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Properties

Label 5610.2.a.be.1.1
Level $5610$
Weight $2$
Character 5610.1
Self dual yes
Analytic conductor $44.796$
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] = [5610,2,Mod(1,5610)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5610, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("5610.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5610.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: \(44.7960755339\)
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 \(\chi\) \(=\) 5610.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} -4.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} +2.00000 q^{19} -1.00000 q^{20} -4.00000 q^{21} -1.00000 q^{22} +6.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} -4.00000 q^{28} -6.00000 q^{29} -1.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +4.00000 q^{35} +1.00000 q^{36} -10.0000 q^{37} +2.00000 q^{38} +2.00000 q^{39} -1.00000 q^{40} -4.00000 q^{42} -4.00000 q^{43} -1.00000 q^{44} -1.00000 q^{45} +6.00000 q^{46} -12.0000 q^{47} +1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} -1.00000 q^{51} +2.00000 q^{52} +1.00000 q^{54} +1.00000 q^{55} -4.00000 q^{56} +2.00000 q^{57} -6.00000 q^{58} -1.00000 q^{60} -10.0000 q^{61} -4.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} -1.00000 q^{66} +14.0000 q^{67} -1.00000 q^{68} +6.00000 q^{69} +4.00000 q^{70} -12.0000 q^{71} +1.00000 q^{72} +2.00000 q^{73} -10.0000 q^{74} +1.00000 q^{75} +2.00000 q^{76} +4.00000 q^{77} +2.00000 q^{78} +2.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -4.00000 q^{84} +1.00000 q^{85} -4.00000 q^{86} -6.00000 q^{87} -1.00000 q^{88} +6.00000 q^{89} -1.00000 q^{90} -8.00000 q^{91} +6.00000 q^{92} -4.00000 q^{93} -12.0000 q^{94} -2.00000 q^{95} +1.00000 q^{96} -16.0000 q^{97} +9.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −1.00000 −0.301511
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −4.00000 −1.06904
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536
\(18\) 1.00000 0.235702
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) −1.00000 −0.223607
\(21\) −4.00000 −0.872872
\(22\) −1.00000 −0.213201
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 1.00000 0.192450
\(28\) −4.00000 −0.755929
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) −1.00000 −0.182574
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) −1.00000 −0.171499
\(35\) 4.00000 0.676123
\(36\) 1.00000 0.166667
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 2.00000 0.324443
\(39\) 2.00000 0.320256
\(40\) −1.00000 −0.158114
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) −4.00000 −0.617213
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −1.00000 −0.150756
\(45\) −1.00000 −0.149071
\(46\) 6.00000 0.884652
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 1.00000 0.144338
\(49\) 9.00000 1.28571
\(50\) 1.00000 0.141421
\(51\) −1.00000 −0.140028
\(52\) 2.00000 0.277350
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 1.00000 0.136083
\(55\) 1.00000 0.134840
\(56\) −4.00000 −0.534522
\(57\) 2.00000 0.264906
\(58\) −6.00000 −0.787839
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −1.00000 −0.129099
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −4.00000 −0.508001
\(63\) −4.00000 −0.503953
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) −1.00000 −0.123091
\(67\) 14.0000 1.71037 0.855186 − 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) −1.00000 −0.121268
\(69\) 6.00000 0.722315
\(70\) 4.00000 0.478091
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 1.00000 0.117851
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −10.0000 −1.16248
\(75\) 1.00000 0.115470
\(76\) 2.00000 0.229416
\(77\) 4.00000 0.455842
\(78\) 2.00000 0.226455
\(79\) 2.00000 0.225018 0.112509 − 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −4.00000 −0.436436
\(85\) 1.00000 0.108465
\(86\) −4.00000 −0.431331
\(87\) −6.00000 −0.643268
\(88\) −1.00000 −0.106600
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −1.00000 −0.105409
\(91\) −8.00000 −0.838628
\(92\) 6.00000 0.625543
\(93\) −4.00000 −0.414781
\(94\) −12.0000 −1.23771
\(95\) −2.00000 −0.205196
\(96\) 1.00000 0.102062
\(97\) −16.0000 −1.62455 −0.812277 − 0.583272i \(-0.801772\pi\)
−0.812277 + 0.583272i \(0.801772\pi\)
\(98\) 9.00000 0.909137
\(99\) −1.00000 −0.100504
