Properties

Label 3744.2.a.be.1.1
Level $3744$
Weight $2$
Character 3744.1
Self dual yes
Analytic conductor $29.896$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.13448.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 7x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 416)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.58874\) 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)
 
\(f(q)\) \(=\) \(q-2.70156 q^{5} -3.36131 q^{7} +O(q^{10})\) \(q-2.70156 q^{5} -3.36131 q^{7} -5.17748 q^{11} +1.00000 q^{13} -6.70156 q^{17} -5.17748 q^{19} +2.29844 q^{25} +2.00000 q^{29} -8.80980 q^{31} +9.08080 q^{35} +2.70156 q^{37} -3.40312 q^{41} +8.53879 q^{43} +3.36131 q^{47} +4.29844 q^{49} +11.4031 q^{53} +13.9873 q^{55} +2.08717 q^{59} -3.40312 q^{61} -2.70156 q^{65} -12.4421 q^{67} -10.6260 q^{71} -6.00000 q^{73} +17.4031 q^{77} +3.09031 q^{79} +1.54515 q^{83} +18.1047 q^{85} +6.00000 q^{89} -3.36131 q^{91} +13.9873 q^{95} -16.8062 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{5} + 4 q^{13} - 14 q^{17} + 22 q^{25} + 8 q^{29} - 2 q^{37} + 12 q^{41} + 30 q^{49} + 20 q^{53} + 12 q^{61} + 2 q^{65} - 24 q^{73} + 44 q^{77} + 34 q^{85} + 24 q^{89} - 16 q^{97}+O(q^{100}) \) 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
\(5\) −2.70156 −1.20818 −0.604088 0.796918i \(-0.706462\pi\)
−0.604088 + 0.796918i \(0.706462\pi\)
\(6\) 0 0
\(7\) −3.36131 −1.27046 −0.635229 0.772324i \(-0.719094\pi\)
−0.635229 + 0.772324i \(0.719094\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.17748 −1.56107 −0.780534 0.625114i \(-0.785053\pi\)
−0.780534 + 0.625114i \(0.785053\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.70156 −1.62537 −0.812684 0.582705i \(-0.801994\pi\)
−0.812684 + 0.582705i \(0.801994\pi\)
\(18\) 0 0
\(19\) −5.17748 −1.18779 −0.593897 0.804541i \(-0.702411\pi\)
−0.593897 + 0.804541i \(0.702411\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 2.29844 0.459688
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −8.80980 −1.58229 −0.791143 0.611631i \(-0.790514\pi\)
−0.791143 + 0.611631i \(0.790514\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 9.08080 1.53494
\(36\) 0 0
\(37\) 2.70156 0.444134 0.222067 0.975031i \(-0.428720\pi\)
0.222067 + 0.975031i \(0.428720\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.40312 −0.531479 −0.265739 0.964045i \(-0.585616\pi\)
−0.265739 + 0.964045i \(0.585616\pi\)
\(42\) 0 0
\(43\) 8.53879 1.30215 0.651077 0.759012i \(-0.274318\pi\)
0.651077 + 0.759012i \(0.274318\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.36131 0.490298 0.245149 0.969485i \(-0.421163\pi\)
0.245149 + 0.969485i \(0.421163\pi\)
\(48\) 0 0
\(49\) 4.29844 0.614063
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 11.4031 1.56634 0.783170 0.621808i \(-0.213602\pi\)
0.783170 + 0.621808i \(0.213602\pi\)
\(54\) 0 0
\(55\) 13.9873 1.88604
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.08717 0.271726 0.135863 0.990728i \(-0.456619\pi\)
0.135863 + 0.990728i \(0.456619\pi\)
\(60\) 0 0
\(61\) −3.40312 −0.435725 −0.217863 0.975979i \(-0.569909\pi\)
−0.217863 + 0.975979i \(0.569909\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.70156 −0.335088
\(66\) 0 0
\(67\) −12.4421 −1.52005 −0.760023 0.649896i \(-0.774812\pi\)
−0.760023 + 0.649896i \(0.774812\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −10.6260 −1.26107 −0.630534 0.776161i \(-0.717164\pi\)
−0.630534 + 0.776161i \(0.717164\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 17.4031 1.98327
\(78\) 0 0
\(79\) 3.09031 0.347687 0.173843 0.984773i \(-0.444381\pi\)
0.173843 + 0.984773i \(0.444381\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.54515 0.169603 0.0848014 0.996398i \(-0.472974\pi\)
0.0848014 + 0.996398i \(0.472974\pi\)
\(84\) 0 0
\(85\) 18.1047 1.96373
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −3.36131 −0.352362
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 13.9873 1.43506
\(96\) 0 0
\(97\) −16.8062 −1.70642 −0.853208 0.521571i \(-0.825346\pi\)
−0.853208 + 0.521571i \(0.825346\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −7.40312 −0.736638 −0.368319 0.929699i \(-0.620067\pi\)
