Properties

Label 3720.2.a.i.1.1
Level $3720$
Weight $2$
Character 3720.1
Self dual yes
Analytic conductor $29.704$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3720,2,Mod(1,3720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3720, 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("3720.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3720.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: \(29.7043495519\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 3720.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} -1.23607 q^{7} +1.00000 q^{9} -3.23607 q^{13} +1.00000 q^{15} +4.47214 q^{17} +6.47214 q^{19} +1.23607 q^{21} -4.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -7.23607 q^{29} -1.00000 q^{31} +1.23607 q^{35} +4.76393 q^{37} +3.23607 q^{39} -6.00000 q^{41} +6.47214 q^{43} -1.00000 q^{45} +8.94427 q^{47} -5.47214 q^{49} -4.47214 q^{51} +8.47214 q^{53} -6.47214 q^{57} +6.76393 q^{59} -12.4721 q^{61} -1.23607 q^{63} +3.23607 q^{65} +3.70820 q^{67} +4.00000 q^{69} -9.23607 q^{71} +12.1803 q^{73} -1.00000 q^{75} -8.94427 q^{79} +1.00000 q^{81} -4.94427 q^{83} -4.47214 q^{85} +7.23607 q^{87} -7.23607 q^{89} +4.00000 q^{91} +1.00000 q^{93} -6.47214 q^{95} -8.47214 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 2 q^{13} + 2 q^{15} + 4 q^{19} - 2 q^{21} - 8 q^{23} + 2 q^{25} - 2 q^{27} - 10 q^{29} - 2 q^{31} - 2 q^{35} + 14 q^{37} + 2 q^{39} - 12 q^{41} + 4 q^{43} - 2 q^{45}+ \cdots - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(7\) −1.23607 −0.467190 −0.233595 0.972334i \(-0.575049\pi\)
−0.233595 + 0.972334i \(0.575049\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −3.23607 −0.897524 −0.448762 0.893651i \(-0.648135\pi\)
−0.448762 + 0.893651i \(0.648135\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 4.47214 1.08465 0.542326 0.840168i \(-0.317544\pi\)
0.542326 + 0.840168i \(0.317544\pi\)
\(18\) 0 0
\(19\) 6.47214 1.48481 0.742405 0.669951i \(-0.233685\pi\)
0.742405 + 0.669951i \(0.233685\pi\)
\(20\) 0 0
\(21\) 1.23607 0.269732
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −7.23607 −1.34370 −0.671852 0.740685i \(-0.734501\pi\)
−0.671852 + 0.740685i \(0.734501\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.23607 0.208934
\(36\) 0 0
\(37\) 4.76393 0.783186 0.391593 0.920139i \(-0.371924\pi\)
0.391593 + 0.920139i \(0.371924\pi\)
\(38\) 0 0
\(39\) 3.23607 0.518186
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 6.47214 0.986991 0.493496 0.869748i \(-0.335719\pi\)
0.493496 + 0.869748i \(0.335719\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 8.94427 1.30466 0.652328 0.757937i \(-0.273792\pi\)
0.652328 + 0.757937i \(0.273792\pi\)
\(48\) 0 0
\(49\) −5.47214 −0.781734
\(50\) 0 0
\(51\) −4.47214 −0.626224
\(52\) 0 0
\(53\) 8.47214 1.16374 0.581869 0.813283i \(-0.302322\pi\)
0.581869 + 0.813283i \(0.302322\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −6.47214 −0.857255
\(58\) 0 0
\(59\) 6.76393 0.880589 0.440294 0.897853i \(-0.354874\pi\)
0.440294 + 0.897853i \(0.354874\pi\)
\(60\) 0 0
\(61\) −12.4721 −1.59689 −0.798447 0.602066i \(-0.794345\pi\)
−0.798447 + 0.602066i \(0.794345\pi\)
\(62\) 0 0
\(63\) −1.23607 −0.155730
\(64\) 0 0
\(65\) 3.23607 0.401385
\(66\) 0 0
\(67\) 3.70820 0.453029 0.226515 0.974008i \(-0.427267\pi\)
0.226515 + 0.974008i \(0.427267\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −9.23607 −1.09612 −0.548060 0.836439i \(-0.684633\pi\)
−0.548060 + 0.836439i \(0.684633\pi\)
\(72\) 0 0
\(73\) 12.1803 1.42560 0.712800 0.701367i \(-0.247427\pi\)
0.712800 + 0.701367i \(0.247427\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.94427 −1.00631 −0.503155 0.864196i \(-0.667827\pi\)
−0.503155 + 0.864196i \(0.667827\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.94427 −0.542704 −0.271352 0.962480i \(-0.587471\pi\)
−0.271352 + 0.962480i \(0.587471\pi\)
\(84\) 0 0
\(85\) −4.47214 −0.485071