\(100\) 1.00000 0.100000
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) −1.00000 −0.0990148
\(103\) −4.00000 −0.394132 −0.197066 − 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 2.00000 0.196116
\(105\) 4.00000 0.390360
\(106\) 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
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 1.00000 0.0953463
\(111\) −10.0000 −0.949158
\(112\) −4.00000 −0.377964
\(113\) 12.0000 1.12887 0.564433 − 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) 2.00000 0.187317
\(115\) −6.00000 −0.559503
\(116\) −6.00000 −0.557086
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) 4.00000 0.366679
\(120\) −1.00000 −0.0912871
\(121\) 1.00000 0.0909091
\(122\) −10.0000 −0.905357
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) −1.00000 −0.0894427
\(126\) −4.00000 −0.356348
\(127\) −16.0000 −1.41977 −0.709885 − 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) −2.00000 −0.175412
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) −1.00000 −0.0870388
\(133\) −8.00000 −0.693688
\(134\) 14.0000 1.20942
\(135\) −1.00000 −0.0860663
\(136\) −1.00000 −0.0857493
\(137\) −6.00000 −0.512615 −0.256307 − 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 6.00000 0.510754
\(139\) −16.0000 −1.35710 −0.678551 − 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 4.00000 0.338062
\(141\) −12.0000 −1.01058
\(142\) −12.0000 −1.00702
\(143\) −2.00000 −0.167248
\(144\) 1.00000 0.0833333
\(145\) 6.00000 0.498273
\(146\) 2.00000 0.165521
\(147\) 9.00000 0.742307
\(148\) −10.0000 −0.821995
\(149\) −12.0000 −0.983078 −0.491539 − 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) 1.00000 0.0816497
\(151\) 8.00000 0.651031 0.325515 − 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 2.00000 0.162221
\(153\) −1.00000 −0.0808452
\(154\) 4.00000 0.322329
\(155\) 4.00000 0.321288
\(156\) 2.00000 0.160128
\(157\) 8.00000 0.638470 0.319235 − 0.947676i \(-0.396574\pi\)
0.319235 + 0.947676i \(0.396574\pi\)
\(158\) 2.00000 0.159111
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −24.0000 −1.89146
\(162\) 1.00000 0.0785674
\(163\) −4.00000 −0.313304 −0.156652 − 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) 1.00000 0.0778499
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 − 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) −4.00000 −0.308607
\(169\) −9.00000 −0.692308
\(170\) 1.00000 0.0766965
\(171\) 2.00000 0.152944
\(172\) −4.00000 −0.304997
\(173\) 6.00000 0.456172 0.228086 − 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −6.00000 −0.454859
\(175\) −4.00000 −0.302372
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −10.0000 −0.743294 −0.371647 − 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) −8.00000 −0.592999
\(183\) −10.0000 −0.739221
\(184\) 6.00000 0.442326
\(185\) 10.0000 0.735215
\(186\) −4.00000 −0.293294
\(187\) 1.00000 0.0731272
\(188\) −12.0000 −0.875190
\(189\) −4.00000 −0.290957
\(190\) −2.00000 −0.145095
\(191\) −24.0000 −1.73658 −0.868290 − 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) 1.00000 0.0721688
\(193\) −22.0000 −1.58359 −0.791797 − 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) −16.0000 −1.14873
\(195\) −2.00000 −0.143223
\(196\) 9.00000 0.642857
\(197\) −6.00000 −0.427482 −0.213741 − 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 20.0000 1.41776 0.708881 − 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 1.00000 0.0707107
\(201\) 14.0000 0.987484
\(202\) 0 0
\(203\) 24.0000 1.68447
\(204\) −1.00000 −0.0700140
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) 6.00000 0.417029
\(208\) 2.00000 0.138675
\(209\) −2.00000 −0.138343
\(210\) 4.00000 0.276026
\(211\) 20.0000 1.37686 0.688428 − 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 0 0
\(213\) −12.0000 −0.822226
\(214\) −12.0000 −0.820303
\(215\) 4.00000 0.272798
\(216\) 1.00000 0.0680414
\(217\) 16.0000 1.08615
\(218\) 2.00000 0.135457
\(219\) 2.00000 0.135147
\(220\) 1.00000 0.0674200
\(221\) −2.00000 −0.134535
\(222\) −10.0000 −0.671156
\(223\) −16.0000 −1.07144 −0.535720 − 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) −4.00000 −0.267261
\(225\) 1.00000 0.0666667
\(226\) 12.0000 0.798228
\(227\) −12.0000 −0.796468 −0.398234 − 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 2.00000 0.132453
\(229\) 2.00000 0.132164 0.0660819 − 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) −6.00000 −0.395628
