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Properties

Label 4598.2.a.i.1.1
Level $4598$
Weight $2$
Character 4598.1
Self dual yes
Analytic conductor $36.715$
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] = [4598,2,Mod(1,4598)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4598, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4598.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.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: \(36.7152148494\)
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 \(\chi\) \(=\) 4598.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -2.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -2.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +2.00000 q^{12} -6.00000 q^{13} +4.00000 q^{14} -4.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} +1.00000 q^{19} -2.00000 q^{20} -8.00000 q^{21} +8.00000 q^{23} -2.00000 q^{24} -1.00000 q^{25} +6.00000 q^{26} -4.00000 q^{27} -4.00000 q^{28} +6.00000 q^{29} +4.00000 q^{30} -10.0000 q^{31} -1.00000 q^{32} +4.00000 q^{34} +8.00000 q^{35} +1.00000 q^{36} +6.00000 q^{37} -1.00000 q^{38} -12.0000 q^{39} +2.00000 q^{40} +6.00000 q^{41} +8.00000 q^{42} -8.00000 q^{43} -2.00000 q^{45} -8.00000 q^{46} +2.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} -8.00000 q^{51} -6.00000 q^{52} +6.00000 q^{53} +4.00000 q^{54} +4.00000 q^{56} +2.00000 q^{57} -6.00000 q^{58} -14.0000 q^{59} -4.00000 q^{60} +12.0000 q^{61} +10.0000 q^{62} -4.00000 q^{63} +1.00000 q^{64} +12.0000 q^{65} +14.0000 q^{67} -4.00000 q^{68} +16.0000 q^{69} -8.00000 q^{70} +6.00000 q^{71} -1.00000 q^{72} -4.00000 q^{73} -6.00000 q^{74} -2.00000 q^{75} +1.00000 q^{76} +12.0000 q^{78} -2.00000 q^{80} -11.0000 q^{81} -6.00000 q^{82} -8.00000 q^{84} +8.00000 q^{85} +8.00000 q^{86} +12.0000 q^{87} -14.0000 q^{89} +2.00000 q^{90} +24.0000 q^{91} +8.00000 q^{92} -20.0000 q^{93} -2.00000 q^{95} -2.00000 q^{96} -2.00000 q^{97} -9.00000 q^{98} +O(q^{100})\)

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 \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) −2.00000 −0.816497
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) 0 0
\(12\) 2.00000 0.577350
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 4.00000 1.06904
\(15\) −4.00000 −1.03280
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 − 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) −1.00000 −0.235702
\(19\) 1.00000 0.229416
\(20\) −2.00000 −0.447214
\(21\) −8.00000 −1.74574
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 − 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) −2.00000 −0.408248
\(25\) −1.00000 −0.200000
\(26\) 6.00000 1.17670
\(27\) −4.00000 −0.769800
\(28\) −4.00000 −0.755929
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 4.00000 0.730297
\(31\) −10.0000 −1.79605 −0.898027 − 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 4.00000 0.685994
\(35\) 8.00000 1.35225
\(36\) 1.00000 0.166667
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −1.00000 −0.162221
\(39\) −12.0000 −1.92154
\(40\) 2.00000 0.316228
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 8.00000 1.23443
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) −8.00000 −1.17954
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 2.00000 0.288675
\(49\) 9.00000 1.28571
\(50\) 1.00000 0.141421
\(51\) −8.00000 −1.12022
\(52\) −6.00000 −0.832050
\(53\) 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
\(56\) 4.00000 0.534522
\(57\) 2.00000 0.264906
\(58\) −6.00000 −0.787839
\(59\) −14.0000 −1.82264 −0.911322 − 0.411693i \(-0.864937\pi\)
−0.911322 + 0.411693i \(0.864937\pi\)
\(60\) −4.00000 −0.516398
\(61\) 12.0000 1.53644 0.768221 − 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) 10.0000 1.27000
\(63\) −4.00000 −0.503953
\(64\) 1.00000 0.125000
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) 14.0000 1.71037 0.855186 − 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) −4.00000 −0.485071
\(69\) 16.0000 1.92617
\(70\) −8.00000 −0.956183
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) −1.00000 −0.117851
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) −6.00000 −0.697486
\(75\) −2.00000 −0.230940
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 12.0000 1.35873
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −2.00000 −0.223607
\(81\) −11.0000 −1.22222
\(82\) −6.00000 −0.662589
