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Properties

Label 1610.2.a.c.1.1
Level $1610$
Weight $2$
Character 1610.1
Self dual yes
Analytic conductor $12.856$
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] = [1610,2,Mod(1,1610)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1610, 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("1610.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1610 = 2 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1610.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: \(12.8559147254\)
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 \(\chi\) \(=\) 1610.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.00000 q^{12} +2.00000 q^{13} +1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} -2.00000 q^{21} -1.00000 q^{23} -2.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} +4.00000 q^{27} +1.00000 q^{28} +6.00000 q^{29} +2.00000 q^{30} +2.00000 q^{31} +1.00000 q^{32} -6.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} -10.0000 q^{37} -4.00000 q^{38} -4.00000 q^{39} -1.00000 q^{40} -6.00000 q^{41} -2.00000 q^{42} -10.0000 q^{43} -1.00000 q^{45} -1.00000 q^{46} -2.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +12.0000 q^{51} +2.00000 q^{52} +6.00000 q^{53} +4.00000 q^{54} +1.00000 q^{56} +8.00000 q^{57} +6.00000 q^{58} -12.0000 q^{59} +2.00000 q^{60} -10.0000 q^{61} +2.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +2.00000 q^{67} -6.00000 q^{68} +2.00000 q^{69} -1.00000 q^{70} +12.0000 q^{71} +1.00000 q^{72} -16.0000 q^{73} -10.0000 q^{74} -2.00000 q^{75} -4.00000 q^{76} -4.00000 q^{78} -10.0000 q^{79} -1.00000 q^{80} -11.0000 q^{81} -6.00000 q^{82} -12.0000 q^{83} -2.00000 q^{84} +6.00000 q^{85} -10.0000 q^{86} -12.0000 q^{87} -6.00000 q^{89} -1.00000 q^{90} +2.00000 q^{91} -1.00000 q^{92} -4.00000 q^{93} +4.00000 q^{95} -2.00000 q^{96} -10.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
\(3\) −2.00000 −1.15470 −0.577350 − 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −2.00000 −0.816497
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −2.00000 −0.577350
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 1.00000 0.267261
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −1.00000 −0.223607
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) −2.00000 −0.408248
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 4.00000 0.769800
\(28\) 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
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) −1.00000 −0.169031
\(36\) 1.00000 0.166667
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −4.00000 −0.648886
\(39\) −4.00000 −0.640513
\(40\) −1.00000 −0.158114
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −2.00000 −0.308607
\(43\) −10.0000 −1.52499 −0.762493 − 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) −1.00000 −0.147442
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −2.00000 −0.288675
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 12.0000 1.68034
\(52\) 2.00000 0.277350
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 8.00000 1.05963
\(58\) 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
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 2.00000 0.254000
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 − 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −6.00000 −0.727607
\(69\) 2.00000 0.240772
\(70\) −1.00000 −0.119523
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 1.00000 0.117851
\(73\) −16.0000 −1.87266 −0.936329 − 0.351123i \(-0.885800\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) −10.0000 −1.16248
\(75\) −2.00000 −0.230940
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) −4.00000 −0.452911
\(79\) −10.0000 −1.12509 −0.562544 − 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) −1.00000 −0.111803
\(81\) −11.0000 −1.22222
\(82\) −6.00000 −0.662589
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) −2.00000 −0.218218
\(85\) 6.00000 0.650791
\(86\) −10.0000 −1.07833
\(87\) −12.0000 −1.28654
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −1.00000 −0.105409
\(91\) 2.00000 0.209657
\(92\) −1.00000 −0.104257
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) −2.00000 −0.204124
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 12.0000 1.19404 0.597022 − 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 12.0000 1.18818