\(231\) 4.00000 0.263181
\(232\) −6.00000 −0.393919
\(233\) −6.00000 −0.393073 −0.196537 − 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 2.00000 0.130744
\(235\) 12.0000 0.782794
\(236\) 0 0
\(237\) 2.00000 0.129914
\(238\) 4.00000 0.259281
\(239\) −12.0000 −0.776215 −0.388108 − 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −4.00000 −0.257663 −0.128831 − 0.991667i \(-0.541123\pi\)
−0.128831 + 0.991667i \(0.541123\pi\)
\(242\) 1.00000 0.0642824
\(243\) 1.00000 0.0641500
\(244\) −10.0000 −0.640184
\(245\) −9.00000 −0.574989
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) −4.00000 −0.254000
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 12.0000 0.757433 0.378717 − 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) −4.00000 −0.251976
\(253\) −6.00000 −0.377217
\(254\) −16.0000 −1.00393
\(255\) 1.00000 0.0626224
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 − 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) −4.00000 −0.249029
\(259\) 40.0000 2.48548
\(260\) −2.00000 −0.124035
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) −1.00000 −0.0615457
\(265\) 0 0
\(266\) −8.00000 −0.490511
\(267\) 6.00000 0.367194
\(268\) 14.0000 0.855186
\(269\) 6.00000 0.365826 0.182913 − 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 20.0000 1.21491 0.607457 − 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) −1.00000 −0.0606339
\(273\) −8.00000 −0.484182
\(274\) −6.00000 −0.362473
\(275\) −1.00000 −0.0603023
\(276\) 6.00000 0.361158
\(277\) −22.0000 −1.32185 −0.660926 − 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) −16.0000 −0.959616
\(279\) −4.00000 −0.239474
\(280\) 4.00000 0.239046
\(281\) 6.00000 0.357930 0.178965 − 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) −12.0000 −0.714590
\(283\) 20.0000 1.18888 0.594438 − 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) −12.0000 −0.712069
\(285\) −2.00000 −0.118470
\(286\) −2.00000 −0.118262
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 1.00000 0.0588235
\(290\) 6.00000 0.352332
\(291\) −16.0000 −0.937937
\(292\) 2.00000 0.117041
\(293\) −18.0000 −1.05157 −0.525786 − 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 9.00000 0.524891
\(295\) 0 0
\(296\) −10.0000 −0.581238
\(297\) −1.00000 −0.0580259
\(298\) −12.0000 −0.695141
\(299\) 12.0000 0.693978
\(300\) 1.00000 0.0577350
\(301\) 16.0000 0.922225
\(302\) 8.00000 0.460348
\(303\) 0 0
\(304\) 2.00000 0.114708
\(305\) 10.0000 0.572598
\(306\) −1.00000 −0.0571662
\(307\) −16.0000 −0.913168 −0.456584 − 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 4.00000 0.227921
\(309\) −4.00000 −0.227552
\(310\) 4.00000 0.227185
\(311\) 24.0000 1.36092 0.680458 − 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 2.00000 0.113228
\(313\) 8.00000 0.452187 0.226093 − 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) 8.00000 0.451466
\(315\) 4.00000 0.225374
\(316\) 2.00000 0.112509
\(317\) 6.00000 0.336994 0.168497 − 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) −1.00000 −0.0559017
\(321\) −12.0000 −0.669775
\(322\) −24.0000 −1.33747
\(323\) −2.00000 −0.111283
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) −4.00000 −0.221540
\(327\) 2.00000 0.110600
\(328\) 0 0
\(329\) 48.0000 2.64633
\(330\) 1.00000 0.0550482
\(331\) 8.00000 0.439720 0.219860 − 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 0 0
\(333\) −10.0000 −0.547997
\(334\) 12.0000 0.656611
\(335\) −14.0000 −0.764902
\(336\) −4.00000 −0.218218
\(337\) −22.0000 −1.19842 −0.599208 − 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) −9.00000 −0.489535
\(339\) 12.0000 0.651751
\(340\) 1.00000 0.0542326
\(341\) 4.00000 0.216612
\(342\) 2.00000 0.108148
\(343\) −8.00000 −0.431959
\(344\) −4.00000 −0.215666
\(345\) −6.00000 −0.323029
\(346\) 6.00000 0.322562
\(347\) −12.0000 −0.644194 −0.322097 − 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −6.00000 −0.321634
\(349\) −16.0000 −0.856460 −0.428230 − 0.903670i \(-0.640863\pi\)
−0.428230 + 0.903670i \(0.640863\pi\)
\(350\) −4.00000 −0.213809
\(351\) 2.00000 0.106752
\(352\) −1.00000 −0.0533002
\(353\) −6.00000 −0.319348 −0.159674 − 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) 6.00000 0.317999
\(357\) 4.00000 0.211702
\(358\) 0 0
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −15.0000 −0.789474
\(362\) −10.0000 −0.525588