\(86\) 0 0
\(87\) 7.23607 0.775788
\(88\) 0 0
\(89\) −7.23607 −0.767022 −0.383511 0.923536i \(-0.625285\pi\)
−0.383511 + 0.923536i \(0.625285\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) 1.00000 0.103695
\(94\) 0 0
\(95\) −6.47214 −0.664027
\(96\) 0 0
\(97\) −8.47214 −0.860215 −0.430108 0.902778i \(-0.641524\pi\)
−0.430108 + 0.902778i \(0.641524\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −10.9443 −1.08900 −0.544498 0.838762i \(-0.683280\pi\)
−0.544498 + 0.838762i \(0.683280\pi\)
\(102\) 0 0
\(103\) −16.6525 −1.64082 −0.820409 0.571778i \(-0.806254\pi\)
−0.820409 + 0.571778i \(0.806254\pi\)
\(104\) 0 0
\(105\) −1.23607 −0.120628
\(106\) 0 0
\(107\) 12.9443 1.25137 0.625685 0.780076i \(-0.284820\pi\)
0.625685 + 0.780076i \(0.284820\pi\)
\(108\) 0 0
\(109\) −3.52786 −0.337908 −0.168954 0.985624i \(-0.554039\pi\)
−0.168954 + 0.985624i \(0.554039\pi\)
\(110\) 0 0
\(111\) −4.76393 −0.452172
\(112\) 0 0
\(113\) 2.94427 0.276974 0.138487 0.990364i \(-0.455776\pi\)
0.138487 + 0.990364i \(0.455776\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 0 0
\(117\) −3.23607 −0.299175
\(118\) 0 0
\(119\) −5.52786 −0.506738
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 6.00000 0.541002
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −11.4164 −1.01304 −0.506521 0.862228i \(-0.669069\pi\)
−0.506521 + 0.862228i \(0.669069\pi\)
\(128\) 0 0
\(129\) −6.47214 −0.569840
\(130\) 0 0
\(131\) −9.23607 −0.806959 −0.403480 0.914989i \(-0.632199\pi\)
−0.403480 + 0.914989i \(0.632199\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −8.94427 −0.753244
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 7.23607 0.600923
\(146\) 0 0
\(147\) 5.47214 0.451334
\(148\) 0 0
\(149\) −18.9443 −1.55198 −0.775988 0.630748i \(-0.782748\pi\)
−0.775988 + 0.630748i \(0.782748\pi\)
\(150\) 0 0
\(151\) 13.8885 1.13023 0.565117 0.825011i \(-0.308831\pi\)
0.565117 + 0.825011i \(0.308831\pi\)
\(152\) 0 0
\(153\) 4.47214 0.361551
\(154\) 0 0
\(155\) 1.00000 0.0803219
\(156\) 0 0
\(157\) −5.41641 −0.432276 −0.216138 0.976363i \(-0.569346\pi\)
−0.216138 + 0.976363i \(0.569346\pi\)
\(158\) 0 0
\(159\) −8.47214 −0.671884
\(160\) 0 0
\(161\) 4.94427 0.389663
\(162\) 0 0
\(163\) 6.76393 0.529792 0.264896 0.964277i \(-0.414662\pi\)
0.264896 + 0.964277i \(0.414662\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −2.52786 −0.194451
\(170\) 0 0
\(171\) 6.47214 0.494937
\(172\) 0 0
\(173\) −22.0000 −1.67263 −0.836315 0.548250i \(-0.815294\pi\)
−0.836315 + 0.548250i \(0.815294\pi\)
\(174\) 0 0
\(175\) −1.23607 −0.0934380
\(176\) 0 0
\(177\) −6.76393 −0.508408
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 21.4164 1.59187 0.795935 0.605383i \(-0.206980\pi\)
0.795935 + 0.605383i \(0.206980\pi\)
\(182\) 0 0
\(183\) 12.4721 0.921967
\(184\) 0 0
\(185\) −4.76393 −0.350251
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 1.23607 0.0899107
\(190\) 0 0
\(191\) −14.7639 −1.06828 −0.534140 0.845396i \(-0.679365\pi\)
−0.534140 + 0.845396i \(0.679365\pi\)
\(192\) 0 0
\(193\) 9.41641 0.677808 0.338904 0.940821i \(-0.389944\pi\)
0.338904 + 0.940821i \(0.389944\pi\)
\(194\) 0 0
\(195\) −3.23607 −0.231740
\(196\) 0 0
\(197\) 2.94427 0.209771 0.104885 0.994484i \(-0.466552\pi\)
0.104885 + 0.994484i \(0.466552\pi\)
\(198\) 0 0
\(199\) 13.8885 0.984533 0.492266 0.870445i \(-0.336169\pi\)
0.492266 + 0.870445i \(0.336169\pi\)
\(200\) 0 0
\(201\) −3.70820 −0.261557
\(202\) 0 0
\(203\) 8.94427 0.627765
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 24.3607 1.67706 0.838529 0.544857i \(-0.183416\pi\)
0.838529 + 0.544857i \(0.183416\pi\)
\(212\) 0 0
\(213\) 9.23607 0.632845
\(214\) 0 0
\(215\) −6.47214 −0.441396
\(216\) 0 0
\(217\) 1.23607 0.0839098
\(218\) 0 0
\(219\) −12.1803 −0.823071
\(220\) 0 0
\(221\) −14.4721 −0.973501
\(222\) 0 0
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −15.4164 −1.02322 −0.511611 0.859217i \(-0.670951\pi\)