\(103\) −16.0000 −1.57653 −0.788263 − 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 2.00000 0.196116
\(105\) 2.00000 0.195180
\(106\) 6.00000 0.582772
\(107\) 18.0000 1.74013 0.870063 − 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) 4.00000 0.384900
\(109\) −4.00000 −0.383131 −0.191565 − 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) 20.0000 1.89832
\(112\) 1.00000 0.0944911
\(113\) −12.0000 −1.12887 −0.564433 − 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 8.00000 0.749269
\(115\) 1.00000 0.0932505
\(116\) 6.00000 0.557086
\(117\) 2.00000 0.184900
\(118\) −12.0000 −1.10469
\(119\) −6.00000 −0.550019
\(120\) 2.00000 0.182574
\(121\) −11.0000 −1.00000
\(122\) −10.0000 −0.905357
\(123\) 12.0000 1.08200
\(124\) 2.00000 0.179605
\(125\) −1.00000 −0.0894427
\(126\) 1.00000 0.0890871
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 20.0000 1.76090
\(130\) −2.00000 −0.175412
\(131\) 12.0000 1.04844 0.524222 − 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) 2.00000 0.172774
\(135\) −4.00000 −0.344265
\(136\) −6.00000 −0.514496
\(137\) 12.0000 1.02523 0.512615 − 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 2.00000 0.170251
\(139\) 20.0000 1.69638 0.848189 − 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) 12.0000 1.00702
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −6.00000 −0.498273
\(146\) −16.0000 −1.32417
\(147\) −2.00000 −0.164957
\(148\) −10.0000 −0.821995
\(149\) −12.0000 −0.983078 −0.491539 − 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) −2.00000 −0.163299
\(151\) 8.00000 0.651031 0.325515 − 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −4.00000 −0.324443
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) −2.00000 −0.160644
\(156\) −4.00000 −0.320256
\(157\) 14.0000 1.11732 0.558661 − 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) −10.0000 −0.795557
\(159\) −12.0000 −0.951662
\(160\) −1.00000 −0.0790569
\(161\) −1.00000 −0.0788110
\(162\) −11.0000 −0.864242
\(163\) −16.0000 −1.25322 −0.626608 − 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −2.00000 −0.154303
\(169\) −9.00000 −0.692308
\(170\) 6.00000 0.460179
\(171\) −4.00000 −0.305888
\(172\) −10.0000 −0.762493
\(173\) 6.00000 0.456172 0.228086 − 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −12.0000 −0.909718
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 24.0000 1.80395
\(178\) −6.00000 −0.449719
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 26.0000 1.93256 0.966282 − 0.257485i \(-0.0828937\pi\)
0.966282 + 0.257485i \(0.0828937\pi\)
\(182\) 2.00000 0.148250
\(183\) 20.0000 1.47844
\(184\) −1.00000 −0.0737210
\(185\) 10.0000 0.735215
\(186\) −4.00000 −0.293294
\(187\) 0 0
\(188\) 0 0
\(189\) 4.00000 0.290957
\(190\) 4.00000 0.290191
\(191\) 18.0000 1.30243 0.651217 − 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) −2.00000 −0.144338
\(193\) −10.0000 −0.719816 −0.359908 − 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) −10.0000 −0.717958
\(195\) 4.00000 0.286446
\(196\) 1.00000 0.0714286
\(197\) −6.00000 −0.427482 −0.213741 − 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 − 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 1.00000 0.0707107
\(201\) −4.00000 −0.282138
\(202\) 12.0000 0.844317
\(203\) 6.00000 0.421117
\(204\) 12.0000 0.840168
\(205\) 6.00000 0.419058
\(206\) −16.0000 −1.11477
\(207\) −1.00000 −0.0695048
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) 2.00000 0.138013
\(211\) −4.00000 −0.275371 −0.137686 − 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 6.00000 0.412082
\(213\) −24.0000 −1.64445
\(214\) 18.0000 1.23045
\(215\) 10.0000 0.681994
\(216\) 4.00000 0.272166
\(217\) 2.00000 0.135769
\(218\) −4.00000 −0.270914
\(219\) 32.0000 2.16236
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 20.0000 1.34231
\(223\) 8.00000 0.535720 0.267860 − 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) −12.0000 −0.798228
\(227\) −12.0000 −0.796468 −0.398234 − 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 8.00000 0.529813
\(229\) 2.00000 0.132164 0.0660819 − 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 1.00000 0.0659380
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) −18.0000 −1.17922 −0.589610 − 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) 20.0000 1.29914
