Properties

Label 3744.2.a.z.1.2
Level $3744$
Weight $2$
Character 3744.1
Self dual yes
Analytic conductor $29.896$
Analytic rank $1$
Dimension $3$
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] = [3744,2,Mod(1,3744)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3744, 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("3744.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3744.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: \(29.8959905168\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1248)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 3744.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.622216 q^{5} +4.42864 q^{7} +O(q^{10})\) \(q-0.622216 q^{5} +4.42864 q^{7} -5.80642 q^{11} +1.00000 q^{13} -2.00000 q^{17} -4.42864 q^{19} +8.85728 q^{23} -4.61285 q^{25} -2.00000 q^{29} -7.18421 q^{31} -2.75557 q^{35} +0.755569 q^{37} -3.37778 q^{41} -7.61285 q^{43} -1.80642 q^{47} +12.6128 q^{49} -4.75557 q^{53} +3.61285 q^{55} -11.0509 q^{59} -8.10171 q^{61} -0.622216 q^{65} +8.04149 q^{67} +7.05086 q^{71} +7.24443 q^{73} -25.7146 q^{77} -12.0000 q^{79} +3.05086 q^{83} +1.24443 q^{85} +1.86665 q^{89} +4.42864 q^{91} +2.75557 q^{95} -0.755569 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{5} - 4 q^{11} + 3 q^{13} - 6 q^{17} + 13 q^{25} - 6 q^{29} - 8 q^{31} - 8 q^{35} + 2 q^{37} - 10 q^{41} + 4 q^{43} + 8 q^{47} + 11 q^{49} - 14 q^{53} - 16 q^{55} - 20 q^{59} + 2 q^{61} - 2 q^{65} - 16 q^{67} + 8 q^{71} + 22 q^{73} - 24 q^{77} - 36 q^{79} - 4 q^{83} + 4 q^{85} + 6 q^{89} + 8 q^{95} - 2 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) −0.622216 −0.278263 −0.139132 0.990274i \(-0.544431\pi\)
−0.139132 + 0.990274i \(0.544431\pi\)
\(6\) 0 0
\(7\) 4.42864 1.67387 0.836934 0.547304i \(-0.184346\pi\)
0.836934 + 0.547304i \(0.184346\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.80642 −1.75070 −0.875351 0.483487i \(-0.839370\pi\)
−0.875351 + 0.483487i \(0.839370\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −4.42864 −1.01600 −0.508000 0.861357i \(-0.669615\pi\)
−0.508000 + 0.861357i \(0.669615\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.85728 1.84687 0.923435 0.383754i \(-0.125369\pi\)
0.923435 + 0.383754i \(0.125369\pi\)
\(24\) 0 0
\(25\) −4.61285 −0.922570
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −7.18421 −1.29032 −0.645161 0.764047i \(-0.723210\pi\)
−0.645161 + 0.764047i \(0.723210\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.75557 −0.465776
\(36\) 0 0
\(37\) 0.755569 0.124215 0.0621074 0.998069i \(-0.480218\pi\)
0.0621074 + 0.998069i \(0.480218\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.37778 −0.527521 −0.263761 0.964588i \(-0.584963\pi\)
−0.263761 + 0.964588i \(0.584963\pi\)
\(42\) 0 0
\(43\) −7.61285 −1.16095 −0.580474 0.814279i \(-0.697133\pi\)
−0.580474 + 0.814279i \(0.697133\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.80642 −0.263494 −0.131747 0.991283i \(-0.542059\pi\)
−0.131747 + 0.991283i \(0.542059\pi\)
\(48\) 0 0
\(49\) 12.6128 1.80184
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.75557 −0.653228 −0.326614 0.945158i \(-0.605908\pi\)
−0.326614 + 0.945158i \(0.605908\pi\)
\(54\) 0 0
\(55\) 3.61285 0.487156
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −11.0509 −1.43870 −0.719349 0.694648i \(-0.755560\pi\)
−0.719349 + 0.694648i \(0.755560\pi\)
\(60\) 0 0
\(61\) −8.10171 −1.03732 −0.518659 0.854981i \(-0.673569\pi\)
−0.518659 + 0.854981i \(0.673569\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.622216 −0.0771764
\(66\) 0 0
\(67\) 8.04149 0.982424 0.491212 0.871040i \(-0.336554\pi\)
0.491212 + 0.871040i \(0.336554\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.05086 0.836783 0.418391 0.908267i \(-0.362594\pi\)
0.418391 + 0.908267i \(0.362594\pi\)
\(72\) 0 0
\(73\) 7.24443 0.847897 0.423948 0.905686i \(-0.360644\pi\)
0.423948 + 0.905686i \(0.360644\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −25.7146 −2.93045
