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Properties

Label 8034.2.a.e.1.1
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8034,2,Mod(1,8034)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8034, 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("8034.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.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: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8034.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +4.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} +4.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -2.00000 q^{20} -4.00000 q^{21} +4.00000 q^{22} +4.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} +4.00000 q^{28} +6.00000 q^{29} +2.00000 q^{30} +1.00000 q^{32} -4.00000 q^{33} +2.00000 q^{34} -8.00000 q^{35} +1.00000 q^{36} +10.0000 q^{37} -1.00000 q^{39} -2.00000 q^{40} -6.00000 q^{41} -4.00000 q^{42} +12.0000 q^{43} +4.00000 q^{44} -2.00000 q^{45} +4.00000 q^{46} -1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} +1.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} -8.00000 q^{55} +4.00000 q^{56} +6.00000 q^{58} -12.0000 q^{59} +2.00000 q^{60} -2.00000 q^{61} +4.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} -4.00000 q^{66} -4.00000 q^{67} +2.00000 q^{68} -4.00000 q^{69} -8.00000 q^{70} +8.00000 q^{71} +1.00000 q^{72} -10.0000 q^{73} +10.0000 q^{74} +1.00000 q^{75} +16.0000 q^{77} -1.00000 q^{78} +16.0000 q^{79} -2.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -12.0000 q^{83} -4.00000 q^{84} -4.00000 q^{85} +12.0000 q^{86} -6.00000 q^{87} +4.00000 q^{88} -14.0000 q^{89} -2.00000 q^{90} +4.00000 q^{91} +4.00000 q^{92} -1.00000 q^{96} -14.0000 q^{97} +9.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) −1.00000 −0.408248
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350
\(14\) 4.00000 1.06904
\(15\) 2.00000 0.516398
\(16\) 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
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −2.00000 −0.447214
\(21\) −4.00000 −0.872872
\(22\) 4.00000 0.852803
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) −1.00000 −0.204124
\(25\) −1.00000 −0.200000
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) 4.00000 0.755929
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 2.00000 0.365148
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.00000 −0.696311
\(34\) 2.00000 0.342997
\(35\) −8.00000 −1.35225
\(36\) 1.00000 0.166667
\(37\) 10.0000 1.64399 0.821995 − 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) −2.00000 −0.316228
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −4.00000 −0.617213
\(43\) 12.0000 1.82998 0.914991 − 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) 4.00000 0.603023
\(45\) −2.00000 −0.298142
\(46\) 4.00000 0.589768
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.00000 1.28571
\(50\) −1.00000 −0.141421
\(51\) −2.00000 −0.280056
\(52\) 1.00000 0.138675
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) −8.00000 −1.07872
\(56\) 4.00000 0.534522
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) −12.0000 −1.56227 −0.781133 − 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 2.00000 0.258199
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) −4.00000 −0.492366
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 2.00000 0.242536
\(69\) −4.00000 −0.481543
\(70\) −8.00000 −0.956183
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 1.00000 0.117851
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 10.0000 1.16248
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 16.0000 1.82337
\(78\) −1.00000 −0.113228
\(79\) 16.0000 1.80014 0.900070 − 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) −4.00000 −0.436436
\(85\) −4.00000 −0.433861
\(86\) 12.0000 1.29399
\(87\) −6.00000 −0.643268
\(88\) 4.00000 0.426401
\(89\) −14.0000 −1.48400 −0.741999 − 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) −2.00000 −0.210819
\(91\) 4.00000 0.419314
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −14.0000 −1.42148 −0.710742 − 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 9.00000 0.909137
\(99\) 4.00000 0.402015
\(100\) −1.00000 −0.100000