\(238\) −6.00000 −0.388922
\(239\) 12.0000 0.776215 0.388108 − 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 2.00000 0.129099
\(241\) −10.0000 −0.644157 −0.322078 − 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) −11.0000 −0.707107
\(243\) 10.0000 0.641500
\(244\) −10.0000 −0.640184
\(245\) −1.00000 −0.0638877
\(246\) 12.0000 0.765092
\(247\) −8.00000 −0.509028
\(248\) 2.00000 0.127000
\(249\) 24.0000 1.52094
\(250\) −1.00000 −0.0632456
\(251\) 12.0000 0.757433 0.378717 − 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) −12.0000 −0.751469
\(256\) 1.00000 0.0625000
\(257\) −24.0000 −1.49708 −0.748539 − 0.663090i \(-0.769245\pi\)
−0.748539 + 0.663090i \(0.769245\pi\)
\(258\) 20.0000 1.24515
\(259\) −10.0000 −0.621370
\(260\) −2.00000 −0.124035
\(261\) 6.00000 0.371391
\(262\) 12.0000 0.741362
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) −4.00000 −0.245256
\(267\) 12.0000 0.734388
\(268\) 2.00000 0.122169
\(269\) −12.0000 −0.731653 −0.365826 − 0.930683i \(-0.619214\pi\)
−0.365826 + 0.930683i \(0.619214\pi\)
\(270\) −4.00000 −0.243432
\(271\) 2.00000 0.121491 0.0607457 − 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) −6.00000 −0.363803
\(273\) −4.00000 −0.242091
\(274\) 12.0000 0.724947
\(275\) 0 0
\(276\) 2.00000 0.120386
\(277\) −10.0000 −0.600842 −0.300421 − 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 20.0000 1.19952
\(279\) 2.00000 0.119737
\(280\) −1.00000 −0.0597614
\(281\) 30.0000 1.78965 0.894825 − 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 0 0
\(283\) −16.0000 −0.951101 −0.475551 − 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 12.0000 0.712069
\(285\) −8.00000 −0.473879
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) −6.00000 −0.352332
\(291\) 20.0000 1.17242
\(292\) −16.0000 −0.936329
\(293\) −6.00000 −0.350524 −0.175262 − 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) −2.00000 −0.116642
\(295\) 12.0000 0.698667
\(296\) −10.0000 −0.581238
\(297\) 0 0
\(298\) −12.0000 −0.695141
\(299\) −2.00000 −0.115663
\(300\) −2.00000 −0.115470
\(301\) −10.0000 −0.576390
\(302\) 8.00000 0.460348
\(303\) −24.0000 −1.37876
\(304\) −4.00000 −0.229416
\(305\) 10.0000 0.572598
\(306\) −6.00000 −0.342997
\(307\) −22.0000 −1.25561 −0.627803 − 0.778372i \(-0.716046\pi\)
−0.627803 + 0.778372i \(0.716046\pi\)
\(308\) 0 0
\(309\) 32.0000 1.82042
\(310\) −2.00000 −0.113592
\(311\) 18.0000 1.02069 0.510343 − 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) −4.00000 −0.226455
\(313\) 26.0000 1.46961 0.734803 − 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) 14.0000 0.790066
\(315\) −1.00000 −0.0563436
\(316\) −10.0000 −0.562544
\(317\) 18.0000 1.01098 0.505490 − 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) −12.0000 −0.672927
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −36.0000 −2.00932
\(322\) −1.00000 −0.0557278
\(323\) 24.0000 1.33540
\(324\) −11.0000 −0.611111
\(325\) 2.00000 0.110940
\(326\) −16.0000 −0.886158
\(327\) 8.00000 0.442401
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 − 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −12.0000 −0.658586
\(333\) −10.0000 −0.547997
\(334\) 0 0
\(335\) −2.00000 −0.109272
\(336\) −2.00000 −0.109109
\(337\) −28.0000 −1.52526 −0.762629 − 0.646837i \(-0.776092\pi\)
−0.762629 + 0.646837i \(0.776092\pi\)
\(338\) −9.00000 −0.489535
\(339\) 24.0000 1.30350
\(340\) 6.00000 0.325396
\(341\) 0 0
\(342\) −4.00000 −0.216295
\(343\) 1.00000 0.0539949
\(344\) −10.0000 −0.539164
\(345\) −2.00000 −0.107676
\(346\) 6.00000 0.322562
\(347\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) −12.0000 −0.643268
\(349\) 8.00000 0.428230 0.214115 − 0.976808i \(-0.431313\pi\)
0.214115 + 0.976808i \(0.431313\pi\)
\(350\) 1.00000 0.0534522
\(351\) 8.00000 0.427008
\(352\) 0 0
\(353\) −12.0000 −0.638696 −0.319348 − 0.947638i \(-0.603464\pi\)
−0.319348 + 0.947638i \(0.603464\pi\)
\(354\) 24.0000 1.27559
\(355\) −12.0000 −0.636894
\(356\) −6.00000 −0.317999
\(357\) 12.0000 0.635107
\(358\) 12.0000 0.634220
\(359\) 18.0000 0.950004 0.475002 − 0.879985i \(-0.342447\pi\)
0.475002 + 0.879985i \(0.342447\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −3.00000 −0.157895
\(362\) 26.0000 1.36653
\(363\) 22.0000 1.15470
\(364\) 2.00000 0.104828
\(365\) 16.0000 0.837478