\(78\) 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
\(83\) 3.05086 0.334875 0.167437 0.985883i \(-0.446451\pi\)
0.167437 + 0.985883i \(0.446451\pi\)
\(84\) 0 0
\(85\) 1.24443 0.134978
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.86665 0.197864 0.0989321 0.995094i \(-0.468457\pi\)
0.0989321 + 0.995094i \(0.468457\pi\)
\(90\) 0 0
\(91\) 4.42864 0.464248
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.75557 0.282715
\(96\) 0 0
\(97\) −0.755569 −0.0767164 −0.0383582 0.999264i \(-0.512213\pi\)
−0.0383582 + 0.999264i \(0.512213\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 15.7146 1.56366 0.781828 0.623494i \(-0.214287\pi\)
0.781828 + 0.623494i \(0.214287\pi\)
\(102\) 0 0
\(103\) −18.1017 −1.78361 −0.891807 0.452416i \(-0.850562\pi\)
−0.891807 + 0.452416i \(0.850562\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −15.3461 −1.48357 −0.741784 0.670639i \(-0.766020\pi\)
−0.741784 + 0.670639i \(0.766020\pi\)
\(108\) 0 0
\(109\) 15.7146 1.50518 0.752591 0.658488i \(-0.228804\pi\)
0.752591 + 0.658488i \(0.228804\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −14.4701 −1.36124 −0.680618 0.732639i \(-0.738288\pi\)
−0.680618 + 0.732639i \(0.738288\pi\)
\(114\) 0 0
\(115\) −5.51114 −0.513916
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −8.85728 −0.811945
\(120\) 0 0
\(121\) 22.7146 2.06496
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 5.98126 0.534981
\(126\) 0 0
\(127\) −20.8573 −1.85078 −0.925392 0.379011i \(-0.876264\pi\)
−0.925392 + 0.379011i \(0.876264\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 10.3684 0.905893 0.452946 0.891538i \(-0.350373\pi\)
0.452946 + 0.891538i \(0.350373\pi\)
\(132\) 0 0
\(133\) −19.6128 −1.70065
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 19.8479 1.69572 0.847861 0.530219i \(-0.177890\pi\)
0.847861 + 0.530219i \(0.177890\pi\)
\(138\) 0 0
\(139\) 6.75557 0.573000 0.286500 0.958080i \(-0.407508\pi\)
0.286500 + 0.958080i \(0.407508\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −5.80642 −0.485558
\(144\) 0 0
\(145\) 1.24443 0.103344
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2.13335 0.174771 0.0873855 0.996175i \(-0.472149\pi\)
0.0873855 + 0.996175i \(0.472149\pi\)
\(150\) 0 0
\(151\) 1.67307 0.136153 0.0680763 0.997680i \(-0.478314\pi\)
0.0680763 + 0.997680i \(0.478314\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.47013 0.359049
\(156\) 0 0
\(157\) 15.1240 1.20703 0.603513 0.797353i \(-0.293767\pi\)
0.603513 + 0.797353i \(0.293767\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 39.2257 3.09142
\(162\) 0 0
\(163\) −15.7748 −1.23558 −0.617788 0.786345i \(-0.711971\pi\)
−0.617788 + 0.786345i \(0.711971\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −7.31756 −0.566250 −0.283125 0.959083i \(-0.591371\pi\)
−0.283125 + 0.959083i \(0.591371\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −7.51114 −0.571061 −0.285531 0.958370i \(-0.592170\pi\)
−0.285531 + 0.958370i \(0.592170\pi\)
\(174\) 0 0
\(175\) −20.4286 −1.54426
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −14.7556 −1.10288 −0.551441 0.834214i \(-0.685922\pi\)
−0.551441 + 0.834214i \(0.685922\pi\)
\(180\) 0 0
\(181\) 4.10171 0.304878 0.152439 0.988313i \(-0.451287\pi\)
0.152439 + 0.988313i \(0.451287\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.470127 −0.0345644
\(186\) 0 0
\(187\) 11.6128 0.849216
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0.266706 0.0192982 0.00964909 0.999953i \(-0.496929\pi\)
0.00964909 + 0.999953i \(0.496929\pi\)
\(192\) 0 0
\(193\) −26.2034 −1.88616 −0.943082 0.332561i \(-0.892087\pi\)
−0.943082 + 0.332561i \(0.892087\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −15.8479 −1.12912 −0.564558 0.825393i \(-0.690954\pi\)
−0.564558 + 0.825393i \(0.690954\pi\)
\(198\) 0 0
\(199\) 10.3684 0.734998 0.367499 0.930024i \(-0.380214\pi\)
0.367499 + 0.930024i \(0.380214\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −8.85728 −0.621659
