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Properties

Label 5610.2.a.u.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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{11} +1.00000 q^{12} -4.00000 q^{13} -2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} -8.00000 q^{19} +1.00000 q^{20} +2.00000 q^{21} -1.00000 q^{22} -8.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +4.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} -2.00000 q^{29} -1.00000 q^{30} -1.00000 q^{32} +1.00000 q^{33} -1.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} +8.00000 q^{38} -4.00000 q^{39} -1.00000 q^{40} +8.00000 q^{41} -2.00000 q^{42} +6.00000 q^{43} +1.00000 q^{44} +1.00000 q^{45} +8.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} +1.00000 q^{51} -4.00000 q^{52} +2.00000 q^{53} -1.00000 q^{54} +1.00000 q^{55} -2.00000 q^{56} -8.00000 q^{57} +2.00000 q^{58} -14.0000 q^{59} +1.00000 q^{60} -14.0000 q^{61} +2.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} -1.00000 q^{66} -14.0000 q^{67} +1.00000 q^{68} -8.00000 q^{69} -2.00000 q^{70} -14.0000 q^{71} -1.00000 q^{72} -2.00000 q^{74} +1.00000 q^{75} -8.00000 q^{76} +2.00000 q^{77} +4.00000 q^{78} +16.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -8.00000 q^{82} +2.00000 q^{84} +1.00000 q^{85} -6.00000 q^{86} -2.00000 q^{87} -1.00000 q^{88} -10.0000 q^{89} -1.00000 q^{90} -8.00000 q^{91} -8.00000 q^{92} +8.00000 q^{94} -8.00000 q^{95} -1.00000 q^{96} -8.00000 q^{97} +3.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511
\(12\) 1.00000 0.288675
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −2.00000 −0.534522
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536
\(18\) −1.00000 −0.235702
\(19\) −8.00000 −1.83533 −0.917663 − 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 1.00000 0.223607
\(21\) 2.00000 0.436436
\(22\) −1.00000 −0.213201
\(23\) −8.00000 −1.66812 −0.834058 − 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 4.00000 0.784465
\(27\) 1.00000 0.192450
\(28\) 2.00000 0.377964
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) −1.00000 −0.182574
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.00000 0.174078
\(34\) −1.00000 −0.171499
\(35\) 2.00000 0.338062
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 8.00000 1.29777
\(39\) −4.00000 −0.640513
\(40\) −1.00000 −0.158114
\(41\) 8.00000 1.24939 0.624695 − 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) −2.00000 −0.308607
\(43\) 6.00000 0.914991 0.457496 − 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 1.00000 0.150756
\(45\) 1.00000 0.149071
\(46\) 8.00000 1.17954
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) 1.00000 0.140028
\(52\) −4.00000 −0.554700
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) −1.00000 −0.136083
\(55\) 1.00000 0.134840
\(56\) −2.00000 −0.267261
\(57\) −8.00000 −1.05963
\(58\) 2.00000 0.262613
\(59\) −14.0000 −1.82264 −0.911322 − 0.411693i \(-0.864937\pi\)
−0.911322 + 0.411693i \(0.864937\pi\)
\(60\) 1.00000 0.129099
\(61\) −14.0000 −1.79252 −0.896258 − 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) −1.00000 −0.123091
\(67\) −14.0000 −1.71037 −0.855186 − 0.518321i \(-0.826557\pi\)
−0.855186 + 0.518321i \(0.826557\pi\)
\(68\) 1.00000 0.121268
\(69\) −8.00000 −0.963087
\(70\) −2.00000 −0.239046
\(71\) −14.0000 −1.66149 −0.830747 − 0.556650i \(-0.812086\pi\)
−0.830747 + 0.556650i \(0.812086\pi\)
\(72\) −1.00000 −0.117851
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) −2.00000 −0.232495
\(75\) 1.00000 0.115470
\(76\) −8.00000 −0.917663
\(77\) 2.00000 0.227921
\(78\) 4.00000 0.452911
\(79\) 16.0000 1.80014 0.900070 − 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −8.00000 −0.883452
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 2.00000 0.218218
\(85\) 1.00000 0.108465
\(86\) −6.00000 −0.646997
\(87\) −2.00000 −0.214423
\(88\) −1.00000 −0.106600
\(89\) −10.0000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) −1.00000 −0.105409
\(91\) −8.00000 −0.838628
\(92\) −8.00000 −0.834058
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) −8.00000 −0.820783
\(96\) −1.00000 −0.102062
\(97\) −8.00000 −0.812277 −0.406138 − 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 3.00000 0.303046
\(99\) 1.00000 0.100504
\(100\) 1.00000 0.100000
\(101\) 12.0000 1.19404 0.597022 − 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) −1.00000 −0.0990148
\(103\) 4.00000 0.394132 0.197066 − 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 4.00000 0.392232
