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Properties

Label 4641.2.a.b.1.1
Level $4641$
Weight $2$
Character 4641.1
Self dual yes
Analytic conductor $37.059$
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] = [4641,2,Mod(1,4641)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4641, 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("4641.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4641 = 3 \cdot 7 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4641.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: \(37.0585715781\)
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 \(\chi\) \(=\) 4641.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} +1.00000 q^{14} +2.00000 q^{15} -1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} +2.00000 q^{20} +1.00000 q^{21} +4.00000 q^{22} -3.00000 q^{24} -1.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +6.00000 q^{29} -2.00000 q^{30} -5.00000 q^{32} +4.00000 q^{33} -1.00000 q^{34} +2.00000 q^{35} -1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{38} -1.00000 q^{39} -6.00000 q^{40} +2.00000 q^{41} -1.00000 q^{42} +12.0000 q^{43} +4.00000 q^{44} -2.00000 q^{45} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -1.00000 q^{51} -1.00000 q^{52} -10.0000 q^{53} +1.00000 q^{54} +8.00000 q^{55} -3.00000 q^{56} +4.00000 q^{57} -6.00000 q^{58} +12.0000 q^{59} -2.00000 q^{60} -10.0000 q^{61} -1.00000 q^{63} +7.00000 q^{64} -2.00000 q^{65} -4.00000 q^{66} +4.00000 q^{67} -1.00000 q^{68} -2.00000 q^{70} +8.00000 q^{71} +3.00000 q^{72} +10.0000 q^{73} +2.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} +4.00000 q^{77} +1.00000 q^{78} -8.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +4.00000 q^{83} -1.00000 q^{84} -2.00000 q^{85} -12.0000 q^{86} -6.00000 q^{87} -12.0000 q^{88} +10.0000 q^{89} +2.00000 q^{90} -1.00000 q^{91} +8.00000 q^{95} +5.00000 q^{96} +2.00000 q^{97} -1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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 −0.353553 − 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) −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
\(7\) −1.00000 −0.377964
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) 1.00000 0.267261
\(15\) 2.00000 0.516398
\(16\) −1.00000 −0.250000
\(17\) 1.00000 0.242536
\(18\) −1.00000 −0.235702
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 2.00000 0.447214
\(21\) 1.00000 0.218218
\(22\) 4.00000 0.852803
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −3.00000 −0.612372
\(25\) −1.00000 −0.200000
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) 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 \(0.5\pi\)
\(32\) −5.00000 −0.883883
\(33\) 4.00000 0.696311
\(34\) −1.00000 −0.171499
\(35\) 2.00000 0.338062
\(36\) −1.00000 −0.166667
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 4.00000 0.648886
\(39\) −1.00000 −0.160128
\(40\) −6.00000 −0.948683
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) −1.00000 −0.154303
\(43\) 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
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −1.00000 −0.140028
\(52\) −1.00000 −0.138675
\(53\) −10.0000 −1.37361 −0.686803 − 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 1.00000 0.136083
\(55\) 8.00000 1.07872
\(56\) −3.00000 −0.400892
\(57\) 4.00000 0.529813
\(58\) −6.00000 −0.787839
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) −2.00000 −0.258199
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 7.00000 0.875000
\(65\) −2.00000 −0.248069
\(66\) −4.00000 −0.492366
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −1.00000 −0.121268
\(69\) 0 0
\(70\) −2.00000 −0.239046
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 3.00000 0.353553
\(73\) 10.0000 1.17041 0.585206 − 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 2.00000 0.232495
\(75\) 1.00000 0.115470
\(76\) 4.00000 0.458831
\(77\) 4.00000 0.455842
\(78\) 1.00000 0.113228
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) −1.00000 −0.109109
\(85\) −2.00000 −0.216930
\(86\) −12.0000 −1.29399
\(87\) −6.00000 −0.643268
\(88\) −12.0000 −1.27920
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 2.00000 0.210819
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.00000 0.820783
\(96\) 5.00000 0.510310