\(204\) 0 0
\(205\) 2.10171 0.146790
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 25.7146 1.77871
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 4.73683 0.323049
\(216\) 0 0
\(217\) −31.8163 −2.15983
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2.00000 −0.134535
\(222\) 0 0
\(223\) −4.69535 −0.314424 −0.157212 0.987565i \(-0.550251\pi\)
−0.157212 + 0.987565i \(0.550251\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.80642 0.385386 0.192693 0.981259i \(-0.438278\pi\)
0.192693 + 0.981259i \(0.438278\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −7.24443 −0.474598 −0.237299 0.971437i \(-0.576262\pi\)
−0.237299 + 0.971437i \(0.576262\pi\)
\(234\) 0 0
\(235\) 1.12399 0.0733207
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −10.6637 −0.689778 −0.344889 0.938644i \(-0.612083\pi\)
−0.344889 + 0.938644i \(0.612083\pi\)
\(240\) 0 0
\(241\) −5.73329 −0.369314 −0.184657 0.982803i \(-0.559117\pi\)
−0.184657 + 0.982803i \(0.559117\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −7.84791 −0.501385
\(246\) 0 0
\(247\) −4.42864 −0.281788
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −23.6128 −1.49043 −0.745215 0.666824i \(-0.767653\pi\)
−0.745215 + 0.666824i \(0.767653\pi\)
\(252\) 0 0
\(253\) −51.4291 −3.23332
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 11.2444 0.701408 0.350704 0.936486i \(-0.385942\pi\)
0.350704 + 0.936486i \(0.385942\pi\)
\(258\) 0 0
\(259\) 3.34614 0.207919
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −4.38715 −0.270523 −0.135262 0.990810i \(-0.543188\pi\)
−0.135262 + 0.990810i \(0.543188\pi\)
\(264\) 0 0
\(265\) 2.95899 0.181769
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −7.51114 −0.457962 −0.228981 0.973431i \(-0.573539\pi\)
−0.228981 + 0.973431i \(0.573539\pi\)
\(270\) 0 0
\(271\) −6.06022 −0.368132 −0.184066 0.982914i \(-0.558926\pi\)
−0.184066 + 0.982914i \(0.558926\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 26.7841 1.61514
\(276\) 0 0
\(277\) 0.488863 0.0293729 0.0146865 0.999892i \(-0.495325\pi\)
0.0146865 + 0.999892i \(0.495325\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −13.0923 −0.781024 −0.390512 0.920598i \(-0.627702\pi\)
−0.390512 + 0.920598i \(0.627702\pi\)
\(282\) 0 0
\(283\) 23.3461 1.38778 0.693892 0.720079i \(-0.255894\pi\)
0.693892 + 0.720079i \(0.255894\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −14.9590 −0.883001
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −13.0923 −0.764863 −0.382431 0.923984i \(-0.624913\pi\)
−0.382431 + 0.923984i \(0.624913\pi\)
\(294\) 0 0
\(295\) 6.87601 0.400337
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 8.85728 0.512230
\(300\) 0 0
\(301\) −33.7146 −1.94327
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 5.04101 0.288647
\(306\) 0 0
\(307\) −32.0415 −1.82870 −0.914352 0.404920i \(-0.867299\pi\)
−0.914352 + 0.404920i \(0.867299\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −24.0830 −1.36562 −0.682810 0.730596i \(-0.739242\pi\)
−0.682810 + 0.730596i \(0.739242\pi\)
\(312\) 0 0
\(313\) 27.7146 1.56652 0.783260 0.621695i \(-0.213556\pi\)
0.783260 + 0.621695i \(0.213556\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 17.0923 0.960002 0.480001 0.877268i \(-0.340636\pi\)
0.480001 + 0.877268i \(0.340636\pi\)
\(318\) 0 0
\(319\) 11.6128 0.650195
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 8.85728 0.492832
\(324\) 0 0
\(325\) −4.61285 −0.255875
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) −7.18421 −0.394880 −0.197440 0.980315i \(-0.563263\pi\)
−0.197440 + 0.980315i \(0.563263\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −5.00354 −0.273373
\(336\) 0 0
\(337\) 0.285442 0.0155490 0.00777451 0.999970i \(-0.497525\pi\)
0.00777451 + 0.999970i \(0.497525\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 41.7146 2.25897
\(342\) 0 0
\(343\) 24.8573 1.34217