\(366\) 20.0000 1.04542
\(367\) 32.0000 1.67039 0.835193 − 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −6.00000 −0.312348
\(370\) 10.0000 0.519875
\(371\) 6.00000 0.311504
\(372\) −4.00000 −0.207390
\(373\) 2.00000 0.103556 0.0517780 − 0.998659i \(-0.483511\pi\)
0.0517780 + 0.998659i \(0.483511\pi\)
\(374\) 0 0
\(375\) 2.00000 0.103280
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 4.00000 0.205738
\(379\) −16.0000 −0.821865 −0.410932 − 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 4.00000 0.205196
\(381\) −16.0000 −0.819705
\(382\) 18.0000 0.920960
\(383\) −24.0000 −1.22634 −0.613171 − 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) −2.00000 −0.102062
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) −10.0000 −0.508329
\(388\) −10.0000 −0.507673
\(389\) −12.0000 −0.608424 −0.304212 − 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(390\) 4.00000 0.202548
\(391\) 6.00000 0.303433
\(392\) 1.00000 0.0505076
\(393\) −24.0000 −1.21064
\(394\) −6.00000 −0.302276
\(395\) 10.0000 0.503155
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 − 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 20.0000 1.00251
\(399\) 8.00000 0.400501
\(400\) 1.00000 0.0500000
\(401\) −18.0000 −0.898877 −0.449439 − 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) −4.00000 −0.199502
\(403\) 4.00000 0.199254
\(404\) 12.0000 0.597022
\(405\) 11.0000 0.546594
\(406\) 6.00000 0.297775
\(407\) 0 0
\(408\) 12.0000 0.594089
\(409\) −34.0000 −1.68119 −0.840596 − 0.541663i \(-0.817795\pi\)
−0.840596 + 0.541663i \(0.817795\pi\)
\(410\) 6.00000 0.296319
\(411\) −24.0000 −1.18383
\(412\) −16.0000 −0.788263
\(413\) −12.0000 −0.590481
\(414\) −1.00000 −0.0491473
\(415\) 12.0000 0.589057
\(416\) 2.00000 0.0980581
\(417\) −40.0000 −1.95881
\(418\) 0 0
\(419\) 36.0000 1.75872 0.879358 − 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 2.00000 0.0975900
\(421\) 32.0000 1.55958 0.779792 − 0.626038i \(-0.215325\pi\)
0.779792 + 0.626038i \(0.215325\pi\)
\(422\) −4.00000 −0.194717
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) −6.00000 −0.291043
\(426\) −24.0000 −1.16280
\(427\) −10.0000 −0.483934
\(428\) 18.0000 0.870063
\(429\) 0 0
\(430\) 10.0000 0.482243
\(431\) −30.0000 −1.44505 −0.722525 − 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) 4.00000 0.192450
\(433\) 14.0000 0.672797 0.336399 − 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 2.00000 0.0960031
\(435\) 12.0000 0.575356
\(436\) −4.00000 −0.191565
\(437\) 4.00000 0.191346
\(438\) 32.0000 1.52902
\(439\) −10.0000 −0.477274 −0.238637 − 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −12.0000 −0.570782
\(443\) 24.0000 1.14027 0.570137 − 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 20.0000 0.949158
\(445\) 6.00000 0.284427
\(446\) 8.00000 0.378811
\(447\) 24.0000 1.13516
\(448\) 1.00000 0.0472456
\(449\) −30.0000 −1.41579 −0.707894 − 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) −12.0000 −0.564433
\(453\) −16.0000 −0.751746
\(454\) −12.0000 −0.563188
\(455\) −2.00000 −0.0937614
\(456\) 8.00000 0.374634
\(457\) 20.0000 0.935561 0.467780 − 0.883845i \(-0.345054\pi\)
0.467780 + 0.883845i \(0.345054\pi\)
\(458\) 2.00000 0.0934539
\(459\) −24.0000 −1.12022
\(460\) 1.00000 0.0466252
\(461\) 36.0000 1.67669 0.838344 − 0.545142i \(-0.183524\pi\)
0.838344 + 0.545142i \(0.183524\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 − 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 6.00000 0.278543
\(465\) 4.00000 0.185496
\(466\) −18.0000 −0.833834
\(467\) 24.0000 1.11059 0.555294 − 0.831654i \(-0.312606\pi\)
0.555294 + 0.831654i \(0.312606\pi\)
\(468\) 2.00000 0.0924500
\(469\) 2.00000 0.0923514
\(470\) 0 0
\(471\) −28.0000 −1.29017
\(472\) −12.0000 −0.552345
\(473\) 0 0
\(474\) 20.0000 0.918630
\(475\) −4.00000 −0.183533
\(476\) −6.00000 −0.275010
\(477\) 6.00000 0.274721
\(478\) 12.0000 0.548867
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 2.00000 0.0912871
\(481\) −20.0000 −0.911922
\(482\) −10.0000 −0.455488
\(483\) 2.00000 0.0910032
\(484\) −11.0000 −0.500000
\(485\) 10.0000 0.454077
\(486\) 10.0000 0.453609
\(487\) −40.0000 −1.81257 −0.906287 − 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) −10.0000 −0.452679
\(489\) 32.0000 1.44709
\(490\) −1.00000 −0.0451754