\(105\) 2.00000 0.195180
\(106\) −2.00000 −0.194257
\(107\) 4.00000 0.386695 0.193347 − 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 1.00000 0.0962250
\(109\) −18.0000 −1.72409 −0.862044 − 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 2.00000 0.189832
\(112\) 2.00000 0.188982
\(113\) 18.0000 1.69330 0.846649 − 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 8.00000 0.749269
\(115\) −8.00000 −0.746004
\(116\) −2.00000 −0.185695
\(117\) −4.00000 −0.369800
\(118\) 14.0000 1.28880
\(119\) 2.00000 0.183340
\(120\) −1.00000 −0.0912871
\(121\) 1.00000 0.0909091
\(122\) 14.0000 1.26750
\(123\) 8.00000 0.721336
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) −2.00000 −0.178174
\(127\) −12.0000 −1.06483 −0.532414 − 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 6.00000 0.528271
\(130\) 4.00000 0.350823
\(131\) 8.00000 0.698963 0.349482 − 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 1.00000 0.0870388
\(133\) −16.0000 −1.38738
\(134\) 14.0000 1.20942
\(135\) 1.00000 0.0860663
\(136\) −1.00000 −0.0857493
\(137\) 10.0000 0.854358 0.427179 − 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 8.00000 0.681005
\(139\) −20.0000 −1.69638 −0.848189 − 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 2.00000 0.169031
\(141\) −8.00000 −0.673722
\(142\) 14.0000 1.17485
\(143\) −4.00000 −0.334497
\(144\) 1.00000 0.0833333
\(145\) −2.00000 −0.166091
\(146\) 0 0
\(147\) −3.00000 −0.247436
\(148\) 2.00000 0.164399
\(149\) 20.0000 1.63846 0.819232 − 0.573462i \(-0.194400\pi\)
0.819232 + 0.573462i \(0.194400\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 16.0000 1.30206 0.651031 − 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 8.00000 0.648886
\(153\) 1.00000 0.0808452
\(154\) −2.00000 −0.161165
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) −16.0000 −1.27289
\(159\) 2.00000 0.158610
\(160\) −1.00000 −0.0790569
\(161\) −16.0000 −1.26098
\(162\) −1.00000 −0.0785674
\(163\) −24.0000 −1.87983 −0.939913 − 0.341415i \(-0.889094\pi\)
−0.939913 + 0.341415i \(0.889094\pi\)
\(164\) 8.00000 0.624695
\(165\) 1.00000 0.0778499
\(166\) 0 0
\(167\) 8.00000 0.619059 0.309529 − 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) −2.00000 −0.154303
\(169\) 3.00000 0.230769
\(170\) −1.00000 −0.0766965
\(171\) −8.00000 −0.611775
\(172\) 6.00000 0.457496
\(173\) 18.0000 1.36851 0.684257 − 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 2.00000 0.151620
\(175\) 2.00000 0.151186
\(176\) 1.00000 0.0753778
\(177\) −14.0000 −1.05230
\(178\) 10.0000 0.749532
\(179\) −10.0000 −0.747435 −0.373718 − 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) 1.00000 0.0745356
\(181\) 14.0000 1.04061 0.520306 − 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 8.00000 0.592999
\(183\) −14.0000 −1.03491
\(184\) 8.00000 0.589768
\(185\) 2.00000 0.147043
\(186\) 0 0
\(187\) 1.00000 0.0731272
\(188\) −8.00000 −0.583460
\(189\) 2.00000 0.145479
\(190\) 8.00000 0.580381
\(191\) −12.0000 −0.868290 −0.434145 − 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 1.00000 0.0721688
\(193\) −12.0000 −0.863779 −0.431889 − 0.901927i \(-0.642153\pi\)
−0.431889 + 0.901927i \(0.642153\pi\)
\(194\) 8.00000 0.574367
\(195\) −4.00000 −0.286446
\(196\) −3.00000 −0.214286
\(197\) −6.00000 −0.427482 −0.213741 − 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 16.0000 1.13421 0.567105 − 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −14.0000 −0.987484
\(202\) −12.0000 −0.844317
\(203\) −4.00000 −0.280745
\(204\) 1.00000 0.0700140
\(205\) 8.00000 0.558744
\(206\) −4.00000 −0.278693
\(207\) −8.00000 −0.556038
\(208\) −4.00000 −0.277350
\(209\) −8.00000 −0.553372
\(210\) −2.00000 −0.138013
\(211\) −4.00000 −0.275371 −0.137686 − 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 2.00000 0.137361
\(213\) −14.0000 −0.959264
\(214\) −4.00000 −0.273434
\(215\) 6.00000 0.409197
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 18.0000 1.21911
\(219\) 0 0
\(220\) 1.00000 0.0674200
\(221\) −4.00000 −0.269069
\(222\) −2.00000 −0.134231
\(223\) −4.00000 −0.267860 −0.133930 − 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) −2.00000 −0.133631
\(225\) 1.00000 0.0666667
\(226\) −18.0000 −1.19734
\(227\) 20.0000 1.32745 0.663723 − 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) −8.00000 −0.529813
\(229\) −18.0000 −1.18947 −0.594737 − 0.803921i \(-0.702744\pi\)
−0.594737 + 0.803921i \(0.702744\pi\)
\(230\) 8.00000 0.527504