−0.511611 + 0.859217i \(0.670951\pi\)
\(228\) 0 0
\(229\) 18.9443 1.25187 0.625936 0.779874i \(-0.284717\pi\)
0.625936 + 0.779874i \(0.284717\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −29.4164 −1.92713 −0.963566 0.267469i \(-0.913813\pi\)
−0.963566 + 0.267469i \(0.913813\pi\)
\(234\) 0 0
\(235\) −8.94427 −0.583460
\(236\) 0 0
\(237\) 8.94427 0.580993
\(238\) 0 0
\(239\) −7.41641 −0.479728 −0.239864 0.970807i \(-0.577103\pi\)
−0.239864 + 0.970807i \(0.577103\pi\)
\(240\) 0 0
\(241\) −1.05573 −0.0680054 −0.0340027 0.999422i \(-0.510825\pi\)
−0.0340027 + 0.999422i \(0.510825\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 5.47214 0.349602
\(246\) 0 0
\(247\) −20.9443 −1.33265
\(248\) 0 0
\(249\) 4.94427 0.313331
\(250\) 0 0
\(251\) −8.00000 −0.504956 −0.252478 0.967603i \(-0.581245\pi\)
−0.252478 + 0.967603i \(0.581245\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 4.47214 0.280056
\(256\) 0 0
\(257\) −21.4164 −1.33592 −0.667959 0.744198i \(-0.732832\pi\)
−0.667959 + 0.744198i \(0.732832\pi\)
\(258\) 0 0
\(259\) −5.88854 −0.365896
\(260\) 0 0
\(261\) −7.23607 −0.447901
\(262\) 0 0
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) −8.47214 −0.520439
\(266\) 0 0
\(267\) 7.23607 0.442840
\(268\) 0 0
\(269\) −30.0689 −1.83333 −0.916666 0.399654i \(-0.869130\pi\)
−0.916666 + 0.399654i \(0.869130\pi\)
\(270\) 0 0
\(271\) −25.8885 −1.57262 −0.786309 0.617834i \(-0.788010\pi\)
−0.786309 + 0.617834i \(0.788010\pi\)
\(272\) 0 0
\(273\) −4.00000 −0.242091
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −3.81966 −0.229501 −0.114751 0.993394i \(-0.536607\pi\)
−0.114751 + 0.993394i \(0.536607\pi\)
\(278\) 0 0
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) −16.4721 −0.982645 −0.491323 0.870978i \(-0.663486\pi\)
−0.491323 + 0.870978i \(0.663486\pi\)
\(282\) 0 0
\(283\) −14.1803 −0.842934 −0.421467 0.906844i \(-0.638485\pi\)
−0.421467 + 0.906844i \(0.638485\pi\)
\(284\) 0 0
\(285\) 6.47214 0.383376
\(286\) 0 0
\(287\) 7.41641 0.437777
\(288\) 0 0
\(289\) 3.00000 0.176471
\(290\) 0 0
\(291\) 8.47214 0.496645
\(292\) 0 0
\(293\) 13.4164 0.783795 0.391897 0.920009i \(-0.371819\pi\)
0.391897 + 0.920009i \(0.371819\pi\)
\(294\) 0 0
\(295\) −6.76393 −0.393811
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 12.9443 0.748587
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 0 0
\(303\) 10.9443 0.628732
\(304\) 0 0
\(305\) 12.4721 0.714152
\(306\) 0 0
\(307\) −6.76393 −0.386038 −0.193019 0.981195i \(-0.561828\pi\)
−0.193019 + 0.981195i \(0.561828\pi\)
\(308\) 0 0
\(309\) 16.6525 0.947326
\(310\) 0 0
\(311\) −14.7639 −0.837186 −0.418593 0.908174i \(-0.637477\pi\)
−0.418593 + 0.908174i \(0.637477\pi\)
\(312\) 0 0
\(313\) −10.6525 −0.602114 −0.301057 0.953606i \(-0.597339\pi\)
−0.301057 + 0.953606i \(0.597339\pi\)
\(314\) 0 0
\(315\) 1.23607 0.0696445
\(316\) 0 0
\(317\) −28.4721 −1.59915 −0.799577 0.600563i \(-0.794943\pi\)
−0.799577 + 0.600563i \(0.794943\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −12.9443 −0.722479
\(322\) 0 0
\(323\) 28.9443 1.61050
\(324\) 0 0
\(325\) −3.23607 −0.179505
\(326\) 0 0
\(327\) 3.52786 0.195091
\(328\) 0 0
\(329\) −11.0557 −0.609522
\(330\) 0 0
\(331\) −28.9443 −1.59092 −0.795461 0.606005i \(-0.792771\pi\)
−0.795461 + 0.606005i \(0.792771\pi\)
\(332\) 0 0
\(333\) 4.76393 0.261062
\(334\) 0 0
\(335\) −3.70820 −0.202601
\(336\) 0 0
\(337\) −32.1803 −1.75297 −0.876487 0.481425i \(-0.840119\pi\)
−0.876487 + 0.481425i \(0.840119\pi\)
\(338\) 0 0
\(339\) −2.94427 −0.159911
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 15.4164 0.832408
\(344\) 0 0
\(345\) −4.00000 −0.215353
\(346\) 0 0
\(347\) −13.8885 −0.745576 −0.372788 0.927917i \(-0.621598\pi\)
−0.372788 + 0.927917i \(0.621598\pi\)
\(348\) 0 0
\(349\) 20.4721 1.09585 0.547924 0.836528i \(-0.315418\pi\)
0.547924 + 0.836528i \(0.315418\pi\)