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −8.00000 −0.872872
\(85\) 8.00000 0.867722
\(86\) 8.00000 0.862662
\(87\) 12.0000 1.28654
\(88\) 0 0
\(89\) −14.0000 −1.48400 −0.741999 − 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 2.00000 0.210819
\(91\) 24.0000 2.51588
\(92\) 8.00000 0.834058
\(93\) −20.0000 −2.07390
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) −2.00000 −0.204124
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −9.00000 −0.909137
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) 8.00000 0.796030 0.398015 − 0.917379i \(-0.369699\pi\)
0.398015 + 0.917379i \(0.369699\pi\)
\(102\) 8.00000 0.792118
\(103\) 14.0000 1.37946 0.689730 − 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 6.00000 0.588348
\(105\) 16.0000 1.56144
\(106\) −6.00000 −0.582772
\(107\) 4.00000 0.386695 0.193347 − 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) −4.00000 −0.384900
\(109\) −18.0000 −1.72409 −0.862044 − 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) 0 0
\(111\) 12.0000 1.13899
\(112\) −4.00000 −0.377964
\(113\) 6.00000 0.564433 0.282216 − 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −2.00000 −0.187317
\(115\) −16.0000 −1.49201
\(116\) 6.00000 0.557086
\(117\) −6.00000 −0.554700
\(118\) 14.0000 1.28880
\(119\) 16.0000 1.46672
\(120\) 4.00000 0.365148
\(121\) 0 0
\(122\) −12.0000 −1.08643
\(123\) 12.0000 1.08200
\(124\) −10.0000 −0.898027
\(125\) 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
\(129\) −16.0000 −1.40872
\(130\) −12.0000 −1.05247
\(131\) −8.00000 −0.698963 −0.349482 − 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) −14.0000 −1.20942
\(135\) 8.00000 0.688530
\(136\) 4.00000 0.342997
\(137\) 6.00000 0.512615 0.256307 − 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) −16.0000 −1.36201
\(139\) −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
\(142\) −6.00000 −0.503509
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −12.0000 −0.996546
\(146\) 4.00000 0.331042
\(147\) 18.0000 1.48461
\(148\) 6.00000 0.493197
\(149\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 2.00000 0.163299
\(151\) 8.00000 0.651031 0.325515 − 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −4.00000 −0.323381
\(154\) 0 0
\(155\) 20.0000 1.60644
\(156\) −12.0000 −0.960769
\(157\) 2.00000 0.159617 0.0798087 − 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) 12.0000 0.951662
\(160\) 2.00000 0.158114
\(161\) −32.0000 −2.52195
\(162\) 11.0000 0.864242
\(163\) 12.0000 0.939913 0.469956 − 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 8.00000 0.617213
\(169\) 23.0000 1.76923
\(170\) −8.00000 −0.613572
\(171\) 1.00000 0.0764719
\(172\) −8.00000 −0.609994
\(173\) 2.00000 0.152057 0.0760286 − 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) −12.0000 −0.909718
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) −28.0000 −2.10461
\(178\) 14.0000 1.04934
\(179\) −6.00000 −0.448461 −0.224231 − 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) −2.00000 −0.149071
\(181\) −10.0000 −0.743294 −0.371647 − 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) −24.0000 −1.77900
\(183\) 24.0000 1.77413
\(184\) −8.00000 −0.589768
\(185\) −12.0000 −0.882258
\(186\) 20.0000 1.46647
\(187\) 0 0
\(188\) 0 0
\(189\) 16.0000 1.16383
\(190\) 2.00000 0.145095
\(191\) 8.00000 0.578860 0.289430 − 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 2.00000 0.144338
\(193\) 22.0000 1.58359 0.791797 − 0.610784i \(-0.209146\pi\)
0.791797 + 0.610784i \(0.209146\pi\)
\(194\) 2.00000 0.143592
\(195\) 24.0000 1.71868
\(196\) 9.00000 0.642857
\(197\) −12.0000 −0.854965 −0.427482 − 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 − 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 1.00000 0.0707107
\(201\) 28.0000 1.97497
\(202\) −8.00000 −0.562878
\(203\) −24.0000 −1.68447
\(204\) −8.00000 −0.560112
\(205\) −12.0000 −0.838116
\(206\) −14.0000 −0.975426
\(207\) 8.00000 0.556038
\(208\) −6.00000 −0.416025
\(209\) 0 0
\(210\) −16.0000 −1.10410
\(211\) 20.0000 1.37686 0.688428 − 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 6.00000 0.412082
\(213\) 12.0000 0.822226
\(214\) −4.00000 −0.273434
\(215\) 16.0000 1.09119
\(216\) 4.00000 0.272166
\(217\) 40.0000 2.71538
\(218\) 18.0000 1.21911
\(219\) −8.00000 −0.540590
\(220\) 0 0
\(221\) 24.0000 1.61441
\(222\) −12.0000 −0.805387
\(223\) 10.0000 0.669650 0.334825 − 0.942280i \(-0.391323\pi\)
