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Properties

Label 4080.2.a.p.1.1
Level $4080$
Weight $2$
Character 4080.1
Self dual yes
Analytic conductor $32.579$
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] = [4080,2,Mod(1,4080)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4080, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4080.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4080.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: \(32.5789640247\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2040)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4080.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -6.00000 q^{13} -1.00000 q^{15} +1.00000 q^{17} +1.00000 q^{19} -3.00000 q^{21} -6.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -5.00000 q^{29} +2.00000 q^{31} -1.00000 q^{33} +3.00000 q^{35} -11.0000 q^{37} +6.00000 q^{39} -9.00000 q^{41} +1.00000 q^{45} -5.00000 q^{47} +2.00000 q^{49} -1.00000 q^{51} -3.00000 q^{53} +1.00000 q^{55} -1.00000 q^{57} +12.0000 q^{59} -10.0000 q^{61} +3.00000 q^{63} -6.00000 q^{65} -4.00000 q^{67} +6.00000 q^{69} +4.00000 q^{71} +3.00000 q^{73} -1.00000 q^{75} +3.00000 q^{77} +2.00000 q^{79} +1.00000 q^{81} -12.0000 q^{83} +1.00000 q^{85} +5.00000 q^{87} +4.00000 q^{89} -18.0000 q^{91} -2.00000 q^{93} +1.00000 q^{95} +14.0000 q^{97} +1.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 3.00000 1.13389 0.566947 − 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 − 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 1.00000 0.242536
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 − 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −5.00000 −0.928477 −0.464238 − 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) 3.00000 0.507093
\(36\) 0 0
\(37\) −11.0000 −1.80839 −0.904194 − 0.427121i \(-0.859528\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) 0 0
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 − 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −5.00000 −0.729325 −0.364662 − 0.931140i \(-0.618816\pi\)
−0.364662 + 0.931140i \(0.618816\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) −1.00000 −0.140028
\(52\) 0 0
\(53\) −3.00000 −0.412082 −0.206041 − 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 3.00000 0.377964
\(64\) 0 0
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 − 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) 3.00000 0.351123 0.175562 − 0.984468i \(-0.443826\pi\)
0.175562 + 0.984468i \(0.443826\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) 2.00000 0.225018 0.112509 − 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 1.00000 0.108465
\(86\) 0 0
\(87\) 5.00000 0.536056
\(88\) 0 0
\(89\) 4.00000 0.423999 0.212000 − 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) 0 0
\(91\) −18.0000 −1.88691
\(92\) 0 0
\(93\) −2.00000 −0.207390
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 − 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) −18.0000 −1.79107 −0.895533 − 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 0 0
\(103\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) −3.00000 −0.292770
\(106\) 0 0
\(107\) 4.00000 0.386695 0.193347 − 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) 6.00000 0.574696 0.287348 − 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) 11.0000 1.04407
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 − 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) −6.00000 −0.559503
\(116\) 0 0
\(117\) −6.00000 −0.554700
\(118\) 0 0
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) 9.00000 0.811503
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 14.0000 1.24230 0.621150 − 0.783692i \(-0.286666\pi\)
