Properties

Label 2352.2.a.d.1.1
Level $2352$
Weight $2$
Character 2352.1
Self dual yes
Analytic conductor $18.781$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2352,2,Mod(1,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2352, 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("2352.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.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: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2352.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.00000 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -2.00000 q^{5} +1.00000 q^{9} +2.00000 q^{11} +1.00000 q^{13} +2.00000 q^{15} -1.00000 q^{19} -1.00000 q^{25} -1.00000 q^{27} +4.00000 q^{29} -9.00000 q^{31} -2.00000 q^{33} +3.00000 q^{37} -1.00000 q^{39} -10.0000 q^{41} -5.00000 q^{43} -2.00000 q^{45} +6.00000 q^{47} +12.0000 q^{53} -4.00000 q^{55} +1.00000 q^{57} +12.0000 q^{59} +10.0000 q^{61} -2.00000 q^{65} +5.00000 q^{67} +6.00000 q^{71} -3.00000 q^{73} +1.00000 q^{75} +1.00000 q^{79} +1.00000 q^{81} -6.00000 q^{83} -4.00000 q^{87} +16.0000 q^{89} +9.00000 q^{93} +2.00000 q^{95} -6.00000 q^{97} +2.00000 q^{99} +O(q^{100})\)

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
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) −9.00000 −1.61645 −0.808224 0.588875i \(-0.799571\pi\)
−0.808224 + 0.588875i \(0.799571\pi\)
\(32\) 0 0
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.00000 0.493197 0.246598 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) −5.00000 −0.762493 −0.381246 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 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
\(55\) −4.00000 −0.539360
\(56\) 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 \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 5.00000 0.610847 0.305424 0.952217i \(-0.401202\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) −3.00000 −0.351123 −0.175562 0.984468i \(-0.556174\pi\)
−0.175562 + 0.984468i \(0.556174\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.00000 0.112509 0.0562544 0.998416i \(-0.482084\pi\)
0.0562544 + 0.998416i \(0.482084\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −4.00000 −0.428845
\(88\) 0 0
\(89\) 16.0000 1.69600 0.847998 0.529999i \(-0.177808\pi\)
0.847998 + 0.529999i \(0.177808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 9.00000 0.933257
\(94\) 0 0
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) 7.00000 0.689730 0.344865 0.938652i \(-0.387925\pi\)
0.344865 + 0.938652i \(0.387925\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) 9.00000 0.862044 0.431022 0.902342i \(-0.358153\pi\)
0.431022 + 0.902342i \(0.358153\pi\)
\(110\) 0 0
\(111\) −3.00000 −0.284747
\(112\) 0 0
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 10.0000 0.901670
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 15.0000 1.33103 0.665517 0.746382i \(-0.268211\pi\)
0.665517 + 0.746382i \(0.268211\pi\)
\(128\) 0 0
\(129\) 5.00000 0.440225
\(130\) 0 0
\(131\) 14.0000 1.22319 0.611593 0.791173i \(-0.290529\pi\)
0.611593 + 0.791173i \(0.290529\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 2.00000 0.172133
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) 3.00000 0.254457 0.127228 0.991873i \(-0.459392\pi\)
0.127228 + 0.991873i \(0.459392\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) 2.00000 0.167248