0.334825 + 0.942280i \(0.391323\pi\)
\(224\) 4.00000 0.267261
\(225\) −1.00000 −0.0666667
\(226\) −6.00000 −0.399114
\(227\) 28.0000 1.85843 0.929213 − 0.369546i \(-0.120487\pi\)
0.929213 + 0.369546i \(0.120487\pi\)
\(228\) 2.00000 0.132453
\(229\) 10.0000 0.660819 0.330409 − 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 16.0000 1.05501
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 8.00000 0.524097 0.262049 − 0.965055i \(-0.415602\pi\)
0.262049 + 0.965055i \(0.415602\pi\)
\(234\) 6.00000 0.392232
\(235\) 0 0
\(236\) −14.0000 −0.911322
\(237\) 0 0
\(238\) −16.0000 −1.03713
\(239\) −8.00000 −0.517477 −0.258738 − 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) −4.00000 −0.258199
\(241\) −14.0000 −0.901819 −0.450910 − 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 0 0
\(243\) −10.0000 −0.641500
\(244\) 12.0000 0.768221
\(245\) −18.0000 −1.14998
\(246\) −12.0000 −0.765092
\(247\) −6.00000 −0.381771
\(248\) 10.0000 0.635001
\(249\) 0 0
\(250\) −12.0000 −0.758947
\(251\) −12.0000 −0.757433 −0.378717 − 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) −4.00000 −0.251976
\(253\) 0 0
\(254\) −16.0000 −1.00393
\(255\) 16.0000 1.00196
\(256\) 1.00000 0.0625000
\(257\) −22.0000 −1.37232 −0.686161 − 0.727450i \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) 16.0000 0.996116
\(259\) −24.0000 −1.49129
\(260\) 12.0000 0.744208
\(261\) 6.00000 0.371391
\(262\) 8.00000 0.494242
\(263\) −8.00000 −0.493301 −0.246651 − 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) 4.00000 0.245256
\(267\) −28.0000 −1.71357
\(268\) 14.0000 0.855186
\(269\) −18.0000 −1.09748 −0.548740 − 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) −8.00000 −0.486864
\(271\) 16.0000 0.971931 0.485965 − 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) −4.00000 −0.242536
\(273\) 48.0000 2.90509
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 16.0000 0.963087
\(277\) −24.0000 −1.44202 −0.721010 − 0.692925i \(-0.756322\pi\)
−0.721010 + 0.692925i \(0.756322\pi\)
\(278\) 4.00000 0.239904
\(279\) −10.0000 −0.598684
\(280\) −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
\(283\) 12.0000 0.713326 0.356663 − 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) 6.00000 0.356034
\(285\) −4.00000 −0.236940
\(286\) 0 0
\(287\) −24.0000 −1.41668
\(288\) −1.00000 −0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 12.0000 0.704664
\(291\) −4.00000 −0.234484
\(292\) −4.00000 −0.234082
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) −18.0000 −1.04978
\(295\) 28.0000 1.63022
\(296\) −6.00000 −0.348743
\(297\) 0 0
\(298\) 0 0
\(299\) −48.0000 −2.77591
\(300\) −2.00000 −0.115470
\(301\) 32.0000 1.84445
\(302\) −8.00000 −0.460348
\(303\) 16.0000 0.919176
\(304\) 1.00000 0.0573539
\(305\) −24.0000 −1.37424
\(306\) 4.00000 0.228665
\(307\) 12.0000 0.684876 0.342438 − 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 28.0000 1.59286
\(310\) −20.0000 −1.13592
\(311\) −12.0000 −0.680458 −0.340229 − 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 12.0000 0.679366
\(313\) 14.0000 0.791327 0.395663 − 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) −2.00000 −0.112867
\(315\) 8.00000 0.450749
\(316\) 0 0
\(317\) 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
\(320\) −2.00000 −0.111803
\(321\) 8.00000 0.446516
\(322\) 32.0000 1.78329
\(323\) −4.00000 −0.222566
\(324\) −11.0000 −0.611111
\(325\) 6.00000 0.332820
\(326\) −12.0000 −0.664619
\(327\) −36.0000 −1.99080
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) −26.0000 −1.42909 −0.714545 − 0.699590i \(-0.753366\pi\)
−0.714545 + 0.699590i \(0.753366\pi\)
\(332\) 0 0
\(333\) 6.00000 0.328798
\(334\) 0 0
\(335\) −28.0000 −1.52980
\(336\) −8.00000 −0.436436
\(337\) −10.0000 −0.544735 −0.272367 − 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) −23.0000 −1.25104
\(339\) 12.0000 0.651751
\(340\) 8.00000 0.433861
\(341\) 0 0
\(342\) −1.00000 −0.0540738
\(343\) −8.00000 −0.431959
\(344\) 8.00000 0.431331
\(345\) −32.0000 −1.72282
\(346\) −2.00000 −0.107521
\(347\) −12.0000 −0.644194 −0.322097 − 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 12.0000 0.643268
\(349\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) −4.00000 −0.213809
\(351\) 24.0000 1.28103
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 − 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 28.0000 1.48818
\(355\) −12.0000 −0.636894