\(231\) 2.00000 0.131590
\(232\) 2.00000 0.131306
\(233\) −14.0000 −0.917170 −0.458585 − 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) 4.00000 0.261488
\(235\) −8.00000 −0.521862
\(236\) −14.0000 −0.911322
\(237\) 16.0000 1.03931
\(238\) −2.00000 −0.129641
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 1.00000 0.0645497
\(241\) −2.00000 −0.128831 −0.0644157 − 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 1.00000 0.0641500
\(244\) −14.0000 −0.896258
\(245\) −3.00000 −0.191663
\(246\) −8.00000 −0.510061
\(247\) 32.0000 2.03611
\(248\) 0 0
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 18.0000 1.13615 0.568075 − 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 2.00000 0.125988
\(253\) −8.00000 −0.502956
\(254\) 12.0000 0.752947
\(255\) 1.00000 0.0626224
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 − 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) −6.00000 −0.373544
\(259\) 4.00000 0.248548
\(260\) −4.00000 −0.248069
\(261\) −2.00000 −0.123797
\(262\) −8.00000 −0.494242
\(263\) 24.0000 1.47990 0.739952 − 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) −1.00000 −0.0615457
\(265\) 2.00000 0.122859
\(266\) 16.0000 0.981023
\(267\) −10.0000 −0.611990
\(268\) −14.0000 −0.855186
\(269\) −10.0000 −0.609711 −0.304855 − 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 8.00000 0.485965 0.242983 − 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 1.00000 0.0606339
\(273\) −8.00000 −0.484182
\(274\) −10.0000 −0.604122
\(275\) 1.00000 0.0603023
\(276\) −8.00000 −0.481543
\(277\) 22.0000 1.32185 0.660926 − 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 20.0000 1.19952
\(279\) 0 0
\(280\) −2.00000 −0.119523
\(281\) 6.00000 0.357930 0.178965 − 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 8.00000 0.476393
\(283\) 4.00000 0.237775 0.118888 − 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −14.0000 −0.830747
\(285\) −8.00000 −0.473879
\(286\) 4.00000 0.236525
\(287\) 16.0000 0.944450
\(288\) −1.00000 −0.0589256
\(289\) 1.00000 0.0588235
\(290\) 2.00000 0.117444
\(291\) −8.00000 −0.468968
\(292\) 0 0
\(293\) 18.0000 1.05157 0.525786 − 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 3.00000 0.174964
\(295\) −14.0000 −0.815112
\(296\) −2.00000 −0.116248
\(297\) 1.00000 0.0580259
\(298\) −20.0000 −1.15857
\(299\) 32.0000 1.85061
\(300\) 1.00000 0.0577350
\(301\) 12.0000 0.691669
\(302\) −16.0000 −0.920697
\(303\) 12.0000 0.689382
\(304\) −8.00000 −0.458831
\(305\) −14.0000 −0.801638
\(306\) −1.00000 −0.0571662
\(307\) −2.00000 −0.114146 −0.0570730 − 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) 2.00000 0.113961
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) 6.00000 0.340229 0.170114 − 0.985424i \(-0.445586\pi\)
0.170114 + 0.985424i \(0.445586\pi\)
\(312\) 4.00000 0.226455
\(313\) −24.0000 −1.35656 −0.678280 − 0.734803i \(-0.737274\pi\)
−0.678280 + 0.734803i \(0.737274\pi\)
\(314\) 0 0
\(315\) 2.00000 0.112687
\(316\) 16.0000 0.900070
\(317\) 2.00000 0.112331 0.0561656 − 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) −2.00000 −0.112154
\(319\) −2.00000 −0.111979
\(320\) 1.00000 0.0559017
\(321\) 4.00000 0.223258
\(322\) 16.0000 0.891645
\(323\) −8.00000 −0.445132
\(324\) 1.00000 0.0555556
\(325\) −4.00000 −0.221880
\(326\) 24.0000 1.32924
\(327\) −18.0000 −0.995402
\(328\) −8.00000 −0.441726
\(329\) −16.0000 −0.882109
\(330\) −1.00000 −0.0550482
\(331\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) 2.00000 0.109599
\(334\) −8.00000 −0.437741
\(335\) −14.0000 −0.764902
\(336\) 2.00000 0.109109
\(337\) −32.0000 −1.74315 −0.871576 − 0.490261i \(-0.836901\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) −3.00000 −0.163178
\(339\) 18.0000 0.977626
\(340\) 1.00000 0.0542326
\(341\) 0 0
\(342\) 8.00000 0.432590
\(343\) −20.0000 −1.07990
\(344\) −6.00000 −0.323498
\(345\) −8.00000 −0.430706
\(346\) −18.0000 −0.967686
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −2.00000 −0.107211
\(349\) −26.0000 −1.39175 −0.695874 − 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) −2.00000 −0.106904
\(351\) −4.00000 −0.213504
\(352\) −1.00000 −0.0533002
\(353\) 14.0000 0.745145 0.372572 − 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 14.0000 0.744092
\(355\) −14.0000 −0.743043
\(356\) −10.0000 −0.529999
\(357\) 2.00000 0.105851
\(358\) 10.0000 0.528516
\(359\) −20.0000 −1.05556 −0.527780 − 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 45.0000 2.36842