\(144\) 0 0
\(145\) −8.00000 −0.664364
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −12.0000 −0.983078 −0.491539 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 18.0000 1.44579
\(156\) 0 0
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 0 0
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) 4.00000 0.311400
\(166\) 0 0
\(167\) 14.0000 1.08335 0.541676 0.840587i \(-0.317790\pi\)
0.541676 + 0.840587i \(0.317790\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) 8.00000 0.608229 0.304114 0.952636i \(-0.401639\pi\)
0.304114 + 0.952636i \(0.401639\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −12.0000 −0.901975
\(178\) 0 0
\(179\) −2.00000 −0.149487 −0.0747435 0.997203i \(-0.523814\pi\)
−0.0747435 + 0.997203i \(0.523814\pi\)
\(180\) 0 0
\(181\) 13.0000 0.966282 0.483141 0.875542i \(-0.339496\pi\)
0.483141 + 0.875542i \(0.339496\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 0 0
\(185\) −6.00000 −0.441129
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −10.0000 −0.723575 −0.361787 0.932261i \(-0.617833\pi\)
−0.361787 + 0.932261i \(0.617833\pi\)
\(192\) 0 0
\(193\) 11.0000 0.791797 0.395899 0.918294i \(-0.370433\pi\)
0.395899 + 0.918294i \(0.370433\pi\)
\(194\) 0 0
\(195\) 2.00000 0.143223
\(196\) 0 0
\(197\) 16.0000 1.13995 0.569976 0.821661i \(-0.306952\pi\)
0.569976 + 0.821661i \(0.306952\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) −5.00000 −0.352673
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 20.0000 1.39686
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2.00000 −0.138343
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) −6.00000 −0.411113
\(214\) 0 0
\(215\) 10.0000 0.681994
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 3.00000 0.202721
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 0 0
\(227\) −18.0000 −1.19470 −0.597351 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) 0 0
\(229\) −19.0000 −1.25556 −0.627778 0.778393i \(-0.716035\pi\)
−0.627778 + 0.778393i \(0.716035\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) 0 0
\(237\) −1.00000 −0.0649570
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1.00000 −0.0636285
\(248\) 0 0
\(249\) 6.00000 0.380235
\(250\) 0 0
\(251\) 8.00000 0.504956 0.252478 0.967603i \(-0.418755\pi\)
0.252478 + 0.967603i \(0.418755\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 26.0000 1.62184 0.810918 0.585160i \(-0.198968\pi\)
0.810918 + 0.585160i \(0.198968\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 4.00000 0.247594
\(262\) 0 0
\(263\) −4.00000 −0.246651 −0.123325 0.992366i \(-0.539356\pi\)
−0.123325 + 0.992366i \(0.539356\pi\)
\(264\) 0 0
\(265\) −24.0000 −1.47431
\(266\) 0 0
\(267\) −16.0000 −0.979184
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2.00000 −0.120605
\(276\) 0 0
\(277\) 13.0000 0.781094 0.390547 0.920583i \(-0.372286\pi\)
0.390547 + 0.920583i \(0.372286\pi\)
\(278\) 0 0
\(279\) −9.00000 −0.538816
\(280\) 0 0
\(281\) −4.00000 −0.238620 −0.119310 0.992857i \(-0.538068\pi\)
−0.119310 + 0.992857i \(0.538068\pi\)
\(282\) 0 0
\(283\) 11.0000 0.653882 0.326941 0.945045i \(-0.393982\pi\)
0.326941 + 0.945045i \(0.393982\pi\)
\(284\) 0 0
\(285\) −2.00000 −0.118470
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 6.00000 0.351726
\(292\) 0 0
\(293\) 8.00000 0.467365 0.233682 0.972313i \(-0.424922\pi\)
0.233682 + 0.972313i \(0.424922\pi\)