\(356\) −14.0000 −0.741999
\(357\) 32.0000 1.69362
\(358\) 6.00000 0.317110
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 2.00000 0.105409
\(361\) 1.00000 0.0526316
\(362\) 10.0000 0.525588
\(363\) 0 0
\(364\) 24.0000 1.25794
\(365\) 8.00000 0.418739
\(366\) −24.0000 −1.25450
\(367\) 24.0000 1.25279 0.626395 − 0.779506i \(-0.284530\pi\)
0.626395 + 0.779506i \(0.284530\pi\)
\(368\) 8.00000 0.417029
\(369\) 6.00000 0.312348
\(370\) 12.0000 0.623850
\(371\) −24.0000 −1.24602
\(372\) −20.0000 −1.03695
\(373\) 26.0000 1.34623 0.673114 − 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 0 0
\(375\) 24.0000 1.23935
\(376\) 0 0
\(377\) −36.0000 −1.85409
\(378\) −16.0000 −0.822951
\(379\) −34.0000 −1.74646 −0.873231 − 0.487306i \(-0.837980\pi\)
−0.873231 + 0.487306i \(0.837980\pi\)
\(380\) −2.00000 −0.102598
\(381\) 32.0000 1.63941
\(382\) −8.00000 −0.409316
\(383\) −30.0000 −1.53293 −0.766464 − 0.642287i \(-0.777986\pi\)
−0.766464 + 0.642287i \(0.777986\pi\)
\(384\) −2.00000 −0.102062
\(385\) 0 0
\(386\) −22.0000 −1.11977
\(387\) −8.00000 −0.406663
\(388\) −2.00000 −0.101535
\(389\) −6.00000 −0.304212 −0.152106 − 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) −24.0000 −1.21529
\(391\) −32.0000 −1.61831
\(392\) −9.00000 −0.454569
\(393\) −16.0000 −0.807093
\(394\) 12.0000 0.604551
\(395\) 0 0
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 − 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) −4.00000 −0.200502
\(399\) −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 \(0.648376\pi\)
\(402\) −28.0000 −1.39651
\(403\) 60.0000 2.98881
\(404\) 8.00000 0.398015
\(405\) 22.0000 1.09319
\(406\) 24.0000 1.19110
\(407\) 0 0
\(408\) 8.00000 0.396059
\(409\) −14.0000 −0.692255 −0.346128 − 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 12.0000 0.592638
\(411\) 12.0000 0.591916
\(412\) 14.0000 0.689730
\(413\) 56.0000 2.75558
\(414\) −8.00000 −0.393179
\(415\) 0 0
\(416\) 6.00000 0.294174
\(417\) −8.00000 −0.391762
\(418\) 0 0
\(419\) −36.0000 −1.75872 −0.879358 − 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 16.0000 0.780720
\(421\) 2.00000 0.0974740 0.0487370 − 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) −20.0000 −0.973585
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 4.00000 0.194029
\(426\) −12.0000 −0.581402
\(427\) −48.0000 −2.32288
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) −16.0000 −0.771589
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −4.00000 −0.192450
\(433\) 22.0000 1.05725 0.528626 − 0.848855i \(-0.322707\pi\)
0.528626 + 0.848855i \(0.322707\pi\)
\(434\) −40.0000 −1.92006
\(435\) −24.0000 −1.15071
\(436\) −18.0000 −0.862044
\(437\) 8.00000 0.382692
\(438\) 8.00000 0.382255
\(439\) −16.0000 −0.763638 −0.381819 − 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) −24.0000 −1.14156
\(443\) 24.0000 1.14027 0.570137 − 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 12.0000 0.569495
\(445\) 28.0000 1.32733
\(446\) −10.0000 −0.473514
\(447\) 0 0
\(448\) −4.00000 −0.188982
\(449\) 2.00000 0.0943858 0.0471929 − 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) 6.00000 0.282216
\(453\) 16.0000 0.751746
\(454\) −28.0000 −1.31411
\(455\) −48.0000 −2.25027
\(456\) −2.00000 −0.0936586
\(457\) −8.00000 −0.374224 −0.187112 − 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) −10.0000 −0.467269
\(459\) 16.0000 0.746816
\(460\) −16.0000 −0.746004
\(461\) 40.0000 1.86299 0.931493 − 0.363760i \(-0.118507\pi\)
0.931493 + 0.363760i \(0.118507\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 − 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) 6.00000 0.278543
\(465\) 40.0000 1.85496
\(466\) −8.00000 −0.370593
\(467\) 12.0000 0.555294 0.277647 − 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) −6.00000 −0.277350
\(469\) −56.0000 −2.58584
\(470\) 0 0
\(471\) 4.00000 0.184310
\(472\) 14.0000 0.644402
\(473\) 0 0
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 16.0000 0.733359
\(477\) 6.00000 0.274721
\(478\) 8.00000 0.365911
\(479\) 12.0000 0.548294 0.274147 − 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) 4.00000 0.182574
\(481\) −36.0000 −1.64146
\(482\) 14.0000 0.637683
\(483\) −64.0000 −2.91210
\(484\) 0 0
\(485\) 4.00000 0.181631
\(486\) 10.0000 0.453609
\(487\) 2.00000 0.0906287 0.0453143 − 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) −12.0000 −0.543214
\(489\) 24.0000 1.08532