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −1.00000 −0.101015
\(99\) −4.00000 −0.402015
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 − 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 1.00000 0.0990148
\(103\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 3.00000 0.294174
\(105\) −2.00000 −0.195180
\(106\) 10.0000 0.971286
\(107\) 4.00000 0.386695 0.193347 − 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 1.00000 0.0962250
\(109\) 6.00000 0.574696 0.287348 − 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) −8.00000 −0.762770
\(111\) 2.00000 0.189832
\(112\) 1.00000 0.0944911
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 1.00000 0.0924500
\(118\) −12.0000 −1.10469
\(119\) −1.00000 −0.0916698
\(120\) 6.00000 0.547723
\(121\) 5.00000 0.454545
\(122\) 10.0000 0.905357
\(123\) −2.00000 −0.180334
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) 1.00000 0.0890871
\(127\) −16.0000 −1.41977 −0.709885 − 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 3.00000 0.265165
\(129\) −12.0000 −1.05654
\(130\) 2.00000 0.175412
\(131\) 12.0000 1.04844 0.524222 − 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) −4.00000 −0.348155
\(133\) 4.00000 0.346844
\(134\) −4.00000 −0.345547
\(135\) 2.00000 0.172133
\(136\) 3.00000 0.257248
\(137\) −6.00000 −0.512615 −0.256307 − 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 − 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) −2.00000 −0.169031
\(141\) 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
\(147\) −1.00000 −0.0824786
\(148\) 2.00000 0.164399
\(149\) 22.0000 1.80231 0.901155 − 0.433497i \(-0.142720\pi\)
0.901155 + 0.433497i \(0.142720\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 8.00000 0.651031 0.325515 − 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −12.0000 −0.973329
\(153\) 1.00000 0.0808452
\(154\) −4.00000 −0.322329
\(155\) 0 0
\(156\) 1.00000 0.0800641
\(157\) 14.0000 1.11732 0.558661 − 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 8.00000 0.636446
\(159\) 10.0000 0.793052
\(160\) 10.0000 0.790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −12.0000 −0.939913 −0.469956 − 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) −2.00000 −0.156174
\(165\) −8.00000 −0.622799
\(166\) −4.00000 −0.310460
\(167\) −8.00000 −0.619059 −0.309529 − 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 3.00000 0.231455
\(169\) 1.00000 0.0769231
\(170\) 2.00000 0.153393
\(171\) −4.00000 −0.305888
\(172\) −12.0000 −0.914991
\(173\) −2.00000 −0.152057 −0.0760286 − 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 6.00000 0.454859
\(175\) 1.00000 0.0755929
\(176\) 4.00000 0.301511
\(177\) −12.0000 −0.901975
\(178\) −10.0000 −0.749532
\(179\) −4.00000 −0.298974 −0.149487 − 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 2.00000 0.149071
\(181\) −18.0000 −1.33793 −0.668965 − 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) 1.00000 0.0741249
\(183\) 10.0000 0.739221
\(184\) 0 0
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) −4.00000 −0.292509
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) −8.00000 −0.580381
\(191\) −24.0000 −1.73658 −0.868290 − 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) −7.00000 −0.505181
\(193\) 10.0000 0.719816 0.359908 − 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) −2.00000 −0.143592
\(195\) 2.00000 0.143223
\(196\) −1.00000 −0.0714286
\(197\) 22.0000 1.56744 0.783718 − 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) 4.00000 0.284268
\(199\) −16.0000 −1.13421 −0.567105 − 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) −3.00000 −0.212132
\(201\) −4.00000 −0.282138
\(202\) −6.00000 −0.422159
\(203\) −6.00000 −0.421117
\(204\) 1.00000 0.0700140
\(205\) −4.00000 −0.279372
\(206\) 0 0
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) 16.0000 1.10674
\(210\) 2.00000 0.138013
\(211\) −4.00000 −0.275371 −0.137686 − 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 10.0000 0.686803
\(213\) −8.00000 −0.548151
\(214\) −4.00000 −0.273434
\(215\) −24.0000 −1.63679
\(216\) −3.00000 −0.204124
\(217\) 0 0
\(218\) −6.00000 −0.406371
\(219\) −10.0000 −0.675737
\(220\) −8.00000 −0.539360
\(221\) 1.00000 0.0672673
\(222\) −2.00000 −0.134231
\(223\) 8.00000 0.535720 0.267860 − 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 5.00000 0.334077