\(350\) 0 0
\(351\) 3.23607 0.172729
\(352\) 0 0
\(353\) −16.4721 −0.876723 −0.438362 0.898799i \(-0.644441\pi\)
−0.438362 + 0.898799i \(0.644441\pi\)
\(354\) 0 0
\(355\) 9.23607 0.490200
\(356\) 0 0
\(357\) 5.52786 0.292566
\(358\) 0 0
\(359\) 27.1246 1.43158 0.715791 0.698314i \(-0.246066\pi\)
0.715791 + 0.698314i \(0.246066\pi\)
\(360\) 0 0
\(361\) 22.8885 1.20466
\(362\) 0 0
\(363\) 11.0000 0.577350
\(364\) 0 0
\(365\) −12.1803 −0.637548
\(366\) 0 0
\(367\) 7.05573 0.368306 0.184153 0.982898i \(-0.441046\pi\)
0.184153 + 0.982898i \(0.441046\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) −10.4721 −0.543686
\(372\) 0 0
\(373\) 30.3607 1.57202 0.786008 0.618216i \(-0.212144\pi\)
0.786008 + 0.618216i \(0.212144\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 23.4164 1.20601
\(378\) 0 0
\(379\) −19.4164 −0.997354 −0.498677 0.866788i \(-0.666181\pi\)
−0.498677 + 0.866788i \(0.666181\pi\)
\(380\) 0 0
\(381\) 11.4164 0.584880
\(382\) 0 0
\(383\) 4.00000 0.204390 0.102195 0.994764i \(-0.467413\pi\)
0.102195 + 0.994764i \(0.467413\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 6.47214 0.328997
\(388\) 0 0
\(389\) −10.2918 −0.521815 −0.260907 0.965364i \(-0.584022\pi\)
−0.260907 + 0.965364i \(0.584022\pi\)
\(390\) 0 0
\(391\) −17.8885 −0.904663
\(392\) 0 0
\(393\) 9.23607 0.465898
\(394\) 0 0
\(395\) 8.94427 0.450035
\(396\) 0 0
\(397\) 38.9443 1.95456 0.977278 0.211959i \(-0.0679844\pi\)
0.977278 + 0.211959i \(0.0679844\pi\)
\(398\) 0 0
\(399\) 8.00000 0.400501
\(400\) 0 0
\(401\) −4.76393 −0.237899 −0.118950 0.992900i \(-0.537953\pi\)
−0.118950 + 0.992900i \(0.537953\pi\)
\(402\) 0 0
\(403\) 3.23607 0.161200
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 10.5836 0.523325 0.261662 0.965159i \(-0.415729\pi\)
0.261662 + 0.965159i \(0.415729\pi\)
\(410\) 0 0
\(411\) 14.0000 0.690569
\(412\) 0 0
\(413\) −8.36068 −0.411402
\(414\) 0 0
\(415\) 4.94427 0.242705
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) −24.6525 −1.20435 −0.602176 0.798363i \(-0.705700\pi\)
−0.602176 + 0.798363i \(0.705700\pi\)
\(420\) 0 0
\(421\) −34.3607 −1.67464 −0.837319 0.546715i \(-0.815878\pi\)
−0.837319 + 0.546715i \(0.815878\pi\)
\(422\) 0 0
\(423\) 8.94427 0.434885
\(424\) 0 0
\(425\) 4.47214 0.216930
\(426\) 0 0
\(427\) 15.4164 0.746052
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 35.1246 1.69189 0.845947 0.533268i \(-0.179036\pi\)
0.845947 + 0.533268i \(0.179036\pi\)
\(432\) 0 0
\(433\) 23.2361 1.11665 0.558327 0.829621i \(-0.311443\pi\)
0.558327 + 0.829621i \(0.311443\pi\)
\(434\) 0 0
\(435\) −7.23607 −0.346943
\(436\) 0 0
\(437\) −25.8885 −1.23842
\(438\) 0 0
\(439\) 30.8328 1.47157 0.735785 0.677215i \(-0.236813\pi\)
0.735785 + 0.677215i \(0.236813\pi\)
\(440\) 0 0
\(441\) −5.47214 −0.260578
\(442\) 0 0
\(443\) 21.5279 1.02282 0.511410 0.859337i \(-0.329123\pi\)
0.511410 + 0.859337i \(0.329123\pi\)
\(444\) 0 0
\(445\) 7.23607 0.343023
\(446\) 0 0
\(447\) 18.9443 0.896033
\(448\) 0 0
\(449\) −25.7082 −1.21325 −0.606623 0.794990i \(-0.707476\pi\)
−0.606623 + 0.794990i \(0.707476\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −13.8885 −0.652541
\(454\) 0 0
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) 15.2361 0.712713 0.356357 0.934350i \(-0.384019\pi\)
0.356357 + 0.934350i \(0.384019\pi\)
\(458\) 0 0
\(459\) −4.47214 −0.208741
\(460\) 0 0
\(461\) 42.0689 1.95934 0.979672 0.200608i \(-0.0642917\pi\)
0.979672 + 0.200608i \(0.0642917\pi\)
\(462\) 0 0
\(463\) −11.4164 −0.530565 −0.265283 0.964171i \(-0.585465\pi\)
−0.265283 + 0.964171i \(0.585465\pi\)
\(464\) 0 0
\(465\) −1.00000 −0.0463739
\(466\) 0 0
\(467\) 4.94427 0.228794 0.114397 0.993435i \(-0.463506\pi\)
0.114397 + 0.993435i \(0.463506\pi\)
\(468\) 0 0
\(469\) −4.58359 −0.211651
\(470\) 0 0
\(471\) 5.41641 0.249575
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 6.47214 0.296962
\(476\) 0 0
