.227349
\(949\) 27.0000 0.876457
\(950\) 0 0
\(951\) 13.0000 0.421554
\(952\) −2.00000 −0.0648204
\(953\) 20.0000 0.647864 0.323932 0.946080i \(-0.394995\pi\)
0.323932 + 0.946080i \(0.394995\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) −5.00000 −0.161627
\(958\) −17.0000 −0.549245
\(959\) 16.0000 0.516667
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −12.0000 −0.386896
\(963\) −12.0000 −0.386695
\(964\) −2.00000 −0.0644157
\(965\) 0 0
\(966\) 1.00000 0.0321745
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) −14.0000 −0.449977
\(969\) 6.00000 0.192748
\(970\) 0 0
\(971\) −31.0000 −0.994837 −0.497419 0.867511i \(-0.665719\pi\)
−0.497419 + 0.867511i \(0.665719\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −10.0000 −0.320585
\(974\) −18.0000 −0.576757
\(975\) 0 0
\(976\) 4.00000 0.128037
\(977\) 24.0000 0.767828 0.383914 0.923369i \(-0.374576\pi\)
0.383914 + 0.923369i \(0.374576\pi\)
\(978\) −4.00000 −0.127906
\(979\) 10.0000 0.319601
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) 0 0
\(983\) 9.00000 0.287055 0.143528 0.989646i \(-0.454155\pi\)
0.143528 + 0.989646i \(0.454155\pi\)
\(984\) −3.00000 −0.0956365
\(985\) 0 0
\(986\) 2.00000 0.0636930
\(987\) 6.00000 0.190982
\(988\) −9.00000 −0.286328
\(989\) 5.00000 0.158991
\(990\) 0 0
\(991\) −20.0000 −0.635321 −0.317660 0.948205i \(-0.602897\pi\)
−0.317660 + 0.948205i \(0.602897\pi\)
\(992\) 0 0
\(993\) −20.0000 −0.634681
\(994\) 2.00000 0.0634361
\(995\) 0 0
\(996\) 1.00000 0.0316862
\(997\) 49.0000 1.55185 0.775923 0.630828i \(-0.217285\pi\)
0.775923 + 0.630828i \(0.217285\pi\)
\(998\) 34.0000 1.07625
\(999\) −4.00000 −0.126554
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.e.1.1 1
5.2 odd 4 3450.2.d.k.2899.1 2
5.3 odd 4 3450.2.d.k.2899.2 2
5.4 even 2 3450.2.a.w.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3450.2.a.e.1.1 1 1.1 even 1 trivial
3450.2.a.w.1.1 yes 1 5.4 even 2
3450.2.d.k.2899.1 2 5.2 odd 4
3450.2.d.k.2899.2 2 5.3 odd 4