\(363\) 1.00000 0.0524864
\(364\) −8.00000 −0.419314
\(365\) −2.00000 −0.104685
\(366\) −10.0000 −0.522708
\(367\) −10.0000 −0.521996 −0.260998 − 0.965339i \(-0.584052\pi\)
−0.260998 + 0.965339i \(0.584052\pi\)
\(368\) 6.00000 0.312772
\(369\) 0 0
\(370\) 10.0000 0.519875
\(371\) 0 0
\(372\) −4.00000 −0.207390
\(373\) 14.0000 0.724893 0.362446 − 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 1.00000 0.0517088
\(375\) −1.00000 −0.0516398
\(376\) −12.0000 −0.618853
\(377\) −12.0000 −0.618031
\(378\) −4.00000 −0.205738
\(379\) 20.0000 1.02733 0.513665 − 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) −2.00000 −0.102598
\(381\) −16.0000 −0.819705
\(382\) −24.0000 −1.22795
\(383\) 12.0000 0.613171 0.306586 − 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 1.00000 0.0510310
\(385\) −4.00000 −0.203859
\(386\) −22.0000 −1.11977
\(387\) −4.00000 −0.203331
\(388\) −16.0000 −0.812277
\(389\) 30.0000 1.52106 0.760530 − 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) −2.00000 −0.101274
\(391\) −6.00000 −0.303433
\(392\) 9.00000 0.454569
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) −2.00000 −0.100631
\(396\) −1.00000 −0.0502519
\(397\) −34.0000 −1.70641 −0.853206 − 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 20.0000 1.00251
\(399\) −8.00000 −0.400501
\(400\) 1.00000 0.0500000
\(401\) 18.0000 0.898877 0.449439 − 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 14.0000 0.698257
\(403\) −8.00000 −0.398508
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 24.0000 1.19110
\(407\) 10.0000 0.495682
\(408\) −1.00000 −0.0495074
\(409\) −10.0000 −0.494468 −0.247234 − 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) −4.00000 −0.197066
\(413\) 0 0
\(414\) 6.00000 0.294884
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) −16.0000 −0.783523
\(418\) −2.00000 −0.0978232
\(419\) 36.0000 1.75872 0.879358 − 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 4.00000 0.195180
\(421\) 38.0000 1.85201 0.926003 − 0.377515i \(-0.123221\pi\)
0.926003 + 0.377515i \(0.123221\pi\)
\(422\) 20.0000 0.973585
\(423\) −12.0000 −0.583460
\(424\) 0 0
\(425\) −1.00000 −0.0485071
\(426\) −12.0000 −0.581402
\(427\) 40.0000 1.93574
\(428\) −12.0000 −0.580042
\(429\) −2.00000 −0.0965609
\(430\) 4.00000 0.192897
\(431\) −6.00000 −0.289010 −0.144505 − 0.989504i \(-0.546159\pi\)
−0.144505 + 0.989504i \(0.546159\pi\)
\(432\) 1.00000 0.0481125
\(433\) 14.0000 0.672797 0.336399 − 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 16.0000 0.768025
\(435\) 6.00000 0.287678
\(436\) 2.00000 0.0957826
\(437\) 12.0000 0.574038
\(438\) 2.00000 0.0955637
\(439\) 26.0000 1.24091 0.620456 − 0.784241i \(-0.286947\pi\)
0.620456 + 0.784241i \(0.286947\pi\)
\(440\) 1.00000 0.0476731
\(441\) 9.00000 0.428571
\(442\) −2.00000 −0.0951303
\(443\) 6.00000 0.285069 0.142534 − 0.989790i \(-0.454475\pi\)
0.142534 + 0.989790i \(0.454475\pi\)
\(444\) −10.0000 −0.474579
\(445\) −6.00000 −0.284427
\(446\) −16.0000 −0.757622
\(447\) −12.0000 −0.567581
\(448\) −4.00000 −0.188982
\(449\) 30.0000 1.41579 0.707894 − 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) 12.0000 0.564433
\(453\) 8.00000 0.375873
\(454\) −12.0000 −0.563188
\(455\) 8.00000 0.375046
\(456\) 2.00000 0.0936586
\(457\) −10.0000 −0.467780 −0.233890 − 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 2.00000 0.0934539
\(459\) −1.00000 −0.0466760
\(460\) −6.00000 −0.279751
\(461\) 24.0000 1.11779 0.558896 − 0.829238i \(-0.311225\pi\)
0.558896 + 0.829238i \(0.311225\pi\)
\(462\) 4.00000 0.186097
\(463\) 8.00000 0.371792 0.185896 − 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) −6.00000 −0.278543
\(465\) 4.00000 0.185496
\(466\) −6.00000 −0.277945
\(467\) 18.0000 0.832941 0.416470 − 0.909149i \(-0.363267\pi\)
0.416470 + 0.909149i \(0.363267\pi\)
\(468\) 2.00000 0.0924500
\(469\) −56.0000 −2.58584
\(470\) 12.0000 0.553519
\(471\) 8.00000 0.368621
\(472\) 0 0
\(473\) 4.00000 0.183920
\(474\) 2.00000 0.0918630
\(475\) 2.00000 0.0917663
\(476\) 4.00000 0.183340
\(477\) 0 0
\(478\) −12.0000 −0.548867
\(479\) 18.0000 0.822441 0.411220 − 0.911536i \(-0.365103\pi\)
0.411220 + 0.911536i \(0.365103\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −20.0000 −0.911922
\(482\) −4.00000 −0.182195
\(483\) −24.0000 −1.09204
\(484\) 1.00000 0.0454545
\(485\) 16.0000 0.726523
\(486\) 1.00000 0.0453609