\(362\) −14.0000 −0.735824
\(363\) 1.00000 0.0524864
\(364\) −8.00000 −0.419314
\(365\) 0 0
\(366\) 14.0000 0.731792
\(367\) 10.0000 0.521996 0.260998 − 0.965339i \(-0.415948\pi\)
0.260998 + 0.965339i \(0.415948\pi\)
\(368\) −8.00000 −0.417029
\(369\) 8.00000 0.416463
\(370\) −2.00000 −0.103975
\(371\) 4.00000 0.207670
\(372\) 0 0
\(373\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) −1.00000 −0.0517088
\(375\) 1.00000 0.0516398
\(376\) 8.00000 0.412568
\(377\) 8.00000 0.412021
\(378\) −2.00000 −0.102869
\(379\) −20.0000 −1.02733 −0.513665 − 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) −8.00000 −0.410391
\(381\) −12.0000 −0.614779
\(382\) 12.0000 0.613973
\(383\) −16.0000 −0.817562 −0.408781 − 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 2.00000 0.101929
\(386\) 12.0000 0.610784
\(387\) 6.00000 0.304997
\(388\) −8.00000 −0.406138
\(389\) −24.0000 −1.21685 −0.608424 − 0.793612i \(-0.708198\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) 4.00000 0.202548
\(391\) −8.00000 −0.404577
\(392\) 3.00000 0.151523
\(393\) 8.00000 0.403547
\(394\) 6.00000 0.302276
\(395\) 16.0000 0.805047
\(396\) 1.00000 0.0502519
\(397\) 34.0000 1.70641 0.853206 − 0.521575i \(-0.174655\pi\)
0.853206 + 0.521575i \(0.174655\pi\)
\(398\) −16.0000 −0.802008
\(399\) −16.0000 −0.801002
\(400\) 1.00000 0.0500000
\(401\) −32.0000 −1.59800 −0.799002 − 0.601329i \(-0.794638\pi\)
−0.799002 + 0.601329i \(0.794638\pi\)
\(402\) 14.0000 0.698257
\(403\) 0 0
\(404\) 12.0000 0.597022
\(405\) 1.00000 0.0496904
\(406\) 4.00000 0.198517
\(407\) 2.00000 0.0991363
\(408\) −1.00000 −0.0495074
\(409\) −26.0000 −1.28562 −0.642809 − 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) −8.00000 −0.395092
\(411\) 10.0000 0.493264
\(412\) 4.00000 0.197066
\(413\) −28.0000 −1.37779
\(414\) 8.00000 0.393179
\(415\) 0 0
\(416\) 4.00000 0.196116
\(417\) −20.0000 −0.979404
\(418\) 8.00000 0.391293
\(419\) −16.0000 −0.781651 −0.390826 − 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 2.00000 0.0975900
\(421\) −26.0000 −1.26716 −0.633581 − 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 4.00000 0.194717
\(423\) −8.00000 −0.388973
\(424\) −2.00000 −0.0971286
\(425\) 1.00000 0.0485071
\(426\) 14.0000 0.678302
\(427\) −28.0000 −1.35501
\(428\) 4.00000 0.193347
\(429\) −4.00000 −0.193122
\(430\) −6.00000 −0.289346
\(431\) 10.0000 0.481683 0.240842 − 0.970564i \(-0.422577\pi\)
0.240842 + 0.970564i \(0.422577\pi\)
\(432\) 1.00000 0.0481125
\(433\) 30.0000 1.44171 0.720854 − 0.693087i \(-0.243750\pi\)
0.720854 + 0.693087i \(0.243750\pi\)
\(434\) 0 0
\(435\) −2.00000 −0.0958927
\(436\) −18.0000 −0.862044
\(437\) 64.0000 3.06154
\(438\) 0 0
\(439\) 16.0000 0.763638 0.381819 − 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) −1.00000 −0.0476731
\(441\) −3.00000 −0.142857
\(442\) 4.00000 0.190261
\(443\) −12.0000 −0.570137 −0.285069 − 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 2.00000 0.0949158
\(445\) −10.0000 −0.474045
\(446\) 4.00000 0.189405
\(447\) 20.0000 0.945968
\(448\) 2.00000 0.0944911
\(449\) 40.0000 1.88772 0.943858 − 0.330350i \(-0.107167\pi\)
0.943858 + 0.330350i \(0.107167\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 8.00000 0.376705
\(452\) 18.0000 0.846649
\(453\) 16.0000 0.751746
\(454\) −20.0000 −0.938647
\(455\) −8.00000 −0.375046
\(456\) 8.00000 0.374634
\(457\) −2.00000 −0.0935561 −0.0467780 − 0.998905i \(-0.514895\pi\)
−0.0467780 + 0.998905i \(0.514895\pi\)
\(458\) 18.0000 0.841085
\(459\) 1.00000 0.0466760
\(460\) −8.00000 −0.373002
\(461\) −24.0000 −1.11779 −0.558896 − 0.829238i \(-0.688775\pi\)
−0.558896 + 0.829238i \(0.688775\pi\)
\(462\) −2.00000 −0.0930484
\(463\) 4.00000 0.185896 0.0929479 − 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) 14.0000 0.648537
\(467\) 8.00000 0.370196 0.185098 − 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) −4.00000 −0.184900
\(469\) −28.0000 −1.29292
\(470\) 8.00000 0.369012
\(471\) 0 0
\(472\) 14.0000 0.644402
\(473\) 6.00000 0.275880
\(474\) −16.0000 −0.734904
\(475\) −8.00000 −0.367065
\(476\) 2.00000 0.0916698
\(477\) 2.00000 0.0915737
\(478\) 0 0
\(479\) 6.00000 0.274147 0.137073 − 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −8.00000 −0.364769
\(482\) 2.00000 0.0910975
\(483\) −16.0000 −0.728025
\(484\) 1.00000 0.0454545
\(485\) −8.00000 −0.363261
\(486\) −1.00000 −0.0453609
\(487\) −2.00000 −0.0906287 −0.0453143 − 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 14.0000 0.633750
