↑

Properties

Label 1872.2.a.l.1.1
Level $1872$
Weight $2$
Character 1872.1
Self dual yes
Analytic conductor $14.948$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1872,2,Mod(1,1872)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1872, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1872.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1872.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: \(14.9479952584\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 104)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1872.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} -5.00000 q^{7} -2.00000 q^{11} -1.00000 q^{13} +3.00000 q^{17} +2.00000 q^{19} +4.00000 q^{23} -4.00000 q^{25} +6.00000 q^{29} +4.00000 q^{31} -5.00000 q^{35} +11.0000 q^{37} -8.00000 q^{41} +1.00000 q^{43} +9.00000 q^{47} +18.0000 q^{49} +12.0000 q^{53} -2.00000 q^{55} +6.00000 q^{59} -1.00000 q^{65} -6.00000 q^{67} +7.00000 q^{71} -2.00000 q^{73} +10.0000 q^{77} -12.0000 q^{79} -16.0000 q^{83} +3.00000 q^{85} +10.0000 q^{89} +5.00000 q^{91} +2.00000 q^{95} -10.0000 q^{97} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000 0.447214 0.223607 − 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) −5.00000 −1.88982 −0.944911 − 0.327327i \(-0.893852\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −5.00000 −0.845154
\(36\) 0 0
\(37\) 11.0000 1.80839 0.904194 − 0.427121i \(-0.140472\pi\)
0.904194 + 0.427121i \(0.140472\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 − 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499 0.0762493 − 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.00000 1.31278 0.656392 − 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 0 0
\(49\) 18.0000 2.57143
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 12.0000 1.64833 0.824163 − 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −6.00000 −0.733017 −0.366508 − 0.930415i \(-0.619447\pi\)
−0.366508 + 0.930415i \(0.619447\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.00000 0.830747 0.415374 − 0.909651i \(-0.363651\pi\)
0.415374 + 0.909651i \(0.363651\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.0000 1.13961
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 − 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −16.0000 −1.75623 −0.878114 − 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 0 0
\(85\) 3.00000 0.325396
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 5.00000 0.524142
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −4.00000 −0.398015 −0.199007 − 0.979998i \(-0.563772\pi\)
−0.199007 + 0.979998i \(0.563772\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 − 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 20.0000 1.93347 0.966736 − 0.255774i \(-0.0823304\pi\)
0.966736 + 0.255774i \(0.0823304\pi\)
\(108\) 0 0
\(109\) −5.00000 −0.478913 −0.239457 − 0.970907i \(-0.576969\pi\)
−0.239457 + 0.970907i \(0.576969\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 − 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −15.0000 −1.37505
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.00000 0.262111 0.131056 − 0.991375i \(-0.458163\pi\)
0.131056 + 0.991375i \(0.458163\pi\)
\(132\) 0 0
\(133\) −10.0000 −0.867110
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.0000 1.02523 0.512615 − 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 0 0
\(139\) −3.00000 −0.254457 −0.127228 − 0.991873i \(-0.540608\pi\)
−0.127228 + 0.991873i \(0.540608\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.00000 0.167248
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −14.0000 −1.14692 −0.573462 − 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) 5.00000 0.406894 0.203447 − 0.979086i \(-0.434786\pi\)
0.203447 + 0.979086i \(0.434786\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 − 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −20.0000 −1.57622