\(225\) −1.00000 −0.0666667
\(226\) 6.00000 0.399114
\(227\) −12.0000 −0.796468 −0.398234 − 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) −4.00000 −0.264906
\(229\) 6.00000 0.396491 0.198246 − 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 18.0000 1.18176
\(233\) −14.0000 −0.917170 −0.458585 − 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) 8.00000 0.519656
\(238\) 1.00000 0.0648204
\(239\) −8.00000 −0.517477 −0.258738 − 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) −2.00000 −0.129099
\(241\) −14.0000 −0.901819 −0.450910 − 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) −5.00000 −0.321412
\(243\) −1.00000 −0.0641500
\(244\) 10.0000 0.640184
\(245\) −2.00000 −0.127775
\(246\) 2.00000 0.127515
\(247\) −4.00000 −0.254514
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) −12.0000 −0.758947
\(251\) −4.00000 −0.252478 −0.126239 − 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) 2.00000 0.125245
\(256\) −17.0000 −1.06250
\(257\) 2.00000 0.124757 0.0623783 − 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 12.0000 0.747087
\(259\) 2.00000 0.124274
\(260\) 2.00000 0.124035
\(261\) 6.00000 0.371391
\(262\) −12.0000 −0.741362
\(263\) −16.0000 −0.986602 −0.493301 − 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 12.0000 0.738549
\(265\) 20.0000 1.22859
\(266\) −4.00000 −0.245256
\(267\) −10.0000 −0.611990
\(268\) −4.00000 −0.244339
\(269\) 14.0000 0.853595 0.426798 − 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) −2.00000 −0.121716
\(271\) 8.00000 0.485965 0.242983 − 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) −1.00000 −0.0606339
\(273\) 1.00000 0.0605228
\(274\) 6.00000 0.362473
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) −26.0000 −1.56219 −0.781094 − 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) 12.0000 0.719712
\(279\) 0 0
\(280\) 6.00000 0.358569
\(281\) 10.0000 0.596550 0.298275 − 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 − 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −8.00000 −0.474713
\(285\) −8.00000 −0.473879
\(286\) 4.00000 0.236525
\(287\) −2.00000 −0.118056
\(288\) −5.00000 −0.294628
\(289\) 1.00000 0.0588235
\(290\) 12.0000 0.704664
\(291\) −2.00000 −0.117242
\(292\) −10.0000 −0.585206
\(293\) −26.0000 −1.51894 −0.759468 − 0.650545i \(-0.774541\pi\)
−0.759468 + 0.650545i \(0.774541\pi\)
\(294\) 1.00000 0.0583212
\(295\) −24.0000 −1.39733
\(296\) −6.00000 −0.348743
\(297\) 4.00000 0.232104
\(298\) −22.0000 −1.27443
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) −12.0000 −0.691669
\(302\) −8.00000 −0.460348
\(303\) −6.00000 −0.344691
\(304\) 4.00000 0.229416
\(305\) 20.0000 1.14520
\(306\) −1.00000 −0.0571662
\(307\) −20.0000 −1.14146 −0.570730 − 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) −4.00000 −0.227921
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) −3.00000 −0.169842
\(313\) −14.0000 −0.791327 −0.395663 − 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) −14.0000 −0.790066
\(315\) 2.00000 0.112687
\(316\) 8.00000 0.450035
\(317\) −18.0000 −1.01098 −0.505490 − 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) −10.0000 −0.560772
\(319\) −24.0000 −1.34374
\(320\) −14.0000 −0.782624
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) −4.00000 −0.222566
\(324\) −1.00000 −0.0555556
\(325\) −1.00000 −0.0554700
\(326\) 12.0000 0.664619
\(327\) −6.00000 −0.331801
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 8.00000 0.440386
\(331\) −20.0000 −1.09930 −0.549650 − 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) −4.00000 −0.219529
\(333\) −2.00000 −0.109599
\(334\) 8.00000 0.437741
\(335\) −8.00000 −0.437087
\(336\) −1.00000 −0.0545545
\(337\) 18.0000 0.980522 0.490261 − 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 6.00000 0.325875
\(340\) 2.00000 0.108465
\(341\) 0 0
\(342\) 4.00000 0.216295
\(343\) −1.00000 −0.0539949
\(344\) 36.0000 1.94099
\(345\) 0 0
\(346\) 2.00000 0.107521
\(347\) −12.0000 −0.644194 −0.322097 − 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 6.00000 0.321634
\(349\) −2.00000 −0.107058 −0.0535288 − 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −1.00000 −0.0533761
\(352\) 20.0000 1.06600
\(353\) 2.00000 0.106449 0.0532246 − 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) 12.0000 0.637793