\(101\) 10.0000 0.995037 0.497519 − 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) −2.00000 −0.198030
\(103\) 1.00000 0.0985329
\(104\) 1.00000 0.0980581
\(105\) 8.00000 0.780720
\(106\) −6.00000 −0.582772
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) −8.00000 −0.762770
\(111\) −10.0000 −0.949158
\(112\) 4.00000 0.377964
\(113\) −2.00000 −0.188144 −0.0940721 − 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) −8.00000 −0.746004
\(116\) 6.00000 0.557086
\(117\) 1.00000 0.0924500
\(118\) −12.0000 −1.10469
\(119\) 8.00000 0.733359
\(120\) 2.00000 0.182574
\(121\) 5.00000 0.454545
\(122\) −2.00000 −0.181071
\(123\) 6.00000 0.541002
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) 4.00000 0.356348
\(127\) 16.0000 1.41977 0.709885 − 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 1.00000 0.0883883
\(129\) −12.0000 −1.05654
\(130\) −2.00000 −0.175412
\(131\) 16.0000 1.39793 0.698963 − 0.715158i \(-0.253645\pi\)
0.698963 + 0.715158i \(0.253645\pi\)
\(132\) −4.00000 −0.348155
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 2.00000 0.172133
\(136\) 2.00000 0.171499
\(137\) −6.00000 −0.512615 −0.256307 − 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) −4.00000 −0.340503
\(139\) −4.00000 −0.339276 −0.169638 − 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −8.00000 −0.676123
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) 4.00000 0.334497
\(144\) 1.00000 0.0833333
\(145\) −12.0000 −0.996546
\(146\) −10.0000 −0.827606
\(147\) −9.00000 −0.742307
\(148\) 10.0000 0.821995
\(149\) −10.0000 −0.819232 −0.409616 − 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 1.00000 0.0816497
\(151\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 16.0000 1.28932
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) −10.0000 −0.798087 −0.399043 − 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 16.0000 1.27289
\(159\) 6.00000 0.475831
\(160\) −2.00000 −0.158114
\(161\) 16.0000 1.26098
\(162\) 1.00000 0.0785674
\(163\) −16.0000 −1.25322 −0.626608 − 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) −6.00000 −0.468521
\(165\) 8.00000 0.622799
\(166\) −12.0000 −0.931381
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −4.00000 −0.308607
\(169\) 1.00000 0.0769231
\(170\) −4.00000 −0.306786
\(171\) 0 0
\(172\) 12.0000 0.914991
\(173\) 10.0000 0.760286 0.380143 − 0.924928i \(-0.375875\pi\)
0.380143 + 0.924928i \(0.375875\pi\)
\(174\) −6.00000 −0.454859
\(175\) −4.00000 −0.302372
\(176\) 4.00000 0.301511
\(177\) 12.0000 0.901975
\(178\) −14.0000 −1.04934
\(179\) −16.0000 −1.19590 −0.597948 − 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) −2.00000 −0.149071
\(181\) −2.00000 −0.148659 −0.0743294 − 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 4.00000 0.296500
\(183\) 2.00000 0.147844
\(184\) 4.00000 0.294884
\(185\) −20.0000 −1.47043
\(186\) 0 0
\(187\) 8.00000 0.585018
\(188\) 0 0
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) 16.0000 1.15772 0.578860 − 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 14.0000 1.00774 0.503871 − 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −14.0000 −1.00514
\(195\) 2.00000 0.143223
\(196\) 9.00000 0.642857
\(197\) −2.00000 −0.142494 −0.0712470 − 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 4.00000 0.284268
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 4.00000 0.282138
\(202\) 10.0000 0.703598
\(203\) 24.0000 1.68447
\(204\) −2.00000 −0.140028
\(205\) 12.0000 0.838116
\(206\) 1.00000 0.0696733
\(207\) 4.00000 0.278019
\(208\) 1.00000 0.0693375
\(209\) 0 0
\(210\) 8.00000 0.552052
\(211\) −20.0000 −1.37686 −0.688428 − 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) −6.00000 −0.412082
\(213\) −8.00000 −0.548151
\(214\) 0 0
\(215\) −24.0000 −1.63679
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 2.00000 0.135457
\(219\) 10.0000 0.675737
\(220\) −8.00000 −0.539360
\(221\) 2.00000 0.134535
\(222\) −10.0000 −0.671156
\(223\) 28.0000 1.87502 0.937509 − 0.347960i \(-0.113126\pi\)
0.937509 + 0.347960i \(0.113126\pi\)
\(224\) 4.00000 0.267261
\(225\) −1.00000 −0.0666667
\(226\) −2.00000 −0.133038
\(227\) −12.0000 −0.796468 −0.398234 − 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) 22.0000 1.45380 0.726900 − 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) −8.00000 −0.527504