\(490\) 18.0000 0.813157
\(491\) 8.00000 0.361035 0.180517 − 0.983572i \(-0.442223\pi\)
0.180517 + 0.983572i \(0.442223\pi\)
\(492\) 12.0000 0.541002
\(493\) −24.0000 −1.08091
\(494\) 6.00000 0.269953
\(495\) 0 0
\(496\) −10.0000 −0.449013
\(497\) −24.0000 −1.07655
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 − 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) 12.0000 0.535586
\(503\) 44.0000 1.96186 0.980932 − 0.194354i \(-0.0622609\pi\)
0.980932 + 0.194354i \(0.0622609\pi\)
\(504\) 4.00000 0.178174
\(505\) −16.0000 −0.711991
\(506\) 0 0
\(507\) 46.0000 2.04293
\(508\) 16.0000 0.709885
\(509\) −18.0000 −0.797836 −0.398918 − 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) −16.0000 −0.708492
\(511\) 16.0000 0.707798
\(512\) −1.00000 −0.0441942
\(513\) −4.00000 −0.176604
\(514\) 22.0000 0.970378
\(515\) −28.0000 −1.23383
\(516\) −16.0000 −0.704361
\(517\) 0 0
\(518\) 24.0000 1.05450
\(519\) 4.00000 0.175581
\(520\) −12.0000 −0.526235
\(521\) 30.0000 1.31432 0.657162 − 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) −6.00000 −0.262613
\(523\) −36.0000 −1.57417 −0.787085 − 0.616844i \(-0.788411\pi\)
−0.787085 + 0.616844i \(0.788411\pi\)
\(524\) −8.00000 −0.349482
\(525\) 8.00000 0.349149
\(526\) 8.00000 0.348817
\(527\) 40.0000 1.74243
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 12.0000 0.521247
\(531\) −14.0000 −0.607548
\(532\) −4.00000 −0.173422
\(533\) −36.0000 −1.55933
\(534\) 28.0000 1.21168
\(535\) −8.00000 −0.345870
\(536\) −14.0000 −0.604708
\(537\) −12.0000 −0.517838
\(538\) 18.0000 0.776035
\(539\) 0 0
\(540\) 8.00000 0.344265
\(541\) 20.0000 0.859867 0.429934 − 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) −16.0000 −0.687259
\(543\) −20.0000 −0.858282
\(544\) 4.00000 0.171499
\(545\) 36.0000 1.54207
\(546\) −48.0000 −2.05421
\(547\) 44.0000 1.88130 0.940652 − 0.339372i \(-0.110215\pi\)
0.940652 + 0.339372i \(0.110215\pi\)
\(548\) 6.00000 0.256307
\(549\) 12.0000 0.512148
\(550\) 0 0
\(551\) 6.00000 0.255609
\(552\) −16.0000 −0.681005
\(553\) 0 0
\(554\) 24.0000 1.01966
\(555\) −24.0000 −1.01874
\(556\) −4.00000 −0.169638
\(557\) −8.00000 −0.338971 −0.169485 − 0.985533i \(-0.554211\pi\)
−0.169485 + 0.985533i \(0.554211\pi\)
\(558\) 10.0000 0.423334
\(559\) 48.0000 2.03018
\(560\) 8.00000 0.338062
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) −4.00000 −0.168580 −0.0842900 − 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) −12.0000 −0.504844
\(566\) −12.0000 −0.504398
\(567\) 44.0000 1.84783
\(568\) −6.00000 −0.251754
\(569\) 30.0000 1.25767 0.628833 − 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 4.00000 0.167542
\(571\) 12.0000 0.502184 0.251092 − 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 0 0
\(573\) 16.0000 0.668410
\(574\) 24.0000 1.00174
\(575\) −8.00000 −0.333623
\(576\) 1.00000 0.0416667
\(577\) 10.0000 0.416305 0.208153 − 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) 1.00000 0.0415945
\(579\) 44.0000 1.82858
\(580\) −12.0000 −0.498273
\(581\) 0 0
\(582\) 4.00000 0.165805
\(583\) 0 0
\(584\) 4.00000 0.165521
\(585\) 12.0000 0.496139
\(586\) −6.00000 −0.247858
\(587\) −16.0000 −0.660391 −0.330195 − 0.943913i \(-0.607115\pi\)
−0.330195 + 0.943913i \(0.607115\pi\)
\(588\) 18.0000 0.742307
\(589\) −10.0000 −0.412043
\(590\) −28.0000 −1.15274
\(591\) −24.0000 −0.987228
\(592\) 6.00000 0.246598
\(593\) −8.00000 −0.328521 −0.164260 − 0.986417i \(-0.552524\pi\)
−0.164260 + 0.986417i \(0.552524\pi\)
\(594\) 0 0
\(595\) −32.0000 −1.31187
\(596\) 0 0
\(597\) 8.00000 0.327418
\(598\) 48.0000 1.96287
\(599\) −18.0000 −0.735460 −0.367730 − 0.929933i \(-0.619865\pi\)
−0.367730 + 0.929933i \(0.619865\pi\)
\(600\) 2.00000 0.0816497
\(601\) 2.00000 0.0815817 0.0407909 − 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) −32.0000 −1.30422
\(603\) 14.0000 0.570124
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) −16.0000 −0.649956
\(607\) 8.00000 0.324710 0.162355 − 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −48.0000 −1.94506
\(610\) 24.0000 0.971732
\(611\) 0 0
\(612\) −4.00000 −0.161690
\(613\) −16.0000 −0.646234 −0.323117 − 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) −12.0000 −0.484281
\(615\) −24.0000 −0.967773
\(616\) 0 0
\(617\) 26.0000 1.04672 0.523360 − 0.852111i \(-0.324678\pi\)
0.523360 + 0.852111i \(0.324678\pi\)
\(618\) −28.0000 −1.12633