\(355\) −16.0000 −0.849192
\(356\) −10.0000 −0.529999
\(357\) 1.00000 0.0529256
\(358\) 4.00000 0.211407
\(359\) −16.0000 −0.844448 −0.422224 − 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) −6.00000 −0.316228
\(361\) −3.00000 −0.157895
\(362\) 18.0000 0.946059
\(363\) −5.00000 −0.262432
\(364\) 1.00000 0.0524142
\(365\) −20.0000 −1.04685
\(366\) −10.0000 −0.522708
\(367\) −24.0000 −1.25279 −0.626395 − 0.779506i \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) 0 0
\(369\) 2.00000 0.104116
\(370\) −4.00000 −0.207950
\(371\) 10.0000 0.519174
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 − 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 4.00000 0.206835
\(375\) −12.0000 −0.619677
\(376\) 0 0
\(377\) 6.00000 0.309016
\(378\) −1.00000 −0.0514344
\(379\) −4.00000 −0.205466 −0.102733 − 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) −8.00000 −0.410391
\(381\) 16.0000 0.819705
\(382\) 24.0000 1.22795
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −3.00000 −0.153093
\(385\) −8.00000 −0.407718
\(386\) −10.0000 −0.508987
\(387\) 12.0000 0.609994
\(388\) −2.00000 −0.101535
\(389\) 22.0000 1.11544 0.557722 − 0.830028i \(-0.311675\pi\)
0.557722 + 0.830028i \(0.311675\pi\)
\(390\) −2.00000 −0.101274
\(391\) 0 0
\(392\) 3.00000 0.151523
\(393\) −12.0000 −0.605320
\(394\) −22.0000 −1.10834
\(395\) 16.0000 0.805047
\(396\) 4.00000 0.201008
\(397\) 30.0000 1.50566 0.752828 − 0.658217i \(-0.228689\pi\)
0.752828 + 0.658217i \(0.228689\pi\)
\(398\) 16.0000 0.802008
\(399\) −4.00000 −0.200250
\(400\) 1.00000 0.0500000
\(401\) 18.0000 0.898877 0.449439 − 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 4.00000 0.199502
\(403\) 0 0
\(404\) −6.00000 −0.298511
\(405\) −2.00000 −0.0993808
\(406\) 6.00000 0.297775
\(407\) 8.00000 0.396545
\(408\) −3.00000 −0.148522
\(409\) −22.0000 −1.08783 −0.543915 − 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 4.00000 0.197546
\(411\) 6.00000 0.295958
\(412\) 0 0
\(413\) −12.0000 −0.590481
\(414\) 0 0
\(415\) −8.00000 −0.392705
\(416\) −5.00000 −0.245145
\(417\) 12.0000 0.587643
\(418\) −16.0000 −0.782586
\(419\) 12.0000 0.586238 0.293119 − 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 2.00000 0.0975900
\(421\) −10.0000 −0.487370 −0.243685 − 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 4.00000 0.194717
\(423\) 0 0
\(424\) −30.0000 −1.45693
\(425\) −1.00000 −0.0485071
\(426\) 8.00000 0.387601
\(427\) 10.0000 0.483934
\(428\) −4.00000 −0.193347
\(429\) 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
\(433\) 18.0000 0.865025 0.432512 − 0.901628i \(-0.357627\pi\)
0.432512 + 0.901628i \(0.357627\pi\)
\(434\) 0 0
\(435\) 12.0000 0.575356
\(436\) −6.00000 −0.287348
\(437\) 0 0
\(438\) 10.0000 0.477818
\(439\) −32.0000 −1.52728 −0.763638 − 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 24.0000 1.14416
\(441\) 1.00000 0.0476190
\(442\) −1.00000 −0.0475651
\(443\) 4.00000 0.190046 0.0950229 − 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) −2.00000 −0.0949158
\(445\) −20.0000 −0.948091
\(446\) −8.00000 −0.378811
\(447\) −22.0000 −1.04056
\(448\) −7.00000 −0.330719
\(449\) −14.0000 −0.660701 −0.330350 − 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) 1.00000 0.0471405
\(451\) −8.00000 −0.376705
\(452\) 6.00000 0.282216
\(453\) −8.00000 −0.375873
\(454\) 12.0000 0.563188
\(455\) 2.00000 0.0937614
\(456\) 12.0000 0.561951
\(457\) 10.0000 0.467780 0.233890 − 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) −6.00000 −0.280362
\(459\) −1.00000 −0.0466760
\(460\) 0 0
\(461\) 30.0000 1.39724 0.698620 − 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 4.00000 0.186097
\(463\) −32.0000 −1.48717 −0.743583 − 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 14.0000 0.648537
\(467\) 20.0000 0.925490 0.462745 − 0.886492i \(-0.346865\pi\)
0.462745 + 0.886492i \(0.346865\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) −14.0000 −0.645086
\(472\) 36.0000 1.65703
\(473\) −48.0000 −2.20704
\(474\) −8.00000 −0.367452
\(475\) 4.00000 0.183533
\(476\) 1.00000 0.0458349
\(477\) −10.0000 −0.457869
\(478\) 8.00000 0.365911
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) −10.0000 −0.456435
\(481\) −2.00000 −0.0911922
\(482\) 14.0000 0.637683
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) −4.00000 −0.181631