\(487\) 26.0000 1.17817 0.589086 − 0.808070i \(-0.299488\pi\)
0.589086 + 0.808070i \(0.299488\pi\)
\(488\) −10.0000 −0.452679
\(489\) −4.00000 −0.180886
\(490\) −9.00000 −0.406579
\(491\) −30.0000 −1.35388 −0.676941 − 0.736038i \(-0.736695\pi\)
−0.676941 + 0.736038i \(0.736695\pi\)
\(492\) 0 0
\(493\) 6.00000 0.270226
\(494\) 4.00000 0.179969
\(495\) 1.00000 0.0449467
\(496\) −4.00000 −0.179605
\(497\) 48.0000 2.15309
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 − 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 12.0000 0.536120
\(502\) 12.0000 0.535586
\(503\) −12.0000 −0.535054 −0.267527 − 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) −4.00000 −0.178174
\(505\) 0 0
\(506\) −6.00000 −0.266733
\(507\) −9.00000 −0.399704
\(508\) −16.0000 −0.709885
\(509\) −42.0000 −1.86162 −0.930809 − 0.365507i \(-0.880896\pi\)
−0.930809 + 0.365507i \(0.880896\pi\)
\(510\) 1.00000 0.0442807
\(511\) −8.00000 −0.353899
\(512\) 1.00000 0.0441942
\(513\) 2.00000 0.0883022
\(514\) 18.0000 0.793946
\(515\) 4.00000 0.176261
\(516\) −4.00000 −0.176090
\(517\) 12.0000 0.527759
\(518\) 40.0000 1.75750
\(519\) 6.00000 0.263371
\(520\) −2.00000 −0.0877058
\(521\) −6.00000 −0.262865 −0.131432 − 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) −6.00000 −0.262613
\(523\) 20.0000 0.874539 0.437269 − 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 0 0
\(525\) −4.00000 −0.174574
\(526\) 0 0
\(527\) 4.00000 0.174243
\(528\) −1.00000 −0.0435194
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) −8.00000 −0.346844
\(533\) 0 0
\(534\) 6.00000 0.259645
\(535\) 12.0000 0.518805
\(536\) 14.0000 0.604708
\(537\) 0 0
\(538\) 6.00000 0.258678
\(539\) −9.00000 −0.387657
\(540\) −1.00000 −0.0430331
\(541\) 14.0000 0.601907 0.300954 − 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 20.0000 0.859074
\(543\) −10.0000 −0.429141
\(544\) −1.00000 −0.0428746
\(545\) −2.00000 −0.0856706
\(546\) −8.00000 −0.342368
\(547\) 44.0000 1.88130 0.940652 − 0.339372i \(-0.110215\pi\)
0.940652 + 0.339372i \(0.110215\pi\)
\(548\) −6.00000 −0.256307
\(549\) −10.0000 −0.426790
\(550\) −1.00000 −0.0426401
\(551\) −12.0000 −0.511217
\(552\) 6.00000 0.255377
\(553\) −8.00000 −0.340195
\(554\) −22.0000 −0.934690
\(555\) 10.0000 0.424476
\(556\) −16.0000 −0.678551
\(557\) 42.0000 1.77960 0.889799 − 0.456354i \(-0.150845\pi\)
0.889799 + 0.456354i \(0.150845\pi\)
\(558\) −4.00000 −0.169334
\(559\) −8.00000 −0.338364
\(560\) 4.00000 0.169031
\(561\) 1.00000 0.0422200
\(562\) 6.00000 0.253095
\(563\) 24.0000 1.01148 0.505740 − 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) −12.0000 −0.505291
\(565\) −12.0000 −0.504844
\(566\) 20.0000 0.840663
\(567\) −4.00000 −0.167984
\(568\) −12.0000 −0.503509
\(569\) 18.0000 0.754599 0.377300 − 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) −2.00000 −0.0837708
\(571\) 8.00000 0.334790 0.167395 − 0.985890i \(-0.446465\pi\)
0.167395 + 0.985890i \(0.446465\pi\)
\(572\) −2.00000 −0.0836242
\(573\) −24.0000 −1.00261
\(574\) 0 0
\(575\) 6.00000 0.250217
\(576\) 1.00000 0.0416667
\(577\) −34.0000 −1.41544 −0.707719 − 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) 1.00000 0.0415945
\(579\) −22.0000 −0.914289
\(580\) 6.00000 0.249136
\(581\) 0 0
\(582\) −16.0000 −0.663221
\(583\) 0 0
\(584\) 2.00000 0.0827606
\(585\) −2.00000 −0.0826898
\(586\) −18.0000 −0.743573
\(587\) 30.0000 1.23823 0.619116 − 0.785299i \(-0.287491\pi\)
0.619116 + 0.785299i \(0.287491\pi\)
\(588\) 9.00000 0.371154
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) −10.0000 −0.410997
\(593\) −18.0000 −0.739171 −0.369586 − 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) −1.00000 −0.0410305
\(595\) −4.00000 −0.163984
\(596\) −12.0000 −0.491539
\(597\) 20.0000 0.818546
\(598\) 12.0000 0.490716
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 1.00000 0.0408248
\(601\) −28.0000 −1.14214 −0.571072 − 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) 16.0000 0.652111
\(603\) 14.0000 0.570124
\(604\) 8.00000 0.325515
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 20.0000 0.811775 0.405887 − 0.913923i \(-0.366962\pi\)
0.405887 + 0.913923i \(0.366962\pi\)
\(608\) 2.00000 0.0811107
\(609\) 24.0000 0.972529
\(610\) 10.0000 0.404888
\(611\) −24.0000 −0.970936
\(612\) −1.00000 −0.0404226
\(613\) −34.0000 −1.37325 −0.686624 − 0.727013i \(-0.740908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) −16.0000 −0.645707