0.621150 + 0.783692i \(0.286666\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 − 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 0 0
\(133\) 3.00000 0.260133
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −13.0000 −1.11066 −0.555332 − 0.831628i \(-0.687409\pi\)
−0.555332 + 0.831628i \(0.687409\pi\)
\(138\) 0 0
\(139\) 2.00000 0.169638 0.0848189 − 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) 0 0
\(141\) 5.00000 0.421076
\(142\) 0 0
\(143\) −6.00000 −0.501745
\(144\) 0 0
\(145\) −5.00000 −0.415227
\(146\) 0 0
\(147\) −2.00000 −0.164957
\(148\) 0 0
\(149\) 14.0000 1.14692 0.573462 − 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 0 0
\(151\) −7.00000 −0.569652 −0.284826 − 0.958579i \(-0.591936\pi\)
−0.284826 + 0.958579i \(0.591936\pi\)
\(152\) 0 0
\(153\) 1.00000 0.0808452
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) 12.0000 0.957704 0.478852 − 0.877896i \(-0.341053\pi\)
0.478852 + 0.877896i \(0.341053\pi\)
\(158\) 0 0
\(159\) 3.00000 0.237915
\(160\) 0 0
\(161\) −18.0000 −1.41860
\(162\) 0 0
\(163\) −19.0000 −1.48819 −0.744097 − 0.668071i \(-0.767120\pi\)
−0.744097 + 0.668071i \(0.767120\pi\)
\(164\) 0 0
\(165\) −1.00000 −0.0778499
\(166\) 0 0
\(167\) −10.0000 −0.773823 −0.386912 − 0.922117i \(-0.626458\pi\)
−0.386912 + 0.922117i \(0.626458\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) −4.00000 −0.304114 −0.152057 − 0.988372i \(-0.548590\pi\)
−0.152057 + 0.988372i \(0.548590\pi\)
\(174\) 0 0
\(175\) 3.00000 0.226779
\(176\) 0 0
\(177\) −12.0000 −0.901975
\(178\) 0 0
\(179\) −18.0000 −1.34538 −0.672692 − 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 − 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) 0 0
\(185\) −11.0000 −0.808736
\(186\) 0 0
\(187\) 1.00000 0.0731272
\(188\) 0 0
\(189\) −3.00000 −0.218218
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 − 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 − 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 0 0
\(195\) 6.00000 0.429669
\(196\) 0 0
\(197\) −16.0000 −1.13995 −0.569976 − 0.821661i \(-0.693048\pi\)
−0.569976 + 0.821661i \(0.693048\pi\)
\(198\) 0 0
\(199\) −18.0000 −1.27599 −0.637993 − 0.770042i \(-0.720235\pi\)
−0.637993 + 0.770042i \(0.720235\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) −15.0000 −1.05279
\(204\) 0 0
\(205\) −9.00000 −0.628587
\(206\) 0 0
\(207\) −6.00000 −0.417029
\(208\) 0 0
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 24.0000 1.65223 0.826114 − 0.563503i \(-0.190547\pi\)
0.826114 + 0.563503i \(0.190547\pi\)
\(212\) 0 0
\(213\) −4.00000 −0.274075
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 6.00000 0.407307
\(218\) 0 0
\(219\) −3.00000 −0.202721
\(220\) 0 0
\(221\) −6.00000 −0.403604
\(222\) 0 0
\(223\) 10.0000 0.669650 0.334825 − 0.942280i \(-0.391323\pi\)
0.334825 + 0.942280i \(0.391323\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −2.00000 −0.132745 −0.0663723 − 0.997795i \(-0.521143\pi\)
−0.0663723 + 0.997795i \(0.521143\pi\)
\(228\) 0 0
\(229\) −13.0000 −0.859064 −0.429532 − 0.903052i \(-0.641321\pi\)
−0.429532 + 0.903052i \(0.641321\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) 0 0
\(233\) −10.0000 −0.655122 −0.327561 − 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) −5.00000 −0.326164
\(236\) 0 0
\(237\) −2.00000 −0.129914
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 − 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) −6.00000 −0.381771
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) −28.0000 −1.76734 −0.883672 − 0.468106i \(-0.844936\pi\)
−0.883672 + 0.468106i \(0.844936\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) 0 0