\(231\) −16.0000 −1.05272
\(232\) 6.00000 0.393919
\(233\) 6.00000 0.393073 0.196537 − 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) −16.0000 −1.03931
\(238\) 8.00000 0.518563
\(239\) −8.00000 −0.517477 −0.258738 − 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 2.00000 0.129099
\(241\) −26.0000 −1.67481 −0.837404 − 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) 5.00000 0.321412
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) −18.0000 −1.14998
\(246\) 6.00000 0.382546
\(247\) 0 0
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 12.0000 0.758947
\(251\) 28.0000 1.76734 0.883672 − 0.468106i \(-0.155064\pi\)
0.883672 + 0.468106i \(0.155064\pi\)
\(252\) 4.00000 0.251976
\(253\) 16.0000 1.00591
\(254\) 16.0000 1.00393
\(255\) 4.00000 0.250490
\(256\) 1.00000 0.0625000
\(257\) −26.0000 −1.62184 −0.810918 − 0.585160i \(-0.801032\pi\)
−0.810918 + 0.585160i \(0.801032\pi\)
\(258\) −12.0000 −0.747087
\(259\) 40.0000 2.48548
\(260\) −2.00000 −0.124035
\(261\) 6.00000 0.371391
\(262\) 16.0000 0.988483
\(263\) 16.0000 0.986602 0.493301 − 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) −4.00000 −0.246183
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 14.0000 0.856786
\(268\) −4.00000 −0.244339
\(269\) −18.0000 −1.09748 −0.548740 − 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 2.00000 0.121716
\(271\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 2.00000 0.121268
\(273\) −4.00000 −0.242091
\(274\) −6.00000 −0.362473
\(275\) −4.00000 −0.241209
\(276\) −4.00000 −0.240772
\(277\) −10.0000 −0.600842 −0.300421 − 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −4.00000 −0.239904
\(279\) 0 0
\(280\) −8.00000 −0.478091
\(281\) −6.00000 −0.357930 −0.178965 − 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 − 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) −24.0000 −1.41668
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) −12.0000 −0.704664
\(291\) 14.0000 0.820695
\(292\) −10.0000 −0.585206
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) −9.00000 −0.524891
\(295\) 24.0000 1.39733
\(296\) 10.0000 0.581238
\(297\) −4.00000 −0.232104
\(298\) −10.0000 −0.579284
\(299\) 4.00000 0.231326
\(300\) 1.00000 0.0577350
\(301\) 48.0000 2.76667
\(302\) 0 0
\(303\) −10.0000 −0.574485
\(304\) 0 0
\(305\) 4.00000 0.229039
\(306\) 2.00000 0.114332
\(307\) 20.0000 1.14146 0.570730 − 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 16.0000 0.911685
\(309\) −1.00000 −0.0568880
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 − 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 26.0000 1.46961 0.734803 − 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) −10.0000 −0.564333
\(315\) −8.00000 −0.450749
\(316\) 16.0000 0.900070
\(317\) 14.0000 0.786318 0.393159 − 0.919470i \(-0.371382\pi\)
0.393159 + 0.919470i \(0.371382\pi\)
\(318\) 6.00000 0.336463
\(319\) 24.0000 1.34374
\(320\) −2.00000 −0.111803
\(321\) 0 0
\(322\) 16.0000 0.891645
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −1.00000 −0.0554700
\(326\) −16.0000 −0.886158
\(327\) −2.00000 −0.110600
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 8.00000 0.440386
\(331\) 4.00000 0.219860 0.109930 − 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) −12.0000 −0.658586
\(333\) 10.0000 0.547997
\(334\) 0 0
\(335\) 8.00000 0.437087
\(336\) −4.00000 −0.218218
\(337\) −14.0000 −0.762629 −0.381314 − 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 1.00000 0.0543928
\(339\) 2.00000 0.108625
\(340\) −4.00000 −0.216930
\(341\) 0 0
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) 12.0000 0.646997
\(345\) 8.00000 0.430706
\(346\) 10.0000 0.537603
\(347\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) −6.00000 −0.321634
\(349\) 34.0000 1.81998 0.909989 − 0.414632i \(-0.136090\pi\)
0.909989 + 0.414632i \(0.136090\pi\)
\(350\) −4.00000 −0.213809
\(351\) −1.00000 −0.0533761
\(352\) 4.00000 0.213201
\(353\) 18.0000 0.958043 0.479022 − 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 12.0000 0.637793
\(355\) −16.0000 −0.849192
\(356\) −14.0000 −0.741999
\(357\) −8.00000 −0.423405
\(358\) −16.0000 −0.845626
\(359\) 8.00000 0.422224 0.211112 − 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) −2.00000 −0.105409