\(615\) 0 0
\(616\) 4.00000 0.161165
\(617\) −12.0000 −0.483102 −0.241551 − 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) −4.00000 −0.160904
\(619\) −28.0000 −1.12542 −0.562708 − 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 4.00000 0.160644
\(621\) 6.00000 0.240772
\(622\) 24.0000 0.962312
\(623\) −24.0000 −0.961540
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) 8.00000 0.319744
\(627\) −2.00000 −0.0798723
\(628\) 8.00000 0.319235
\(629\) 10.0000 0.398726
\(630\) 4.00000 0.159364
\(631\) 32.0000 1.27390 0.636950 − 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 2.00000 0.0795557
\(633\) 20.0000 0.794929
\(634\) 6.00000 0.238290
\(635\) 16.0000 0.634941
\(636\) 0 0
\(637\) 18.0000 0.713186
\(638\) 6.00000 0.237542
\(639\) −12.0000 −0.474713
\(640\) −1.00000 −0.0395285
\(641\) 42.0000 1.65890 0.829450 − 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) −12.0000 −0.473602
\(643\) −40.0000 −1.57745 −0.788723 − 0.614749i \(-0.789257\pi\)
−0.788723 + 0.614749i \(0.789257\pi\)
\(644\) −24.0000 −0.945732
\(645\) 4.00000 0.157500
\(646\) −2.00000 −0.0786889
\(647\) 24.0000 0.943537 0.471769 − 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 2.00000 0.0784465
\(651\) 16.0000 0.627089
\(652\) −4.00000 −0.156652
\(653\) −6.00000 −0.234798 −0.117399 − 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 2.00000 0.0782062
\(655\) 0 0
\(656\) 0 0
\(657\) 2.00000 0.0780274
\(658\) 48.0000 1.87123
\(659\) 18.0000 0.701180 0.350590 − 0.936529i \(-0.385981\pi\)
0.350590 + 0.936529i \(0.385981\pi\)
\(660\) 1.00000 0.0389249
\(661\) 14.0000 0.544537 0.272268 − 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 8.00000 0.310929
\(663\) −2.00000 −0.0776736
\(664\) 0 0
\(665\) 8.00000 0.310227
\(666\) −10.0000 −0.387492
\(667\) −36.0000 −1.39393
\(668\) 12.0000 0.464294
\(669\) −16.0000 −0.618596
\(670\) −14.0000 −0.540867
\(671\) 10.0000 0.386046
\(672\) −4.00000 −0.154303
\(673\) −46.0000 −1.77317 −0.886585 − 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) −22.0000 −0.847408
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) 18.0000 0.691796 0.345898 − 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 12.0000 0.460857
\(679\) 64.0000 2.45609
\(680\) 1.00000 0.0383482
\(681\) −12.0000 −0.459841
\(682\) 4.00000 0.153168
\(683\) 12.0000 0.459167 0.229584 − 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 2.00000 0.0764719
\(685\) 6.00000 0.229248
\(686\) −8.00000 −0.305441
\(687\) 2.00000 0.0763048
\(688\) −4.00000 −0.152499
\(689\) 0 0
\(690\) −6.00000 −0.228416
\(691\) −28.0000 −1.06517 −0.532585 − 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 6.00000 0.228086
\(693\) 4.00000 0.151947
\(694\) −12.0000 −0.455514
\(695\) 16.0000 0.606915
\(696\) −6.00000 −0.227429
\(697\) 0 0
\(698\) −16.0000 −0.605609
\(699\) −6.00000 −0.226941
\(700\) −4.00000 −0.151186
\(701\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 2.00000 0.0754851
\(703\) −20.0000 −0.754314
\(704\) −1.00000 −0.0376889
\(705\) 12.0000 0.451946
\(706\) −6.00000 −0.225813
\(707\) 0 0
\(708\) 0 0
\(709\) −34.0000 −1.27690 −0.638448 − 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) 12.0000 0.450352
\(711\) 2.00000 0.0750059
\(712\) 6.00000 0.224860
\(713\) −24.0000 −0.898807
\(714\) 4.00000 0.149696
\(715\) 2.00000 0.0747958
\(716\) 0 0
\(717\) −12.0000 −0.448148
\(718\) 0 0
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 16.0000 0.595871
\(722\) −15.0000 −0.558242
\(723\) −4.00000 −0.148762
\(724\) −10.0000 −0.371647
\(725\) −6.00000 −0.222834
\(726\) 1.00000 0.0371135
\(727\) −28.0000 −1.03846 −0.519231 − 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) −8.00000 −0.296500
\(729\) 1.00000 0.0370370
\(730\) −2.00000 −0.0740233
\(731\) 4.00000 0.147945
\(732\) −10.0000 −0.369611
\(733\) 38.0000 1.40356 0.701781 − 0.712393i \(-0.252388\pi\)
0.701781 + 0.712393i \(0.252388\pi\)
\(734\) −10.0000 −0.369107
\(735\) −9.00000 −0.331970
\(736\) 6.00000 0.221163
\(737\) −14.0000 −0.515697
\(738\) 0 0
\(739\) −34.0000 −1.25071 −0.625355 − 0.780340i \(-0.715046\pi\)
−0.625355 + 0.780340i \(0.715046\pi\)
\(740\) 10.0000 0.367607
\(741\) 4.00000 0.146944
\(742\) 0 0
\(743\) 24.0000 0.880475 0.440237 − 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) −4.00000 −0.146647
\(745\) 12.0000 0.439646
\(746\) 14.0000 0.512576
\(747\) 0 0