\(294\) 0 0
\(295\) −24.0000 −1.39733
\(296\) 0 0
\(297\) −2.00000 −0.116052
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −2.00000 −0.114897
\(304\) 0 0
\(305\) −20.0000 −1.14520
\(306\) 0 0
\(307\) 17.0000 0.970241 0.485121 0.874447i \(-0.338776\pi\)
0.485121 + 0.874447i \(0.338776\pi\)
\(308\) 0 0
\(309\) −7.00000 −0.398216
\(310\) 0 0
\(311\) 6.00000 0.340229 0.170114 0.985424i \(-0.445586\pi\)
0.170114 + 0.985424i \(0.445586\pi\)
\(312\) 0 0
\(313\) −1.00000 −0.0565233 −0.0282617 0.999601i \(-0.508997\pi\)
−0.0282617 + 0.999601i \(0.508997\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 24.0000 1.34797 0.673987 0.738743i \(-0.264580\pi\)
0.673987 + 0.738743i \(0.264580\pi\)
\(318\) 0 0
\(319\) 8.00000 0.447914
\(320\) 0 0
\(321\) −8.00000 −0.446516
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 0 0
\(327\) −9.00000 −0.497701
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 25.0000 1.37412 0.687062 0.726599i \(-0.258900\pi\)
0.687062 + 0.726599i \(0.258900\pi\)
\(332\) 0 0
\(333\) 3.00000 0.164399
\(334\) 0 0
\(335\) −10.0000 −0.546358
\(336\) 0 0
\(337\) 13.0000 0.708155 0.354078 0.935216i \(-0.384795\pi\)
0.354078 + 0.935216i \(0.384795\pi\)
\(338\) 0 0
\(339\) −10.0000 −0.543125
\(340\) 0 0
\(341\) −18.0000 −0.974755
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −32.0000 −1.71785 −0.858925 0.512101i \(-0.828867\pi\)
−0.858925 + 0.512101i \(0.828867\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) 34.0000 1.80964 0.904819 0.425797i \(-0.140006\pi\)
0.904819 + 0.425797i \(0.140006\pi\)
\(354\) 0 0
\(355\) −12.0000 −0.636894
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −20.0000 −1.05556 −0.527780 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) 0 0
\(367\) 9.00000 0.469796 0.234898 0.972020i \(-0.424524\pi\)
0.234898 + 0.972020i \(0.424524\pi\)
\(368\) 0 0
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 23.0000 1.19089 0.595447 0.803394i \(-0.296975\pi\)
0.595447 + 0.803394i \(0.296975\pi\)
\(374\) 0 0
\(375\) −12.0000 −0.619677
\(376\) 0 0
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) −3.00000 −0.154100 −0.0770498 0.997027i \(-0.524550\pi\)
−0.0770498 + 0.997027i \(0.524550\pi\)
\(380\) 0 0
\(381\) −15.0000 −0.768473
\(382\) 0 0
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −5.00000 −0.254164
\(388\) 0 0
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −14.0000 −0.706207
\(394\) 0 0
\(395\) −2.00000 −0.100631
\(396\) 0 0
\(397\) −9.00000 −0.451697 −0.225849 0.974162i \(-0.572515\pi\)
−0.225849 + 0.974162i \(0.572515\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −36.0000 −1.79775 −0.898877 0.438201i \(-0.855616\pi\)
−0.898877 + 0.438201i \(0.855616\pi\)
\(402\) 0 0
\(403\) −9.00000 −0.448322
\(404\) 0 0
\(405\) −2.00000 −0.0993808
\(406\) 0 0
\(407\) 6.00000 0.297409
\(408\) 0 0
\(409\) 5.00000 0.247234 0.123617 0.992330i \(-0.460551\pi\)
0.123617 + 0.992330i \(0.460551\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) 0 0
\(417\) −3.00000 −0.146911
\(418\) 0 0
\(419\) −30.0000 −1.46560 −0.732798 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) 0 0
\(421\) −7.00000 −0.341159 −0.170580 0.985344i \(-0.554564\pi\)
−0.170580 + 0.985344i \(0.554564\pi\)
\(422\) 0 0
\(423\) 6.00000 0.291730
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −2.00000 −0.0965609
\(430\) 0 0