\(619\) 24.0000 0.964641 0.482321 − 0.875995i \(-0.339794\pi\)
0.482321 + 0.875995i \(0.339794\pi\)
\(620\) 20.0000 0.803219
\(621\) −32.0000 −1.28412
\(622\) 12.0000 0.481156
\(623\) 56.0000 2.24359
\(624\) −12.0000 −0.480384
\(625\) −19.0000 −0.760000
\(626\) −14.0000 −0.559553
\(627\) 0 0
\(628\) 2.00000 0.0798087
\(629\) −24.0000 −0.956943
\(630\) −8.00000 −0.318728
\(631\) −36.0000 −1.43314 −0.716569 − 0.697517i \(-0.754288\pi\)
−0.716569 + 0.697517i \(0.754288\pi\)
\(632\) 0 0
\(633\) 40.0000 1.58986
\(634\) −18.0000 −0.714871
\(635\) −32.0000 −1.26988
\(636\) 12.0000 0.475831
\(637\) −54.0000 −2.13956
\(638\) 0 0
\(639\) 6.00000 0.237356
\(640\) 2.00000 0.0790569
\(641\) 18.0000 0.710957 0.355479 − 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) −8.00000 −0.315735
\(643\) 28.0000 1.10421 0.552106 − 0.833774i \(-0.313824\pi\)
0.552106 + 0.833774i \(0.313824\pi\)
\(644\) −32.0000 −1.26098
\(645\) 32.0000 1.26000
\(646\) 4.00000 0.157378
\(647\) 28.0000 1.10079 0.550397 − 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) 11.0000 0.432121
\(649\) 0 0
\(650\) −6.00000 −0.235339
\(651\) 80.0000 3.13545
\(652\) 12.0000 0.469956
\(653\) 34.0000 1.33052 0.665261 − 0.746611i \(-0.268320\pi\)
0.665261 + 0.746611i \(0.268320\pi\)
\(654\) 36.0000 1.40771
\(655\) 16.0000 0.625172
\(656\) 6.00000 0.234261
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 − 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) −14.0000 −0.544537 −0.272268 − 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) 26.0000 1.01052
\(663\) 48.0000 1.86417
\(664\) 0 0
\(665\) 8.00000 0.310227
\(666\) −6.00000 −0.232495
\(667\) 48.0000 1.85857
\(668\) 0 0
\(669\) 20.0000 0.773245
\(670\) 28.0000 1.08173
\(671\) 0 0
\(672\) 8.00000 0.308607
\(673\) −14.0000 −0.539660 −0.269830 − 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 10.0000 0.385186
\(675\) 4.00000 0.153960
\(676\) 23.0000 0.884615
\(677\) 22.0000 0.845529 0.422764 − 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) −12.0000 −0.460857
\(679\) 8.00000 0.307012
\(680\) −8.00000 −0.306786
\(681\) 56.0000 2.14592
\(682\) 0 0
\(683\) −14.0000 −0.535695 −0.267848 − 0.963461i \(-0.586312\pi\)
−0.267848 + 0.963461i \(0.586312\pi\)
\(684\) 1.00000 0.0382360
\(685\) −12.0000 −0.458496
\(686\) 8.00000 0.305441
\(687\) 20.0000 0.763048
\(688\) −8.00000 −0.304997
\(689\) −36.0000 −1.37149
\(690\) 32.0000 1.21822
\(691\) −12.0000 −0.456502 −0.228251 − 0.973602i \(-0.573301\pi\)
−0.228251 + 0.973602i \(0.573301\pi\)
\(692\) 2.00000 0.0760286
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 8.00000 0.303457
\(696\) −12.0000 −0.454859
\(697\) −24.0000 −0.909065
\(698\) 0 0
\(699\) 16.0000 0.605176
\(700\) 4.00000 0.151186
\(701\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) −24.0000 −0.905822
\(703\) 6.00000 0.226294
\(704\) 0 0
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) −32.0000 −1.20348
\(708\) −28.0000 −1.05230
\(709\) 38.0000 1.42712 0.713560 − 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) 12.0000 0.450352
\(711\) 0 0
\(712\) 14.0000 0.524672
\(713\) −80.0000 −2.99602
\(714\) −32.0000 −1.19757
\(715\) 0 0
\(716\) −6.00000 −0.224231
\(717\) −16.0000 −0.597531
\(718\) 0 0
\(719\) −28.0000 −1.04422 −0.522112 − 0.852877i \(-0.674856\pi\)
−0.522112 + 0.852877i \(0.674856\pi\)
\(720\) −2.00000 −0.0745356
\(721\) −56.0000 −2.08555
\(722\) −1.00000 −0.0372161
\(723\) −28.0000 −1.04133
\(724\) −10.0000 −0.371647
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) 28.0000 1.03846 0.519231 − 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) −24.0000 −0.889499
\(729\) 13.0000 0.481481
\(730\) −8.00000 −0.296093
\(731\) 32.0000 1.18356
\(732\) 24.0000 0.887066
\(733\) 36.0000 1.32969 0.664845 − 0.746981i \(-0.268498\pi\)
0.664845 + 0.746981i \(0.268498\pi\)
\(734\) −24.0000 −0.885856
\(735\) −36.0000 −1.32788
\(736\) −8.00000 −0.294884
\(737\) 0 0
\(738\) −6.00000 −0.220863
\(739\) 40.0000 1.47142 0.735712 − 0.677295i \(-0.236848\pi\)
0.735712 + 0.677295i \(0.236848\pi\)
\(740\) −12.0000 −0.441129
\(741\) −12.0000 −0.440831
\(742\) 24.0000 0.881068
\(743\) 8.00000 0.293492 0.146746 − 0.989174i \(-0.453120\pi\)
0.146746 + 0.989174i \(0.453120\pi\)
\(744\) 20.0000 0.733236
\(745\) 0 0
\(746\) −26.0000 −0.951928
\(747\) 0 0
\(748\) 0 0
\(749\) −16.0000 −0.584627
\(750\) −24.0000 −0.876356
\(751\) 6.00000 0.218943 0.109472 − 0.993990i \(-0.465084\pi\)