\(748\) 1.00000 0.0365636
\(749\) 48.0000 1.75388
\(750\) −1.00000 −0.0365148
\(751\) −4.00000 −0.145962 −0.0729810 − 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) −12.0000 −0.437595
\(753\) 12.0000 0.437304
\(754\) −12.0000 −0.437014
\(755\) −8.00000 −0.291150
\(756\) −4.00000 −0.145479
\(757\) 20.0000 0.726912 0.363456 − 0.931611i \(-0.381597\pi\)
0.363456 + 0.931611i \(0.381597\pi\)
\(758\) 20.0000 0.726433
\(759\) −6.00000 −0.217786
\(760\) −2.00000 −0.0725476
\(761\) −30.0000 −1.08750 −0.543750 − 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) −16.0000 −0.579619
\(763\) −8.00000 −0.289619
\(764\) −24.0000 −0.868290
\(765\) 1.00000 0.0361551
\(766\) 12.0000 0.433578
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 2.00000 0.0721218 0.0360609 − 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) −4.00000 −0.144150
\(771\) 18.0000 0.648254
\(772\) −22.0000 −0.791797
\(773\) −36.0000 −1.29483 −0.647415 − 0.762138i \(-0.724150\pi\)
−0.647415 + 0.762138i \(0.724150\pi\)
\(774\) −4.00000 −0.143777
\(775\) −4.00000 −0.143684
\(776\) −16.0000 −0.574367
\(777\) 40.0000 1.43499
\(778\) 30.0000 1.07555
\(779\) 0 0
\(780\) −2.00000 −0.0716115
\(781\) 12.0000 0.429394
\(782\) −6.00000 −0.214560
\(783\) −6.00000 −0.214423
\(784\) 9.00000 0.321429
\(785\) −8.00000 −0.285532
\(786\) 0 0
\(787\) −4.00000 −0.142585 −0.0712923 − 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) −6.00000 −0.213741
\(789\) 0 0
\(790\) −2.00000 −0.0711568
\(791\) −48.0000 −1.70668
\(792\) −1.00000 −0.0355335
\(793\) −20.0000 −0.710221
\(794\) −34.0000 −1.20661
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) 12.0000 0.425062 0.212531 − 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) −8.00000 −0.283197
\(799\) 12.0000 0.424529
\(800\) 1.00000 0.0353553
\(801\) 6.00000 0.212000
\(802\) 18.0000 0.635602
\(803\) −2.00000 −0.0705785
\(804\) 14.0000 0.493742
\(805\) 24.0000 0.845889
\(806\) −8.00000 −0.281788
\(807\) 6.00000 0.211210
\(808\) 0 0
\(809\) −36.0000 −1.26569 −0.632846 − 0.774277i \(-0.718114\pi\)
−0.632846 + 0.774277i \(0.718114\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −28.0000 −0.983213 −0.491606 − 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 24.0000 0.842235
\(813\) 20.0000 0.701431
\(814\) 10.0000 0.350500
\(815\) 4.00000 0.140114
\(816\) −1.00000 −0.0350070
\(817\) −8.00000 −0.279885
\(818\) −10.0000 −0.349642
\(819\) −8.00000 −0.279543
\(820\) 0 0
\(821\) 30.0000 1.04701 0.523504 − 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) −6.00000 −0.209274
\(823\) −46.0000 −1.60346 −0.801730 − 0.597687i \(-0.796087\pi\)
−0.801730 + 0.597687i \(0.796087\pi\)
\(824\) −4.00000 −0.139347
\(825\) −1.00000 −0.0348155
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 − 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 6.00000 0.208514
\(829\) 2.00000 0.0694629 0.0347314 − 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) −22.0000 −0.763172
\(832\) 2.00000 0.0693375
\(833\) −9.00000 −0.311832
\(834\) −16.0000 −0.554035
\(835\) −12.0000 −0.415277
\(836\) −2.00000 −0.0691714
\(837\) −4.00000 −0.138260
\(838\) 36.0000 1.24360
\(839\) 24.0000 0.828572 0.414286 − 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 4.00000 0.138013
\(841\) 7.00000 0.241379
\(842\) 38.0000 1.30957
\(843\) 6.00000 0.206651
\(844\) 20.0000 0.688428
\(845\) 9.00000 0.309609
\(846\) −12.0000 −0.412568
\(847\) −4.00000 −0.137442
\(848\) 0 0
\(849\) 20.0000 0.686398
\(850\) −1.00000 −0.0342997
\(851\) −60.0000 −2.05677
\(852\) −12.0000 −0.411113
\(853\) −10.0000 −0.342393 −0.171197 − 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) 40.0000 1.36877
\(855\) −2.00000 −0.0683986
\(856\) −12.0000 −0.410152
\(857\) 6.00000 0.204956 0.102478 − 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) −2.00000 −0.0682789
\(859\) −40.0000 −1.36478 −0.682391 − 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) −6.00000 −0.204361
\(863\) −24.0000 −0.816970 −0.408485 − 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 1.00000 0.0340207
\(865\) −6.00000 −0.204006
\(866\) 14.0000 0.475739
\(867\) 1.00000 0.0339618
\(868\) 16.0000 0.543075
\(869\) −2.00000 −0.0678454
\(870\) 6.00000 0.203419
\(871\) 28.0000 0.948744
\(872\) 2.00000 0.0677285
\(873\) −16.0000 −0.541518
\(874\) 12.0000 0.405906
\(875\) 4.00000 0.135225
\(876\) 2.00000 0.0675737