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 0 0
\(433\) 31.0000 1.48976 0.744882 0.667196i \(-0.232506\pi\)
0.744882 + 0.667196i \(0.232506\pi\)
\(434\) 0 0
\(435\) 8.00000 0.383571
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) −32.0000 −1.51695
\(446\) 0 0
\(447\) 12.0000 0.567581
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) −20.0000 −0.941763
\(452\) 0 0
\(453\) −16.0000 −0.751746
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −11.0000 −0.514558 −0.257279 0.966337i \(-0.582826\pi\)
−0.257279 + 0.966337i \(0.582826\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 20.0000 0.931493 0.465746 0.884918i \(-0.345786\pi\)
0.465746 + 0.884918i \(0.345786\pi\)
\(462\) 0 0
\(463\) 17.0000 0.790057 0.395029 0.918669i \(-0.370735\pi\)
0.395029 + 0.918669i \(0.370735\pi\)
\(464\) 0 0
\(465\) −18.0000 −0.834730
\(466\) 0 0
\(467\) −6.00000 −0.277647 −0.138823 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) 0 0
\(473\) −10.0000 −0.459800
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) 0 0
\(479\) 28.0000 1.27935 0.639676 0.768644i \(-0.279068\pi\)
0.639676 + 0.768644i \(0.279068\pi\)
\(480\) 0 0
\(481\) 3.00000 0.136788
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 12.0000 0.544892
\(486\) 0 0
\(487\) −31.0000 −1.40474 −0.702372 0.711810i \(-0.747876\pi\)
−0.702372 + 0.711810i \(0.747876\pi\)
\(488\) 0 0
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −4.00000 −0.179787
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −37.0000 −1.65635 −0.828174 0.560471i \(-0.810620\pi\)
−0.828174 + 0.560471i \(0.810620\pi\)
\(500\) 0 0
\(501\) −14.0000 −0.625474
\(502\) 0 0
\(503\) 42.0000 1.87269 0.936344 0.351085i \(-0.114187\pi\)
0.936344 + 0.351085i \(0.114187\pi\)
\(504\) 0 0
\(505\) −4.00000 −0.177998
\(506\) 0 0
\(507\) 12.0000 0.532939
\(508\) 0 0
\(509\) 2.00000 0.0886484 0.0443242 0.999017i \(-0.485887\pi\)
0.0443242 + 0.999017i \(0.485887\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 1.00000 0.0441511
\(514\) 0 0
\(515\) −14.0000 −0.616914
\(516\) 0 0
\(517\) 12.0000 0.527759
\(518\) 0 0
\(519\) −8.00000 −0.351161
\(520\) 0 0
\(521\) 12.0000 0.525730 0.262865 0.964833i \(-0.415333\pi\)
0.262865 + 0.964833i \(0.415333\pi\)
\(522\) 0 0
\(523\) −31.0000 −1.35554 −0.677768 0.735276i \(-0.737052\pi\)
−0.677768 + 0.735276i \(0.737052\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) −10.0000 −0.433148
\(534\) 0 0
\(535\) −16.0000 −0.691740
\(536\) 0 0
\(537\) 2.00000 0.0863064
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −19.0000 −0.816874 −0.408437 0.912787i \(-0.633926\pi\)
−0.408437 + 0.912787i \(0.633926\pi\)
\(542\) 0 0
\(543\) −13.0000 −0.557883
\(544\) 0 0
\(545\) −18.0000 −0.771035
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 0 0
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) −4.00000 −0.170406
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 6.00000 0.254686
\(556\) 0 0
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 0 0
\(559\) −5.00000 −0.211477
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 26.0000 1.09577 0.547885 0.836554i \(-0.315433\pi\)
0.547885 + 0.836554i \(0.315433\pi\)
\(564\) 0 0
\(565\) −20.0000 −0.841406
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −26.0000 −1.08998 −0.544988 0.838444i \(-0.683466\pi\)
−0.544988 + 0.838444i \(0.683466\pi\)
\(570\) 0 0