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 24.2034 1.29931 0.649654 0.760230i \(-0.274914\pi\)
0.649654 + 0.760230i \(0.274914\pi\)
\(348\) 0 0
\(349\) −12.7556 −0.682790 −0.341395 0.939920i \(-0.610899\pi\)
−0.341395 + 0.939920i \(0.610899\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −21.0923 −1.12263 −0.561316 0.827602i \(-0.689705\pi\)
−0.561316 + 0.827602i \(0.689705\pi\)
\(354\) 0 0
\(355\) −4.38715 −0.232846
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 15.1338 0.798733 0.399366 0.916791i \(-0.369230\pi\)
0.399366 + 0.916791i \(0.369230\pi\)
\(360\) 0 0
\(361\) 0.612848 0.0322551
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −4.50760 −0.235938
\(366\) 0 0
\(367\) −18.9590 −0.989651 −0.494826 0.868992i \(-0.664768\pi\)
−0.494826 + 0.868992i \(0.664768\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −21.0607 −1.09342
\(372\) 0 0
\(373\) 2.97773 0.154181 0.0770904 0.997024i \(-0.475437\pi\)
0.0770904 + 0.997024i \(0.475437\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.00000 −0.103005
\(378\) 0 0
\(379\) −14.1432 −0.726487 −0.363244 0.931694i \(-0.618331\pi\)
−0.363244 + 0.931694i \(0.618331\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 25.8064 1.31865 0.659323 0.751860i \(-0.270843\pi\)
0.659323 + 0.751860i \(0.270843\pi\)
\(384\) 0 0
\(385\) 16.0000 0.815436
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 12.9590 0.657047 0.328523 0.944496i \(-0.393449\pi\)
0.328523 + 0.944496i \(0.393449\pi\)
\(390\) 0 0
\(391\) −17.7146 −0.895864
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 7.46659 0.375685
\(396\) 0 0
\(397\) 3.24443 0.162833 0.0814167 0.996680i \(-0.474056\pi\)
0.0814167 + 0.996680i \(0.474056\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 30.6035 1.52826 0.764132 0.645059i \(-0.223167\pi\)
0.764132 + 0.645059i \(0.223167\pi\)
\(402\) 0 0
\(403\) −7.18421 −0.357871
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −4.38715 −0.217463
\(408\) 0 0
\(409\) −8.75557 −0.432935 −0.216468 0.976290i \(-0.569454\pi\)
−0.216468 + 0.976290i \(0.569454\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −48.9403 −2.40819
\(414\) 0 0
\(415\) −1.89829 −0.0931834
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 23.8796 1.16659 0.583296 0.812259i \(-0.301763\pi\)
0.583296 + 0.812259i \(0.301763\pi\)
\(420\) 0 0
\(421\) −8.28544 −0.403808 −0.201904 0.979405i \(-0.564713\pi\)
−0.201904 + 0.979405i \(0.564713\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 9.22570 0.447512
\(426\) 0 0
\(427\) −35.8796 −1.73633
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 14.0098 0.674830 0.337415 0.941356i \(-0.390447\pi\)
0.337415 + 0.941356i \(0.390447\pi\)
\(432\) 0 0
\(433\) 27.1240 1.30350 0.651748 0.758436i \(-0.274036\pi\)
0.651748 + 0.758436i \(0.274036\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −39.2257 −1.87642
\(438\) 0 0
\(439\) 4.59057 0.219096 0.109548 0.993982i \(-0.465060\pi\)
0.109548 + 0.993982i \(0.465060\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 10.3684 0.492618 0.246309 0.969191i \(-0.420782\pi\)
0.246309 + 0.969191i \(0.420782\pi\)
\(444\) 0 0
\(445\) −1.16146 −0.0550583
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −10.3368 −0.487823 −0.243911 0.969798i \(-0.578431\pi\)
−0.243911 + 0.969798i \(0.578431\pi\)
\(450\) 0 0
\(451\) 19.6128 0.923533
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −2.75557 −0.129183
\(456\) 0 0
\(457\) 5.52987 0.258677 0.129338 0.991601i \(-0.458715\pi\)
0.129338 + 0.991601i \(0.458715\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 17.8666 0.832133 0.416066 0.909334i \(-0.363408\pi\)
0.416066 + 0.909334i \(0.363408\pi\)
\(462\) 0 0
\(463\) 38.6766 1.79745 0.898727 0.438508i \(-0.144493\pi\)
0.898727 + 0.438508i \(0.144493\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −10.1017 −0.467451 −0.233726 0.972303i \(-0.575092\pi\)
−0.233726 + 0.972303i \(0.575092\pi\)
\(468\) 0 0
\(469\) 35.6128 1.64445