\(491\) 12.0000 0.541552 0.270776 − 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 12.0000 0.541002
\(493\) −36.0000 −1.62136
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) 12.0000 0.538274
\(498\) 24.0000 1.07547
\(499\) 20.0000 0.895323 0.447661 − 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 12.0000 0.535586
\(503\) −24.0000 −1.07011 −0.535054 − 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 1.00000 0.0445435
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) 18.0000 0.799408
\(508\) 8.00000 0.354943
\(509\) −24.0000 −1.06378 −0.531891 − 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) −12.0000 −0.531369
\(511\) −16.0000 −0.707798
\(512\) 1.00000 0.0441942
\(513\) −16.0000 −0.706417
\(514\) −24.0000 −1.05859
\(515\) 16.0000 0.705044
\(516\) 20.0000 0.880451
\(517\) 0 0
\(518\) −10.0000 −0.439375
\(519\) −12.0000 −0.526742
\(520\) −2.00000 −0.0877058
\(521\) 6.00000 0.262865 0.131432 − 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 6.00000 0.262613
\(523\) 8.00000 0.349816 0.174908 − 0.984585i \(-0.444037\pi\)
0.174908 + 0.984585i \(0.444037\pi\)
\(524\) 12.0000 0.524222
\(525\) −2.00000 −0.0872872
\(526\) 0 0
\(527\) −12.0000 −0.522728
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −6.00000 −0.260623
\(531\) −12.0000 −0.520756
\(532\) −4.00000 −0.173422
\(533\) −12.0000 −0.519778
\(534\) 12.0000 0.519291
\(535\) −18.0000 −0.778208
\(536\) 2.00000 0.0863868
\(537\) −24.0000 −1.03568
\(538\) −12.0000 −0.517357
\(539\) 0 0
\(540\) −4.00000 −0.172133
\(541\) −22.0000 −0.945854 −0.472927 − 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 2.00000 0.0859074
\(543\) −52.0000 −2.23153
\(544\) −6.00000 −0.257248
\(545\) 4.00000 0.171341
\(546\) −4.00000 −0.171184
\(547\) 8.00000 0.342055 0.171028 − 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 12.0000 0.512615
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) 2.00000 0.0851257
\(553\) −10.0000 −0.425243
\(554\) −10.0000 −0.424859
\(555\) −20.0000 −0.848953
\(556\) 20.0000 0.848189
\(557\) −30.0000 −1.27114 −0.635570 − 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 2.00000 0.0846668
\(559\) −20.0000 −0.845910
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 30.0000 1.26547
\(563\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 12.0000 0.504844
\(566\) −16.0000 −0.672530
\(567\) −11.0000 −0.461957
\(568\) 12.0000 0.503509
\(569\) 30.0000 1.25767 0.628833 − 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) −8.00000 −0.335083
\(571\) −16.0000 −0.669579 −0.334790 − 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) 0 0
\(573\) −36.0000 −1.50392
\(574\) −6.00000 −0.250435
\(575\) −1.00000 −0.0417029
\(576\) 1.00000 0.0416667
\(577\) 8.00000 0.333044 0.166522 − 0.986038i \(-0.446746\pi\)
0.166522 + 0.986038i \(0.446746\pi\)
\(578\) 19.0000 0.790296
\(579\) 20.0000 0.831172
\(580\) −6.00000 −0.249136
\(581\) −12.0000 −0.497844
\(582\) 20.0000 0.829027
\(583\) 0 0
\(584\) −16.0000 −0.662085
\(585\) −2.00000 −0.0826898
\(586\) −6.00000 −0.247858
\(587\) 18.0000 0.742940 0.371470 − 0.928445i \(-0.378854\pi\)
0.371470 + 0.928445i \(0.378854\pi\)
\(588\) −2.00000 −0.0824786
\(589\) −8.00000 −0.329634
\(590\) 12.0000 0.494032
\(591\) 12.0000 0.493614
\(592\) −10.0000 −0.410997
\(593\) −36.0000 −1.47834 −0.739171 − 0.673517i \(-0.764783\pi\)
−0.739171 + 0.673517i \(0.764783\pi\)
\(594\) 0 0
\(595\) 6.00000 0.245976
\(596\) −12.0000 −0.491539
\(597\) −40.0000 −1.63709
\(598\) −2.00000 −0.0817861
\(599\) −12.0000 −0.490307 −0.245153 − 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) −2.00000 −0.0816497
\(601\) −10.0000 −0.407909 −0.203954 − 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) −10.0000 −0.407570
\(603\) 2.00000 0.0814463
\(604\) 8.00000 0.325515
\(605\) 11.0000 0.447214
\(606\) −24.0000 −0.974933
\(607\) 20.0000 0.811775 0.405887 − 0.913923i \(-0.366962\pi\)
0.405887 + 0.913923i \(0.366962\pi\)
\(608\) −4.00000 −0.162221
\(609\) −12.0000 −0.486265
\(610\) 10.0000 0.404888
\(611\) 0 0
\(612\) −6.00000 −0.242536
\(613\) 38.0000 1.53481 0.767403 − 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) −22.0000 −0.887848
\(615\) −12.0000 −0.483887
\(616\) 0 0
\(617\) −36.0000 −1.44931 −0.724653 − 0.689114i \(-0.758000\pi\)
−0.724653 + 0.689114i \(0.758000\pi\)
\(618\) 32.0000 1.28723