\(255\) −1.00000 −0.0626224
\(256\) 0 0
\(257\) −14.0000 −0.873296 −0.436648 − 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) −33.0000 −2.05052
\(260\) 0 0
\(261\) −5.00000 −0.309492
\(262\) 0 0
\(263\) 21.0000 1.29492 0.647458 − 0.762101i \(-0.275832\pi\)
0.647458 + 0.762101i \(0.275832\pi\)
\(264\) 0 0
\(265\) −3.00000 −0.184289
\(266\) 0 0
\(267\) −4.00000 −0.244796
\(268\) 0 0
\(269\) −15.0000 −0.914566 −0.457283 − 0.889321i \(-0.651177\pi\)
−0.457283 + 0.889321i \(0.651177\pi\)
\(270\) 0 0
\(271\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 18.0000 1.08941
\(274\) 0 0
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 − 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 0 0
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) −16.0000 −0.954480 −0.477240 − 0.878773i \(-0.658363\pi\)
−0.477240 + 0.878773i \(0.658363\pi\)
\(282\) 0 0
\(283\) −5.00000 −0.297219 −0.148610 − 0.988896i \(-0.547480\pi\)
−0.148610 + 0.988896i \(0.547480\pi\)
\(284\) 0 0
\(285\) −1.00000 −0.0592349
\(286\) 0 0
\(287\) −27.0000 −1.59376
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) −14.0000 −0.820695
\(292\) 0 0
\(293\) 1.00000 0.0584206 0.0292103 − 0.999573i \(-0.490701\pi\)
0.0292103 + 0.999573i \(0.490701\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) 36.0000 2.08193
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 18.0000 1.03407
\(304\) 0 0
\(305\) −10.0000 −0.572598
\(306\) 0 0
\(307\) −32.0000 −1.82634 −0.913168 − 0.407583i \(-0.866372\pi\)
−0.913168 + 0.407583i \(0.866372\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1.00000 0.0567048 0.0283524 − 0.999598i \(-0.490974\pi\)
0.0283524 + 0.999598i \(0.490974\pi\)
\(312\) 0 0
\(313\) −29.0000 −1.63918 −0.819588 − 0.572953i \(-0.805798\pi\)
−0.819588 + 0.572953i \(0.805798\pi\)
\(314\) 0 0
\(315\) 3.00000 0.169031
\(316\) 0 0
\(317\) −22.0000 −1.23564 −0.617822 − 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) 0 0
\(319\) −5.00000 −0.279946
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 1.00000 0.0556415
\(324\) 0 0
\(325\) −6.00000 −0.332820
\(326\) 0 0
\(327\) −6.00000 −0.331801
\(328\) 0 0
\(329\) −15.0000 −0.826977
\(330\) 0 0
\(331\) 13.0000 0.714545 0.357272 − 0.934000i \(-0.383707\pi\)
0.357272 + 0.934000i \(0.383707\pi\)
\(332\) 0 0
\(333\) −11.0000 −0.602796
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) 23.0000 1.25289 0.626445 − 0.779466i \(-0.284509\pi\)
0.626445 + 0.779466i \(0.284509\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 0 0
\(345\) 6.00000 0.323029
\(346\) 0 0
\(347\) −12.0000 −0.644194 −0.322097 − 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) 35.0000 1.87351 0.936754 − 0.349990i \(-0.113815\pi\)
0.936754 + 0.349990i \(0.113815\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) 0 0
\(353\) 11.0000 0.585471 0.292735 − 0.956193i \(-0.405434\pi\)
0.292735 + 0.956193i \(0.405434\pi\)
\(354\) 0 0
\(355\) 4.00000 0.212298
\(356\) 0 0
\(357\) −3.00000 −0.158777
\(358\) 0 0
\(359\) −36.0000 −1.90001 −0.950004 − 0.312239i \(-0.898921\pi\)
−0.950004 + 0.312239i \(0.898921\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 10.0000 0.524864
\(364\) 0 0
\(365\) 3.00000 0.157027
\(366\) 0 0
\(367\) −16.0000 −0.835193 −0.417597 − 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 0 0
\(369\) −9.00000 −0.468521
\(370\) 0 0
\(371\) −9.00000 −0.467257
\(372\) 0 0
\(373\) 4.00000 0.207112 0.103556 − 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 30.0000 1.54508
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 − 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) −14.0000 −0.717242
\(382\) 0 0
\(383\) −15.0000 −0.766464 −0.383232 − 0.923652i \(-0.625189\pi\)