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 44.2034 2.03248
\(474\) 0 0
\(475\) 20.4286 0.937330
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 30.8671 1.41035 0.705177 0.709031i \(-0.250867\pi\)
0.705177 + 0.709031i \(0.250867\pi\)
\(480\) 0 0
\(481\) 0.755569 0.0344510
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0.470127 0.0213474
\(486\) 0 0
\(487\) 25.7560 1.16712 0.583559 0.812071i \(-0.301660\pi\)
0.583559 + 0.812071i \(0.301660\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −27.8163 −1.25533 −0.627665 0.778483i \(-0.715989\pi\)
−0.627665 + 0.778483i \(0.715989\pi\)
\(492\) 0 0
\(493\) 4.00000 0.180151
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 31.2257 1.40066
\(498\) 0 0
\(499\) 0.815792 0.0365199 0.0182599 0.999833i \(-0.494187\pi\)
0.0182599 + 0.999833i \(0.494187\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −26.8385 −1.19667 −0.598336 0.801245i \(-0.704171\pi\)
−0.598336 + 0.801245i \(0.704171\pi\)
\(504\) 0 0
\(505\) −9.77784 −0.435108
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 6.60348 0.292694 0.146347 0.989233i \(-0.453248\pi\)
0.146347 + 0.989233i \(0.453248\pi\)
\(510\) 0 0
\(511\) 32.0830 1.41927
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 11.2632 0.496314
\(516\) 0 0
\(517\) 10.4889 0.461300
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −35.9813 −1.57637 −0.788184 0.615440i \(-0.788978\pi\)
−0.788184 + 0.615440i \(0.788978\pi\)
\(522\) 0 0
\(523\) −34.9590 −1.52865 −0.764325 0.644831i \(-0.776928\pi\)
−0.764325 + 0.644831i \(0.776928\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 14.3684 0.625898
\(528\) 0 0
\(529\) 55.4514 2.41093
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −3.37778 −0.146308
\(534\) 0 0
\(535\) 9.54861 0.412822
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −73.2355 −3.15448
\(540\) 0 0
\(541\) 19.5111 0.838849 0.419425 0.907790i \(-0.362232\pi\)
0.419425 + 0.907790i \(0.362232\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −9.77784 −0.418837
\(546\) 0 0
\(547\) −4.26671 −0.182431 −0.0912156 0.995831i \(-0.529075\pi\)
−0.0912156 + 0.995831i \(0.529075\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 8.85728 0.377333
\(552\) 0 0
\(553\) −53.1437 −2.25990
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −5.86665 −0.248578 −0.124289 0.992246i \(-0.539665\pi\)
−0.124289 + 0.992246i \(0.539665\pi\)
\(558\) 0 0
\(559\) −7.61285 −0.321989
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 27.4924 1.15867 0.579333 0.815091i \(-0.303313\pi\)
0.579333 + 0.815091i \(0.303313\pi\)
\(564\) 0 0
\(565\) 9.00354 0.378782
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 13.4924 0.565631 0.282815 0.959174i \(-0.408732\pi\)
0.282815 + 0.959174i \(0.408732\pi\)
\(570\) 0 0
\(571\) 23.8796 0.999328 0.499664 0.866219i \(-0.333457\pi\)
0.499664 + 0.866219i \(0.333457\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −40.8573 −1.70387
\(576\) 0 0
\(577\) 32.4514 1.35097 0.675485 0.737374i \(-0.263935\pi\)
0.675485 + 0.737374i \(0.263935\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 13.5111 0.560536
\(582\) 0 0
\(583\) 27.6128 1.14361
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 17.3363 0.715546 0.357773 0.933809i \(-0.383536\pi\)
0.357773 + 0.933809i \(0.383536\pi\)
\(588\) 0 0
\(589\) 31.8163 1.31097
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 18.8069 0.772307 0.386153 0.922435i \(-0.373804\pi\)
0.386153 + 0.922435i \(0.373804\pi\)
\(594\) 0 0
\(595\) 5.51114 0.225935
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −31.8163 −1.29998 −0.649989 0.759944i \(-0.725226\pi\)
−0.649989 + 0.759944i \(0.725226\pi\)
\(600\) 0 0
\(601\) −4.28544 −0.174807 −0.0874034 0.996173i \(-0.527857\pi\)
−0.0874034 + 0.996173i \(0.527857\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −14.1334 −0.574603
\(606\) 0 0
\(607\) 17.2444 0.699930 0.349965 0.936763i \(-0.386193\pi\)