\(361\) −19.0000 −1.00000
\(362\) −2.00000 −0.105118
\(363\) −5.00000 −0.262432
\(364\) 4.00000 0.209657
\(365\) 20.0000 1.04685
\(366\) 2.00000 0.104542
\(367\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 4.00000 0.208514
\(369\) −6.00000 −0.312348
\(370\) −20.0000 −1.03975
\(371\) −24.0000 −1.24602
\(372\) 0 0
\(373\) 22.0000 1.13912 0.569558 − 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 8.00000 0.413670
\(375\) −12.0000 −0.619677
\(376\) 0 0
\(377\) 6.00000 0.309016
\(378\) −4.00000 −0.205738
\(379\) 28.0000 1.43826 0.719132 − 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 16.0000 0.818631
\(383\) 24.0000 1.22634 0.613171 − 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −32.0000 −1.63087
\(386\) 14.0000 0.712581
\(387\) 12.0000 0.609994
\(388\) −14.0000 −0.710742
\(389\) −30.0000 −1.52106 −0.760530 − 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 2.00000 0.101274
\(391\) 8.00000 0.404577
\(392\) 9.00000 0.454569
\(393\) −16.0000 −0.807093
\(394\) −2.00000 −0.100759
\(395\) −32.0000 −1.61009
\(396\) 4.00000 0.201008
\(397\) −38.0000 −1.90717 −0.953583 − 0.301131i \(-0.902636\pi\)
−0.953583 + 0.301131i \(0.902636\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −30.0000 −1.49813 −0.749064 − 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 4.00000 0.199502
\(403\) 0 0
\(404\) 10.0000 0.497519
\(405\) −2.00000 −0.0993808
\(406\) 24.0000 1.19110
\(407\) 40.0000 1.98273
\(408\) −2.00000 −0.0990148
\(409\) 34.0000 1.68119 0.840596 − 0.541663i \(-0.182205\pi\)
0.840596 + 0.541663i \(0.182205\pi\)
\(410\) 12.0000 0.592638
\(411\) 6.00000 0.295958
\(412\) 1.00000 0.0492665
\(413\) −48.0000 −2.36193
\(414\) 4.00000 0.196589
\(415\) 24.0000 1.17811
\(416\) 1.00000 0.0490290
\(417\) 4.00000 0.195881
\(418\) 0 0
\(419\) 24.0000 1.17248 0.586238 − 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 8.00000 0.390360
\(421\) 30.0000 1.46211 0.731055 − 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) −20.0000 −0.973585
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) −2.00000 −0.0970143
\(426\) −8.00000 −0.387601
\(427\) −8.00000 −0.387147
\(428\) 0 0
\(429\) −4.00000 −0.193122
\(430\) −24.0000 −1.15738
\(431\) 16.0000 0.770693 0.385346 − 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 26.0000 1.24948 0.624740 − 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 0 0
\(435\) 12.0000 0.575356
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) 10.0000 0.477818
\(439\) −8.00000 −0.381819 −0.190910 − 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) −8.00000 −0.381385
\(441\) 9.00000 0.428571
\(442\) 2.00000 0.0951303
\(443\) 28.0000 1.33032 0.665160 − 0.746701i \(-0.268363\pi\)
0.665160 + 0.746701i \(0.268363\pi\)
\(444\) −10.0000 −0.474579
\(445\) 28.0000 1.32733
\(446\) 28.0000 1.32584
\(447\) 10.0000 0.472984
\(448\) 4.00000 0.188982
\(449\) −22.0000 −1.03824 −0.519122 − 0.854700i \(-0.673741\pi\)
−0.519122 + 0.854700i \(0.673741\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −24.0000 −1.13012
\(452\) −2.00000 −0.0940721
\(453\) 0 0
\(454\) −12.0000 −0.563188
\(455\) −8.00000 −0.375046
\(456\) 0 0
\(457\) 6.00000 0.280668 0.140334 − 0.990104i \(-0.455182\pi\)
0.140334 + 0.990104i \(0.455182\pi\)
\(458\) 22.0000 1.02799
\(459\) −2.00000 −0.0933520
\(460\) −8.00000 −0.373002
\(461\) −18.0000 −0.838344 −0.419172 − 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) −16.0000 −0.744387
\(463\) −8.00000 −0.371792 −0.185896 − 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 24.0000 1.11059 0.555294 − 0.831654i \(-0.312606\pi\)
0.555294 + 0.831654i \(0.312606\pi\)
\(468\) 1.00000 0.0462250
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) −12.0000 −0.552345
\(473\) 48.0000 2.20704
\(474\) −16.0000 −0.734904
\(475\) 0 0
\(476\) 8.00000 0.366679
\(477\) −6.00000 −0.274721
\(478\) −8.00000 −0.365911
\(479\) −32.0000 −1.46212 −0.731059 − 0.682315i \(-0.760973\pi\)
−0.731059 + 0.682315i \(0.760973\pi\)
\(480\) 2.00000 0.0912871
\(481\) 10.0000 0.455961
\(482\) −26.0000 −1.18427
\(483\) −16.0000 −0.728025
\(484\) 5.00000 0.227273
\(485\) 28.0000 1.27141
\(486\) −1.00000 −0.0453609
\(487\) 40.0000 1.81257 0.906287 − 0.422664i \(-0.138905\pi\)