\(477\) 8.47214 0.387912
\(478\) 0 0
\(479\) −12.2918 −0.561626 −0.280813 0.959762i \(-0.590604\pi\)
−0.280813 + 0.959762i \(0.590604\pi\)
\(480\) 0 0
\(481\) −15.4164 −0.702928
\(482\) 0 0
\(483\) −4.94427 −0.224972
\(484\) 0 0
\(485\) 8.47214 0.384700
\(486\) 0 0
\(487\) 9.52786 0.431749 0.215874 0.976421i \(-0.430740\pi\)
0.215874 + 0.976421i \(0.430740\pi\)
\(488\) 0 0
\(489\) −6.76393 −0.305876
\(490\) 0 0
\(491\) 0.583592 0.0263371 0.0131686 0.999913i \(-0.495808\pi\)
0.0131686 + 0.999913i \(0.495808\pi\)
\(492\) 0 0
\(493\) −32.3607 −1.45745
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 11.4164 0.512096
\(498\) 0 0
\(499\) 32.9443 1.47479 0.737394 0.675463i \(-0.236056\pi\)
0.737394 + 0.675463i \(0.236056\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −18.8328 −0.839714 −0.419857 0.907590i \(-0.637920\pi\)
−0.419857 + 0.907590i \(0.637920\pi\)
\(504\) 0 0
\(505\) 10.9443 0.487014
\(506\) 0 0
\(507\) 2.52786 0.112266
\(508\) 0 0
\(509\) 37.1246 1.64552 0.822760 0.568389i \(-0.192433\pi\)
0.822760 + 0.568389i \(0.192433\pi\)
\(510\) 0 0
\(511\) −15.0557 −0.666026
\(512\) 0 0
\(513\) −6.47214 −0.285752
\(514\) 0 0
\(515\) 16.6525 0.733796
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 22.0000 0.965693
\(520\) 0 0
\(521\) −26.9443 −1.18045 −0.590225 0.807239i \(-0.700961\pi\)
−0.590225 + 0.807239i \(0.700961\pi\)
\(522\) 0 0
\(523\) 40.3607 1.76485 0.882425 0.470454i \(-0.155910\pi\)
0.882425 + 0.470454i \(0.155910\pi\)
\(524\) 0 0
\(525\) 1.23607 0.0539464
\(526\) 0 0
\(527\) −4.47214 −0.194809
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 6.76393 0.293530
\(532\) 0 0
\(533\) 19.4164 0.841018
\(534\) 0 0
\(535\) −12.9443 −0.559630
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −39.3050 −1.68985 −0.844926 0.534883i \(-0.820356\pi\)
−0.844926 + 0.534883i \(0.820356\pi\)
\(542\) 0 0
\(543\) −21.4164 −0.919066
\(544\) 0 0
\(545\) 3.52786 0.151117
\(546\) 0 0
\(547\) 16.6525 0.712008 0.356004 0.934484i \(-0.384139\pi\)
0.356004 + 0.934484i \(0.384139\pi\)
\(548\) 0 0
\(549\) −12.4721 −0.532298
\(550\) 0 0
\(551\) −46.8328 −1.99515
\(552\) 0 0
\(553\) 11.0557 0.470137
\(554\) 0 0
\(555\) 4.76393 0.202218
\(556\) 0 0
\(557\) −17.4164 −0.737957 −0.368978 0.929438i \(-0.620292\pi\)
−0.368978 + 0.929438i \(0.620292\pi\)
\(558\) 0 0
\(559\) −20.9443 −0.885848
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 32.0000 1.34864 0.674320 0.738440i \(-0.264437\pi\)
0.674320 + 0.738440i \(0.264437\pi\)
\(564\) 0 0
\(565\) −2.94427 −0.123866
\(566\) 0 0
\(567\) −1.23607 −0.0519100
\(568\) 0 0
\(569\) −6.65248 −0.278886 −0.139443 0.990230i \(-0.544531\pi\)
−0.139443 + 0.990230i \(0.544531\pi\)
\(570\) 0 0
\(571\) −24.9443 −1.04389 −0.521943 0.852981i \(-0.674793\pi\)
−0.521943 + 0.852981i \(0.674793\pi\)
\(572\) 0 0
\(573\) 14.7639 0.616772
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) −9.05573 −0.376995 −0.188497 0.982074i \(-0.560362\pi\)
−0.188497 + 0.982074i \(0.560362\pi\)
\(578\) 0 0
\(579\) −9.41641 −0.391333
\(580\) 0 0
\(581\) 6.11146 0.253546
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 3.23607 0.133795
\(586\) 0 0
\(587\) −24.9443 −1.02956 −0.514780 0.857322i \(-0.672126\pi\)
−0.514780 + 0.857322i \(0.672126\pi\)
\(588\) 0 0
\(589\) −6.47214 −0.266680
\(590\) 0 0
\(591\) −2.94427 −0.121111
\(592\) 0 0
\(593\) −8.83282 −0.362720 −0.181360 0.983417i \(-0.558050\pi\)
−0.181360 + 0.983417i \(0.558050\pi\)
\(594\) 0 0
\(595\) 5.52786 0.226620
\(596\) 0 0
\(597\) −13.8885 −0.568420
\(598\) 0 0
\(599\) 19.1246 0.781410 0.390705 0.920516i \(-0.372231\pi\)
0.390705 + 0.920516i \(0.372231\pi\)
\(600\) 0 0
\(601\) 36.4721 1.48773 0.743865 0.668330i \(-0.232991\pi\)
0.743865 + 0.668330i \(0.232991\pi\)
\(602\) 0 0
\(603\) 3.70820 0.151010
\(604\) 0 0
\(605\) 11.0000 0.447214
\(606\) 0 0
\(607\) −5.59675 −0.227165 −0.113582 0.993529i \(-0.536233\pi\)