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 − 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.00000 0.619059 0.309529 − 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 12.0000 0.912343 0.456172 − 0.889892i \(-0.349220\pi\)
0.456172 + 0.889892i \(0.349220\pi\)
\(174\) 0 0
\(175\) 20.0000 1.51186
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3.00000 0.224231 0.112115 − 0.993695i \(-0.464237\pi\)
0.112115 + 0.993695i \(0.464237\pi\)
\(180\) 0 0
\(181\) 16.0000 1.18927 0.594635 − 0.803996i \(-0.297296\pi\)
0.594635 + 0.803996i \(0.297296\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 11.0000 0.808736
\(186\) 0 0
\(187\) −6.00000 −0.438763
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.0000 0.723575 0.361787 − 0.932261i \(-0.382167\pi\)
0.361787 + 0.932261i \(0.382167\pi\)
\(192\) 0 0
\(193\) 16.0000 1.15171 0.575853 − 0.817554i \(-0.304670\pi\)
0.575853 + 0.817554i \(0.304670\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −17.0000 −1.21120 −0.605600 − 0.795769i \(-0.707067\pi\)
−0.605600 + 0.795769i \(0.707067\pi\)
\(198\) 0 0
\(199\) 18.0000 1.27599 0.637993 − 0.770042i \(-0.279765\pi\)
0.637993 + 0.770042i \(0.279765\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −30.0000 −2.10559
\(204\) 0 0
\(205\) −8.00000 −0.558744
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 5.00000 0.344214 0.172107 − 0.985078i \(-0.444942\pi\)
0.172107 + 0.985078i \(0.444942\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.00000 0.0681994
\(216\) 0 0
\(217\) −20.0000 −1.35769
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −3.00000 −0.201802
\(222\) 0 0
\(223\) 17.0000 1.13840 0.569202 − 0.822198i \(-0.307252\pi\)
0.569202 + 0.822198i \(0.307252\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −16.0000 −1.06196 −0.530979 − 0.847385i \(-0.678176\pi\)
−0.530979 + 0.847385i \(0.678176\pi\)
\(228\) 0 0
\(229\) −15.0000 −0.991228 −0.495614 − 0.868543i \(-0.665057\pi\)
−0.495614 + 0.868543i \(0.665057\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −13.0000 −0.851658 −0.425829 − 0.904804i \(-0.640018\pi\)
−0.425829 + 0.904804i \(0.640018\pi\)
\(234\) 0 0
\(235\) 9.00000 0.587095
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −3.00000 −0.194054 −0.0970269 − 0.995282i \(-0.530933\pi\)
−0.0970269 + 0.995282i \(0.530933\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 − 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 18.0000 1.14998
\(246\) 0 0
\(247\) −2.00000 −0.127257
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 8.00000 0.504956 0.252478 − 0.967603i \(-0.418755\pi\)
0.252478 + 0.967603i \(0.418755\pi\)
\(252\) 0 0
\(253\) −8.00000 −0.502956
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 31.0000 1.93373 0.966863 − 0.255294i \(-0.0821723\pi\)
0.966863 + 0.255294i \(0.0821723\pi\)
\(258\) 0 0
\(259\) −55.0000 −3.41753
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 20.0000 1.23325 0.616626 − 0.787256i \(-0.288499\pi\)
0.616626 + 0.787256i \(0.288499\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −16.0000 −0.975537 −0.487769 − 0.872973i \(-0.662189\pi\)
−0.487769 + 0.872973i \(0.662189\pi\)
\(270\) 0 0
\(271\) 7.00000 0.425220 0.212610 − 0.977137i \(-0.431804\pi\)
0.212610 + 0.977137i \(0.431804\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 8.00000 0.482418
\(276\) 0 0
\(277\) −12.0000 −0.721010 −0.360505 − 0.932757i \(-0.617396\pi\)
−0.360505 + 0.932757i \(0.617396\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 − 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 − 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 40.0000 2.36113
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −15.0000 −0.876309 −0.438155 − 0.898900i \(-0.644368\pi\)
−0.438155 + 0.898900i \(0.644368\pi\)
\(294\) 0 0
\(295\) 6.00000 0.349334
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −4.00000 −0.231326
\(300\) 0 0
\(301\) −5.00000 −0.288195