\(619\) −28.0000 −1.12542 −0.562708 − 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) −2.00000 −0.0803219
\(621\) −4.00000 −0.160514
\(622\) 18.0000 0.721734
\(623\) −6.00000 −0.240385
\(624\) −4.00000 −0.160128
\(625\) 1.00000 0.0400000
\(626\) 26.0000 1.03917
\(627\) 0 0
\(628\) 14.0000 0.558661
\(629\) 60.0000 2.39236
\(630\) −1.00000 −0.0398410
\(631\) 2.00000 0.0796187 0.0398094 − 0.999207i \(-0.487325\pi\)
0.0398094 + 0.999207i \(0.487325\pi\)
\(632\) −10.0000 −0.397779
\(633\) 8.00000 0.317971
\(634\) 18.0000 0.714871
\(635\) −8.00000 −0.317470
\(636\) −12.0000 −0.475831
\(637\) 2.00000 0.0792429
\(638\) 0 0
\(639\) 12.0000 0.474713
\(640\) −1.00000 −0.0395285
\(641\) −30.0000 −1.18493 −0.592464 − 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) −36.0000 −1.42081
\(643\) 44.0000 1.73519 0.867595 − 0.497271i \(-0.165665\pi\)
0.867595 + 0.497271i \(0.165665\pi\)
\(644\) −1.00000 −0.0394055
\(645\) −20.0000 −0.787499
\(646\) 24.0000 0.944267
\(647\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −11.0000 −0.432121
\(649\) 0 0
\(650\) 2.00000 0.0784465
\(651\) −4.00000 −0.156772
\(652\) −16.0000 −0.626608
\(653\) 6.00000 0.234798 0.117399 − 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 8.00000 0.312825
\(655\) −12.0000 −0.468879
\(656\) −6.00000 −0.234261
\(657\) −16.0000 −0.624219
\(658\) 0 0
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 − 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) −4.00000 −0.155464
\(663\) 24.0000 0.932083
\(664\) −12.0000 −0.465690
\(665\) 4.00000 0.155113
\(666\) −10.0000 −0.387492
\(667\) −6.00000 −0.232321
\(668\) 0 0
\(669\) −16.0000 −0.618596
\(670\) −2.00000 −0.0772667
\(671\) 0 0
\(672\) −2.00000 −0.0771517
\(673\) 38.0000 1.46479 0.732396 − 0.680879i \(-0.238402\pi\)
0.732396 + 0.680879i \(0.238402\pi\)
\(674\) −28.0000 −1.07852
\(675\) 4.00000 0.153960
\(676\) −9.00000 −0.346154
\(677\) 6.00000 0.230599 0.115299 − 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 24.0000 0.921714
\(679\) −10.0000 −0.383765
\(680\) 6.00000 0.230089
\(681\) 24.0000 0.919682
\(682\) 0 0
\(683\) 24.0000 0.918334 0.459167 − 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) −4.00000 −0.152944
\(685\) −12.0000 −0.458496
\(686\) 1.00000 0.0381802
\(687\) −4.00000 −0.152610
\(688\) −10.0000 −0.381246
\(689\) 12.0000 0.457164
\(690\) −2.00000 −0.0761387
\(691\) −40.0000 −1.52167 −0.760836 − 0.648944i \(-0.775211\pi\)
−0.760836 + 0.648944i \(0.775211\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) 0 0
\(695\) −20.0000 −0.758643
\(696\) −12.0000 −0.454859
\(697\) 36.0000 1.36360
\(698\) 8.00000 0.302804
\(699\) 36.0000 1.36165
\(700\) 1.00000 0.0377964
\(701\) −48.0000 −1.81293 −0.906467 − 0.422276i \(-0.861231\pi\)
−0.906467 + 0.422276i \(0.861231\pi\)
\(702\) 8.00000 0.301941
\(703\) 40.0000 1.50863
\(704\) 0 0
\(705\) 0 0
\(706\) −12.0000 −0.451626
\(707\) 12.0000 0.451306
\(708\) 24.0000 0.901975
\(709\) −28.0000 −1.05156 −0.525781 − 0.850620i \(-0.676227\pi\)
−0.525781 + 0.850620i \(0.676227\pi\)
\(710\) −12.0000 −0.450352
\(711\) −10.0000 −0.375029
\(712\) −6.00000 −0.224860
\(713\) −2.00000 −0.0749006
\(714\) 12.0000 0.449089
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) −24.0000 −0.896296
\(718\) 18.0000 0.671754
\(719\) 42.0000 1.56634 0.783168 − 0.621810i \(-0.213603\pi\)
0.783168 + 0.621810i \(0.213603\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −16.0000 −0.595871
\(722\) −3.00000 −0.111648
\(723\) 20.0000 0.743808
\(724\) 26.0000 0.966282
\(725\) 6.00000 0.222834
\(726\) 22.0000 0.816497
\(727\) −16.0000 −0.593407 −0.296704 − 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 2.00000 0.0741249
\(729\) 13.0000 0.481481
\(730\) 16.0000 0.592187
\(731\) 60.0000 2.21918
\(732\) 20.0000 0.739221
\(733\) −34.0000 −1.25582 −0.627909 − 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) 32.0000 1.18114
\(735\) 2.00000 0.0737711
\(736\) −1.00000 −0.0368605
\(737\) 0 0
\(738\) −6.00000 −0.220863
\(739\) −28.0000 −1.03000 −0.514998 − 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) 10.0000 0.367607
\(741\) 16.0000 0.587775
\(742\) 6.00000 0.220267
\(743\) −24.0000 −0.880475 −0.440237 − 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) −4.00000 −0.146647
\(745\) 12.0000 0.439646
\(746\) 2.00000 0.0732252
\(747\) −12.0000 −0.439057
\(748\) 0 0
\(749\) 18.0000 0.657706
\(750\) 2.00000 0.0730297