−0.113582 + 0.993529i \(0.536233\pi\)
\(608\) 0 0
\(609\) −8.94427 −0.362440
\(610\) 0 0
\(611\) −28.9443 −1.17096
\(612\) 0 0
\(613\) −5.70820 −0.230552 −0.115276 0.993333i \(-0.536775\pi\)
−0.115276 + 0.993333i \(0.536775\pi\)
\(614\) 0 0
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) 36.8328 1.48283 0.741417 0.671045i \(-0.234154\pi\)
0.741417 + 0.671045i \(0.234154\pi\)
\(618\) 0 0
\(619\) 43.7771 1.75955 0.879775 0.475391i \(-0.157693\pi\)
0.879775 + 0.475391i \(0.157693\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 0 0
\(623\) 8.94427 0.358345
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 21.3050 0.849484
\(630\) 0 0
\(631\) −3.05573 −0.121647 −0.0608233 0.998149i \(-0.519373\pi\)
−0.0608233 + 0.998149i \(0.519373\pi\)
\(632\) 0 0
\(633\) −24.3607 −0.968250
\(634\) 0 0
\(635\) 11.4164 0.453046
\(636\) 0 0
\(637\) 17.7082 0.701625
\(638\) 0 0
\(639\) −9.23607 −0.365373
\(640\) 0 0
\(641\) 0.180340 0.00712300 0.00356150 0.999994i \(-0.498866\pi\)
0.00356150 + 0.999994i \(0.498866\pi\)
\(642\) 0 0
\(643\) −27.4164 −1.08120 −0.540599 0.841281i \(-0.681802\pi\)
−0.540599 + 0.841281i \(0.681802\pi\)
\(644\) 0 0
\(645\) 6.47214 0.254840
\(646\) 0 0
\(647\) −19.0557 −0.749158 −0.374579 0.927195i \(-0.622213\pi\)
−0.374579 + 0.927195i \(0.622213\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −1.23607 −0.0484453
\(652\) 0 0
\(653\) 29.4164 1.15115 0.575576 0.817748i \(-0.304778\pi\)
0.575576 + 0.817748i \(0.304778\pi\)
\(654\) 0 0
\(655\) 9.23607 0.360883
\(656\) 0 0
\(657\) 12.1803 0.475200
\(658\) 0 0
\(659\) 25.2361 0.983058 0.491529 0.870861i \(-0.336438\pi\)
0.491529 + 0.870861i \(0.336438\pi\)
\(660\) 0 0
\(661\) −6.94427 −0.270101 −0.135050 0.990839i \(-0.543120\pi\)
−0.135050 + 0.990839i \(0.543120\pi\)
\(662\) 0 0
\(663\) 14.4721 0.562051
\(664\) 0 0
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) 28.9443 1.12073
\(668\) 0 0
\(669\) 4.00000 0.154649
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 12.7639 0.492013 0.246007 0.969268i \(-0.420881\pi\)
0.246007 + 0.969268i \(0.420881\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −33.4164 −1.28430 −0.642148 0.766580i \(-0.721957\pi\)
−0.642148 + 0.766580i \(0.721957\pi\)
\(678\) 0 0
\(679\) 10.4721 0.401884
\(680\) 0 0
\(681\) 15.4164 0.590758
\(682\) 0 0
\(683\) −7.41641 −0.283781 −0.141890 0.989882i \(-0.545318\pi\)
−0.141890 + 0.989882i \(0.545318\pi\)
\(684\) 0 0
\(685\) 14.0000 0.534913
\(686\) 0 0
\(687\) −18.9443 −0.722769
\(688\) 0 0
\(689\) −27.4164 −1.04448
\(690\) 0 0
\(691\) −12.0000 −0.456502 −0.228251 0.973602i \(-0.573301\pi\)
−0.228251 + 0.973602i \(0.573301\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) −26.8328 −1.01637
\(698\) 0 0
\(699\) 29.4164 1.11263
\(700\) 0 0
\(701\) −35.5279 −1.34187 −0.670934 0.741517i \(-0.734107\pi\)
−0.670934 + 0.741517i \(0.734107\pi\)
\(702\) 0 0
\(703\) 30.8328 1.16288
\(704\) 0 0
\(705\) 8.94427 0.336861
\(706\) 0 0
\(707\) 13.5279 0.508768
\(708\) 0 0
\(709\) 14.0000 0.525781 0.262891 0.964826i \(-0.415324\pi\)
0.262891 + 0.964826i \(0.415324\pi\)
\(710\) 0 0
\(711\) −8.94427 −0.335436
\(712\) 0 0
\(713\) 4.00000 0.149801
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 7.41641 0.276971
\(718\) 0 0
\(719\) −16.0000 −0.596699 −0.298350 0.954457i \(-0.596436\pi\)
−0.298350 + 0.954457i \(0.596436\pi\)
\(720\) 0 0
\(721\) 20.5836 0.766573
\(722\) 0 0
\(723\) 1.05573 0.0392630
\(724\) 0 0
\(725\) −7.23607 −0.268741
\(726\) 0 0
\(727\) −2.40325 −0.0891317 −0.0445658 0.999006i \(-0.514190\pi\)
−0.0445658 + 0.999006i \(0.514190\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 28.9443 1.07054
\(732\) 0 0
\(733\) −25.7771 −0.952098 −0.476049 0.879419i \(-0.657932\pi\)
−0.476049 + 0.879419i \(0.657932\pi\)
\(734\) 0 0
\(735\) −5.47214 −0.201843
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 20.9443 0.770447 0.385224 0.922823i \(-0.374124\pi\)