−0.383232 + 0.923652i \(0.625189\pi\)
\(384\) 0 0
\(385\) 3.00000 0.152894
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −36.0000 −1.82527 −0.912636 − 0.408773i \(-0.865957\pi\)
−0.912636 + 0.408773i \(0.865957\pi\)
\(390\) 0 0
\(391\) −6.00000 −0.303433
\(392\) 0 0
\(393\) −8.00000 −0.403547
\(394\) 0 0
\(395\) 2.00000 0.100631
\(396\) 0 0
\(397\) −7.00000 −0.351320 −0.175660 − 0.984451i \(-0.556206\pi\)
−0.175660 + 0.984451i \(0.556206\pi\)
\(398\) 0 0
\(399\) −3.00000 −0.150188
\(400\) 0 0
\(401\) 1.00000 0.0499376 0.0249688 − 0.999688i \(-0.492051\pi\)
0.0249688 + 0.999688i \(0.492051\pi\)
\(402\) 0 0
\(403\) −12.0000 −0.597763
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −11.0000 −0.545250
\(408\) 0 0
\(409\) 17.0000 0.840596 0.420298 − 0.907386i \(-0.361926\pi\)
0.420298 + 0.907386i \(0.361926\pi\)
\(410\) 0 0
\(411\) 13.0000 0.641243
\(412\) 0 0
\(413\) 36.0000 1.77144
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 0 0
\(417\) −2.00000 −0.0979404
\(418\) 0 0
\(419\) 39.0000 1.90527 0.952637 − 0.304109i \(-0.0983586\pi\)
0.952637 + 0.304109i \(0.0983586\pi\)
\(420\) 0 0
\(421\) −7.00000 −0.341159 −0.170580 − 0.985344i \(-0.554564\pi\)
−0.170580 + 0.985344i \(0.554564\pi\)
\(422\) 0 0
\(423\) −5.00000 −0.243108
\(424\) 0 0
\(425\) 1.00000 0.0485071
\(426\) 0 0
\(427\) −30.0000 −1.45180
\(428\) 0 0
\(429\) 6.00000 0.289683
\(430\) 0 0
\(431\) 7.00000 0.337178 0.168589 − 0.985686i \(-0.446079\pi\)
0.168589 + 0.985686i \(0.446079\pi\)
\(432\) 0 0
\(433\) 6.00000 0.288342 0.144171 − 0.989553i \(-0.453949\pi\)
0.144171 + 0.989553i \(0.453949\pi\)
\(434\) 0 0
\(435\) 5.00000 0.239732
\(436\) 0 0
\(437\) −6.00000 −0.287019
\(438\) 0 0
\(439\) 14.0000 0.668184 0.334092 − 0.942541i \(-0.391570\pi\)
0.334092 + 0.942541i \(0.391570\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) 4.00000 0.190046 0.0950229 − 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 0 0
\(445\) 4.00000 0.189618
\(446\) 0 0
\(447\) −14.0000 −0.662177
\(448\) 0 0
\(449\) 42.0000 1.98210 0.991051 − 0.133482i \(-0.0426157\pi\)
0.991051 + 0.133482i \(0.0426157\pi\)
\(450\) 0 0
\(451\) −9.00000 −0.423793
\(452\) 0 0
\(453\) 7.00000 0.328889
\(454\) 0 0
\(455\) −18.0000 −0.843853
\(456\) 0 0
\(457\) 32.0000 1.49690 0.748448 − 0.663193i \(-0.230799\pi\)
0.748448 + 0.663193i \(0.230799\pi\)
\(458\) 0 0
\(459\) −1.00000 −0.0466760
\(460\) 0 0
\(461\) −4.00000 −0.186299 −0.0931493 − 0.995652i \(-0.529693\pi\)
−0.0931493 + 0.995652i \(0.529693\pi\)
\(462\) 0 0
\(463\) 34.0000 1.58011 0.790057 − 0.613033i \(-0.210051\pi\)
0.790057 + 0.613033i \(0.210051\pi\)
\(464\) 0 0
\(465\) −2.00000 −0.0927478
\(466\) 0 0
\(467\) −11.0000 −0.509019 −0.254510 − 0.967070i \(-0.581914\pi\)
−0.254510 + 0.967070i \(0.581914\pi\)
\(468\) 0 0
\(469\) −12.0000 −0.554109
\(470\) 0 0
\(471\) −12.0000 −0.552931
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) −3.00000 −0.137361
\(478\) 0 0
\(479\) 8.00000 0.365529 0.182765 − 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 0 0
\(481\) 66.0000 3.00934
\(482\) 0 0
\(483\) 18.0000 0.819028
\(484\) 0 0
\(485\) 14.0000 0.635707
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 − 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 0 0
\(489\) 19.0000 0.859210
\(490\) 0 0
\(491\) 30.0000 1.35388 0.676941 − 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) 0 0
\(493\) −5.00000 −0.225189
\(494\) 0 0
\(495\) 1.00000 0.0449467
\(496\) 0 0
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) −16.0000 −0.716258 −0.358129 − 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) 0 0
\(501\) 10.0000 0.446767
\(502\) 0 0
\(503\) −14.0000 −0.624229 −0.312115 − 0.950044i \(-0.601037\pi\)