\(486\) 1.00000 0.0453609
\(487\) −40.0000 −1.81257 −0.906287 − 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) −30.0000 −1.35804
\(489\) 12.0000 0.542659
\(490\) 2.00000 0.0903508
\(491\) 4.00000 0.180517 0.0902587 − 0.995918i \(-0.471231\pi\)
0.0902587 + 0.995918i \(0.471231\pi\)
\(492\) 2.00000 0.0901670
\(493\) 6.00000 0.270226
\(494\) 4.00000 0.179969
\(495\) 8.00000 0.359573
\(496\) 0 0
\(497\) −8.00000 −0.358849
\(498\) 4.00000 0.179244
\(499\) 36.0000 1.61158 0.805791 − 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) −12.0000 −0.536656
\(501\) 8.00000 0.357414
\(502\) 4.00000 0.178529
\(503\) 16.0000 0.713405 0.356702 − 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) −3.00000 −0.133631
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 16.0000 0.709885
\(509\) −2.00000 −0.0886484 −0.0443242 − 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) −2.00000 −0.0885615
\(511\) −10.0000 −0.442374
\(512\) 11.0000 0.486136
\(513\) 4.00000 0.176604
\(514\) −2.00000 −0.0882162
\(515\) 0 0
\(516\) 12.0000 0.528271
\(517\) 0 0
\(518\) −2.00000 −0.0878750
\(519\) 2.00000 0.0877903
\(520\) −6.00000 −0.263117
\(521\) −6.00000 −0.262865 −0.131432 − 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) −6.00000 −0.262613
\(523\) 4.00000 0.174908 0.0874539 − 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) −12.0000 −0.524222
\(525\) −1.00000 −0.0436436
\(526\) 16.0000 0.697633
\(527\) 0 0
\(528\) −4.00000 −0.174078
\(529\) −23.0000 −1.00000
\(530\) −20.0000 −0.868744
\(531\) 12.0000 0.520756
\(532\) −4.00000 −0.173422
\(533\) 2.00000 0.0866296
\(534\) 10.0000 0.432742
\(535\) −8.00000 −0.345870
\(536\) 12.0000 0.518321
\(537\) 4.00000 0.172613
\(538\) −14.0000 −0.603583
\(539\) −4.00000 −0.172292
\(540\) −2.00000 −0.0860663
\(541\) 6.00000 0.257960 0.128980 − 0.991647i \(-0.458830\pi\)
0.128980 + 0.991647i \(0.458830\pi\)
\(542\) −8.00000 −0.343629
\(543\) 18.0000 0.772454
\(544\) −5.00000 −0.214373
\(545\) −12.0000 −0.514024
\(546\) −1.00000 −0.0427960
\(547\) −20.0000 −0.855138 −0.427569 − 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 6.00000 0.256307
\(549\) −10.0000 −0.426790
\(550\) −4.00000 −0.170561
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 26.0000 1.10463
\(555\) −4.00000 −0.169791
\(556\) 12.0000 0.508913
\(557\) 46.0000 1.94908 0.974541 − 0.224208i \(-0.0719796\pi\)
0.974541 + 0.224208i \(0.0719796\pi\)
\(558\) 0 0
\(559\) 12.0000 0.507546
\(560\) −2.00000 −0.0845154
\(561\) 4.00000 0.168880
\(562\) −10.0000 −0.421825
\(563\) 20.0000 0.842900 0.421450 − 0.906852i \(-0.361521\pi\)
0.421450 + 0.906852i \(0.361521\pi\)
\(564\) 0 0
\(565\) 12.0000 0.504844
\(566\) −4.00000 −0.168133
\(567\) −1.00000 −0.0419961
\(568\) 24.0000 1.00702
\(569\) 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
\(573\) 24.0000 1.00261
\(574\) 2.00000 0.0834784
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) 18.0000 0.749350 0.374675 − 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −10.0000 −0.415586
\(580\) 12.0000 0.498273
\(581\) −4.00000 −0.165948
\(582\) 2.00000 0.0829027
\(583\) 40.0000 1.65663
\(584\) 30.0000 1.24141
\(585\) −2.00000 −0.0826898
\(586\) 26.0000 1.07405
\(587\) 44.0000 1.81607 0.908037 − 0.418890i \(-0.137581\pi\)
0.908037 + 0.418890i \(0.137581\pi\)
\(588\) 1.00000 0.0412393
\(589\) 0 0
\(590\) 24.0000 0.988064
\(591\) −22.0000 −0.904959
\(592\) 2.00000 0.0821995
\(593\) −30.0000 −1.23195 −0.615976 − 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) −4.00000 −0.164122
\(595\) 2.00000 0.0819920
\(596\) −22.0000 −0.901155
\(597\) 16.0000 0.654836
\(598\) 0 0
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 3.00000 0.122474
\(601\) −30.0000 −1.22373 −0.611863 − 0.790964i \(-0.709580\pi\)
−0.611863 + 0.790964i \(0.709580\pi\)
\(602\) 12.0000 0.489083
\(603\) 4.00000 0.162893
\(604\) −8.00000 −0.325515
\(605\) −10.0000 −0.406558
\(606\) 6.00000 0.243733
\(607\) 8.00000 0.324710 0.162355 − 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 20.0000 0.811107
\(609\) 6.00000 0.243132
\(610\) −20.0000 −0.809776
\(611\) 0 0
\(612\) −1.00000 −0.0404226
\(613\) −26.0000 −1.05013 −0.525065 − 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 20.0000 0.807134
\(615\) 4.00000 0.161296
\(616\) 12.0000 0.483494