0.906287 + 0.422664i \(0.138905\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 16.0000 0.723545
\(490\) −18.0000 −0.813157
\(491\) −24.0000 −1.08310 −0.541552 − 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) 6.00000 0.270501
\(493\) 12.0000 0.540453
\(494\) 0 0
\(495\) −8.00000 −0.359573
\(496\) 0 0
\(497\) 32.0000 1.43540
\(498\) 12.0000 0.537733
\(499\) −20.0000 −0.895323 −0.447661 − 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) 28.0000 1.24970
\(503\) −12.0000 −0.535054 −0.267527 − 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 4.00000 0.178174
\(505\) −20.0000 −0.889988
\(506\) 16.0000 0.711287
\(507\) −1.00000 −0.0444116
\(508\) 16.0000 0.709885
\(509\) 14.0000 0.620539 0.310270 − 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 4.00000 0.177123
\(511\) −40.0000 −1.76950
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −26.0000 −1.14681
\(515\) −2.00000 −0.0881305
\(516\) −12.0000 −0.528271
\(517\) 0 0
\(518\) 40.0000 1.75750
\(519\) −10.0000 −0.438951
\(520\) −2.00000 −0.0877058
\(521\) −42.0000 −1.84005 −0.920027 − 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) 6.00000 0.262613
\(523\) 28.0000 1.22435 0.612177 − 0.790721i \(-0.290294\pi\)
0.612177 + 0.790721i \(0.290294\pi\)
\(524\) 16.0000 0.698963
\(525\) 4.00000 0.174574
\(526\) 16.0000 0.697633
\(527\) 0 0
\(528\) −4.00000 −0.174078
\(529\) −7.00000 −0.304348
\(530\) 12.0000 0.521247
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) −6.00000 −0.259889
\(534\) 14.0000 0.605839
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 16.0000 0.690451
\(538\) −18.0000 −0.776035
\(539\) 36.0000 1.55063
\(540\) 2.00000 0.0860663
\(541\) −10.0000 −0.429934 −0.214967 − 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 0 0
\(543\) 2.00000 0.0858282
\(544\) 2.00000 0.0857493
\(545\) −4.00000 −0.171341
\(546\) −4.00000 −0.171184
\(547\) −20.0000 −0.855138 −0.427569 − 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) −6.00000 −0.256307
\(549\) −2.00000 −0.0853579
\(550\) −4.00000 −0.170561
\(551\) 0 0
\(552\) −4.00000 −0.170251
\(553\) 64.0000 2.72156
\(554\) −10.0000 −0.424859
\(555\) 20.0000 0.848953
\(556\) −4.00000 −0.169638
\(557\) −18.0000 −0.762684 −0.381342 − 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) 12.0000 0.507546
\(560\) −8.00000 −0.338062
\(561\) −8.00000 −0.337760
\(562\) −6.00000 −0.253095
\(563\) 36.0000 1.51722 0.758610 − 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 0 0
\(565\) 4.00000 0.168281
\(566\) 20.0000 0.840663
\(567\) 4.00000 0.167984
\(568\) 8.00000 0.335673
\(569\) 30.0000 1.25767 0.628833 − 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) 36.0000 1.50655 0.753277 − 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(572\) 4.00000 0.167248
\(573\) −16.0000 −0.668410
\(574\) −24.0000 −1.00174
\(575\) −4.00000 −0.166812
\(576\) 1.00000 0.0416667
\(577\) 38.0000 1.58196 0.790980 − 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) −13.0000 −0.540729
\(579\) −14.0000 −0.581820
\(580\) −12.0000 −0.498273
\(581\) −48.0000 −1.99138
\(582\) 14.0000 0.580319
\(583\) −24.0000 −0.993978
\(584\) −10.0000 −0.413803
\(585\) −2.00000 −0.0826898
\(586\) 6.00000 0.247858
\(587\) −44.0000 −1.81607 −0.908037 − 0.418890i \(-0.862419\pi\)
−0.908037 + 0.418890i \(0.862419\pi\)
\(588\) −9.00000 −0.371154
\(589\) 0 0
\(590\) 24.0000 0.988064
\(591\) 2.00000 0.0822690
\(592\) 10.0000 0.410997
\(593\) −22.0000 −0.903432 −0.451716 − 0.892162i \(-0.649188\pi\)
−0.451716 + 0.892162i \(0.649188\pi\)
\(594\) −4.00000 −0.164122
\(595\) −16.0000 −0.655936
\(596\) −10.0000 −0.409616
\(597\) 0 0
\(598\) 4.00000 0.163572
\(599\) −16.0000 −0.653742 −0.326871 − 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 1.00000 0.0408248
\(601\) 2.00000 0.0815817 0.0407909 − 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 48.0000 1.95633
\(603\) −4.00000 −0.162893
\(604\) 0 0
\(605\) −10.0000 −0.406558
\(606\) −10.0000 −0.406222
\(607\) −8.00000 −0.324710 −0.162355 − 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 0 0
\(609\) −24.0000 −0.972529
\(610\) 4.00000 0.161955
\(611\) 0 0
\(612\) 2.00000 0.0808452
\(613\) −2.00000 −0.0807792 −0.0403896 − 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 20.0000 0.807134
\(615\) −12.0000 −0.483887