−0.312115 + 0.950044i \(0.601037\pi\)
\(504\) 0 0
\(505\) −18.0000 −0.800989
\(506\) 0 0
\(507\) −23.0000 −1.02147
\(508\) 0 0
\(509\) 22.0000 0.975133 0.487566 − 0.873086i \(-0.337885\pi\)
0.487566 + 0.873086i \(0.337885\pi\)
\(510\) 0 0
\(511\) 9.00000 0.398137
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −5.00000 −0.219900
\(518\) 0 0
\(519\) 4.00000 0.175581
\(520\) 0 0
\(521\) 9.00000 0.394297 0.197149 − 0.980374i \(-0.436832\pi\)
0.197149 + 0.980374i \(0.436832\pi\)
\(522\) 0 0
\(523\) −2.00000 −0.0874539 −0.0437269 − 0.999044i \(-0.513923\pi\)
−0.0437269 + 0.999044i \(0.513923\pi\)
\(524\) 0 0
\(525\) −3.00000 −0.130931
\(526\) 0 0
\(527\) 2.00000 0.0871214
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) 54.0000 2.33900
\(534\) 0 0
\(535\) 4.00000 0.172935
\(536\) 0 0
\(537\) 18.0000 0.776757
\(538\) 0 0
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 − 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) 0 0
\(543\) −14.0000 −0.600798
\(544\) 0 0
\(545\) 6.00000 0.257012
\(546\) 0 0
\(547\) −9.00000 −0.384812 −0.192406 − 0.981315i \(-0.561629\pi\)
−0.192406 + 0.981315i \(0.561629\pi\)
\(548\) 0 0
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) −5.00000 −0.213007
\(552\) 0 0
\(553\) 6.00000 0.255146
\(554\) 0 0
\(555\) 11.0000 0.466924
\(556\) 0 0
\(557\) 6.00000 0.254228 0.127114 − 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −1.00000 −0.0422200
\(562\) 0 0
\(563\) 11.0000 0.463595 0.231797 − 0.972764i \(-0.425539\pi\)
0.231797 + 0.972764i \(0.425539\pi\)
\(564\) 0 0
\(565\) 6.00000 0.252422
\(566\) 0 0
\(567\) 3.00000 0.125988
\(568\) 0 0
\(569\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 30.0000 1.25546 0.627730 − 0.778431i \(-0.283984\pi\)
0.627730 + 0.778431i \(0.283984\pi\)
\(572\) 0 0
\(573\) −12.0000 −0.501307
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) 0 0
\(577\) 32.0000 1.33218 0.666089 − 0.745873i \(-0.267967\pi\)
0.666089 + 0.745873i \(0.267967\pi\)
\(578\) 0 0
\(579\) 2.00000 0.0831172
\(580\) 0 0
\(581\) −36.0000 −1.49353
\(582\) 0 0
\(583\) −3.00000 −0.124247
\(584\) 0 0
\(585\) −6.00000 −0.248069
\(586\) 0 0
\(587\) −23.0000 −0.949312 −0.474656 − 0.880172i \(-0.657427\pi\)
−0.474656 + 0.880172i \(0.657427\pi\)
\(588\) 0 0
\(589\) 2.00000 0.0824086
\(590\) 0 0
\(591\) 16.0000 0.658152
\(592\) 0 0
\(593\) 3.00000 0.123195 0.0615976 − 0.998101i \(-0.480380\pi\)
0.0615976 + 0.998101i \(0.480380\pi\)
\(594\) 0 0
\(595\) 3.00000 0.122988
\(596\) 0 0
\(597\) 18.0000 0.736691
\(598\) 0 0
\(599\) −14.0000 −0.572024 −0.286012 − 0.958226i \(-0.592330\pi\)
−0.286012 + 0.958226i \(0.592330\pi\)
\(600\) 0 0
\(601\) −46.0000 −1.87638 −0.938190 − 0.346122i \(-0.887498\pi\)
−0.938190 + 0.346122i \(0.887498\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) 0 0
\(605\) −10.0000 −0.406558
\(606\) 0 0
\(607\) 35.0000 1.42061 0.710303 − 0.703896i \(-0.248558\pi\)
0.710303 + 0.703896i \(0.248558\pi\)
\(608\) 0 0
\(609\) 15.0000 0.607831
\(610\) 0 0
\(611\) 30.0000 1.21367
\(612\) 0 0
\(613\) −20.0000 −0.807792 −0.403896 − 0.914805i \(-0.632344\pi\)
−0.403896 + 0.914805i \(0.632344\pi\)
\(614\) 0 0
\(615\) 9.00000 0.362915
\(616\) 0 0
\(617\) −34.0000 −1.36879 −0.684394 − 0.729112i \(-0.739933\pi\)
−0.684394 + 0.729112i \(0.739933\pi\)
\(618\) 0 0
\(619\) 34.0000 1.36658 0.683288 − 0.730149i \(-0.260549\pi\)
0.683288 + 0.730149i \(0.260549\pi\)
\(620\) 0 0
\(621\) 6.00000 0.240772
\(622\) 0 0
\(623\) 12.0000 0.480770
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −1.00000 −0.0399362
\(628\) 0 0
\(629\) −11.0000 −0.438599
\(630\) 0 0
\(631\) −23.0000 −0.915616 −0.457808 − 0.889051i \(-0.651365\pi\)