\(751\) 14.0000 0.510867 0.255434 − 0.966827i \(-0.417782\pi\)
0.255434 + 0.966827i \(0.417782\pi\)
\(752\) 0 0
\(753\) −24.0000 −0.874609
\(754\) 12.0000 0.437014
\(755\) −8.00000 −0.291150
\(756\) 4.00000 0.145479
\(757\) −34.0000 −1.23575 −0.617876 − 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) 4.00000 0.145095
\(761\) −42.0000 −1.52250 −0.761249 − 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) −16.0000 −0.579619
\(763\) −4.00000 −0.144810
\(764\) 18.0000 0.651217
\(765\) 6.00000 0.216930
\(766\) −24.0000 −0.867155
\(767\) −24.0000 −0.866590
\(768\) −2.00000 −0.0721688
\(769\) −46.0000 −1.65880 −0.829401 − 0.558653i \(-0.811318\pi\)
−0.829401 + 0.558653i \(0.811318\pi\)
\(770\) 0 0
\(771\) 48.0000 1.72868
\(772\) −10.0000 −0.359908
\(773\) 6.00000 0.215805 0.107903 − 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) −10.0000 −0.359443
\(775\) 2.00000 0.0718421
\(776\) −10.0000 −0.358979
\(777\) 20.0000 0.717496
\(778\) −12.0000 −0.430221
\(779\) 24.0000 0.859889
\(780\) 4.00000 0.143223
\(781\) 0 0
\(782\) 6.00000 0.214560
\(783\) 24.0000 0.857690
\(784\) 1.00000 0.0357143
\(785\) −14.0000 −0.499681
\(786\) −24.0000 −0.856052
\(787\) 32.0000 1.14068 0.570338 − 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) −6.00000 −0.213741
\(789\) 0 0
\(790\) 10.0000 0.355784
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) −20.0000 −0.710221
\(794\) 14.0000 0.496841
\(795\) 12.0000 0.425596
\(796\) 20.0000 0.708881
\(797\) 18.0000 0.637593 0.318796 − 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 8.00000 0.283197
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) −6.00000 −0.212000
\(802\) −18.0000 −0.635602
\(803\) 0 0
\(804\) −4.00000 −0.141069
\(805\) 1.00000 0.0352454
\(806\) 4.00000 0.140894
\(807\) 24.0000 0.844840
\(808\) 12.0000 0.422159
\(809\) 6.00000 0.210949 0.105474 − 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 11.0000 0.386501
\(811\) −16.0000 −0.561836 −0.280918 − 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 6.00000 0.210559
\(813\) −4.00000 −0.140286
\(814\) 0 0
\(815\) 16.0000 0.560456
\(816\) 12.0000 0.420084
\(817\) 40.0000 1.39942
\(818\) −34.0000 −1.18878
\(819\) 2.00000 0.0698857
\(820\) 6.00000 0.209529
\(821\) 30.0000 1.04701 0.523504 − 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) −24.0000 −0.837096
\(823\) −40.0000 −1.39431 −0.697156 − 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) −16.0000 −0.557386
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) −30.0000 −1.04320 −0.521601 − 0.853189i \(-0.674665\pi\)
−0.521601 + 0.853189i \(0.674665\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 56.0000 1.94496 0.972480 − 0.232986i \(-0.0748495\pi\)
0.972480 + 0.232986i \(0.0748495\pi\)
\(830\) 12.0000 0.416526
\(831\) 20.0000 0.693792
\(832\) 2.00000 0.0693375
\(833\) −6.00000 −0.207888
\(834\) −40.0000 −1.38509
\(835\) 0 0
\(836\) 0 0
\(837\) 8.00000 0.276520
\(838\) 36.0000 1.24360
\(839\) −48.0000 −1.65714 −0.828572 − 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 2.00000 0.0690066
\(841\) 7.00000 0.241379
\(842\) 32.0000 1.10279
\(843\) −60.0000 −2.06651
\(844\) −4.00000 −0.137686
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) −11.0000 −0.377964
\(848\) 6.00000 0.206041
\(849\) 32.0000 1.09824
\(850\) −6.00000 −0.205798
\(851\) 10.0000 0.342796
\(852\) −24.0000 −0.822226
\(853\) 38.0000 1.30110 0.650548 − 0.759465i \(-0.274539\pi\)
0.650548 + 0.759465i \(0.274539\pi\)
\(854\) −10.0000 −0.342193
\(855\) 4.00000 0.136797
\(856\) 18.0000 0.615227
\(857\) −48.0000 −1.63965 −0.819824 − 0.572615i \(-0.805929\pi\)
−0.819824 + 0.572615i \(0.805929\pi\)
\(858\) 0 0
\(859\) −40.0000 −1.36478 −0.682391 − 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) 10.0000 0.340997
\(861\) 12.0000 0.408959
\(862\) −30.0000 −1.02180
\(863\) 48.0000 1.63394 0.816970 − 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) 4.00000 0.136083
\(865\) −6.00000 −0.204006
\(866\) 14.0000 0.475739
\(867\) −38.0000 −1.29055
\(868\) 2.00000 0.0678844
\(869\) 0 0
\(870\) 12.0000 0.406838
\(871\) 4.00000 0.135535
\(872\) −4.00000 −0.135457
\(873\) −10.0000 −0.338449
\(874\) 4.00000 0.135302
\(875\) −1.00000 −0.0338062
\(876\) 32.0000 1.08118
\(877\) 14.0000 0.472746 0.236373 − 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) −10.0000 −0.337484