\(489\) −24.0000 −1.08532
\(490\) 3.00000 0.135526
\(491\) −6.00000 −0.270776 −0.135388 − 0.990793i \(-0.543228\pi\)
−0.135388 + 0.990793i \(0.543228\pi\)
\(492\) 8.00000 0.360668
\(493\) −2.00000 −0.0900755
\(494\) −32.0000 −1.43975
\(495\) 1.00000 0.0449467
\(496\) 0 0
\(497\) −28.0000 −1.25597
\(498\) 0 0
\(499\) 20.0000 0.895323 0.447661 − 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 1.00000 0.0447214
\(501\) 8.00000 0.357414
\(502\) −18.0000 −0.803379
\(503\) −20.0000 −0.891756 −0.445878 − 0.895094i \(-0.647108\pi\)
−0.445878 + 0.895094i \(0.647108\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 12.0000 0.533993
\(506\) 8.00000 0.355643
\(507\) 3.00000 0.133235
\(508\) −12.0000 −0.532414
\(509\) 24.0000 1.06378 0.531891 − 0.846813i \(-0.321482\pi\)
0.531891 + 0.846813i \(0.321482\pi\)
\(510\) −1.00000 −0.0442807
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −8.00000 −0.353209
\(514\) −6.00000 −0.264649
\(515\) 4.00000 0.176261
\(516\) 6.00000 0.264135
\(517\) −8.00000 −0.351840
\(518\) −4.00000 −0.175750
\(519\) 18.0000 0.790112
\(520\) 4.00000 0.175412
\(521\) −8.00000 −0.350486 −0.175243 − 0.984525i \(-0.556071\pi\)
−0.175243 + 0.984525i \(0.556071\pi\)
\(522\) 2.00000 0.0875376
\(523\) 10.0000 0.437269 0.218635 − 0.975807i \(-0.429840\pi\)
0.218635 + 0.975807i \(0.429840\pi\)
\(524\) 8.00000 0.349482
\(525\) 2.00000 0.0872872
\(526\) −24.0000 −1.04645
\(527\) 0 0
\(528\) 1.00000 0.0435194
\(529\) 41.0000 1.78261
\(530\) −2.00000 −0.0868744
\(531\) −14.0000 −0.607548
\(532\) −16.0000 −0.693688
\(533\) −32.0000 −1.38607
\(534\) 10.0000 0.432742
\(535\) 4.00000 0.172935
\(536\) 14.0000 0.604708
\(537\) −10.0000 −0.431532
\(538\) 10.0000 0.431131
\(539\) −3.00000 −0.129219
\(540\) 1.00000 0.0430331
\(541\) −46.0000 −1.97769 −0.988847 − 0.148933i \(-0.952416\pi\)
−0.988847 + 0.148933i \(0.952416\pi\)
\(542\) −8.00000 −0.343629
\(543\) 14.0000 0.600798
\(544\) −1.00000 −0.0428746
\(545\) −18.0000 −0.771035
\(546\) 8.00000 0.342368
\(547\) −12.0000 −0.513083 −0.256541 − 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 10.0000 0.427179
\(549\) −14.0000 −0.597505
\(550\) −1.00000 −0.0426401
\(551\) 16.0000 0.681623
\(552\) 8.00000 0.340503
\(553\) 32.0000 1.36078
\(554\) −22.0000 −0.934690
\(555\) 2.00000 0.0848953
\(556\) −20.0000 −0.848189
\(557\) −18.0000 −0.762684 −0.381342 − 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) 2.00000 0.0845154
\(561\) 1.00000 0.0422200
\(562\) −6.00000 −0.253095
\(563\) −24.0000 −1.01148 −0.505740 − 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) −8.00000 −0.336861
\(565\) 18.0000 0.757266
\(566\) −4.00000 −0.168133
\(567\) 2.00000 0.0839921
\(568\) 14.0000 0.587427
\(569\) 10.0000 0.419222 0.209611 − 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 8.00000 0.335083
\(571\) −12.0000 −0.502184 −0.251092 − 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) −4.00000 −0.167248
\(573\) −12.0000 −0.501307
\(574\) −16.0000 −0.667827
\(575\) −8.00000 −0.333623
\(576\) 1.00000 0.0416667
\(577\) −18.0000 −0.749350 −0.374675 − 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −12.0000 −0.498703
\(580\) −2.00000 −0.0830455
\(581\) 0 0
\(582\) 8.00000 0.331611
\(583\) 2.00000 0.0828315
\(584\) 0 0
\(585\) −4.00000 −0.165380
\(586\) −18.0000 −0.743573
\(587\) 4.00000 0.165098 0.0825488 − 0.996587i \(-0.473694\pi\)
0.0825488 + 0.996587i \(0.473694\pi\)
\(588\) −3.00000 −0.123718
\(589\) 0 0
\(590\) 14.0000 0.576371
\(591\) −6.00000 −0.246807
\(592\) 2.00000 0.0821995
\(593\) 30.0000 1.23195 0.615976 − 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 2.00000 0.0819920
\(596\) 20.0000 0.819232
\(597\) 16.0000 0.654836
\(598\) −32.0000 −1.30858
\(599\) 12.0000 0.490307 0.245153 − 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 10.0000 0.407909 0.203954 − 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) −12.0000 −0.489083
\(603\) −14.0000 −0.570124
\(604\) 16.0000 0.651031
\(605\) 1.00000 0.0406558
\(606\) −12.0000 −0.487467
\(607\) 10.0000 0.405887 0.202944 − 0.979190i \(-0.434949\pi\)
0.202944 + 0.979190i \(0.434949\pi\)
\(608\) 8.00000 0.324443
\(609\) −4.00000 −0.162088
\(610\) 14.0000 0.566843
\(611\) 32.0000 1.29458
\(612\) 1.00000 0.0404226
\(613\) 40.0000 1.61558 0.807792 − 0.589467i \(-0.200662\pi\)
0.807792 + 0.589467i \(0.200662\pi\)
\(614\) 2.00000 0.0807134
\(615\) 8.00000 0.322591
\(616\) −2.00000 −0.0805823
\(617\) 30.0000 1.20775 0.603877 − 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) −4.00000 −0.160904