−0.457808 + 0.889051i \(0.651365\pi\)
\(632\) 0 0
\(633\) −24.0000 −0.953914
\(634\) 0 0
\(635\) 14.0000 0.555573
\(636\) 0 0
\(637\) −12.0000 −0.475457
\(638\) 0 0
\(639\) 4.00000 0.158238
\(640\) 0 0
\(641\) 33.0000 1.30342 0.651711 − 0.758468i \(-0.274052\pi\)
0.651711 + 0.758468i \(0.274052\pi\)
\(642\) 0 0
\(643\) −49.0000 −1.93237 −0.966186 − 0.257847i \(-0.916987\pi\)
−0.966186 + 0.257847i \(0.916987\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 48.0000 1.88707 0.943537 − 0.331266i \(-0.107476\pi\)
0.943537 + 0.331266i \(0.107476\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) 0 0
\(651\) −6.00000 −0.235159
\(652\) 0 0
\(653\) −22.0000 −0.860927 −0.430463 − 0.902608i \(-0.641650\pi\)
−0.430463 + 0.902608i \(0.641650\pi\)
\(654\) 0 0
\(655\) 8.00000 0.312586
\(656\) 0 0
\(657\) 3.00000 0.117041
\(658\) 0 0
\(659\) −10.0000 −0.389545 −0.194772 − 0.980848i \(-0.562397\pi\)
−0.194772 + 0.980848i \(0.562397\pi\)
\(660\) 0 0
\(661\) 39.0000 1.51692 0.758462 − 0.651717i \(-0.225951\pi\)
0.758462 + 0.651717i \(0.225951\pi\)
\(662\) 0 0
\(663\) 6.00000 0.233021
\(664\) 0 0
\(665\) 3.00000 0.116335
\(666\) 0 0
\(667\) 30.0000 1.16160
\(668\) 0 0
\(669\) −10.0000 −0.386622
\(670\) 0 0
\(671\) −10.0000 −0.386046
\(672\) 0 0
\(673\) 22.0000 0.848038 0.424019 − 0.905653i \(-0.360619\pi\)
0.424019 + 0.905653i \(0.360619\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −46.0000 −1.76792 −0.883962 − 0.467559i \(-0.845134\pi\)
−0.883962 + 0.467559i \(0.845134\pi\)
\(678\) 0 0
\(679\) 42.0000 1.61181
\(680\) 0 0
\(681\) 2.00000 0.0766402
\(682\) 0 0
\(683\) −16.0000 −0.612223 −0.306111 − 0.951996i \(-0.599028\pi\)
−0.306111 + 0.951996i \(0.599028\pi\)
\(684\) 0 0
\(685\) −13.0000 −0.496704
\(686\) 0 0
\(687\) 13.0000 0.495981
\(688\) 0 0
\(689\) 18.0000 0.685745
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 − 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 0 0
\(693\) 3.00000 0.113961
\(694\) 0 0
\(695\) 2.00000 0.0758643
\(696\) 0 0
\(697\) −9.00000 −0.340899
\(698\) 0 0
\(699\) 10.0000 0.378235
\(700\) 0 0
\(701\) −24.0000 −0.906467 −0.453234 − 0.891392i \(-0.649730\pi\)
−0.453234 + 0.891392i \(0.649730\pi\)
\(702\) 0 0
\(703\) −11.0000 −0.414873
\(704\) 0 0
\(705\) 5.00000 0.188311
\(706\) 0 0
\(707\) −54.0000 −2.03088
\(708\) 0 0
\(709\) 26.0000 0.976450 0.488225 − 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 0 0
\(711\) 2.00000 0.0750059
\(712\) 0 0
\(713\) −12.0000 −0.449404
\(714\) 0 0
\(715\) −6.00000 −0.224387
\(716\) 0 0
\(717\) 6.00000 0.224074
\(718\) 0 0
\(719\) −33.0000 −1.23069 −0.615346 − 0.788257i \(-0.710984\pi\)
−0.615346 + 0.788257i \(0.710984\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −10.0000 −0.371904
\(724\) 0 0
\(725\) −5.00000 −0.185695
\(726\) 0 0
\(727\) −10.0000 −0.370879 −0.185440 − 0.982656i \(-0.559371\pi\)
−0.185440 + 0.982656i \(0.559371\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 34.0000 1.25582 0.627909 − 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) 0 0
\(735\) −2.00000 −0.0737711
\(736\) 0 0
\(737\) −4.00000 −0.147342
\(738\) 0 0
\(739\) 28.0000 1.03000 0.514998 − 0.857191i \(-0.327793\pi\)
0.514998 + 0.857191i \(0.327793\pi\)
\(740\) 0 0
\(741\) 6.00000 0.220416
\(742\) 0 0
\(743\) −14.0000 −0.513610 −0.256805 − 0.966463i \(-0.582670\pi\)
−0.256805 + 0.966463i \(0.582670\pi\)
\(744\) 0 0
\(745\) 14.0000 0.512920
\(746\) 0 0
\(747\) −12.0000 −0.439057
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) −2.00000 −0.0729810 −0.0364905 − 0.999334i \(-0.511618\pi\)
−0.0364905 + 0.999334i \(0.511618\pi\)
\(752\) 0 0
\(753\) 28.0000 1.02038
\(754\) 0 0
\(755\) −7.00000 −0.254756
\(756\) 0 0