0.109472 + 0.993990i \(0.465084\pi\)
\(752\) 0 0
\(753\) −24.0000 −0.874609
\(754\) 36.0000 1.31104
\(755\) −16.0000 −0.582300
\(756\) 16.0000 0.581914
\(757\) 30.0000 1.09037 0.545184 − 0.838316i \(-0.316460\pi\)
0.545184 + 0.838316i \(0.316460\pi\)
\(758\) 34.0000 1.23494
\(759\) 0 0
\(760\) 2.00000 0.0725476
\(761\) −20.0000 −0.724999 −0.362500 − 0.931984i \(-0.618077\pi\)
−0.362500 + 0.931984i \(0.618077\pi\)
\(762\) −32.0000 −1.15924
\(763\) 72.0000 2.60658
\(764\) 8.00000 0.289430
\(765\) 8.00000 0.289241
\(766\) 30.0000 1.08394
\(767\) 84.0000 3.03306
\(768\) 2.00000 0.0721688
\(769\) −24.0000 −0.865462 −0.432731 − 0.901523i \(-0.642450\pi\)
−0.432731 + 0.901523i \(0.642450\pi\)
\(770\) 0 0
\(771\) −44.0000 −1.58462
\(772\) 22.0000 0.791797
\(773\) 14.0000 0.503545 0.251773 − 0.967786i \(-0.418987\pi\)
0.251773 + 0.967786i \(0.418987\pi\)
\(774\) 8.00000 0.287554
\(775\) 10.0000 0.359211
\(776\) 2.00000 0.0717958
\(777\) −48.0000 −1.72199
\(778\) 6.00000 0.215110
\(779\) 6.00000 0.214972
\(780\) 24.0000 0.859338
\(781\) 0 0
\(782\) 32.0000 1.14432
\(783\) −24.0000 −0.857690
\(784\) 9.00000 0.321429
\(785\) −4.00000 −0.142766
\(786\) 16.0000 0.570701
\(787\) −12.0000 −0.427754 −0.213877 − 0.976861i \(-0.568609\pi\)
−0.213877 + 0.976861i \(0.568609\pi\)
\(788\) −12.0000 −0.427482
\(789\) −16.0000 −0.569615
\(790\) 0 0
\(791\) −24.0000 −0.853342
\(792\) 0 0
\(793\) −72.0000 −2.55679
\(794\) 14.0000 0.496841
\(795\) −24.0000 −0.851192
\(796\) 4.00000 0.141776
\(797\) 10.0000 0.354218 0.177109 − 0.984191i \(-0.443325\pi\)
0.177109 + 0.984191i \(0.443325\pi\)
\(798\) 8.00000 0.283197
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) −14.0000 −0.494666
\(802\) 18.0000 0.635602
\(803\) 0 0
\(804\) 28.0000 0.987484
\(805\) 64.0000 2.25570
\(806\) −60.0000 −2.11341
\(807\) −36.0000 −1.26726
\(808\) −8.00000 −0.281439
\(809\) 40.0000 1.40633 0.703163 − 0.711029i \(-0.251771\pi\)
0.703163 + 0.711029i \(0.251771\pi\)
\(810\) −22.0000 −0.773001
\(811\) −36.0000 −1.26413 −0.632065 − 0.774915i \(-0.717793\pi\)
−0.632065 + 0.774915i \(0.717793\pi\)
\(812\) −24.0000 −0.842235
\(813\) 32.0000 1.12229
\(814\) 0 0
\(815\) −24.0000 −0.840683
\(816\) −8.00000 −0.280056
\(817\) −8.00000 −0.279885
\(818\) 14.0000 0.489499
\(819\) 24.0000 0.838628
\(820\) −12.0000 −0.419058
\(821\) 40.0000 1.39601 0.698005 − 0.716093i \(-0.254071\pi\)
0.698005 + 0.716093i \(0.254071\pi\)
\(822\) −12.0000 −0.418548
\(823\) −4.00000 −0.139431 −0.0697156 − 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) −14.0000 −0.487713
\(825\) 0 0
\(826\) −56.0000 −1.94849
\(827\) −44.0000 −1.53003 −0.765015 − 0.644013i \(-0.777268\pi\)
−0.765015 + 0.644013i \(0.777268\pi\)
\(828\) 8.00000 0.278019
\(829\) −46.0000 −1.59765 −0.798823 − 0.601566i \(-0.794544\pi\)
−0.798823 + 0.601566i \(0.794544\pi\)
\(830\) 0 0
\(831\) −48.0000 −1.66510
\(832\) −6.00000 −0.208013
\(833\) −36.0000 −1.24733
\(834\) 8.00000 0.277017
\(835\) 0 0
\(836\) 0 0
\(837\) 40.0000 1.38260
\(838\) 36.0000 1.24360
\(839\) −38.0000 −1.31191 −0.655953 − 0.754802i \(-0.727733\pi\)
−0.655953 + 0.754802i \(0.727733\pi\)
\(840\) −16.0000 −0.552052
\(841\) 7.00000 0.241379
\(842\) −2.00000 −0.0689246
\(843\) −12.0000 −0.413302
\(844\) 20.0000 0.688428
\(845\) −46.0000 −1.58245
\(846\) 0 0
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) 24.0000 0.823678
\(850\) −4.00000 −0.137199
\(851\) 48.0000 1.64542
\(852\) 12.0000 0.411113
\(853\) −4.00000 −0.136957 −0.0684787 − 0.997653i \(-0.521815\pi\)
−0.0684787 + 0.997653i \(0.521815\pi\)
\(854\) 48.0000 1.64253
\(855\) −2.00000 −0.0683986
\(856\) −4.00000 −0.136717
\(857\) 10.0000 0.341593 0.170797 − 0.985306i \(-0.445366\pi\)
0.170797 + 0.985306i \(0.445366\pi\)
\(858\) 0 0
\(859\) 24.0000 0.818869 0.409435 − 0.912339i \(-0.365726\pi\)
0.409435 + 0.912339i \(0.365726\pi\)
\(860\) 16.0000 0.545595
\(861\) −48.0000 −1.63584
\(862\) 0 0
\(863\) 30.0000 1.02121 0.510606 − 0.859815i \(-0.329421\pi\)
0.510606 + 0.859815i \(0.329421\pi\)
\(864\) 4.00000 0.136083
\(865\) −4.00000 −0.136004
\(866\) −22.0000 −0.747590
\(867\) −2.00000 −0.0679236
\(868\) 40.0000 1.35769
\(869\) 0 0
\(870\) 24.0000 0.813676
\(871\) −84.0000 −2.84623
\(872\) 18.0000 0.609557
\(873\) −2.00000 −0.0676897
\(874\) −8.00000 −0.270604
\(875\) −48.0000 −1.62270
\(876\) −8.00000 −0.270295
\(877\) 18.0000 0.607817 0.303908 − 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) 16.0000 0.539974