0.349965 + 0.936763i \(0.386193\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1.80642 −0.0730801
\(612\) 0 0
\(613\) 23.9813 0.968594 0.484297 0.874904i \(-0.339075\pi\)
0.484297 + 0.874904i \(0.339075\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −6.40006 −0.257657 −0.128828 0.991667i \(-0.541122\pi\)
−0.128828 + 0.991667i \(0.541122\pi\)
\(618\) 0 0
\(619\) −8.04149 −0.323215 −0.161607 0.986855i \(-0.551668\pi\)
−0.161607 + 0.986855i \(0.551668\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 8.26671 0.331199
\(624\) 0 0
\(625\) 19.3426 0.773704
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1.51114 −0.0602530
\(630\) 0 0
\(631\) 23.1842 0.922949 0.461474 0.887154i \(-0.347321\pi\)
0.461474 + 0.887154i \(0.347321\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 12.9777 0.515005
\(636\) 0 0
\(637\) 12.6128 0.499739
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 3.51114 0.138682 0.0693408 0.997593i \(-0.477910\pi\)
0.0693408 + 0.997593i \(0.477910\pi\)
\(642\) 0 0
\(643\) 49.7560 1.96219 0.981093 0.193535i \(-0.0619952\pi\)
0.981093 + 0.193535i \(0.0619952\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.9403 0.980503 0.490251 0.871581i \(-0.336905\pi\)
0.490251 + 0.871581i \(0.336905\pi\)
\(648\) 0 0
\(649\) 64.1659 2.51873
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −11.7146 −0.458426 −0.229213 0.973376i \(-0.573615\pi\)
−0.229213 + 0.973376i \(0.573615\pi\)
\(654\) 0 0
\(655\) −6.45139 −0.252077
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.51114 0.0588656 0.0294328 0.999567i \(-0.490630\pi\)
0.0294328 + 0.999567i \(0.490630\pi\)
\(660\) 0 0
\(661\) −24.1847 −0.940675 −0.470338 0.882487i \(-0.655868\pi\)
−0.470338 + 0.882487i \(0.655868\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 12.2034 0.473228
\(666\) 0 0
\(667\) −17.7146 −0.685910
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 47.0420 1.81603
\(672\) 0 0
\(673\) −8.83854 −0.340701 −0.170350 0.985384i \(-0.554490\pi\)
−0.170350 + 0.985384i \(0.554490\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −39.2444 −1.50829 −0.754143 0.656710i \(-0.771947\pi\)
−0.754143 + 0.656710i \(0.771947\pi\)
\(678\) 0 0
\(679\) −3.34614 −0.128413
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 30.1561 1.15389 0.576946 0.816783i \(-0.304244\pi\)
0.576946 + 0.816783i \(0.304244\pi\)
\(684\) 0 0
\(685\) −12.3497 −0.471857
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −4.75557 −0.181173
\(690\) 0 0
\(691\) 21.5526 0.819900 0.409950 0.912108i \(-0.365546\pi\)
0.409950 + 0.912108i \(0.365546\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.20342 −0.159445
\(696\) 0 0
\(697\) 6.75557 0.255885
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0.755569 0.0285374 0.0142687 0.999898i \(-0.495458\pi\)
0.0142687 + 0.999898i \(0.495458\pi\)
\(702\) 0 0
\(703\) −3.34614 −0.126202
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 69.5941 2.61736
\(708\) 0 0
\(709\) −1.22570 −0.0460320 −0.0230160 0.999735i \(-0.507327\pi\)
−0.0230160 + 0.999735i \(0.507327\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −63.6325 −2.38306
\(714\) 0 0
\(715\) 3.61285 0.135113
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −11.3461 −0.423140 −0.211570 0.977363i \(-0.567858\pi\)
−0.211570 + 0.977363i \(0.567858\pi\)
\(720\) 0 0
\(721\) −80.1659 −2.98554
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 9.22570 0.342634
\(726\) 0 0
\(727\) −49.0607 −1.81956 −0.909780 0.415090i \(-0.863750\pi\)
−0.909780 + 0.415090i \(0.863750\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 15.2257 0.563143
\(732\) 0 0
\(733\) −53.4291 −1.97345 −0.986725 0.162402i \(-0.948076\pi\)
−0.986725 + 0.162402i \(0.948076\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −46.6923 −1.71993
\(738\) 0 0
\(739\) −21.0192 −0.773204 −0.386602 0.922247i \(-0.626351\pi\)
−0.386602 + 0.922247i \(0.626351\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 4.56199 0.167363 0.0836816 0.996493i \(-0.473332\pi\)