\(571\) 19.0000 0.795125 0.397563 0.917575i \(-0.369856\pi\)
0.397563 + 0.917575i \(0.369856\pi\)
\(572\) 0 0
\(573\) 10.0000 0.417756
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −17.0000 −0.707719 −0.353860 0.935299i \(-0.615131\pi\)
−0.353860 + 0.935299i \(0.615131\pi\)
\(578\) 0 0
\(579\) −11.0000 −0.457144
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 24.0000 0.993978
\(584\) 0 0
\(585\) −2.00000 −0.0826898
\(586\) 0 0
\(587\) −16.0000 −0.660391 −0.330195 0.943913i \(-0.607115\pi\)
−0.330195 + 0.943913i \(0.607115\pi\)
\(588\) 0 0
\(589\) 9.00000 0.370839
\(590\) 0 0
\(591\) −16.0000 −0.658152
\(592\) 0 0
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) −9.00000 −0.367118 −0.183559 0.983009i \(-0.558762\pi\)
−0.183559 + 0.983009i \(0.558762\pi\)
\(602\) 0 0
\(603\) 5.00000 0.203616
\(604\) 0 0
\(605\) 14.0000 0.569181
\(606\) 0 0
\(607\) −23.0000 −0.933541 −0.466771 0.884378i \(-0.654583\pi\)
−0.466771 + 0.884378i \(0.654583\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 6.00000 0.242734
\(612\) 0 0
\(613\) 34.0000 1.37325 0.686624 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) 0 0
\(615\) −20.0000 −0.806478
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) 29.0000 1.16561 0.582804 0.812613i \(-0.301955\pi\)
0.582804 + 0.812613i \(0.301955\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 2.00000 0.0798723
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 0 0
\(633\) 4.00000 0.158986
\(634\) 0 0
\(635\) −30.0000 −1.19051
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 6.00000 0.237356
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 19.0000 0.749287 0.374643 0.927169i \(-0.377765\pi\)
0.374643 + 0.927169i \(0.377765\pi\)
\(644\) 0 0
\(645\) −10.0000 −0.393750
\(646\) 0 0
\(647\) −2.00000 −0.0786281 −0.0393141 0.999227i \(-0.512517\pi\)
−0.0393141 + 0.999227i \(0.512517\pi\)
\(648\) 0 0
\(649\) 24.0000 0.942082
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 0 0
\(655\) −28.0000 −1.09405
\(656\) 0 0
\(657\) −3.00000 −0.117041
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) −41.0000 −1.59472 −0.797358 0.603507i \(-0.793769\pi\)
−0.797358 + 0.603507i \(0.793769\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) 20.0000 0.772091
\(672\) 0 0
\(673\) −41.0000 −1.58043 −0.790217 0.612827i \(-0.790032\pi\)
−0.790217 + 0.612827i \(0.790032\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 12.0000 0.461197 0.230599 0.973049i \(-0.425932\pi\)
0.230599 + 0.973049i \(0.425932\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 18.0000 0.689761
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) 24.0000 0.916993
\(686\) 0 0
\(687\) 19.0000 0.724895
\(688\) 0 0
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 37.0000 1.40755 0.703773 0.710425i \(-0.251497\pi\)
0.703773 + 0.710425i \(0.251497\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.00000 −0.227593
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) −3.00000 −0.113147
\(704\) 0 0
\(705\) 12.0000 0.451946
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 30.0000 1.12667 0.563337 0.826227i \(-0.309517\pi\)
0.563337 + 0.826227i \(0.309517\pi\)
\(710\) 0 0
\(711\) 1.00000 0.0375029
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) 0 0
\(717\) 6.00000 0.224074
\(718\) 0 0
\(719\) 18.0000 0.671287 0.335643 0.941989i \(-0.391046\pi\)