\(757\) −44.0000 −1.59921 −0.799604 − 0.600528i \(-0.794957\pi\)
−0.799604 + 0.600528i \(0.794957\pi\)
\(758\) 0 0
\(759\) 6.00000 0.217786
\(760\) 0 0
\(761\) 26.0000 0.942499 0.471250 − 0.882000i \(-0.343803\pi\)
0.471250 + 0.882000i \(0.343803\pi\)
\(762\) 0 0
\(763\) 18.0000 0.651644
\(764\) 0 0
\(765\) 1.00000 0.0361551
\(766\) 0 0
\(767\) −72.0000 −2.59977
\(768\) 0 0
\(769\) 27.0000 0.973645 0.486822 − 0.873501i \(-0.338156\pi\)
0.486822 + 0.873501i \(0.338156\pi\)
\(770\) 0 0
\(771\) 14.0000 0.504198
\(772\) 0 0
\(773\) 3.00000 0.107903 0.0539513 − 0.998544i \(-0.482818\pi\)
0.0539513 + 0.998544i \(0.482818\pi\)
\(774\) 0 0
\(775\) 2.00000 0.0718421
\(776\) 0 0
\(777\) 33.0000 1.18387
\(778\) 0 0
\(779\) −9.00000 −0.322458
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) 0 0
\(783\) 5.00000 0.178685
\(784\) 0 0
\(785\) 12.0000 0.428298
\(786\) 0 0
\(787\) 35.0000 1.24762 0.623808 − 0.781578i \(-0.285585\pi\)
0.623808 + 0.781578i \(0.285585\pi\)
\(788\) 0 0
\(789\) −21.0000 −0.747620
\(790\) 0 0
\(791\) 18.0000 0.640006
\(792\) 0 0
\(793\) 60.0000 2.13066
\(794\) 0 0
\(795\) 3.00000 0.106399
\(796\) 0 0
\(797\) −29.0000 −1.02723 −0.513616 − 0.858020i \(-0.671695\pi\)
−0.513616 + 0.858020i \(0.671695\pi\)
\(798\) 0 0
\(799\) −5.00000 −0.176887
\(800\) 0 0
\(801\) 4.00000 0.141333
\(802\) 0 0
\(803\) 3.00000 0.105868
\(804\) 0 0
\(805\) −18.0000 −0.634417
\(806\) 0 0
\(807\) 15.0000 0.528025
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 − 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 − 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −19.0000 −0.665541
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −18.0000 −0.628971
\(820\) 0 0
\(821\) −34.0000 −1.18661 −0.593304 − 0.804978i \(-0.702177\pi\)
−0.593304 + 0.804978i \(0.702177\pi\)
\(822\) 0 0
\(823\) 57.0000 1.98690 0.993448 − 0.114289i \(-0.0364590\pi\)
0.993448 + 0.114289i \(0.0364590\pi\)
\(824\) 0 0
\(825\) −1.00000 −0.0348155
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 − 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) 13.0000 0.451509 0.225754 − 0.974184i \(-0.427515\pi\)
0.225754 + 0.974184i \(0.427515\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) 0 0
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) −10.0000 −0.346064
\(836\) 0 0
\(837\) −2.00000 −0.0691301
\(838\) 0 0
\(839\) 35.0000 1.20833 0.604167 − 0.796858i \(-0.293506\pi\)
0.604167 + 0.796858i \(0.293506\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 0 0
\(843\) 16.0000 0.551069
\(844\) 0 0
\(845\) 23.0000 0.791224
\(846\) 0 0
\(847\) −30.0000 −1.03081
\(848\) 0 0
\(849\) 5.00000 0.171600
\(850\) 0 0
\(851\) 66.0000 2.26245
\(852\) 0 0
\(853\) 2.00000 0.0684787 0.0342393 − 0.999414i \(-0.489099\pi\)
0.0342393 + 0.999414i \(0.489099\pi\)
\(854\) 0 0
\(855\) 1.00000 0.0341993
\(856\) 0 0
\(857\) −40.0000 −1.36637 −0.683187 − 0.730243i \(-0.739407\pi\)
−0.683187 + 0.730243i \(0.739407\pi\)
\(858\) 0 0
\(859\) 23.0000 0.784750 0.392375 − 0.919805i \(-0.371654\pi\)
0.392375 + 0.919805i \(0.371654\pi\)
\(860\) 0 0
\(861\) 27.0000 0.920158
\(862\) 0 0
\(863\) −35.0000 −1.19141 −0.595707 − 0.803202i \(-0.703128\pi\)
−0.595707 + 0.803202i \(0.703128\pi\)
\(864\) 0 0
\(865\) −4.00000 −0.136004
\(866\) 0 0
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) 2.00000 0.0678454
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) 0 0
\(873\) 14.0000 0.473828
\(874\) 0 0
\(875\) 3.00000 0.101419
\(876\) 0 0
\(877\) 13.0000 0.438979 0.219489 − 0.975615i \(-0.429561\pi\)
0.219489 + 0.975615i \(0.429561\pi\)
\(878\) 0 0
\(879\) −1.00000 −0.0337292
\(880\) 0 0
\(881\) 11.0000 0.370599 0.185300 − 0.982682i \(-0.440674\pi\)