\(616\) 16.0000 0.644658
\(617\) −38.0000 −1.52982 −0.764911 − 0.644136i \(-0.777217\pi\)
−0.764911 + 0.644136i \(0.777217\pi\)
\(618\) −1.00000 −0.0402259
\(619\) 8.00000 0.321547 0.160774 − 0.986991i \(-0.448601\pi\)
0.160774 + 0.986991i \(0.448601\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) −12.0000 −0.481156
\(623\) −56.0000 −2.24359
\(624\) −1.00000 −0.0400320
\(625\) −19.0000 −0.760000
\(626\) 26.0000 1.03917
\(627\) 0 0
\(628\) −10.0000 −0.399043
\(629\) 20.0000 0.797452
\(630\) −8.00000 −0.318728
\(631\) −28.0000 −1.11466 −0.557331 − 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) 16.0000 0.636446
\(633\) 20.0000 0.794929
\(634\) 14.0000 0.556011
\(635\) −32.0000 −1.26988
\(636\) 6.00000 0.237915
\(637\) 9.00000 0.356593
\(638\) 24.0000 0.950169
\(639\) 8.00000 0.316475
\(640\) −2.00000 −0.0790569
\(641\) 18.0000 0.710957 0.355479 − 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) −48.0000 −1.89294 −0.946468 − 0.322799i \(-0.895376\pi\)
−0.946468 + 0.322799i \(0.895376\pi\)
\(644\) 16.0000 0.630488
\(645\) 24.0000 0.944999
\(646\) 0 0
\(647\) −20.0000 −0.786281 −0.393141 − 0.919478i \(-0.628611\pi\)
−0.393141 + 0.919478i \(0.628611\pi\)
\(648\) 1.00000 0.0392837
\(649\) −48.0000 −1.88416
\(650\) −1.00000 −0.0392232
\(651\) 0 0
\(652\) −16.0000 −0.626608
\(653\) −14.0000 −0.547862 −0.273931 − 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) −2.00000 −0.0782062
\(655\) −32.0000 −1.25034
\(656\) −6.00000 −0.234261
\(657\) −10.0000 −0.390137
\(658\) 0 0
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 8.00000 0.311400
\(661\) 18.0000 0.700119 0.350059 − 0.936727i \(-0.386161\pi\)
0.350059 + 0.936727i \(0.386161\pi\)
\(662\) 4.00000 0.155464
\(663\) −2.00000 −0.0776736
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) 10.0000 0.387492
\(667\) 24.0000 0.929284
\(668\) 0 0
\(669\) −28.0000 −1.08254
\(670\) 8.00000 0.309067
\(671\) −8.00000 −0.308837
\(672\) −4.00000 −0.154303
\(673\) 2.00000 0.0770943 0.0385472 − 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) −14.0000 −0.539260
\(675\) 1.00000 0.0384900
\(676\) 1.00000 0.0384615
\(677\) 22.0000 0.845529 0.422764 − 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 2.00000 0.0768095
\(679\) −56.0000 −2.14908
\(680\) −4.00000 −0.153393
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 − 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) 8.00000 0.305441
\(687\) −22.0000 −0.839352
\(688\) 12.0000 0.457496
\(689\) −6.00000 −0.228582
\(690\) 8.00000 0.304555
\(691\) 28.0000 1.06517 0.532585 − 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 10.0000 0.380143
\(693\) 16.0000 0.607790
\(694\) 0 0
\(695\) 8.00000 0.303457
\(696\) −6.00000 −0.227429
\(697\) −12.0000 −0.454532
\(698\) 34.0000 1.28692
\(699\) −6.00000 −0.226941
\(700\) −4.00000 −0.151186
\(701\) −42.0000 −1.58632 −0.793159 − 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 0 0
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) 40.0000 1.50435
\(708\) 12.0000 0.450988
\(709\) −10.0000 −0.375558 −0.187779 − 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) −16.0000 −0.600469
\(711\) 16.0000 0.600047
\(712\) −14.0000 −0.524672
\(713\) 0 0
\(714\) −8.00000 −0.299392
\(715\) −8.00000 −0.299183
\(716\) −16.0000 −0.597948
\(717\) 8.00000 0.298765
\(718\) 8.00000 0.298557
\(719\) 8.00000 0.298350 0.149175 − 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 4.00000 0.148968
\(722\) −19.0000 −0.707107
\(723\) 26.0000 0.966950
\(724\) −2.00000 −0.0743294
\(725\) −6.00000 −0.222834
\(726\) −5.00000 −0.185567
\(727\) 40.0000 1.48352 0.741759 − 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) 4.00000 0.148250
\(729\) 1.00000 0.0370370
\(730\) 20.0000 0.740233
\(731\) 24.0000 0.887672
\(732\) 2.00000 0.0739221
\(733\) −22.0000 −0.812589 −0.406294 − 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 0 0
\(735\) 18.0000 0.663940
\(736\) 4.00000 0.147442
\(737\) −16.0000 −0.589368
\(738\) −6.00000 −0.220863
\(739\) −16.0000 −0.588570 −0.294285 − 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) −20.0000 −0.735215
\(741\) 0 0
\(742\) −24.0000 −0.881068
\(743\) −48.0000 −1.76095 −0.880475 − 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) 0 0
\(745\) 20.0000 0.732743