\(877\) −34.0000 −1.14810 −0.574049 − 0.818821i \(-0.694628\pi\)
−0.574049 + 0.818821i \(0.694628\pi\)
\(878\) 26.0000 0.877457
\(879\) −18.0000 −0.607125
\(880\) 1.00000 0.0337100
\(881\) 30.0000 1.01073 0.505363 − 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 9.00000 0.303046
\(883\) −34.0000 −1.14419 −0.572096 − 0.820187i \(-0.693869\pi\)
−0.572096 + 0.820187i \(0.693869\pi\)
\(884\) −2.00000 −0.0672673
\(885\) 0 0
\(886\) 6.00000 0.201574
\(887\) −48.0000 −1.61168 −0.805841 − 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) −10.0000 −0.335578
\(889\) 64.0000 2.14649
\(890\) −6.00000 −0.201120
\(891\) −1.00000 −0.0335013
\(892\) −16.0000 −0.535720
\(893\) −24.0000 −0.803129
\(894\) −12.0000 −0.401340
\(895\) 0 0
\(896\) −4.00000 −0.133631
\(897\) 12.0000 0.400668
\(898\) 30.0000 1.00111
\(899\) 24.0000 0.800445
\(900\) 1.00000 0.0333333
\(901\) 0 0
\(902\) 0 0
\(903\) 16.0000 0.532447
\(904\) 12.0000 0.399114
\(905\) 10.0000 0.332411
\(906\) 8.00000 0.265782
\(907\) 8.00000 0.265636 0.132818 − 0.991140i \(-0.457597\pi\)
0.132818 + 0.991140i \(0.457597\pi\)
\(908\) −12.0000 −0.398234
\(909\) 0 0
\(910\) 8.00000 0.265197
\(911\) −12.0000 −0.397578 −0.198789 − 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 2.00000 0.0662266
\(913\) 0 0
\(914\) −10.0000 −0.330771
\(915\) 10.0000 0.330590
\(916\) 2.00000 0.0660819
\(917\) 0 0
\(918\) −1.00000 −0.0330049
\(919\) −16.0000 −0.527791 −0.263896 − 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) −6.00000 −0.197814
\(921\) −16.0000 −0.527218
\(922\) 24.0000 0.790398
\(923\) −24.0000 −0.789970
\(924\) 4.00000 0.131590
\(925\) −10.0000 −0.328798
\(926\) 8.00000 0.262896
\(927\) −4.00000 −0.131377
\(928\) −6.00000 −0.196960
\(929\) 42.0000 1.37798 0.688988 − 0.724773i \(-0.258055\pi\)
0.688988 + 0.724773i \(0.258055\pi\)
\(930\) 4.00000 0.131165
\(931\) 18.0000 0.589926
\(932\) −6.00000 −0.196537
\(933\) 24.0000 0.785725
\(934\) 18.0000 0.588978
\(935\) −1.00000 −0.0327035
\(936\) 2.00000 0.0653720
\(937\) 50.0000 1.63343 0.816714 − 0.577042i \(-0.195793\pi\)
0.816714 + 0.577042i \(0.195793\pi\)
\(938\) −56.0000 −1.82846
\(939\) 8.00000 0.261070
\(940\) 12.0000 0.391397
\(941\) −42.0000 −1.36916 −0.684580 − 0.728937i \(-0.740015\pi\)
−0.684580 + 0.728937i \(0.740015\pi\)
\(942\) 8.00000 0.260654
\(943\) 0 0
\(944\) 0 0
\(945\) 4.00000 0.130120
\(946\) 4.00000 0.130051
\(947\) 12.0000 0.389948 0.194974 − 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 2.00000 0.0649570
\(949\) 4.00000 0.129845
\(950\) 2.00000 0.0648886
\(951\) 6.00000 0.194563
\(952\) 4.00000 0.129641
\(953\) −54.0000 −1.74923 −0.874616 − 0.484817i \(-0.838886\pi\)
−0.874616 + 0.484817i \(0.838886\pi\)
\(954\) 0 0
\(955\) 24.0000 0.776622
\(956\) −12.0000 −0.388108
\(957\) 6.00000 0.193952
\(958\) 18.0000 0.581554
\(959\) 24.0000 0.775000
\(960\) −1.00000 −0.0322749
\(961\) −15.0000 −0.483871
\(962\) −20.0000 −0.644826
\(963\) −12.0000 −0.386695
\(964\) −4.00000 −0.128831
\(965\) 22.0000 0.708205
\(966\) −24.0000 −0.772187
\(967\) 32.0000 1.02905 0.514525 − 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 1.00000 0.0321412
\(969\) −2.00000 −0.0642493
\(970\) 16.0000 0.513729
\(971\) 36.0000 1.15529 0.577647 − 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 1.00000 0.0320750
\(973\) 64.0000 2.05175
\(974\) 26.0000 0.833094
\(975\) 2.00000 0.0640513
\(976\) −10.0000 −0.320092
\(977\) 42.0000 1.34370 0.671850 − 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) −4.00000 −0.127906
\(979\) −6.00000 −0.191761
\(980\) −9.00000 −0.287494
\(981\) 2.00000 0.0638551
\(982\) −30.0000 −0.957338
\(983\) 18.0000 0.574111 0.287055 − 0.957914i \(-0.407324\pi\)
0.287055 + 0.957914i \(0.407324\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 6.00000 0.191079
\(987\) 48.0000 1.52786
\(988\) 4.00000 0.127257
\(989\) −24.0000 −0.763156
\(990\) 1.00000 0.0317821
\(991\) −16.0000 −0.508257 −0.254128 − 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) −4.00000 −0.127000
\(993\) 8.00000 0.253872
\(994\) 48.0000 1.52247
\(995\) −20.0000 −0.634043
\(996\) 0 0
\(997\) −46.0000 −1.45683 −0.728417 − 0.685134i \(-0.759744\pi\)
−0.728417 + 0.685134i \(0.759744\pi\)
\(998\) −4.00000 −0.126618
\(999\) −10.0000 −0.316386
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 5610.2.a.be.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5610.2.a.be.1.1 ✓ 1 1.1 even 1 trivial