0.385224 + 0.922823i \(0.374124\pi\)
\(740\) 0 0
\(741\) 20.9443 0.769407
\(742\) 0 0
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 0 0
\(745\) 18.9443 0.694064
\(746\) 0 0
\(747\) −4.94427 −0.180901
\(748\) 0 0
\(749\) −16.0000 −0.584627
\(750\) 0 0
\(751\) 1.88854 0.0689139 0.0344570 0.999406i \(-0.489030\pi\)
0.0344570 + 0.999406i \(0.489030\pi\)
\(752\) 0 0
\(753\) 8.00000 0.291536
\(754\) 0 0
\(755\) −13.8885 −0.505456
\(756\) 0 0
\(757\) −13.1246 −0.477022 −0.238511 0.971140i \(-0.576659\pi\)
−0.238511 + 0.971140i \(0.576659\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 40.1803 1.45654 0.728268 0.685292i \(-0.240326\pi\)
0.728268 + 0.685292i \(0.240326\pi\)
\(762\) 0 0
\(763\) 4.36068 0.157867
\(764\) 0 0
\(765\) −4.47214 −0.161690
\(766\) 0 0
\(767\) −21.8885 −0.790350
\(768\) 0 0
\(769\) 8.47214 0.305513 0.152757 0.988264i \(-0.451185\pi\)
0.152757 + 0.988264i \(0.451185\pi\)
\(770\) 0 0
\(771\) 21.4164 0.771293
\(772\) 0 0
\(773\) 23.3050 0.838221 0.419110 0.907935i \(-0.362342\pi\)
0.419110 + 0.907935i \(0.362342\pi\)
\(774\) 0 0
\(775\) −1.00000 −0.0359211
\(776\) 0 0
\(777\) 5.88854 0.211250
\(778\) 0 0
\(779\) −38.8328 −1.39133
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 7.23607 0.258596
\(784\) 0 0
\(785\) 5.41641 0.193320
\(786\) 0 0
\(787\) 12.0000 0.427754 0.213877 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(788\) 0 0
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) −3.63932 −0.129399
\(792\) 0 0
\(793\) 40.3607 1.43325
\(794\) 0 0
\(795\) 8.47214 0.300476
\(796\) 0 0
\(797\) 28.8328 1.02131 0.510655 0.859785i \(-0.329403\pi\)
0.510655 + 0.859785i \(0.329403\pi\)
\(798\) 0 0
\(799\) 40.0000 1.41510
\(800\) 0 0
\(801\) −7.23607 −0.255674
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −4.94427 −0.174263
\(806\) 0 0
\(807\) 30.0689 1.05847
\(808\) 0 0
\(809\) −1.70820 −0.0600573 −0.0300286 0.999549i \(-0.509560\pi\)
−0.0300286 + 0.999549i \(0.509560\pi\)
\(810\) 0 0
\(811\) −32.3607 −1.13634 −0.568169 0.822912i \(-0.692348\pi\)
−0.568169 + 0.822912i \(0.692348\pi\)
\(812\) 0 0
\(813\) 25.8885 0.907951
\(814\) 0 0
\(815\) −6.76393 −0.236930
\(816\) 0 0
\(817\) 41.8885 1.46549
\(818\) 0 0
\(819\) 4.00000 0.139771
\(820\) 0 0
\(821\) −4.18034 −0.145895 −0.0729474 0.997336i \(-0.523241\pi\)
−0.0729474 + 0.997336i \(0.523241\pi\)
\(822\) 0 0
\(823\) −8.36068 −0.291435 −0.145717 0.989326i \(-0.546549\pi\)
−0.145717 + 0.989326i \(0.546549\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 16.0000 0.556375 0.278187 0.960527i \(-0.410266\pi\)
0.278187 + 0.960527i \(0.410266\pi\)
\(828\) 0 0
\(829\) 31.3050 1.08727 0.543633 0.839323i \(-0.317048\pi\)
0.543633 + 0.839323i \(0.317048\pi\)
\(830\) 0 0
\(831\) 3.81966 0.132503
\(832\) 0 0
\(833\) −24.4721 −0.847909
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1.00000 0.0345651
\(838\) 0 0
\(839\) 1.23607 0.0426738 0.0213369 0.999772i \(-0.493208\pi\)
0.0213369 + 0.999772i \(0.493208\pi\)
\(840\) 0 0
\(841\) 23.3607 0.805541
\(842\) 0 0
\(843\) 16.4721 0.567330
\(844\) 0 0
\(845\) 2.52786 0.0869612
\(846\) 0 0
\(847\) 13.5967 0.467190
\(848\) 0 0
\(849\) 14.1803 0.486668
\(850\) 0 0
\(851\) −19.0557 −0.653222
\(852\) 0 0
\(853\) 37.7771 1.29346 0.646731 0.762718i \(-0.276135\pi\)
0.646731 + 0.762718i \(0.276135\pi\)
\(854\) 0 0
\(855\) −6.47214 −0.221342
\(856\) 0 0
\(857\) −2.00000 −0.0683187 −0.0341593 0.999416i \(-0.510875\pi\)
−0.0341593 + 0.999416i \(0.510875\pi\)
\(858\) 0 0
\(859\) 53.8885 1.83865 0.919327 0.393495i \(-0.128734\pi\)
0.919327 + 0.393495i \(0.128734\pi\)
\(860\) 0 0
\(861\) −7.41641 −0.252751
\(862\) 0 0
\(863\) 32.0000 1.08929 0.544646 0.838666i \(-0.316664\pi\)
0.544646 + 0.838666i \(0.316664\pi\)
\(864\) 0 0
\(865\) 22.0000 0.748022
\(866\) 0 0
\(867\) −3.00000 −0.101885
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) 0 0