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 2.00000 0.114146 0.0570730 − 0.998370i \(-0.481823\pi\)
0.0570730 + 0.998370i \(0.481823\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −22.0000 −1.24751 −0.623753 − 0.781622i \(-0.714393\pi\)
−0.623753 + 0.781622i \(0.714393\pi\)
\(312\) 0 0
\(313\) −9.00000 −0.508710 −0.254355 − 0.967111i \(-0.581863\pi\)
−0.254355 + 0.967111i \(0.581863\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.00000 0.112331 0.0561656 − 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 0 0
\(319\) −12.0000 −0.671871
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 6.00000 0.333849
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −45.0000 −2.48093
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 − 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −6.00000 −0.327815
\(336\) 0 0
\(337\) 15.0000 0.817102 0.408551 − 0.912735i \(-0.366034\pi\)
0.408551 + 0.912735i \(0.366034\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −8.00000 −0.433224
\(342\) 0 0
\(343\) −55.0000 −2.96972
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3.00000 0.161048 0.0805242 − 0.996753i \(-0.474341\pi\)
0.0805242 + 0.996753i \(0.474341\pi\)
\(348\) 0 0
\(349\) −9.00000 −0.481759 −0.240879 − 0.970555i \(-0.577436\pi\)
−0.240879 + 0.970555i \(0.577436\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 28.0000 1.49029 0.745145 − 0.666903i \(-0.232380\pi\)
0.745145 + 0.666903i \(0.232380\pi\)
\(354\) 0 0
\(355\) 7.00000 0.371521
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −2.00000 −0.104685
\(366\) 0 0
\(367\) 18.0000 0.939592 0.469796 − 0.882775i \(-0.344327\pi\)
0.469796 + 0.882775i \(0.344327\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −60.0000 −3.11504
\(372\) 0 0
\(373\) −4.00000 −0.207112 −0.103556 − 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −6.00000 −0.309016
\(378\) 0 0
\(379\) −24.0000 −1.23280 −0.616399 − 0.787434i \(-0.711409\pi\)
−0.616399 + 0.787434i \(0.711409\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 15.0000 0.766464 0.383232 − 0.923652i \(-0.374811\pi\)
0.383232 + 0.923652i \(0.374811\pi\)
\(384\) 0 0
\(385\) 10.0000 0.509647
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 6.00000 0.304212 0.152106 − 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) 10.0000 0.501886 0.250943 − 0.968002i \(-0.419259\pi\)
0.250943 + 0.968002i \(0.419259\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −22.0000 −1.09050
\(408\) 0 0
\(409\) −4.00000 −0.197787 −0.0988936 − 0.995098i \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −30.0000 −1.47620
\(414\) 0 0
\(415\) −16.0000 −0.785409
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −39.0000 −1.90527 −0.952637 − 0.304109i \(-0.901641\pi\)
−0.952637 + 0.304109i \(0.901641\pi\)
\(420\) 0 0
\(421\) 11.0000 0.536107 0.268054 − 0.963404i \(-0.413620\pi\)
0.268054 + 0.963404i \(0.413620\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −12.0000 −0.582086
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −11.0000 −0.529851 −0.264926 − 0.964269i \(-0.585347\pi\)
−0.264926 + 0.964269i \(0.585347\pi\)
\(432\) 0 0
\(433\) −17.0000 −0.816968 −0.408484 − 0.912766i \(-0.633942\pi\)
−0.408484 + 0.912766i \(0.633942\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 8.00000 0.382692
\(438\) 0 0
\(439\) −10.0000 −0.477274 −0.238637 − 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −27.0000 −1.28281 −0.641404 − 0.767203i \(-0.721648\pi\)
−0.641404 + 0.767203i \(0.721648\pi\)
\(444\) 0 0
\(445\) 10.0000 0.474045
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 − 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 16.0000 0.753411
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 5.00000 0.234404
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 − 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −3.00000 −0.139724 −0.0698620 − 0.997557i \(-0.522256\pi\)