\(617\) −6.00000 −0.241551 −0.120775 − 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 − 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −10.0000 −0.400642
\(624\) 1.00000 0.0400320
\(625\) −19.0000 −0.760000
\(626\) 14.0000 0.559553
\(627\) −16.0000 −0.638978
\(628\) −14.0000 −0.558661
\(629\) −2.00000 −0.0797452
\(630\) −2.00000 −0.0796819
\(631\) −40.0000 −1.59237 −0.796187 − 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) −24.0000 −0.954669
\(633\) 4.00000 0.158986
\(634\) 18.0000 0.714871
\(635\) 32.0000 1.26988
\(636\) −10.0000 −0.396526
\(637\) 1.00000 0.0396214
\(638\) 24.0000 0.950169
\(639\) 8.00000 0.316475
\(640\) −6.00000 −0.237171
\(641\) 10.0000 0.394976 0.197488 − 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) 4.00000 0.157867
\(643\) 4.00000 0.157745 0.0788723 − 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 0 0
\(645\) 24.0000 0.944999
\(646\) 4.00000 0.157378
\(647\) 8.00000 0.314512 0.157256 − 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) 3.00000 0.117851
\(649\) −48.0000 −1.88416
\(650\) 1.00000 0.0392232
\(651\) 0 0
\(652\) 12.0000 0.469956
\(653\) 6.00000 0.234798 0.117399 − 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 6.00000 0.234619
\(655\) −24.0000 −0.937758
\(656\) −2.00000 −0.0780869
\(657\) 10.0000 0.390137
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 − 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 8.00000 0.311400
\(661\) −10.0000 −0.388955 −0.194477 − 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 20.0000 0.777322
\(663\) −1.00000 −0.0388368
\(664\) 12.0000 0.465690
\(665\) −8.00000 −0.310227
\(666\) 2.00000 0.0774984
\(667\) 0 0
\(668\) 8.00000 0.309529
\(669\) −8.00000 −0.309298
\(670\) 8.00000 0.309067
\(671\) 40.0000 1.54418
\(672\) −5.00000 −0.192879
\(673\) −14.0000 −0.539660 −0.269830 − 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) −18.0000 −0.693334
\(675\) 1.00000 0.0384900
\(676\) −1.00000 −0.0384615
\(677\) −26.0000 −0.999261 −0.499631 − 0.866239i \(-0.666531\pi\)
−0.499631 + 0.866239i \(0.666531\pi\)
\(678\) −6.00000 −0.230429
\(679\) −2.00000 −0.0767530
\(680\) −6.00000 −0.230089
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) −4.00000 −0.153056 −0.0765279 − 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 4.00000 0.152944
\(685\) 12.0000 0.458496
\(686\) 1.00000 0.0381802
\(687\) −6.00000 −0.228914
\(688\) −12.0000 −0.457496
\(689\) −10.0000 −0.380970
\(690\) 0 0
\(691\) −12.0000 −0.456502 −0.228251 − 0.973602i \(-0.573301\pi\)
−0.228251 + 0.973602i \(0.573301\pi\)
\(692\) 2.00000 0.0760286
\(693\) 4.00000 0.151947
\(694\) 12.0000 0.455514
\(695\) 24.0000 0.910372
\(696\) −18.0000 −0.682288
\(697\) 2.00000 0.0757554
\(698\) 2.00000 0.0757011
\(699\) 14.0000 0.529529
\(700\) −1.00000 −0.0377964
\(701\) 30.0000 1.13308 0.566542 − 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 1.00000 0.0377426
\(703\) 8.00000 0.301726
\(704\) −28.0000 −1.05529
\(705\) 0 0
\(706\) −2.00000 −0.0752710
\(707\) −6.00000 −0.225653
\(708\) 12.0000 0.450988
\(709\) −50.0000 −1.87779 −0.938895 − 0.344204i \(-0.888149\pi\)
−0.938895 + 0.344204i \(0.888149\pi\)
\(710\) 16.0000 0.600469
\(711\) −8.00000 −0.300023
\(712\) 30.0000 1.12430
\(713\) 0 0
\(714\) −1.00000 −0.0374241
\(715\) 8.00000 0.299183
\(716\) 4.00000 0.149487
\(717\) 8.00000 0.298765
\(718\) 16.0000 0.597115
\(719\) 24.0000 0.895049 0.447524 − 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 2.00000 0.0745356
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) 14.0000 0.520666
\(724\) 18.0000 0.668965
\(725\) −6.00000 −0.222834
\(726\) 5.00000 0.185567
\(727\) 32.0000 1.18681 0.593407 − 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) −3.00000 −0.111187
\(729\) 1.00000 0.0370370
\(730\) 20.0000 0.740233
\(731\) 12.0000 0.443836
\(732\) −10.0000 −0.369611
\(733\) 30.0000 1.10808 0.554038 − 0.832492i \(-0.313086\pi\)
0.554038 + 0.832492i \(0.313086\pi\)
\(734\) 24.0000 0.885856
\(735\) 2.00000 0.0737711
\(736\) 0 0
\(737\) −16.0000 −0.589368
\(738\) −2.00000 −0.0736210
\(739\) 4.00000 0.147142 0.0735712 − 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) −4.00000 −0.147043
\(741\) 4.00000 0.146944
\(742\) −10.0000 −0.367112
\(743\) 8.00000 0.293492 0.146746 − 0.989174i \(-0.453120\pi\)
0.146746 + 0.989174i \(0.453120\pi\)