0.335643 + 0.941989i \(0.391046\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −14.0000 −0.520666
\(724\) 0 0
\(725\) −4.00000 −0.148556
\(726\) 0 0
\(727\) 13.0000 0.482143 0.241072 0.970507i \(-0.422501\pi\)
0.241072 + 0.970507i \(0.422501\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −15.0000 −0.554038 −0.277019 0.960864i \(-0.589346\pi\)
−0.277019 + 0.960864i \(0.589346\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 10.0000 0.368355
\(738\) 0 0
\(739\) 15.0000 0.551784 0.275892 0.961189i \(-0.411027\pi\)
0.275892 + 0.961189i \(0.411027\pi\)
\(740\) 0 0
\(741\) 1.00000 0.0367359
\(742\) 0 0
\(743\) −42.0000 −1.54083 −0.770415 0.637542i \(-0.779951\pi\)
−0.770415 + 0.637542i \(0.779951\pi\)
\(744\) 0 0
\(745\) 24.0000 0.879292
\(746\) 0 0
\(747\) −6.00000 −0.219529
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −13.0000 −0.474377 −0.237188 0.971464i \(-0.576226\pi\)
−0.237188 + 0.971464i \(0.576226\pi\)
\(752\) 0 0
\(753\) −8.00000 −0.291536
\(754\) 0 0
\(755\) −32.0000 −1.16460
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −48.0000 −1.74000 −0.869999 0.493053i \(-0.835881\pi\)
−0.869999 + 0.493053i \(0.835881\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 12.0000 0.433295
\(768\) 0 0
\(769\) −49.0000 −1.76699 −0.883493 0.468445i \(-0.844814\pi\)
−0.883493 + 0.468445i \(0.844814\pi\)
\(770\) 0 0
\(771\) −26.0000 −0.936367
\(772\) 0 0
\(773\) −34.0000 −1.22290 −0.611448 0.791285i \(-0.709412\pi\)
−0.611448 + 0.791285i \(0.709412\pi\)
\(774\) 0 0
\(775\) 9.00000 0.323290
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 10.0000 0.358287
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) 0 0
\(783\) −4.00000 −0.142948
\(784\) 0 0
\(785\) 28.0000 0.999363
\(786\) 0 0
\(787\) −40.0000 −1.42585 −0.712923 0.701242i \(-0.752629\pi\)
−0.712923 + 0.701242i \(0.752629\pi\)
\(788\) 0 0
\(789\) 4.00000 0.142404
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 10.0000 0.355110
\(794\) 0 0
\(795\) 24.0000 0.851192
\(796\) 0 0
\(797\) −8.00000 −0.283375 −0.141687 0.989911i \(-0.545253\pi\)
−0.141687 + 0.989911i \(0.545253\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 16.0000 0.565332
\(802\) 0 0
\(803\) −6.00000 −0.211735
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −6.00000 −0.211210
\(808\) 0 0
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) −32.0000 −1.12367 −0.561836 0.827249i \(-0.689905\pi\)
−0.561836 + 0.827249i \(0.689905\pi\)
\(812\) 0 0
\(813\) 16.0000 0.561144
\(814\) 0 0
\(815\) 8.00000 0.280228
\(816\) 0 0
\(817\) 5.00000 0.174928
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 2.00000 0.0698005 0.0349002 0.999391i \(-0.488889\pi\)
0.0349002 + 0.999391i \(0.488889\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) 2.00000 0.0696311
\(826\) 0 0
\(827\) 30.0000 1.04320 0.521601 0.853189i \(-0.325335\pi\)
0.521601 + 0.853189i \(0.325335\pi\)
\(828\) 0 0
\(829\) 41.0000 1.42399 0.711994 0.702185i \(-0.247792\pi\)
0.711994 + 0.702185i \(0.247792\pi\)
\(830\) 0 0
\(831\) −13.0000 −0.450965
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −28.0000 −0.968980
\(836\) 0 0
\(837\) 9.00000 0.311086
\(838\) 0 0
\(839\) 44.0000 1.51905 0.759524 0.650479i \(-0.225432\pi\)
0.759524 + 0.650479i \(0.225432\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) 4.00000 0.137767
\(844\) 0 0
\(845\) 24.0000 0.825625