\(879\) 12.0000 0.404750
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 − 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 1.00000 0.0336718
\(883\) −4.00000 −0.134611 −0.0673054 − 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) −12.0000 −0.403604
\(885\) −24.0000 −0.806751
\(886\) 24.0000 0.806296
\(887\) −36.0000 −1.20876 −0.604381 − 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) 20.0000 0.671156
\(889\) 8.00000 0.268311
\(890\) 6.00000 0.201120
\(891\) 0 0
\(892\) 8.00000 0.267860
\(893\) 0 0
\(894\) 24.0000 0.802680
\(895\) −12.0000 −0.401116
\(896\) 1.00000 0.0334077
\(897\) 4.00000 0.133556
\(898\) −30.0000 −1.00111
\(899\) 12.0000 0.400222
\(900\) 1.00000 0.0333333
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) 20.0000 0.665558
\(904\) −12.0000 −0.399114
\(905\) −26.0000 −0.864269
\(906\) −16.0000 −0.531564
\(907\) −22.0000 −0.730498 −0.365249 − 0.930910i \(-0.619016\pi\)
−0.365249 + 0.930910i \(0.619016\pi\)
\(908\) −12.0000 −0.398234
\(909\) 12.0000 0.398015
\(910\) −2.00000 −0.0662994
\(911\) −6.00000 −0.198789 −0.0993944 − 0.995048i \(-0.531691\pi\)
−0.0993944 + 0.995048i \(0.531691\pi\)
\(912\) 8.00000 0.264906
\(913\) 0 0
\(914\) 20.0000 0.661541
\(915\) −20.0000 −0.661180
\(916\) 2.00000 0.0660819
\(917\) 12.0000 0.396275
\(918\) −24.0000 −0.792118
\(919\) 50.0000 1.64935 0.824674 − 0.565608i \(-0.191359\pi\)
0.824674 + 0.565608i \(0.191359\pi\)
\(920\) 1.00000 0.0329690
\(921\) 44.0000 1.44985
\(922\) 36.0000 1.18560
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) 8.00000 0.262896
\(927\) −16.0000 −0.525509
\(928\) 6.00000 0.196960
\(929\) −18.0000 −0.590561 −0.295280 − 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 4.00000 0.131165
\(931\) −4.00000 −0.131095
\(932\) −18.0000 −0.589610
\(933\) −36.0000 −1.17859
\(934\) 24.0000 0.785304
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 14.0000 0.457360 0.228680 − 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) 2.00000 0.0653023
\(939\) −52.0000 −1.69696
\(940\) 0 0
\(941\) 42.0000 1.36916 0.684580 − 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) −28.0000 −0.912289
\(943\) 6.00000 0.195387
\(944\) −12.0000 −0.390567
\(945\) −4.00000 −0.130120
\(946\) 0 0
\(947\) 36.0000 1.16984 0.584921 − 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) 20.0000 0.649570
\(949\) −32.0000 −1.03876
\(950\) −4.00000 −0.129777
\(951\) −36.0000 −1.16738
\(952\) −6.00000 −0.194461
\(953\) 36.0000 1.16615 0.583077 − 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) 6.00000 0.194257
\(955\) −18.0000 −0.582466
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) 0 0
\(959\) 12.0000 0.387500
\(960\) 2.00000 0.0645497
\(961\) −27.0000 −0.870968
\(962\) −20.0000 −0.644826
\(963\) 18.0000 0.580042
\(964\) −10.0000 −0.322078
\(965\) 10.0000 0.321911
\(966\) 2.00000 0.0643489
\(967\) −16.0000 −0.514525 −0.257263 − 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) −11.0000 −0.353553
\(969\) −48.0000 −1.54198
\(970\) 10.0000 0.321081
\(971\) 12.0000 0.385098 0.192549 − 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 10.0000 0.320750
\(973\) 20.0000 0.641171
\(974\) −40.0000 −1.28168
\(975\) −4.00000 −0.128103
\(976\) −10.0000 −0.320092
\(977\) 24.0000 0.767828 0.383914 − 0.923369i \(-0.374576\pi\)
0.383914 + 0.923369i \(0.374576\pi\)
\(978\) 32.0000 1.02325
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −4.00000 −0.127710
\(982\) 12.0000 0.382935
\(983\) −48.0000 −1.53096 −0.765481 − 0.643458i \(-0.777499\pi\)
−0.765481 + 0.643458i \(0.777499\pi\)
\(984\) 12.0000 0.382546
\(985\) 6.00000 0.191176
\(986\) −36.0000 −1.14647
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 10.0000 0.317982
\(990\) 0 0
\(991\) −28.0000 −0.889449 −0.444725 − 0.895667i \(-0.646698\pi\)
−0.444725 + 0.895667i \(0.646698\pi\)
\(992\) 2.00000 0.0635001
\(993\) 8.00000 0.253872
\(994\) 12.0000 0.380617
\(995\) −20.0000 −0.634043
\(996\) 24.0000 0.760469
\(997\) 2.00000 0.0633406 0.0316703 − 0.999498i \(-0.489917\pi\)
0.0316703 + 0.999498i \(0.489917\pi\)
\(998\) 20.0000 0.633089
\(999\) −40.0000 −1.26554
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 1610.2.a.c.1.1 ✓ 1
5.4 even 2 8050.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1610.2.a.c.1.1 ✓ 1 1.1 even 1 trivial
8050.2.a.l.1.1 1 5.4 even 2