\(744\) 0 0
\(745\) −44.0000 −1.61204
\(746\) 10.0000 0.366126
\(747\) 4.00000 0.146352
\(748\) 4.00000 0.146254
\(749\) −4.00000 −0.146157
\(750\) 12.0000 0.438178
\(751\) 40.0000 1.45962 0.729810 − 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 0 0
\(753\) 4.00000 0.145768
\(754\) −6.00000 −0.218507
\(755\) −16.0000 −0.582300
\(756\) −1.00000 −0.0363696
\(757\) −10.0000 −0.363456 −0.181728 − 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) 24.0000 0.870572
\(761\) 10.0000 0.362500 0.181250 − 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) −16.0000 −0.579619
\(763\) −6.00000 −0.217215
\(764\) 24.0000 0.868290
\(765\) −2.00000 −0.0723102
\(766\) 0 0
\(767\) 12.0000 0.433295
\(768\) 17.0000 0.613435
\(769\) 18.0000 0.649097 0.324548 − 0.945869i \(-0.394788\pi\)
0.324548 + 0.945869i \(0.394788\pi\)
\(770\) 8.00000 0.288300
\(771\) −2.00000 −0.0720282
\(772\) −10.0000 −0.359908
\(773\) 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
\(776\) 6.00000 0.215387
\(777\) −2.00000 −0.0717496
\(778\) −22.0000 −0.788738
\(779\) −8.00000 −0.286630
\(780\) −2.00000 −0.0716115
\(781\) −32.0000 −1.14505
\(782\) 0 0
\(783\) −6.00000 −0.214423
\(784\) −1.00000 −0.0357143
\(785\) −28.0000 −0.999363
\(786\) 12.0000 0.428026
\(787\) 20.0000 0.712923 0.356462 − 0.934310i \(-0.383983\pi\)
0.356462 + 0.934310i \(0.383983\pi\)
\(788\) −22.0000 −0.783718
\(789\) 16.0000 0.569615
\(790\) −16.0000 −0.569254
\(791\) 6.00000 0.213335
\(792\) −12.0000 −0.426401
\(793\) −10.0000 −0.355110
\(794\) −30.0000 −1.06466
\(795\) −20.0000 −0.709327
\(796\) 16.0000 0.567105
\(797\) 30.0000 1.06265 0.531327 − 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 4.00000 0.141598
\(799\) 0 0
\(800\) 5.00000 0.176777
\(801\) 10.0000 0.353333
\(802\) −18.0000 −0.635602
\(803\) −40.0000 −1.41157
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 0 0
\(807\) −14.0000 −0.492823
\(808\) 18.0000 0.633238
\(809\) −30.0000 −1.05474 −0.527372 − 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 2.00000 0.0702728
\(811\) −4.00000 −0.140459 −0.0702295 − 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) 6.00000 0.210559
\(813\) −8.00000 −0.280572
\(814\) −8.00000 −0.280400
\(815\) 24.0000 0.840683
\(816\) 1.00000 0.0350070
\(817\) −48.0000 −1.67931
\(818\) 22.0000 0.769212
\(819\) −1.00000 −0.0349428
\(820\) 4.00000 0.139686
\(821\) −10.0000 −0.349002 −0.174501 − 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) −6.00000 −0.209274
\(823\) −16.0000 −0.557725 −0.278862 − 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 0 0
\(825\) −4.00000 −0.139262
\(826\) 12.0000 0.417533
\(827\) −4.00000 −0.139094 −0.0695468 − 0.997579i \(-0.522155\pi\)
−0.0695468 + 0.997579i \(0.522155\pi\)
\(828\) 0 0
\(829\) −18.0000 −0.625166 −0.312583 − 0.949890i \(-0.601194\pi\)
−0.312583 + 0.949890i \(0.601194\pi\)
\(830\) 8.00000 0.277684
\(831\) 26.0000 0.901930
\(832\) 7.00000 0.242681
\(833\) 1.00000 0.0346479
\(834\) −12.0000 −0.415526
\(835\) 16.0000 0.553703
\(836\) −16.0000 −0.553372
\(837\) 0 0
\(838\) −12.0000 −0.414533
\(839\) −8.00000 −0.276191 −0.138095 − 0.990419i \(-0.544098\pi\)
−0.138095 + 0.990419i \(0.544098\pi\)
\(840\) −6.00000 −0.207020
\(841\) 7.00000 0.241379
\(842\) 10.0000 0.344623
\(843\) −10.0000 −0.344418
\(844\) 4.00000 0.137686
\(845\) −2.00000 −0.0688021
\(846\) 0 0
\(847\) −5.00000 −0.171802
\(848\) 10.0000 0.343401
\(849\) −4.00000 −0.137280
\(850\) 1.00000 0.0342997
\(851\) 0 0
\(852\) 8.00000 0.274075
\(853\) −10.0000 −0.342393 −0.171197 − 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) −10.0000 −0.342193
\(855\) 8.00000 0.273594
\(856\) 12.0000 0.410152
\(857\) −38.0000 −1.29806 −0.649028 − 0.760765i \(-0.724824\pi\)
−0.649028 + 0.760765i \(0.724824\pi\)
\(858\) −4.00000 −0.136558
\(859\) 20.0000 0.682391 0.341196 − 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 24.0000 0.818393
\(861\) 2.00000 0.0681598
\(862\) −16.0000 −0.544962
\(863\) 24.0000 0.816970 0.408485 − 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 5.00000 0.170103
\(865\) 4.00000 0.136004
\(866\) −18.0000 −0.611665
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) 32.0000 1.08553
\(870\) −12.0000 −0.406838
\(871\) 4.00000 0.135535
\(872\) 18.0000 0.609557
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) −12.0000 −0.405674