\(619\) −28.0000 −1.12542 −0.562708 − 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) −6.00000 −0.240578
\(623\) −20.0000 −0.801283
\(624\) −4.00000 −0.160128
\(625\) 1.00000 0.0400000
\(626\) 24.0000 0.959233
\(627\) −8.00000 −0.319489
\(628\) 0 0
\(629\) 2.00000 0.0797452
\(630\) −2.00000 −0.0796819
\(631\) 32.0000 1.27390 0.636950 − 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) −16.0000 −0.636446
\(633\) −4.00000 −0.158986
\(634\) −2.00000 −0.0794301
\(635\) −12.0000 −0.476205
\(636\) 2.00000 0.0793052
\(637\) 12.0000 0.475457
\(638\) 2.00000 0.0791808
\(639\) −14.0000 −0.553831
\(640\) −1.00000 −0.0395285
\(641\) −40.0000 −1.57991 −0.789953 − 0.613168i \(-0.789895\pi\)
−0.789953 + 0.613168i \(0.789895\pi\)
\(642\) −4.00000 −0.157867
\(643\) 16.0000 0.630978 0.315489 − 0.948929i \(-0.397831\pi\)
0.315489 + 0.948929i \(0.397831\pi\)
\(644\) −16.0000 −0.630488
\(645\) 6.00000 0.236250
\(646\) 8.00000 0.314756
\(647\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −14.0000 −0.549548
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) −24.0000 −0.939913
\(653\) −18.0000 −0.704394 −0.352197 − 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 18.0000 0.703856
\(655\) 8.00000 0.312586
\(656\) 8.00000 0.312348
\(657\) 0 0
\(658\) 16.0000 0.623745
\(659\) 18.0000 0.701180 0.350590 − 0.936529i \(-0.385981\pi\)
0.350590 + 0.936529i \(0.385981\pi\)
\(660\) 1.00000 0.0389249
\(661\) 10.0000 0.388955 0.194477 − 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) 0 0
\(663\) −4.00000 −0.155347
\(664\) 0 0
\(665\) −16.0000 −0.620453
\(666\) −2.00000 −0.0774984
\(667\) 16.0000 0.619522
\(668\) 8.00000 0.309529
\(669\) −4.00000 −0.154649
\(670\) 14.0000 0.540867
\(671\) −14.0000 −0.540464
\(672\) −2.00000 −0.0771517
\(673\) −28.0000 −1.07932 −0.539660 − 0.841883i \(-0.681447\pi\)
−0.539660 + 0.841883i \(0.681447\pi\)
\(674\) 32.0000 1.23259
\(675\) 1.00000 0.0384900
\(676\) 3.00000 0.115385
\(677\) −22.0000 −0.845529 −0.422764 − 0.906240i \(-0.638940\pi\)
−0.422764 + 0.906240i \(0.638940\pi\)
\(678\) −18.0000 −0.691286
\(679\) −16.0000 −0.614024
\(680\) −1.00000 −0.0383482
\(681\) 20.0000 0.766402
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 − 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) −8.00000 −0.305888
\(685\) 10.0000 0.382080
\(686\) 20.0000 0.763604
\(687\) −18.0000 −0.686743
\(688\) 6.00000 0.228748
\(689\) −8.00000 −0.304776
\(690\) 8.00000 0.304555
\(691\) −12.0000 −0.456502 −0.228251 − 0.973602i \(-0.573301\pi\)
−0.228251 + 0.973602i \(0.573301\pi\)
\(692\) 18.0000 0.684257
\(693\) 2.00000 0.0759737
\(694\) −12.0000 −0.455514
\(695\) −20.0000 −0.758643
\(696\) 2.00000 0.0758098
\(697\) 8.00000 0.303022
\(698\) 26.0000 0.984115
\(699\) −14.0000 −0.529529
\(700\) 2.00000 0.0755929
\(701\) −24.0000 −0.906467 −0.453234 − 0.891392i \(-0.649730\pi\)
−0.453234 + 0.891392i \(0.649730\pi\)
\(702\) 4.00000 0.150970
\(703\) −16.0000 −0.603451
\(704\) 1.00000 0.0376889
\(705\) −8.00000 −0.301297
\(706\) −14.0000 −0.526897
\(707\) 24.0000 0.902613
\(708\) −14.0000 −0.526152
\(709\) −46.0000 −1.72757 −0.863783 − 0.503864i \(-0.831911\pi\)
−0.863783 + 0.503864i \(0.831911\pi\)
\(710\) 14.0000 0.525411
\(711\) 16.0000 0.600047
\(712\) 10.0000 0.374766
\(713\) 0 0
\(714\) −2.00000 −0.0748481
\(715\) −4.00000 −0.149592
\(716\) −10.0000 −0.373718
\(717\) 0 0
\(718\) 20.0000 0.746393
\(719\) −30.0000 −1.11881 −0.559406 − 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 1.00000 0.0372678
\(721\) 8.00000 0.297936
\(722\) −45.0000 −1.67473
\(723\) −2.00000 −0.0743808
\(724\) 14.0000 0.520306
\(725\) −2.00000 −0.0742781
\(726\) −1.00000 −0.0371135
\(727\) 28.0000 1.03846 0.519231 − 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) 8.00000 0.296500
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 6.00000 0.221918
\(732\) −14.0000 −0.517455
\(733\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) −10.0000 −0.369107
\(735\) −3.00000 −0.110657
\(736\) 8.00000 0.294884
\(737\) −14.0000 −0.515697
\(738\) −8.00000 −0.294484
\(739\) −28.0000 −1.03000 −0.514998 − 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) 2.00000 0.0735215
\(741\) 32.0000 1.17555
\(742\) −4.00000 −0.146845
\(743\) −24.0000 −0.880475 −0.440237 − 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 20.0000 0.732743
\(746\) 0 0
\(747\) 0 0
\(748\) 1.00000 0.0365636
\(749\) 8.00000 0.292314
\(750\) −1.00000 −0.0365148