\(873\) −8.47214 −0.286738
\(874\) 0 0
\(875\) 1.23607 0.0417867
\(876\) 0 0
\(877\) 14.9443 0.504632 0.252316 0.967645i \(-0.418808\pi\)
0.252316 + 0.967645i \(0.418808\pi\)
\(878\) 0 0
\(879\) −13.4164 −0.452524
\(880\) 0 0
\(881\) −14.0689 −0.473993 −0.236996 0.971511i \(-0.576163\pi\)
−0.236996 + 0.971511i \(0.576163\pi\)
\(882\) 0 0
\(883\) −34.8328 −1.17222 −0.586109 0.810232i \(-0.699341\pi\)
−0.586109 + 0.810232i \(0.699341\pi\)
\(884\) 0 0
\(885\) 6.76393 0.227367
\(886\) 0 0
\(887\) −19.4164 −0.651939 −0.325970 0.945380i \(-0.605691\pi\)
−0.325970 + 0.945380i \(0.605691\pi\)
\(888\) 0 0
\(889\) 14.1115 0.473283
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 57.8885 1.93717
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −12.9443 −0.432197
\(898\) 0 0
\(899\) 7.23607 0.241336
\(900\) 0 0
\(901\) 37.8885 1.26225
\(902\) 0 0
\(903\) 8.00000 0.266223
\(904\) 0 0
\(905\) −21.4164 −0.711905
\(906\) 0 0
\(907\) −50.5410 −1.67819 −0.839094 0.543987i \(-0.816914\pi\)
−0.839094 + 0.543987i \(0.816914\pi\)
\(908\) 0 0
\(909\) −10.9443 −0.362999
\(910\) 0 0
\(911\) −9.88854 −0.327622 −0.163811 0.986492i \(-0.552379\pi\)
−0.163811 + 0.986492i \(0.552379\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −12.4721 −0.412316
\(916\) 0 0
\(917\) 11.4164 0.377003
\(918\) 0 0
\(919\) −33.3050 −1.09863 −0.549314 0.835616i \(-0.685111\pi\)
−0.549314 + 0.835616i \(0.685111\pi\)
\(920\) 0 0
\(921\) 6.76393 0.222879
\(922\) 0 0
\(923\) 29.8885 0.983793
\(924\) 0 0
\(925\) 4.76393 0.156637
\(926\) 0 0
\(927\) −16.6525 −0.546939
\(928\) 0 0
\(929\) 23.5967 0.774184 0.387092 0.922041i \(-0.373480\pi\)
0.387092 + 0.922041i \(0.373480\pi\)
\(930\) 0 0
\(931\) −35.4164 −1.16073
\(932\) 0 0
\(933\) 14.7639 0.483349
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 2.58359 0.0844023 0.0422011 0.999109i \(-0.486563\pi\)
0.0422011 + 0.999109i \(0.486563\pi\)
\(938\) 0 0
\(939\) 10.6525 0.347630
\(940\) 0 0
\(941\) 33.4853 1.09159 0.545795 0.837919i \(-0.316228\pi\)
0.545795 + 0.837919i \(0.316228\pi\)
\(942\) 0 0
\(943\) 24.0000 0.781548
\(944\) 0 0
\(945\) −1.23607 −0.0402093
\(946\) 0 0
\(947\) −4.94427 −0.160667 −0.0803336 0.996768i \(-0.525599\pi\)
−0.0803336 + 0.996768i \(0.525599\pi\)
\(948\) 0 0
\(949\) −39.4164 −1.27951
\(950\) 0 0
\(951\) 28.4721 0.923272
\(952\) 0 0
\(953\) −26.9443 −0.872811 −0.436405 0.899750i \(-0.643749\pi\)
−0.436405 + 0.899750i \(0.643749\pi\)
\(954\) 0 0
\(955\) 14.7639 0.477750
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 17.3050 0.558806
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) 12.9443 0.417123
\(964\) 0 0
\(965\) −9.41641 −0.303125
\(966\) 0 0
\(967\) −44.0000 −1.41494 −0.707472 0.706741i \(-0.750165\pi\)
−0.707472 + 0.706741i \(0.750165\pi\)
\(968\) 0 0
\(969\) −28.9443 −0.929824
\(970\) 0 0
\(971\) −19.7082 −0.632466 −0.316233 0.948681i \(-0.602418\pi\)
−0.316233 + 0.948681i \(0.602418\pi\)
\(972\) 0 0
\(973\) −4.94427 −0.158506
\(974\) 0 0
\(975\) 3.23607 0.103637
\(976\) 0 0
\(977\) 2.94427 0.0941956 0.0470978 0.998890i \(-0.485003\pi\)
0.0470978 + 0.998890i \(0.485003\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −3.52786 −0.112636
\(982\) 0 0
\(983\) −11.7771 −0.375631 −0.187815 0.982204i \(-0.560141\pi\)
−0.187815 + 0.982204i \(0.560141\pi\)
\(984\) 0 0
\(985\) −2.94427 −0.0938123
\(986\) 0 0
\(987\) 11.0557 0.351908
\(988\) 0 0
\(989\) −25.8885 −0.823208
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) 28.9443 0.918519
\(994\) 0 0
\(995\) −13.8885 −0.440296
\(996\) 0 0
\(997\) −11.5279 −0.365091 −0.182546 0.983197i \(-0.558434\pi\)
−0.182546 + 0.983197i \(0.558434\pi\)
\(998\) 0 0
\(999\) −4.76393 −0.150724
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 3720.2.a.i.1.1 2
4.3 odd 2 7440.2.a.bi.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3720.2.a.i.1.1 2 1.1 even 1 trivial
7440.2.a.bi.1.2 2 4.3 odd 2