−0.0698620 + 0.997557i \(0.522256\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 − 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −20.0000 −0.925490 −0.462745 − 0.886492i \(-0.653135\pi\)
−0.462745 + 0.886492i \(0.653135\pi\)
\(468\) 0 0
\(469\) 30.0000 1.38527
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −2.00000 −0.0919601
\(474\) 0 0
\(475\) −8.00000 −0.367065
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 9.00000 0.411220 0.205610 − 0.978634i \(-0.434082\pi\)
0.205610 + 0.978634i \(0.434082\pi\)
\(480\) 0 0
\(481\) −11.0000 −0.501557
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −10.0000 −0.454077
\(486\) 0 0
\(487\) −8.00000 −0.362515 −0.181257 − 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 15.0000 0.676941 0.338470 − 0.940977i \(-0.390091\pi\)
0.338470 + 0.940977i \(0.390091\pi\)
\(492\) 0 0
\(493\) 18.0000 0.810679
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −35.0000 −1.56996
\(498\) 0 0
\(499\) 24.0000 1.07439 0.537194 − 0.843459i \(-0.319484\pi\)
0.537194 + 0.843459i \(0.319484\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 26.0000 1.15928 0.579641 − 0.814872i \(-0.303193\pi\)
0.579641 + 0.814872i \(0.303193\pi\)
\(504\) 0 0
\(505\) −4.00000 −0.177998
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −2.00000 −0.0886484 −0.0443242 − 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) 10.0000 0.442374
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) −18.0000 −0.791639
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −15.0000 −0.657162 −0.328581 − 0.944476i \(-0.606570\pi\)
−0.328581 + 0.944476i \(0.606570\pi\)
\(522\) 0 0
\(523\) −20.0000 −0.874539 −0.437269 − 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 8.00000 0.346518
\(534\) 0 0
\(535\) 20.0000 0.864675
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −36.0000 −1.55063
\(540\) 0 0
\(541\) 9.00000 0.386940 0.193470 − 0.981106i \(-0.438026\pi\)
0.193470 + 0.981106i \(0.438026\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −5.00000 −0.214176
\(546\) 0 0
\(547\) 39.0000 1.66752 0.833760 − 0.552127i \(-0.186184\pi\)
0.833760 + 0.552127i \(0.186184\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) 60.0000 2.55146
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 23.0000 0.974541 0.487271 − 0.873251i \(-0.337993\pi\)
0.487271 + 0.873251i \(0.337993\pi\)
\(558\) 0 0
\(559\) −1.00000 −0.0422955
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 39.0000 1.64365 0.821827 − 0.569737i \(-0.192955\pi\)
0.821827 + 0.569737i \(0.192955\pi\)
\(564\) 0 0
\(565\) −2.00000 −0.0841406
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 17.0000 0.712677 0.356339 − 0.934357i \(-0.384025\pi\)
0.356339 + 0.934357i \(0.384025\pi\)
\(570\) 0 0
\(571\) −37.0000 −1.54840 −0.774201 − 0.632940i \(-0.781848\pi\)
−0.774201 + 0.632940i \(0.781848\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −16.0000 −0.667246
\(576\) 0 0
\(577\) −38.0000 −1.58196 −0.790980 − 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 80.0000 3.31896
\(582\) 0 0
\(583\) −24.0000 −0.993978
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 12.0000 0.495293 0.247647 − 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −2.00000 −0.0821302 −0.0410651 − 0.999156i \(-0.513075\pi\)
−0.0410651 + 0.999156i \(0.513075\pi\)
\(594\) 0 0
\(595\) −15.0000 −0.614940
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −2.00000 −0.0817178 −0.0408589 − 0.999165i \(-0.513009\pi\)
−0.0408589 + 0.999165i \(0.513009\pi\)
\(600\) 0 0
\(601\) −3.00000 −0.122373 −0.0611863 − 0.998126i \(-0.519488\pi\)
−0.0611863 + 0.998126i \(0.519488\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −7.00000 −0.284590
\(606\) 0 0
\(607\) 10.0000 0.405887 0.202944 − 0.979190i \(-0.434949\pi\)
0.202944 + 0.979190i \(0.434949\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −9.00000 −0.364101
\(612\) 0 0
\(613\) −6.00000 −0.242338 −0.121169 − 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −32.0000 −1.28827 −0.644136 − 0.764911i \(-0.722783\pi\)