\(746\) 22.0000 0.805477
\(747\) −12.0000 −0.439057
\(748\) 8.00000 0.292509
\(749\) 0 0
\(750\) −12.0000 −0.438178
\(751\) 24.0000 0.875772 0.437886 − 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) 0 0
\(753\) −28.0000 −1.02038
\(754\) 6.00000 0.218507
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) 6.00000 0.218074 0.109037 − 0.994038i \(-0.465223\pi\)
0.109037 + 0.994038i \(0.465223\pi\)
\(758\) 28.0000 1.01701
\(759\) −16.0000 −0.580763
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 − 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) −16.0000 −0.579619
\(763\) 8.00000 0.289619
\(764\) 16.0000 0.578860
\(765\) −4.00000 −0.144620
\(766\) 24.0000 0.867155
\(767\) −12.0000 −0.433295
\(768\) −1.00000 −0.0360844
\(769\) 14.0000 0.504853 0.252426 − 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) −32.0000 −1.15320
\(771\) 26.0000 0.936367
\(772\) 14.0000 0.503871
\(773\) 6.00000 0.215805 0.107903 − 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 12.0000 0.431331
\(775\) 0 0
\(776\) −14.0000 −0.502571
\(777\) −40.0000 −1.43499
\(778\) −30.0000 −1.07555
\(779\) 0 0
\(780\) 2.00000 0.0716115
\(781\) 32.0000 1.14505
\(782\) 8.00000 0.286079
\(783\) −6.00000 −0.214423
\(784\) 9.00000 0.321429
\(785\) 20.0000 0.713831
\(786\) −16.0000 −0.570701
\(787\) −40.0000 −1.42585 −0.712923 − 0.701242i \(-0.752629\pi\)
−0.712923 + 0.701242i \(0.752629\pi\)
\(788\) −2.00000 −0.0712470
\(789\) −16.0000 −0.569615
\(790\) −32.0000 −1.13851
\(791\) −8.00000 −0.284447
\(792\) 4.00000 0.142134
\(793\) −2.00000 −0.0710221
\(794\) −38.0000 −1.34857
\(795\) −12.0000 −0.425596
\(796\) 0 0
\(797\) −18.0000 −0.637593 −0.318796 − 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) −14.0000 −0.494666
\(802\) −30.0000 −1.05934
\(803\) −40.0000 −1.41157
\(804\) 4.00000 0.141069
\(805\) −32.0000 −1.12785
\(806\) 0 0
\(807\) 18.0000 0.633630
\(808\) 10.0000 0.351799
\(809\) −2.00000 −0.0703163 −0.0351581 − 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) −2.00000 −0.0702728
\(811\) 44.0000 1.54505 0.772524 − 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) 24.0000 0.842235
\(813\) 0 0
\(814\) 40.0000 1.40200
\(815\) 32.0000 1.12091
\(816\) −2.00000 −0.0700140
\(817\) 0 0
\(818\) 34.0000 1.18878
\(819\) 4.00000 0.139771
\(820\) 12.0000 0.419058
\(821\) −10.0000 −0.349002 −0.174501 − 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) 6.00000 0.209274
\(823\) 24.0000 0.836587 0.418294 − 0.908312i \(-0.362628\pi\)
0.418294 + 0.908312i \(0.362628\pi\)
\(824\) 1.00000 0.0348367
\(825\) 4.00000 0.139262
\(826\) −48.0000 −1.67013
\(827\) −36.0000 −1.25184 −0.625921 − 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 4.00000 0.139010
\(829\) 14.0000 0.486240 0.243120 − 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 24.0000 0.833052
\(831\) 10.0000 0.346896
\(832\) 1.00000 0.0346688
\(833\) 18.0000 0.623663
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 24.0000 0.829066
\(839\) −48.0000 −1.65714 −0.828572 − 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 8.00000 0.276026
\(841\) 7.00000 0.241379
\(842\) 30.0000 1.03387
\(843\) 6.00000 0.206651
\(844\) −20.0000 −0.688428
\(845\) −2.00000 −0.0688021
\(846\) 0 0
\(847\) 20.0000 0.687208
\(848\) −6.00000 −0.206041
\(849\) −20.0000 −0.686398
\(850\) −2.00000 −0.0685994
\(851\) 40.0000 1.37118
\(852\) −8.00000 −0.274075
\(853\) 54.0000 1.84892 0.924462 − 0.381273i \(-0.124514\pi\)
0.924462 + 0.381273i \(0.124514\pi\)
\(854\) −8.00000 −0.273754
\(855\) 0 0
\(856\) 0 0
\(857\) 50.0000 1.70797 0.853984 − 0.520300i \(-0.174180\pi\)
0.853984 + 0.520300i \(0.174180\pi\)
\(858\) −4.00000 −0.136558
\(859\) 28.0000 0.955348 0.477674 − 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) −24.0000 −0.818393
\(861\) 24.0000 0.817918
\(862\) 16.0000 0.544962
\(863\) −24.0000 −0.816970 −0.408485 − 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −20.0000 −0.680020
\(866\) 26.0000 0.883516
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) 64.0000 2.17105
\(870\) 12.0000 0.406838
\(871\) −4.00000 −0.135535
\(872\) 2.00000 0.0677285
\(873\) −14.0000 −0.473828
\(874\) 0 0
\(875\) 48.0000 1.62270
\(876\) 10.0000 0.337869
\(877\) 18.0000 0.607817 0.303908 − 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) −8.00000 −0.269987