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −11.0000 −0.377519
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 35.0000 1.19838 0.599189 0.800608i \(-0.295490\pi\)
0.599189 + 0.800608i \(0.295490\pi\)
\(854\) 0 0
\(855\) 2.00000 0.0683986
\(856\) 0 0
\(857\) −32.0000 −1.09310 −0.546550 0.837427i \(-0.684059\pi\)
−0.546550 + 0.837427i \(0.684059\pi\)
\(858\) 0 0
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 54.0000 1.83818 0.919091 0.394046i \(-0.128925\pi\)
0.919091 + 0.394046i \(0.128925\pi\)
\(864\) 0 0
\(865\) −16.0000 −0.544016
\(866\) 0 0
\(867\) 17.0000 0.577350
\(868\) 0 0
\(869\) 2.00000 0.0678454
\(870\) 0 0
\(871\) 5.00000 0.169419
\(872\) 0 0
\(873\) −6.00000 −0.203069
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −38.0000 −1.28317 −0.641584 0.767052i \(-0.721723\pi\)
−0.641584 + 0.767052i \(0.721723\pi\)
\(878\) 0 0
\(879\) −8.00000 −0.269833
\(880\) 0 0
\(881\) 24.0000 0.808581 0.404290 0.914631i \(-0.367519\pi\)
0.404290 + 0.914631i \(0.367519\pi\)
\(882\) 0 0
\(883\) 13.0000 0.437485 0.218742 0.975783i \(-0.429805\pi\)
0.218742 + 0.975783i \(0.429805\pi\)
\(884\) 0 0
\(885\) 24.0000 0.806751
\(886\) 0 0
\(887\) 34.0000 1.14161 0.570804 0.821086i \(-0.306632\pi\)
0.570804 + 0.821086i \(0.306632\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 0 0
\(893\) −6.00000 −0.200782
\(894\) 0 0
\(895\) 4.00000 0.133705
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −36.0000 −1.20067
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −26.0000 −0.864269
\(906\) 0 0
\(907\) 37.0000 1.22856 0.614282 0.789086i \(-0.289446\pi\)
0.614282 + 0.789086i \(0.289446\pi\)
\(908\) 0 0
\(909\) 2.00000 0.0663358
\(910\) 0 0
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) 0 0
\(915\) 20.0000 0.661180
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −23.0000 −0.758700 −0.379350 0.925253i \(-0.623852\pi\)
−0.379350 + 0.925253i \(0.623852\pi\)
\(920\) 0 0
\(921\) −17.0000 −0.560169
\(922\) 0 0
\(923\) 6.00000 0.197492
\(924\) 0 0
\(925\) −3.00000 −0.0986394
\(926\) 0 0
\(927\) 7.00000 0.229910
\(928\) 0 0
\(929\) 14.0000 0.459325 0.229663 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −6.00000 −0.196431
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 15.0000 0.490029 0.245014 0.969519i \(-0.421207\pi\)
0.245014 + 0.969519i \(0.421207\pi\)
\(938\) 0 0
\(939\) 1.00000 0.0326338
\(940\) 0 0
\(941\) −4.00000 −0.130396 −0.0651981 0.997872i \(-0.520768\pi\)
−0.0651981 + 0.997872i \(0.520768\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 10.0000 0.324956 0.162478 0.986712i \(-0.448051\pi\)
0.162478 + 0.986712i \(0.448051\pi\)
\(948\) 0 0
\(949\) −3.00000 −0.0973841
\(950\) 0 0
\(951\) −24.0000 −0.778253
\(952\) 0 0
\(953\) 44.0000 1.42530 0.712650 0.701520i \(-0.247495\pi\)
0.712650 + 0.701520i \(0.247495\pi\)
\(954\) 0 0
\(955\) 20.0000 0.647185
\(956\) 0 0
\(957\) −8.00000 −0.258603
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 50.0000 1.61290
\(962\) 0 0
\(963\) 8.00000 0.257796
\(964\) 0 0
\(965\) −22.0000 −0.708205
\(966\) 0 0
\(967\) −19.0000 −0.610999 −0.305499 0.952192i \(-0.598823\pi\)
−0.305499 + 0.952192i \(0.598823\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 1.00000 0.0320256
\(976\) 0 0
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) 32.0000 1.02272
\(980\) 0 0
\(981\) 9.00000 0.287348
\(982\) 0 0
\(983\) −36.0000 −1.14822 −0.574111 0.818778i \(-0.694652\pi\)