−0.644136 + 0.764911i \(0.722783\pi\)
\(618\) 0 0
\(619\) −28.0000 −1.12542 −0.562708 − 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −50.0000 −2.00321
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 33.0000 1.31580
\(630\) 0 0
\(631\) −15.0000 −0.597141 −0.298570 − 0.954388i \(-0.596510\pi\)
−0.298570 + 0.954388i \(0.596510\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 8.00000 0.317470
\(636\) 0 0
\(637\) −18.0000 −0.713186
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 34.0000 1.34292 0.671460 − 0.741041i \(-0.265668\pi\)
0.671460 + 0.741041i \(0.265668\pi\)
\(642\) 0 0
\(643\) 26.0000 1.02534 0.512670 − 0.858586i \(-0.328656\pi\)
0.512670 + 0.858586i \(0.328656\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 42.0000 1.65119 0.825595 − 0.564263i \(-0.190840\pi\)
0.825595 + 0.564263i \(0.190840\pi\)
\(648\) 0 0
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −24.0000 −0.939193 −0.469596 − 0.882881i \(-0.655601\pi\)
−0.469596 + 0.882881i \(0.655601\pi\)
\(654\) 0 0
\(655\) 3.00000 0.117220
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 28.0000 1.09073 0.545363 − 0.838200i \(-0.316392\pi\)
0.545363 + 0.838200i \(0.316392\pi\)
\(660\) 0 0
\(661\) 38.0000 1.47803 0.739014 − 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −10.0000 −0.387783
\(666\) 0 0
\(667\) 24.0000 0.929284
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 21.0000 0.809491 0.404745 − 0.914429i \(-0.367360\pi\)
0.404745 + 0.914429i \(0.367360\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 36.0000 1.38359 0.691796 − 0.722093i \(-0.256820\pi\)
0.691796 + 0.722093i \(0.256820\pi\)
\(678\) 0 0
\(679\) 50.0000 1.91882
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 − 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −40.0000 −1.52167 −0.760836 − 0.648944i \(-0.775211\pi\)
−0.760836 + 0.648944i \(0.775211\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −3.00000 −0.113796
\(696\) 0 0
\(697\) −24.0000 −0.909065
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 4.00000 0.151078 0.0755390 − 0.997143i \(-0.475932\pi\)
0.0755390 + 0.997143i \(0.475932\pi\)
\(702\) 0 0
\(703\) 22.0000 0.829746
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 20.0000 0.752177
\(708\) 0 0
\(709\) 22.0000 0.826227 0.413114 − 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 16.0000 0.599205
\(714\) 0 0
\(715\) 2.00000 0.0747958
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −30.0000 −1.11881 −0.559406 − 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) −40.0000 −1.48968
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −24.0000 −0.891338
\(726\) 0 0
\(727\) 14.0000 0.519231 0.259616 − 0.965712i \(-0.416404\pi\)
0.259616 + 0.965712i \(0.416404\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 3.00000 0.110959
\(732\) 0 0
\(733\) 37.0000 1.36663 0.683313 − 0.730125i \(-0.260538\pi\)
0.683313 + 0.730125i \(0.260538\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 12.0000 0.442026
\(738\) 0 0
\(739\) −52.0000 −1.91285 −0.956425 − 0.291977i \(-0.905687\pi\)
−0.956425 + 0.291977i \(0.905687\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 29.0000 1.06391 0.531953 − 0.846774i \(-0.321458\pi\)
0.531953 + 0.846774i \(0.321458\pi\)
\(744\) 0 0
\(745\) −14.0000 −0.512920
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −100.000 −3.65392
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 − 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 5.00000 0.181969
\(756\) 0 0
\(757\) −52.0000 −1.88997 −0.944986 − 0.327111i \(-0.893925\pi\)
−0.944986 + 0.327111i \(0.893925\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 − 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) 25.0000 0.905061
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −6.00000 −0.216647
\(768\) 0 0
\(769\) 16.0000 0.576975 0.288487 − 0.957484i \(-0.406848\pi\)