\(751\) −28.0000 −1.02173 −0.510867 − 0.859660i \(-0.670676\pi\)
−0.510867 + 0.859660i \(0.670676\pi\)
\(752\) −8.00000 −0.291730
\(753\) 18.0000 0.655956
\(754\) −8.00000 −0.291343
\(755\) 16.0000 0.582300
\(756\) 2.00000 0.0727393
\(757\) 48.0000 1.74459 0.872295 − 0.488980i \(-0.162631\pi\)
0.872295 + 0.488980i \(0.162631\pi\)
\(758\) 20.0000 0.726433
\(759\) −8.00000 −0.290382
\(760\) 8.00000 0.290191
\(761\) 34.0000 1.23250 0.616250 − 0.787551i \(-0.288651\pi\)
0.616250 + 0.787551i \(0.288651\pi\)
\(762\) 12.0000 0.434714
\(763\) −36.0000 −1.30329
\(764\) −12.0000 −0.434145
\(765\) 1.00000 0.0361551
\(766\) 16.0000 0.578103
\(767\) 56.0000 2.02204
\(768\) 1.00000 0.0360844
\(769\) −26.0000 −0.937584 −0.468792 − 0.883309i \(-0.655311\pi\)
−0.468792 + 0.883309i \(0.655311\pi\)
\(770\) −2.00000 −0.0720750
\(771\) 6.00000 0.216085
\(772\) −12.0000 −0.431889
\(773\) −6.00000 −0.215805 −0.107903 − 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) −6.00000 −0.215666
\(775\) 0 0
\(776\) 8.00000 0.287183
\(777\) 4.00000 0.143499
\(778\) 24.0000 0.860442
\(779\) −64.0000 −2.29304
\(780\) −4.00000 −0.143223
\(781\) −14.0000 −0.500959
\(782\) 8.00000 0.286079
\(783\) −2.00000 −0.0714742
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) −8.00000 −0.285351
\(787\) −32.0000 −1.14068 −0.570338 − 0.821410i \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(788\) −6.00000 −0.213741
\(789\) 24.0000 0.854423
\(790\) −16.0000 −0.569254
\(791\) 36.0000 1.28001
\(792\) −1.00000 −0.0355335
\(793\) 56.0000 1.98862
\(794\) −34.0000 −1.20661
\(795\) 2.00000 0.0709327
\(796\) 16.0000 0.567105
\(797\) 30.0000 1.06265 0.531327 − 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 16.0000 0.566394
\(799\) −8.00000 −0.283020
\(800\) −1.00000 −0.0353553
\(801\) −10.0000 −0.353333
\(802\) 32.0000 1.12996
\(803\) 0 0
\(804\) −14.0000 −0.493742
\(805\) −16.0000 −0.563926
\(806\) 0 0
\(807\) −10.0000 −0.352017
\(808\) −12.0000 −0.422159
\(809\) 40.0000 1.40633 0.703163 − 0.711029i \(-0.251771\pi\)
0.703163 + 0.711029i \(0.251771\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 52.0000 1.82597 0.912983 − 0.407997i \(-0.133772\pi\)
0.912983 + 0.407997i \(0.133772\pi\)
\(812\) −4.00000 −0.140372
\(813\) 8.00000 0.280572
\(814\) −2.00000 −0.0701000
\(815\) −24.0000 −0.840683
\(816\) 1.00000 0.0350070
\(817\) −48.0000 −1.67931
\(818\) 26.0000 0.909069
\(819\) −8.00000 −0.279543
\(820\) 8.00000 0.279372
\(821\) 10.0000 0.349002 0.174501 − 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) −10.0000 −0.348790
\(823\) 26.0000 0.906303 0.453152 − 0.891434i \(-0.350300\pi\)
0.453152 + 0.891434i \(0.350300\pi\)
\(824\) −4.00000 −0.139347
\(825\) 1.00000 0.0348155
\(826\) 28.0000 0.974245
\(827\) −44.0000 −1.53003 −0.765015 − 0.644013i \(-0.777268\pi\)
−0.765015 + 0.644013i \(0.777268\pi\)
\(828\) −8.00000 −0.278019
\(829\) −22.0000 −0.764092 −0.382046 − 0.924143i \(-0.624780\pi\)
−0.382046 + 0.924143i \(0.624780\pi\)
\(830\) 0 0
\(831\) 22.0000 0.763172
\(832\) −4.00000 −0.138675
\(833\) −3.00000 −0.103944
\(834\) 20.0000 0.692543
\(835\) 8.00000 0.276851
\(836\) −8.00000 −0.276686
\(837\) 0 0
\(838\) 16.0000 0.552711
\(839\) 6.00000 0.207143 0.103572 − 0.994622i \(-0.466973\pi\)
0.103572 + 0.994622i \(0.466973\pi\)
\(840\) −2.00000 −0.0690066
\(841\) −25.0000 −0.862069
\(842\) 26.0000 0.896019
\(843\) 6.00000 0.206651
\(844\) −4.00000 −0.137686
\(845\) 3.00000 0.103203
\(846\) 8.00000 0.275046
\(847\) 2.00000 0.0687208
\(848\) 2.00000 0.0686803
\(849\) 4.00000 0.137280
\(850\) −1.00000 −0.0342997
\(851\) −16.0000 −0.548473
\(852\) −14.0000 −0.479632
\(853\) 14.0000 0.479351 0.239675 − 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) 28.0000 0.958140
\(855\) −8.00000 −0.273594
\(856\) −4.00000 −0.136717
\(857\) −22.0000 −0.751506 −0.375753 − 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(858\) 4.00000 0.136558
\(859\) −56.0000 −1.91070 −0.955348 − 0.295484i \(-0.904519\pi\)
−0.955348 + 0.295484i \(0.904519\pi\)
\(860\) 6.00000 0.204598
\(861\) 16.0000 0.545279
\(862\) −10.0000 −0.340601
\(863\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 18.0000 0.612018
\(866\) −30.0000 −1.01944
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) 16.0000 0.542763
\(870\) 2.00000 0.0678064
\(871\) 56.0000 1.89749
\(872\) 18.0000 0.609557
\(873\) −8.00000 −0.270759
\(874\) −64.0000 −2.16483
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) 6.00000 0.202606 0.101303 − 0.994856i \(-0.467699\pi\)