−0.574111 + 0.818778i \(0.694652\pi\)
\(984\) 0 0
\(985\) −32.0000 −1.01960
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −17.0000 −0.540023 −0.270011 0.962857i \(-0.587027\pi\)
−0.270011 + 0.962857i \(0.587027\pi\)
\(992\) 0 0
\(993\) −25.0000 −0.793351
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 19.0000 0.601736 0.300868 0.953666i \(-0.402724\pi\)
0.300868 + 0.953666i \(0.402724\pi\)
\(998\) 0 0
\(999\) −3.00000 −0.0949158
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 2352.2.a.d.1.1 1
3.2 odd 2 7056.2.a.bp.1.1 1
4.3 odd 2 147.2.a.c.1.1 1
7.2 even 3 336.2.q.f.193.1 2
7.3 odd 6 2352.2.q.c.961.1 2
7.4 even 3 336.2.q.f.289.1 2
7.5 odd 6 2352.2.q.c.1537.1 2
7.6 odd 2 2352.2.a.w.1.1 1
8.3 odd 2 9408.2.a.bg.1.1 1
8.5 even 2 9408.2.a.cv.1.1 1
12.11 even 2 441.2.a.b.1.1 1
20.19 odd 2 3675.2.a.a.1.1 1
21.2 odd 6 1008.2.s.d.865.1 2
21.11 odd 6 1008.2.s.d.289.1 2
21.20 even 2 7056.2.a.m.1.1 1
28.3 even 6 147.2.e.a.79.1 2
28.11 odd 6 21.2.e.a.16.1 yes 2
28.19 even 6 147.2.e.a.67.1 2
28.23 odd 6 21.2.e.a.4.1 2
28.27 even 2 147.2.a.b.1.1 1
56.11 odd 6 1344.2.q.m.961.1 2
56.13 odd 2 9408.2.a.k.1.1 1
56.27 even 2 9408.2.a.bz.1.1 1
56.37 even 6 1344.2.q.c.193.1 2
56.51 odd 6 1344.2.q.m.193.1 2
56.53 even 6 1344.2.q.c.961.1 2
84.11 even 6 63.2.e.b.37.1 2
84.23 even 6 63.2.e.b.46.1 2
84.47 odd 6 441.2.e.e.361.1 2
84.59 odd 6 441.2.e.e.226.1 2
84.83 odd 2 441.2.a.a.1.1 1
140.23 even 12 525.2.r.e.424.1 4
140.39 odd 6 525.2.i.e.226.1 2
140.67 even 12 525.2.r.e.499.1 4
140.79 odd 6 525.2.i.e.151.1 2
140.107 even 12 525.2.r.e.424.2 4
140.123 even 12 525.2.r.e.499.2 4
140.139 even 2 3675.2.a.c.1.1 1
252.11 even 6 567.2.h.a.352.1 2
252.23 even 6 567.2.h.a.298.1 2
252.67 odd 6 567.2.g.a.541.1 2
252.79 odd 6 567.2.g.a.109.1 2
252.95 even 6 567.2.g.f.541.1 2
252.151 odd 6 567.2.h.f.352.1 2
252.191 even 6 567.2.g.f.109.1 2
252.247 odd 6 567.2.h.f.298.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.2.e.a.4.1 2 28.23 odd 6
21.2.e.a.16.1 yes 2 28.11 odd 6
63.2.e.b.37.1 2 84.11 even 6
63.2.e.b.46.1 2 84.23 even 6
147.2.a.b.1.1 1 28.27 even 2
147.2.a.c.1.1 1 4.3 odd 2
147.2.e.a.67.1 2 28.19 even 6
147.2.e.a.79.1 2 28.3 even 6
336.2.q.f.193.1 2 7.2 even 3
336.2.q.f.289.1 2 7.4 even 3
441.2.a.a.1.1 1 84.83 odd 2
441.2.a.b.1.1 1 12.11 even 2
441.2.e.e.226.1 2 84.59 odd 6
441.2.e.e.361.1 2 84.47 odd 6
525.2.i.e.151.1 2 140.79 odd 6
525.2.i.e.226.1 2 140.39 odd 6
525.2.r.e.424.1 4 140.23 even 12
525.2.r.e.424.2 4 140.107 even 12
525.2.r.e.499.1 4 140.67 even 12
525.2.r.e.499.2 4 140.123 even 12
567.2.g.a.109.1 2 252.79 odd 6
567.2.g.a.541.1 2 252.67 odd 6
567.2.g.f.109.1 2 252.191 even 6
567.2.g.f.541.1 2 252.95 even 6
567.2.h.a.298.1 2 252.23 even 6
567.2.h.a.352.1 2 252.11 even 6
567.2.h.f.298.1 2 252.247 odd 6
567.2.h.f.352.1 2 252.151 odd 6
1008.2.s.d.289.1 2 21.11 odd 6
1008.2.s.d.865.1 2 21.2 odd 6
1344.2.q.c.193.1 2 56.37 even 6
1344.2.q.c.961.1 2 56.53 even 6
1344.2.q.m.193.1 2 56.51 odd 6
1344.2.q.m.961.1 2 56.11 odd 6
2352.2.a.d.1.1 1 1.1 even 1 trivial
2352.2.a.w.1.1 1 7.6 odd 2
2352.2.q.c.961.1 2 7.3 odd 6
2352.2.q.c.1537.1 2 7.5 odd 6
3675.2.a.a.1.1 1 20.19 odd 2
3675.2.a.c.1.1 1 140.139 even 2
7056.2.a.m.1.1 1 21.20 even 2
7056.2.a.bp.1.1 1 3.2 odd 2
9408.2.a.k.1.1 1 56.13 odd 2
9408.2.a.bg.1.1 1 8.3 odd 2
9408.2.a.bz.1.1 1 56.27 even 2
9408.2.a.cv.1.1 1 8.5 even 2