0.0836816 + 0.996493i \(0.473332\pi\)
\(744\) 0 0
\(745\) −1.32741 −0.0486324
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −67.9625 −2.48330
\(750\) 0 0
\(751\) −21.8983 −0.799080 −0.399540 0.916716i \(-0.630830\pi\)
−0.399540 + 0.916716i \(0.630830\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1.04101 −0.0378863
\(756\) 0 0
\(757\) −36.8385 −1.33892 −0.669460 0.742848i \(-0.733474\pi\)
−0.669460 + 0.742848i \(0.733474\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −38.8069 −1.40675 −0.703375 0.710819i \(-0.748324\pi\)
−0.703375 + 0.710819i \(0.748324\pi\)
\(762\) 0 0
\(763\) 69.5941 2.51948
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −11.0509 −0.399023
\(768\) 0 0
\(769\) 24.6923 0.890426 0.445213 0.895425i \(-0.353128\pi\)
0.445213 + 0.895425i \(0.353128\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −7.58120 −0.272677 −0.136338 0.990662i \(-0.543533\pi\)
−0.136338 + 0.990662i \(0.543533\pi\)
\(774\) 0 0
\(775\) 33.1397 1.19041
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 14.9590 0.535961
\(780\) 0 0
\(781\) −40.9403 −1.46496
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −9.41038 −0.335871
\(786\) 0 0
\(787\) −13.0192 −0.464085 −0.232042 0.972706i \(-0.574541\pi\)
−0.232042 + 0.972706i \(0.574541\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −64.0830 −2.27853
\(792\) 0 0
\(793\) −8.10171 −0.287700
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −23.2444 −0.823360 −0.411680 0.911328i \(-0.635058\pi\)
−0.411680 + 0.911328i \(0.635058\pi\)
\(798\) 0 0
\(799\) 3.61285 0.127813
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −42.0642 −1.48441
\(804\) 0 0
\(805\) −24.4068 −0.860228
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 36.9590 1.29941 0.649704 0.760187i \(-0.274893\pi\)
0.649704 + 0.760187i \(0.274893\pi\)
\(810\) 0 0
\(811\) 8.54909 0.300199 0.150099 0.988671i \(-0.452041\pi\)
0.150099 + 0.988671i \(0.452041\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 9.81532 0.343816
\(816\) 0 0
\(817\) 33.7146 1.17952
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −2.07007 −0.0722459 −0.0361229 0.999347i \(-0.511501\pi\)
−0.0361229 + 0.999347i \(0.511501\pi\)
\(822\) 0 0
\(823\) −40.5531 −1.41359 −0.706796 0.707417i \(-0.749860\pi\)
−0.706796 + 0.707417i \(0.749860\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −36.4987 −1.26918 −0.634592 0.772847i \(-0.718832\pi\)
−0.634592 + 0.772847i \(0.718832\pi\)
\(828\) 0 0
\(829\) 40.8385 1.41838 0.709191 0.705017i \(-0.249061\pi\)
0.709191 + 0.705017i \(0.249061\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −25.2257 −0.874019
\(834\) 0 0
\(835\) 4.55310 0.157567
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1.72345 0.0595001 0.0297500 0.999557i \(-0.490529\pi\)
0.0297500 + 0.999557i \(0.490529\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −0.622216 −0.0214049
\(846\) 0 0
\(847\) 100.595 3.45647
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 6.69228 0.229409
\(852\) 0 0
\(853\) 29.4924 1.00980 0.504900 0.863178i \(-0.331529\pi\)
0.504900 + 0.863178i \(0.331529\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −8.01874 −0.273915 −0.136957 0.990577i \(-0.543732\pi\)
−0.136957 + 0.990577i \(0.543732\pi\)
\(858\) 0 0
\(859\) 55.4291 1.89122 0.945609 0.325307i \(-0.105468\pi\)
0.945609 + 0.325307i \(0.105468\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −25.6227 −0.872207 −0.436103 0.899897i \(-0.643642\pi\)
−0.436103 + 0.899897i \(0.643642\pi\)
\(864\) 0 0
\(865\) 4.67355 0.158905
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 69.6771 2.36363
\(870\) 0 0
\(871\) 8.04149 0.272475
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 26.4889 0.895487
\(876\) 0 0
\(877\) 28.1847 0.951729 0.475865 0.879519i \(-0.342135\pi\)
0.475865 + 0.879519i \(0.342135\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 9.52987 0.321070 0.160535 0.987030i \(-0.448678\pi\)