0.101303 + 0.994856i \(0.467699\pi\)
\(878\) −16.0000 −0.539974
\(879\) 18.0000 0.607125
\(880\) 1.00000 0.0337100
\(881\) 32.0000 1.07811 0.539054 − 0.842271i \(-0.318782\pi\)
0.539054 + 0.842271i \(0.318782\pi\)
\(882\) 3.00000 0.101015
\(883\) 6.00000 0.201916 0.100958 − 0.994891i \(-0.467809\pi\)
0.100958 + 0.994891i \(0.467809\pi\)
\(884\) −4.00000 −0.134535
\(885\) −14.0000 −0.470605
\(886\) 12.0000 0.403148
\(887\) 28.0000 0.940148 0.470074 − 0.882627i \(-0.344227\pi\)
0.470074 + 0.882627i \(0.344227\pi\)
\(888\) −2.00000 −0.0671156
\(889\) −24.0000 −0.804934
\(890\) 10.0000 0.335201
\(891\) 1.00000 0.0335013
\(892\) −4.00000 −0.133930
\(893\) 64.0000 2.14168
\(894\) −20.0000 −0.668900
\(895\) −10.0000 −0.334263
\(896\) −2.00000 −0.0668153
\(897\) 32.0000 1.06845
\(898\) −40.0000 −1.33482
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) 2.00000 0.0666297
\(902\) −8.00000 −0.266371
\(903\) 12.0000 0.399335
\(904\) −18.0000 −0.598671
\(905\) 14.0000 0.465376
\(906\) −16.0000 −0.531564
\(907\) 32.0000 1.06254 0.531271 − 0.847202i \(-0.321714\pi\)
0.531271 + 0.847202i \(0.321714\pi\)
\(908\) 20.0000 0.663723
\(909\) 12.0000 0.398015
\(910\) 8.00000 0.265197
\(911\) −46.0000 −1.52405 −0.762024 − 0.647549i \(-0.775794\pi\)
−0.762024 + 0.647549i \(0.775794\pi\)
\(912\) −8.00000 −0.264906
\(913\) 0 0
\(914\) 2.00000 0.0661541
\(915\) −14.0000 −0.462826
\(916\) −18.0000 −0.594737
\(917\) 16.0000 0.528367
\(918\) −1.00000 −0.0330049
\(919\) 16.0000 0.527791 0.263896 − 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 8.00000 0.263752
\(921\) −2.00000 −0.0659022
\(922\) 24.0000 0.790398
\(923\) 56.0000 1.84326
\(924\) 2.00000 0.0657952
\(925\) 2.00000 0.0657596
\(926\) −4.00000 −0.131448
\(927\) 4.00000 0.131377
\(928\) 2.00000 0.0656532
\(929\) 20.0000 0.656179 0.328089 − 0.944647i \(-0.393595\pi\)
0.328089 + 0.944647i \(0.393595\pi\)
\(930\) 0 0
\(931\) 24.0000 0.786568
\(932\) −14.0000 −0.458585
\(933\) 6.00000 0.196431
\(934\) −8.00000 −0.261768
\(935\) 1.00000 0.0327035
\(936\) 4.00000 0.130744
\(937\) −2.00000 −0.0653372 −0.0326686 − 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 28.0000 0.914232
\(939\) −24.0000 −0.783210
\(940\) −8.00000 −0.260931
\(941\) −38.0000 −1.23876 −0.619382 − 0.785090i \(-0.712617\pi\)
−0.619382 + 0.785090i \(0.712617\pi\)
\(942\) 0 0
\(943\) −64.0000 −2.08413
\(944\) −14.0000 −0.455661
\(945\) 2.00000 0.0650600
\(946\) −6.00000 −0.195077
\(947\) −4.00000 −0.129983 −0.0649913 − 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 16.0000 0.519656
\(949\) 0 0
\(950\) 8.00000 0.259554
\(951\) 2.00000 0.0648544
\(952\) −2.00000 −0.0648204
\(953\) −6.00000 −0.194359 −0.0971795 − 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) −2.00000 −0.0647524
\(955\) −12.0000 −0.388311
\(956\) 0 0
\(957\) −2.00000 −0.0646508
\(958\) −6.00000 −0.193851
\(959\) 20.0000 0.645834
\(960\) 1.00000 0.0322749
\(961\) −31.0000 −1.00000
\(962\) 8.00000 0.257930
\(963\) 4.00000 0.128898
\(964\) −2.00000 −0.0644157
\(965\) −12.0000 −0.386294
\(966\) 16.0000 0.514792
\(967\) −16.0000 −0.514525 −0.257263 − 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) −1.00000 −0.0321412
\(969\) −8.00000 −0.256997
\(970\) 8.00000 0.256865
\(971\) 14.0000 0.449281 0.224641 − 0.974442i \(-0.427879\pi\)
0.224641 + 0.974442i \(0.427879\pi\)
\(972\) 1.00000 0.0320750
\(973\) −40.0000 −1.28234
\(974\) 2.00000 0.0640841
\(975\) −4.00000 −0.128103
\(976\) −14.0000 −0.448129
\(977\) 46.0000 1.47167 0.735835 − 0.677161i \(-0.236790\pi\)
0.735835 + 0.677161i \(0.236790\pi\)
\(978\) 24.0000 0.767435
\(979\) −10.0000 −0.319601
\(980\) −3.00000 −0.0958315
\(981\) −18.0000 −0.574696
\(982\) 6.00000 0.191468
\(983\) 36.0000 1.14822 0.574111 − 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) −8.00000 −0.255031
\(985\) −6.00000 −0.191176
\(986\) 2.00000 0.0636930
\(987\) −16.0000 −0.509286
\(988\) 32.0000 1.01806
\(989\) −48.0000 −1.52631
\(990\) −1.00000 −0.0317821
\(991\) −48.0000 −1.52477 −0.762385 − 0.647124i \(-0.775972\pi\)
−0.762385 + 0.647124i \(0.775972\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 28.0000 0.888106
\(995\) 16.0000 0.507234
\(996\) 0 0
\(997\) −6.00000 −0.190022 −0.0950110 − 0.995476i \(-0.530289\pi\)
−0.0950110 + 0.995476i \(0.530289\pi\)
\(998\) −20.0000 −0.633089
\(999\) 2.00000 0.0632772
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.u.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5610.2.a.u.1.1 ✓ 1 1.1 even 1 trivial