\(879\) −6.00000 −0.202375
\(880\) −8.00000 −0.269680
\(881\) −34.0000 −1.14549 −0.572745 − 0.819734i \(-0.694121\pi\)
−0.572745 + 0.819734i \(0.694121\pi\)
\(882\) 9.00000 0.303046
\(883\) −52.0000 −1.74994 −0.874970 − 0.484178i \(-0.839119\pi\)
−0.874970 + 0.484178i \(0.839119\pi\)
\(884\) 2.00000 0.0672673
\(885\) −24.0000 −0.806751
\(886\) 28.0000 0.940678
\(887\) −52.0000 −1.74599 −0.872995 − 0.487730i \(-0.837825\pi\)
−0.872995 + 0.487730i \(0.837825\pi\)
\(888\) −10.0000 −0.335578
\(889\) 64.0000 2.14649
\(890\) 28.0000 0.938562
\(891\) 4.00000 0.134005
\(892\) 28.0000 0.937509
\(893\) 0 0
\(894\) 10.0000 0.334450
\(895\) 32.0000 1.06964
\(896\) 4.00000 0.133631
\(897\) −4.00000 −0.133556
\(898\) −22.0000 −0.734150
\(899\) 0 0
\(900\) −1.00000 −0.0333333
\(901\) −12.0000 −0.399778
\(902\) −24.0000 −0.799113
\(903\) −48.0000 −1.59734
\(904\) −2.00000 −0.0665190
\(905\) 4.00000 0.132964
\(906\) 0 0
\(907\) −28.0000 −0.929725 −0.464862 − 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) −12.0000 −0.398234
\(909\) 10.0000 0.331679
\(910\) −8.00000 −0.265197
\(911\) −24.0000 −0.795155 −0.397578 − 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 0 0
\(913\) −48.0000 −1.58857
\(914\) 6.00000 0.198462
\(915\) −4.00000 −0.132236
\(916\) 22.0000 0.726900
\(917\) 64.0000 2.11347
\(918\) −2.00000 −0.0660098
\(919\) −24.0000 −0.791687 −0.395843 − 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) −8.00000 −0.263752
\(921\) −20.0000 −0.659022
\(922\) −18.0000 −0.592798
\(923\) 8.00000 0.263323
\(924\) −16.0000 −0.526361
\(925\) −10.0000 −0.328798
\(926\) −8.00000 −0.262896
\(927\) 1.00000 0.0328443
\(928\) 6.00000 0.196960
\(929\) −14.0000 −0.459325 −0.229663 − 0.973270i \(-0.573762\pi\)
−0.229663 + 0.973270i \(0.573762\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 6.00000 0.196537
\(933\) 12.0000 0.392862
\(934\) 24.0000 0.785304
\(935\) −16.0000 −0.523256
\(936\) 1.00000 0.0326860
\(937\) −38.0000 −1.24141 −0.620703 − 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) −16.0000 −0.522419
\(939\) −26.0000 −0.848478
\(940\) 0 0
\(941\) 46.0000 1.49956 0.749779 − 0.661689i \(-0.230160\pi\)
0.749779 + 0.661689i \(0.230160\pi\)
\(942\) 10.0000 0.325818
\(943\) −24.0000 −0.781548
\(944\) −12.0000 −0.390567
\(945\) 8.00000 0.260240
\(946\) 48.0000 1.56061
\(947\) −4.00000 −0.129983 −0.0649913 − 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) −16.0000 −0.519656
\(949\) −10.0000 −0.324614
\(950\) 0 0
\(951\) −14.0000 −0.453981
\(952\) 8.00000 0.259281
\(953\) −30.0000 −0.971795 −0.485898 − 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) −6.00000 −0.194257
\(955\) −32.0000 −1.03550
\(956\) −8.00000 −0.258738
\(957\) −24.0000 −0.775810
\(958\) −32.0000 −1.03387
\(959\) −24.0000 −0.775000
\(960\) 2.00000 0.0645497
\(961\) −31.0000 −1.00000
\(962\) 10.0000 0.322413
\(963\) 0 0
\(964\) −26.0000 −0.837404
\(965\) −28.0000 −0.901352
\(966\) −16.0000 −0.514792
\(967\) −8.00000 −0.257263 −0.128631 − 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 5.00000 0.160706
\(969\) 0 0
\(970\) 28.0000 0.899026
\(971\) −12.0000 −0.385098 −0.192549 − 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −16.0000 −0.512936
\(974\) 40.0000 1.28168
\(975\) 1.00000 0.0320256
\(976\) −2.00000 −0.0640184
\(977\) 2.00000 0.0639857 0.0319928 − 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) 16.0000 0.511624
\(979\) −56.0000 −1.78977
\(980\) −18.0000 −0.574989
\(981\) 2.00000 0.0638551
\(982\) −24.0000 −0.765871
\(983\) 16.0000 0.510321 0.255160 − 0.966899i \(-0.417872\pi\)
0.255160 + 0.966899i \(0.417872\pi\)
\(984\) 6.00000 0.191273
\(985\) 4.00000 0.127451
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 0 0
\(989\) 48.0000 1.52631
\(990\) −8.00000 −0.254257
\(991\) −40.0000 −1.27064 −0.635321 − 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 0 0
\(993\) −4.00000 −0.126936
\(994\) 32.0000 1.01498
\(995\) 0 0
\(996\) 12.0000 0.380235
\(997\) −10.0000 −0.316703 −0.158352 − 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) −20.0000 −0.633089
\(999\) −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 8034.2.a.e.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.e.1.1 ✓ 1 1.1 even 1 trivial