0.185300 + 0.982682i \(0.440674\pi\)
\(882\) 0 0
\(883\) 8.00000 0.269221 0.134611 − 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) 0 0
\(885\) −12.0000 −0.403376
\(886\) 0 0
\(887\) 12.0000 0.402921 0.201460 − 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) 0 0
\(889\) 42.0000 1.40863
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) −5.00000 −0.167319
\(894\) 0 0
\(895\) −18.0000 −0.601674
\(896\) 0 0
\(897\) −36.0000 −1.20201
\(898\) 0 0
\(899\) −10.0000 −0.333519
\(900\) 0 0
\(901\) −3.00000 −0.0999445
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 14.0000 0.465376
\(906\) 0 0
\(907\) −37.0000 −1.22856 −0.614282 − 0.789086i \(-0.710554\pi\)
−0.614282 + 0.789086i \(0.710554\pi\)
\(908\) 0 0
\(909\) −18.0000 −0.597022
\(910\) 0 0
\(911\) 45.0000 1.49092 0.745458 − 0.666552i \(-0.232231\pi\)
0.745458 + 0.666552i \(0.232231\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) 0 0
\(915\) 10.0000 0.330590
\(916\) 0 0
\(917\) 24.0000 0.792550
\(918\) 0 0
\(919\) 49.0000 1.61636 0.808180 − 0.588935i \(-0.200453\pi\)
0.808180 + 0.588935i \(0.200453\pi\)
\(920\) 0 0
\(921\) 32.0000 1.05444
\(922\) 0 0
\(923\) −24.0000 −0.789970
\(924\) 0 0
\(925\) −11.0000 −0.361678
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 27.0000 0.885841 0.442921 − 0.896561i \(-0.353942\pi\)
0.442921 + 0.896561i \(0.353942\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) 0 0
\(933\) −1.00000 −0.0327385
\(934\) 0 0
\(935\) 1.00000 0.0327035
\(936\) 0 0
\(937\) 42.0000 1.37208 0.686040 − 0.727564i \(-0.259347\pi\)
0.686040 + 0.727564i \(0.259347\pi\)
\(938\) 0 0
\(939\) 29.0000 0.946379
\(940\) 0 0
\(941\) −50.0000 −1.62995 −0.814977 − 0.579494i \(-0.803250\pi\)
−0.814977 + 0.579494i \(0.803250\pi\)
\(942\) 0 0
\(943\) 54.0000 1.75848
\(944\) 0 0
\(945\) −3.00000 −0.0975900
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 − 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) −18.0000 −0.584305
\(950\) 0 0
\(951\) 22.0000 0.713399
\(952\) 0 0
\(953\) −5.00000 −0.161966 −0.0809829 − 0.996715i \(-0.525806\pi\)
−0.0809829 + 0.996715i \(0.525806\pi\)
\(954\) 0 0
\(955\) 12.0000 0.388311
\(956\) 0 0
\(957\) 5.00000 0.161627
\(958\) 0 0
\(959\) −39.0000 −1.25938
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 4.00000 0.128898
\(964\) 0 0
\(965\) −2.00000 −0.0643823
\(966\) 0 0
\(967\) 12.0000 0.385894 0.192947 − 0.981209i \(-0.438195\pi\)
0.192947 + 0.981209i \(0.438195\pi\)
\(968\) 0 0
\(969\) −1.00000 −0.0321246
\(970\) 0 0
\(971\) 36.0000 1.15529 0.577647 − 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 0 0
\(973\) 6.00000 0.192351
\(974\) 0 0
\(975\) 6.00000 0.192154
\(976\) 0 0
\(977\) −14.0000 −0.447900 −0.223950 − 0.974601i \(-0.571895\pi\)
−0.223950 + 0.974601i \(0.571895\pi\)
\(978\) 0 0
\(979\) 4.00000 0.127841
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) 0 0
\(983\) 10.0000 0.318950 0.159475 − 0.987202i \(-0.449020\pi\)
0.159475 + 0.987202i \(0.449020\pi\)
\(984\) 0 0
\(985\) −16.0000 −0.509802
\(986\) 0 0
\(987\) 15.0000 0.477455
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −46.0000 −1.46124 −0.730619 − 0.682785i \(-0.760768\pi\)
−0.730619 + 0.682785i \(0.760768\pi\)
\(992\) 0 0
\(993\) −13.0000 −0.412543
\(994\) 0 0
\(995\) −18.0000 −0.570638
\(996\) 0 0
\(997\) 27.0000 0.855099 0.427549 − 0.903992i \(-0.359377\pi\)
0.427549 + 0.903992i \(0.359377\pi\)
\(998\) 0 0
\(999\) 11.0000 0.348025
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 4080.2.a.p.1.1 1
4.3 odd 2 2040.2.a.m.1.1 ✓ 1
12.11 even 2 6120.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2040.2.a.m.1.1 ✓ 1 4.3 odd 2
4080.2.a.p.1.1 1 1.1 even 1 trivial
6120.2.a.c.1.1 1 12.11 even 2