0.160535 + 0.987030i \(0.448678\pi\)
\(882\) 0 0
\(883\) 17.8350 0.600196 0.300098 0.953908i \(-0.402981\pi\)
0.300098 + 0.953908i \(0.402981\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −7.73329 −0.259659 −0.129829 0.991536i \(-0.541443\pi\)
−0.129829 + 0.991536i \(0.541443\pi\)
\(888\) 0 0
\(889\) −92.3694 −3.09797
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 8.00000 0.267710
\(894\) 0 0
\(895\) 9.18115 0.306892
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 14.3684 0.479214
\(900\) 0 0
\(901\) 9.51114 0.316862
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2.55215 −0.0848363
\(906\) 0 0
\(907\) −36.0830 −1.19812 −0.599058 0.800706i \(-0.704458\pi\)
−0.599058 + 0.800706i \(0.704458\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 58.9215 1.95216 0.976078 0.217418i \(-0.0697636\pi\)
0.976078 + 0.217418i \(0.0697636\pi\)
\(912\) 0 0
\(913\) −17.7146 −0.586266
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 45.9180 1.51635
\(918\) 0 0
\(919\) 12.5334 0.413439 0.206720 0.978400i \(-0.433721\pi\)
0.206720 + 0.978400i \(0.433721\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 7.05086 0.232082
\(924\) 0 0
\(925\) −3.48532 −0.114597
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 45.0291 1.47736 0.738678 0.674059i \(-0.235451\pi\)
0.738678 + 0.674059i \(0.235451\pi\)
\(930\) 0 0
\(931\) −55.8578 −1.83066
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −7.22570 −0.236306
\(936\) 0 0
\(937\) 0.876015 0.0286182 0.0143091 0.999898i \(-0.495445\pi\)
0.0143091 + 0.999898i \(0.495445\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −0.888922 −0.0289780 −0.0144890 0.999895i \(-0.504612\pi\)
−0.0144890 + 0.999895i \(0.504612\pi\)
\(942\) 0 0
\(943\) −29.9180 −0.974263
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −43.9911 −1.42952 −0.714759 0.699370i \(-0.753464\pi\)
−0.714759 + 0.699370i \(0.753464\pi\)
\(948\) 0 0
\(949\) 7.24443 0.235164
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 18.4701 0.598306 0.299153 0.954205i \(-0.403296\pi\)
0.299153 + 0.954205i \(0.403296\pi\)
\(954\) 0 0
\(955\) −0.165949 −0.00536998
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 87.8992 2.83841
\(960\) 0 0
\(961\) 20.6128 0.664931
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 16.3042 0.524850
\(966\) 0 0
\(967\) 50.2895 1.61720 0.808600 0.588359i \(-0.200226\pi\)
0.808600 + 0.588359i \(0.200226\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 35.2257 1.13045 0.565223 0.824938i \(-0.308790\pi\)
0.565223 + 0.824938i \(0.308790\pi\)
\(972\) 0 0
\(973\) 29.9180 0.959126
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 58.8069 1.88140 0.940700 0.339240i \(-0.110170\pi\)
0.940700 + 0.339240i \(0.110170\pi\)
\(978\) 0 0
\(979\) −10.8385 −0.346401
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −47.2168 −1.50598 −0.752991 0.658031i \(-0.771390\pi\)
−0.752991 + 0.658031i \(0.771390\pi\)
\(984\) 0 0
\(985\) 9.86082 0.314192
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −67.4291 −2.14412
\(990\) 0 0
\(991\) −35.4924 −1.12745 −0.563727 0.825961i \(-0.690633\pi\)
−0.563727 + 0.825961i \(0.690633\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −6.45139 −0.204523
\(996\) 0 0
\(997\) 49.4291 1.56544 0.782718 0.622377i \(-0.213833\pi\)
0.782718 + 0.622377i \(0.213833\pi\)
\(998\) 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 3744.2.a.z.1.2 3
3.2 odd 2 1248.2.a.p.1.2 yes 3
4.3 odd 2 3744.2.a.ba.1.2 3
8.3 odd 2 7488.2.a.cx.1.2 3
8.5 even 2 7488.2.a.cy.1.2 3
12.11 even 2 1248.2.a.o.1.2 3
24.5 odd 2 2496.2.a.bk.1.2 3
24.11 even 2 2496.2.a.bl.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1248.2.a.o.1.2 3 12.11 even 2
1248.2.a.p.1.2 yes 3 3.2 odd 2
2496.2.a.bk.1.2 3 24.5 odd 2
2496.2.a.bl.1.2 3 24.11 even 2
3744.2.a.z.1.2 3 1.1 even 1 trivial
3744.2.a.ba.1.2 3 4.3 odd 2
7488.2.a.cx.1.2 3 8.3 odd 2
7488.2.a.cy.1.2 3 8.5 even 2