0.288487 + 0.957484i \(0.406848\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −27.0000 −0.971123 −0.485561 − 0.874203i \(-0.661385\pi\)
−0.485561 + 0.874203i \(0.661385\pi\)
\(774\) 0 0
\(775\) −16.0000 −0.574737
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −16.0000 −0.573259
\(780\) 0 0
\(781\) −14.0000 −0.500959
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −2.00000 −0.0713831
\(786\) 0 0
\(787\) 24.0000 0.855508 0.427754 − 0.903895i \(-0.359305\pi\)
0.427754 + 0.903895i \(0.359305\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 10.0000 0.355559
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −38.0000 −1.34603 −0.673015 − 0.739629i \(-0.735001\pi\)
−0.673015 + 0.739629i \(0.735001\pi\)
\(798\) 0 0
\(799\) 27.0000 0.955191
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4.00000 0.141157
\(804\) 0 0
\(805\) −20.0000 −0.704907
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −15.0000 −0.527372 −0.263686 − 0.964609i \(-0.584938\pi\)
−0.263686 + 0.964609i \(0.584938\pi\)
\(810\) 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
\(815\) −4.00000 −0.140114
\(816\) 0 0
\(817\) 2.00000 0.0699711
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 41.0000 1.43091 0.715455 − 0.698659i \(-0.246219\pi\)
0.715455 + 0.698659i \(0.246219\pi\)
\(822\) 0 0
\(823\) 26.0000 0.906303 0.453152 − 0.891434i \(-0.350300\pi\)
0.453152 + 0.891434i \(0.350300\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −10.0000 −0.347734 −0.173867 − 0.984769i \(-0.555626\pi\)
−0.173867 + 0.984769i \(0.555626\pi\)
\(828\) 0 0
\(829\) −6.00000 −0.208389 −0.104194 − 0.994557i \(-0.533226\pi\)
−0.104194 + 0.994557i \(0.533226\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 54.0000 1.87099
\(834\) 0 0
\(835\) 8.00000 0.276851
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) 35.0000 1.20261
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 44.0000 1.50830
\(852\) 0 0
\(853\) −47.0000 −1.60925 −0.804625 − 0.593784i \(-0.797633\pi\)
−0.804625 + 0.593784i \(0.797633\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −30.0000 −1.02478 −0.512390 − 0.858753i \(-0.671240\pi\)
−0.512390 + 0.858753i \(0.671240\pi\)
\(858\) 0 0
\(859\) −44.0000 −1.50126 −0.750630 − 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −7.00000 −0.238283 −0.119141 − 0.992877i \(-0.538014\pi\)
−0.119141 + 0.992877i \(0.538014\pi\)
\(864\) 0 0
\(865\) 12.0000 0.408012
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 24.0000 0.814144
\(870\) 0 0
\(871\) 6.00000 0.203302
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 45.0000 1.52128
\(876\) 0 0
\(877\) −7.00000 −0.236373 −0.118187 − 0.992991i \(-0.537708\pi\)
−0.118187 + 0.992991i \(0.537708\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 27.0000 0.909653 0.454827 − 0.890580i \(-0.349701\pi\)
0.454827 + 0.890580i \(0.349701\pi\)
\(882\) 0 0
\(883\) 51.0000 1.71629 0.858143 − 0.513410i \(-0.171618\pi\)
0.858143 + 0.513410i \(0.171618\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −48.0000 −1.61168 −0.805841 − 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 0 0
\(889\) −40.0000 −1.34156
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 18.0000 0.602347
\(894\) 0 0
\(895\) 3.00000 0.100279
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 16.0000 0.531858
\(906\) 0 0
\(907\) 5.00000 0.166022 0.0830111 − 0.996549i \(-0.473546\pi\)
0.0830111 + 0.996549i \(0.473546\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 26.0000 0.861418 0.430709 − 0.902491i \(-0.358263\pi\)
0.430709 + 0.902491i \(0.358263\pi\)
\(912\) 0 0
\(913\) 32.0000 1.05905
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −15.0000 −0.495344
\(918\) 0 0
\(919\) 24.0000 0.791687 0.395843 − 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −7.00000 −0.230408
\(924\) 0 0
\(925\) −44.0000 −1.44671
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 20.0000 0.656179 0.328089 − 0.944647i \(-0.393595\pi\)