\(879\) 12.0000 0.404750
\(880\) 0 0
\(881\) 46.0000 1.54978 0.774890 − 0.632096i \(-0.217805\pi\)
0.774890 + 0.632096i \(0.217805\pi\)
\(882\) −9.00000 −0.303046
\(883\) −24.0000 −0.807664 −0.403832 − 0.914833i \(-0.632322\pi\)
−0.403832 + 0.914833i \(0.632322\pi\)
\(884\) 24.0000 0.807207
\(885\) 56.0000 1.88242
\(886\) −24.0000 −0.806296
\(887\) −48.0000 −1.61168 −0.805841 − 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) −12.0000 −0.402694
\(889\) −64.0000 −2.14649
\(890\) −28.0000 −0.938562
\(891\) 0 0
\(892\) 10.0000 0.334825
\(893\) 0 0
\(894\) 0 0
\(895\) 12.0000 0.401116
\(896\) 4.00000 0.133631
\(897\) −96.0000 −3.20535
\(898\) −2.00000 −0.0667409
\(899\) −60.0000 −2.00111
\(900\) −1.00000 −0.0333333
\(901\) −24.0000 −0.799556
\(902\) 0 0
\(903\) 64.0000 2.12979
\(904\) −6.00000 −0.199557
\(905\) 20.0000 0.664822
\(906\) −16.0000 −0.531564
\(907\) 26.0000 0.863316 0.431658 − 0.902037i \(-0.357929\pi\)
0.431658 + 0.902037i \(0.357929\pi\)
\(908\) 28.0000 0.929213
\(909\) 8.00000 0.265343
\(910\) 48.0000 1.59118
\(911\) 46.0000 1.52405 0.762024 − 0.647549i \(-0.224206\pi\)
0.762024 + 0.647549i \(0.224206\pi\)
\(912\) 2.00000 0.0662266
\(913\) 0 0
\(914\) 8.00000 0.264616
\(915\) −48.0000 −1.58683
\(916\) 10.0000 0.330409
\(917\) 32.0000 1.05673
\(918\) −16.0000 −0.528079
\(919\) −40.0000 −1.31948 −0.659739 − 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 16.0000 0.527504
\(921\) 24.0000 0.790827
\(922\) −40.0000 −1.31733
\(923\) −36.0000 −1.18495
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) 4.00000 0.131448
\(927\) 14.0000 0.459820
\(928\) −6.00000 −0.196960
\(929\) 14.0000 0.459325 0.229663 − 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) −40.0000 −1.31165
\(931\) 9.00000 0.294963
\(932\) 8.00000 0.262049
\(933\) −24.0000 −0.785725
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) 6.00000 0.196116
\(937\) 12.0000 0.392023 0.196011 − 0.980602i \(-0.437201\pi\)
0.196011 + 0.980602i \(0.437201\pi\)
\(938\) 56.0000 1.82846
\(939\) 28.0000 0.913745
\(940\) 0 0
\(941\) 42.0000 1.36916 0.684580 − 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) −4.00000 −0.130327
\(943\) 48.0000 1.56310
\(944\) −14.0000 −0.455661
\(945\) −32.0000 −1.04096
\(946\) 0 0
\(947\) −40.0000 −1.29983 −0.649913 − 0.760009i \(-0.725195\pi\)
−0.649913 + 0.760009i \(0.725195\pi\)
\(948\) 0 0
\(949\) 24.0000 0.779073
\(950\) 1.00000 0.0324443
\(951\) 36.0000 1.16738
\(952\) −16.0000 −0.518563
\(953\) 38.0000 1.23094 0.615470 − 0.788160i \(-0.288966\pi\)
0.615470 + 0.788160i \(0.288966\pi\)
\(954\) −6.00000 −0.194257
\(955\) −16.0000 −0.517748
\(956\) −8.00000 −0.258738
\(957\) 0 0
\(958\) −12.0000 −0.387702
\(959\) −24.0000 −0.775000
\(960\) −4.00000 −0.129099
\(961\) 69.0000 2.22581
\(962\) 36.0000 1.16069
\(963\) 4.00000 0.128898
\(964\) −14.0000 −0.450910
\(965\) −44.0000 −1.41641
\(966\) 64.0000 2.05917
\(967\) 4.00000 0.128631 0.0643157 − 0.997930i \(-0.479514\pi\)
0.0643157 + 0.997930i \(0.479514\pi\)
\(968\) 0 0
\(969\) −8.00000 −0.256997
\(970\) −4.00000 −0.128432
\(971\) 50.0000 1.60458 0.802288 − 0.596937i \(-0.203616\pi\)
0.802288 + 0.596937i \(0.203616\pi\)
\(972\) −10.0000 −0.320750
\(973\) 16.0000 0.512936
\(974\) −2.00000 −0.0640841
\(975\) 12.0000 0.384308
\(976\) 12.0000 0.384111
\(977\) −18.0000 −0.575871 −0.287936 − 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) −24.0000 −0.767435
\(979\) 0 0
\(980\) −18.0000 −0.574989
\(981\) −18.0000 −0.574696
\(982\) −8.00000 −0.255290
\(983\) −18.0000 −0.574111 −0.287055 − 0.957914i \(-0.592676\pi\)
−0.287055 + 0.957914i \(0.592676\pi\)
\(984\) −12.0000 −0.382546
\(985\) 24.0000 0.764704
\(986\) 24.0000 0.764316
\(987\) 0 0
\(988\) −6.00000 −0.190885
\(989\) −64.0000 −2.03508
\(990\) 0 0
\(991\) −54.0000 −1.71537 −0.857683 − 0.514178i \(-0.828097\pi\)
−0.857683 + 0.514178i \(0.828097\pi\)
\(992\) 10.0000 0.317500
\(993\) −52.0000 −1.65017
\(994\) 24.0000 0.761234
\(995\) −8.00000 −0.253617
\(996\) 0 0
\(997\) −8.00000 −0.253363 −0.126681 − 0.991943i \(-0.540433\pi\)
−0.126681 + 0.991943i \(0.540433\pi\)
\(998\) 4.00000 0.126618
\(999\) −24.0000 −0.759326
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 4598.2.a.i.1.1 ✓ 1
11.10 odd 2 4598.2.a.r.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4598.2.a.i.1.1 ✓ 1 1.1 even 1 trivial
4598.2.a.r.1.1 yes 1 11.10 odd 2