\(876\) 10.0000 0.337869
\(877\) −26.0000 −0.877958 −0.438979 − 0.898497i \(-0.644660\pi\)
−0.438979 + 0.898497i \(0.644660\pi\)
\(878\) 32.0000 1.07995
\(879\) 26.0000 0.876958
\(880\) −8.00000 −0.269680
\(881\) −30.0000 −1.01073 −0.505363 − 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 4.00000 0.134611 0.0673054 − 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) −1.00000 −0.0336336
\(885\) 24.0000 0.806751
\(886\) −4.00000 −0.134383
\(887\) −48.0000 −1.61168 −0.805841 − 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 6.00000 0.201347
\(889\) 16.0000 0.536623
\(890\) 20.0000 0.670402
\(891\) −4.00000 −0.134005
\(892\) −8.00000 −0.267860
\(893\) 0 0
\(894\) 22.0000 0.735790
\(895\) 8.00000 0.267411
\(896\) −3.00000 −0.100223
\(897\) 0 0
\(898\) 14.0000 0.467186
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) −10.0000 −0.333148
\(902\) 8.00000 0.266371
\(903\) 12.0000 0.399335
\(904\) −18.0000 −0.598671
\(905\) 36.0000 1.19668
\(906\) 8.00000 0.265782
\(907\) −12.0000 −0.398453 −0.199227 − 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) 12.0000 0.398234
\(909\) 6.00000 0.199007
\(910\) −2.00000 −0.0662994
\(911\) −24.0000 −0.795155 −0.397578 − 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) −4.00000 −0.132453
\(913\) −16.0000 −0.529523
\(914\) −10.0000 −0.330771
\(915\) −20.0000 −0.661180
\(916\) −6.00000 −0.198246
\(917\) −12.0000 −0.396275
\(918\) 1.00000 0.0330049
\(919\) −8.00000 −0.263896 −0.131948 − 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) −30.0000 −0.987997
\(923\) 8.00000 0.263323
\(924\) 4.00000 0.131590
\(925\) 2.00000 0.0657596
\(926\) 32.0000 1.05159
\(927\) 0 0
\(928\) −30.0000 −0.984798
\(929\) 42.0000 1.37798 0.688988 − 0.724773i \(-0.258055\pi\)
0.688988 + 0.724773i \(0.258055\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) 14.0000 0.458585
\(933\) 0 0
\(934\) −20.0000 −0.654420
\(935\) 8.00000 0.261628
\(936\) 3.00000 0.0980581
\(937\) 26.0000 0.849383 0.424691 − 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 4.00000 0.130605
\(939\) 14.0000 0.456873
\(940\) 0 0
\(941\) 6.00000 0.195594 0.0977972 − 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) 14.0000 0.456145
\(943\) 0 0
\(944\) −12.0000 −0.390567
\(945\) −2.00000 −0.0650600
\(946\) 48.0000 1.56061
\(947\) −44.0000 −1.42981 −0.714904 − 0.699223i \(-0.753530\pi\)
−0.714904 + 0.699223i \(0.753530\pi\)
\(948\) −8.00000 −0.259828
\(949\) 10.0000 0.324614
\(950\) −4.00000 −0.129777
\(951\) 18.0000 0.583690
\(952\) −3.00000 −0.0972306
\(953\) −22.0000 −0.712650 −0.356325 − 0.934362i \(-0.615970\pi\)
−0.356325 + 0.934362i \(0.615970\pi\)
\(954\) 10.0000 0.323762
\(955\) 48.0000 1.55324
\(956\) 8.00000 0.258738
\(957\) 24.0000 0.775810
\(958\) 0 0
\(959\) 6.00000 0.193750
\(960\) 14.0000 0.451848
\(961\) −31.0000 −1.00000
\(962\) 2.00000 0.0644826
\(963\) 4.00000 0.128898
\(964\) 14.0000 0.450910
\(965\) −20.0000 −0.643823
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 − 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 15.0000 0.482118
\(969\) 4.00000 0.128499
\(970\) 4.00000 0.128432
\(971\) −52.0000 −1.66876 −0.834380 − 0.551190i \(-0.814174\pi\)
−0.834380 + 0.551190i \(0.814174\pi\)
\(972\) 1.00000 0.0320750
\(973\) 12.0000 0.384702
\(974\) 40.0000 1.28168
\(975\) 1.00000 0.0320256
\(976\) 10.0000 0.320092
\(977\) 34.0000 1.08776 0.543878 − 0.839164i \(-0.316955\pi\)
0.543878 + 0.839164i \(0.316955\pi\)
\(978\) −12.0000 −0.383718
\(979\) −40.0000 −1.27841
\(980\) 2.00000 0.0638877
\(981\) 6.00000 0.191565
\(982\) −4.00000 −0.127645
\(983\) −24.0000 −0.765481 −0.382741 − 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) −6.00000 −0.191273
\(985\) −44.0000 −1.40196
\(986\) −6.00000 −0.191079
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) 0 0
\(990\) −8.00000 −0.254257
\(991\) 8.00000 0.254128 0.127064 − 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 0 0
\(993\) 20.0000 0.634681
\(994\) 8.00000 0.253745
\(995\) 32.0000 1.01447
\(996\) 4.00000 0.126745
\(997\) 30.0000 0.950110 0.475055 − 0.879956i \(-0.342428\pi\)
0.475055 + 0.879956i \(0.342428\pi\)
\(998\) −36.0000 −1.13956
\(999\) 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 4641.2.a.b.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4641.2.a.b.1.1 ✓ 1 1.1 even 1 trivial