0.328089 + 0.944647i \(0.393595\pi\)
\(930\) 0 0
\(931\) 36.0000 1.17985
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −6.00000 −0.196221
\(936\) 0 0
\(937\) 22.0000 0.718709 0.359354 − 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 7.00000 0.228193 0.114097 − 0.993470i \(-0.463603\pi\)
0.114097 + 0.993470i \(0.463603\pi\)
\(942\) 0 0
\(943\) −32.0000 −1.04206
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −18.0000 −0.584921 −0.292461 − 0.956278i \(-0.594474\pi\)
−0.292461 + 0.956278i \(0.594474\pi\)
\(948\) 0 0
\(949\) 2.00000 0.0649227
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 9.00000 0.291539 0.145769 − 0.989319i \(-0.453434\pi\)
0.145769 + 0.989319i \(0.453434\pi\)
\(954\) 0 0
\(955\) 10.0000 0.323592
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −60.0000 −1.93750
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 16.0000 0.515058
\(966\) 0 0
\(967\) 13.0000 0.418052 0.209026 − 0.977910i \(-0.432971\pi\)
0.209026 + 0.977910i \(0.432971\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 45.0000 1.44412 0.722059 − 0.691831i \(-0.243196\pi\)
0.722059 + 0.691831i \(0.243196\pi\)
\(972\) 0 0
\(973\) 15.0000 0.480878
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 30.0000 0.959785 0.479893 − 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 0 0
\(979\) −20.0000 −0.639203
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 5.00000 0.159475 0.0797376 − 0.996816i \(-0.474592\pi\)
0.0797376 + 0.996816i \(0.474592\pi\)
\(984\) 0 0
\(985\) −17.0000 −0.541665
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) −58.0000 −1.84243 −0.921215 − 0.389053i \(-0.872802\pi\)
−0.921215 + 0.389053i \(0.872802\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 18.0000 0.570638
\(996\) 0 0
\(997\) 14.0000 0.443384 0.221692 − 0.975117i \(-0.428842\pi\)
0.221692 + 0.975117i \(0.428842\pi\)
\(998\) 0 0
\(999\) 0 0
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 1872.2.a.l.1.1 1
3.2 odd 2 208.2.a.b.1.1 1
4.3 odd 2 936.2.a.f.1.1 1
8.3 odd 2 7488.2.a.x.1.1 1
8.5 even 2 7488.2.a.u.1.1 1
12.11 even 2 104.2.a.a.1.1 ✓ 1
15.14 odd 2 5200.2.a.bb.1.1 1
24.5 odd 2 832.2.a.h.1.1 1
24.11 even 2 832.2.a.c.1.1 1
39.5 even 4 2704.2.f.e.337.2 2
39.8 even 4 2704.2.f.e.337.1 2
39.38 odd 2 2704.2.a.d.1.1 1
48.5 odd 4 3328.2.b.t.1665.2 2
48.11 even 4 3328.2.b.a.1665.1 2
48.29 odd 4 3328.2.b.t.1665.1 2
48.35 even 4 3328.2.b.a.1665.2 2
60.23 odd 4 2600.2.d.f.1249.2 2
60.47 odd 4 2600.2.d.f.1249.1 2
60.59 even 2 2600.2.a.e.1.1 1
84.83 odd 2 5096.2.a.c.1.1 1
156.11 odd 12 1352.2.o.a.1161.1 4
156.23 even 6 1352.2.i.c.529.1 2
156.35 even 6 1352.2.i.b.1329.1 2
156.47 odd 4 1352.2.f.b.337.1 2
156.59 odd 12 1352.2.o.a.361.1 4
156.71 odd 12 1352.2.o.a.361.2 4
156.83 odd 4 1352.2.f.b.337.2 2
156.95 even 6 1352.2.i.c.1329.1 2
156.107 even 6 1352.2.i.b.529.1 2
156.119 odd 12 1352.2.o.a.1161.2 4
156.155 even 2 1352.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
104.2.a.a.1.1 ✓ 1 12.11 even 2
208.2.a.b.1.1 1 3.2 odd 2
832.2.a.c.1.1 1 24.11 even 2
832.2.a.h.1.1 1 24.5 odd 2
936.2.a.f.1.1 1 4.3 odd 2
1352.2.a.b.1.1 1 156.155 even 2
1352.2.f.b.337.1 2 156.47 odd 4
1352.2.f.b.337.2 2 156.83 odd 4
1352.2.i.b.529.1 2 156.107 even 6
1352.2.i.b.1329.1 2 156.35 even 6
1352.2.i.c.529.1 2 156.23 even 6
1352.2.i.c.1329.1 2 156.95 even 6
1352.2.o.a.361.1 4 156.59 odd 12
1352.2.o.a.361.2 4 156.71 odd 12
1352.2.o.a.1161.1 4 156.11 odd 12
1352.2.o.a.1161.2 4 156.119 odd 12
1872.2.a.l.1.1 1 1.1 even 1 trivial
2600.2.a.e.1.1 1 60.59 even 2
2600.2.d.f.1249.1 2 60.47 odd 4
2600.2.d.f.1249.2 2 60.23 odd 4
2704.2.a.d.1.1 1 39.38 odd 2
2704.2.f.e.337.1 2 39.8 even 4
2704.2.f.e.337.2 2 39.5 even 4
3328.2.b.a.1665.1 2 48.11 even 4
3328.2.b.a.1665.2 2 48.35 even 4
3328.2.b.t.1665.1 2 48.29 odd 4
3328.2.b.t.1665.2 2 48.5 odd 4
5096.2.a.c.1.1 1 84.83 odd 2
5200.2.a.bb.1.1 1 15.14 odd 2
7488.2.a.u.1.1 1 8.5 even 2
7488.2.a.x.1.1 1 8.3 odd 2