−0.368319 + 0.929699i \(0.620067\pi\)
\(102\) 0 0
\(103\) 7.26464 0.715806 0.357903 0.933759i \(-0.383492\pi\)
0.357903 + 0.933759i \(0.383492\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.09031 0.298751 0.149376 0.988781i \(-0.452274\pi\)
0.149376 + 0.988781i \(0.452274\pi\)
\(108\) 0 0
\(109\) 10.7016 1.02502 0.512512 0.858680i \(-0.328715\pi\)
0.512512 + 0.858680i \(0.328715\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 22.5261 2.06496
\(120\) 0 0
\(121\) 15.8062 1.43693
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 7.29844 0.652792
\(126\) 0 0
\(127\) 13.4453 1.19307 0.596537 0.802586i \(-0.296543\pi\)
0.596537 + 0.802586i \(0.296543\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −18.8937 −1.65075 −0.825377 0.564582i \(-0.809037\pi\)
−0.825377 + 0.564582i \(0.809037\pi\)
\(132\) 0 0
\(133\) 17.4031 1.50904
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 18.2094 1.55573 0.777866 0.628430i \(-0.216302\pi\)
0.777866 + 0.628430i \(0.216302\pi\)
\(138\) 0 0
\(139\) −9.08080 −0.770223 −0.385112 0.922870i \(-0.625837\pi\)
−0.385112 + 0.922870i \(0.625837\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −5.17748 −0.432962
\(144\) 0 0
\(145\) −5.40312 −0.448705
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 12.8062 1.04913 0.524564 0.851371i \(-0.324228\pi\)
0.524564 + 0.851371i \(0.324228\pi\)
\(150\) 0 0
\(151\) −6.99364 −0.569134 −0.284567 0.958656i \(-0.591850\pi\)
−0.284567 + 0.958656i \(0.591850\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 23.8002 1.91168
\(156\) 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
\(161\) 0 0
\(162\) 0 0
\(163\) −11.9001 −0.932088 −0.466044 0.884762i \(-0.654321\pi\)
−0.466044 + 0.884762i \(0.654321\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −22.2551 −1.72215 −0.861074 0.508480i \(-0.830208\pi\)
−0.861074 + 0.508480i \(0.830208\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 3.40312 0.258735 0.129367 0.991597i \(-0.458705\pi\)
0.129367 + 0.991597i \(0.458705\pi\)
\(174\) 0 0
\(175\) −7.72577 −0.584014
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −8.53879 −0.638219 −0.319110 0.947718i \(-0.603384\pi\)
−0.319110 + 0.947718i \(0.603384\pi\)
\(180\) 0 0
\(181\) 15.4031 1.14491 0.572453 0.819938i \(-0.305992\pi\)
0.572453 + 0.819938i \(0.305992\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −7.29844 −0.536592
\(186\) 0 0
\(187\) 34.6972 2.53731
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −13.9873 −1.01208 −0.506042 0.862509i \(-0.668892\pi\)
−0.506042 + 0.862509i \(0.668892\pi\)
\(192\) 0 0
\(193\) 8.59688 0.618817 0.309408 0.950929i \(-0.399869\pi\)
0.309408 + 0.950929i \(0.399869\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 22.7016 1.61742 0.808710 0.588208i \(-0.200166\pi\)
0.808710 + 0.588208i \(0.200166\pi\)
\(198\) 0 0
\(199\) −10.8970 −0.772465 −0.386233 0.922401i \(-0.626224\pi\)
−0.386233 + 0.922401i \(0.626224\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −6.72263 −0.471836
\(204\) 0 0
\(205\) 9.19375 0.642119
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 26.8062 1.85423
\(210\) 0 0
\(211\) 12.7131 0.875207 0.437604 0.899168i \(-0.355827\pi\)
0.437604 + 0.899168i \(0.355827\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −23.0681 −1.57323
\(216\) 0 0
\(217\) 29.6125 2.01023
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −6.70156 −0.450796
\(222\) 0 0
\(223\) 17.8906 1.19804 0.599021 0.800733i \(-0.295556\pi\)
0.599021 + 0.800733i \(0.295556\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 8.80980 0.584727 0.292363 0.956307i \(-0.405558\pi\)
0.292363 + 0.956307i \(0.405558\pi\)
\(228\) 0 0
\(229\) 28.1047 1.85721 0.928605 0.371070i \(-0.121009\pi\)
0.928605 + 0.371070i \(0.121009\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3.89531 −0.255190 −0.127595 0.991826i \(-0.540726\pi\)
−0.127595 + 0.991826i \(0.540726\pi\)
\(234\) 0 0
\(235\) −9.08080 −0.592366
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −14.2583 −0.922291 −0.461146 0.887324i \(-0.652561\pi\)
−0.461146 + 0.887324i \(0.652561\pi\)
\(240\) 0 0
\(241\) 20.8062 1.34025 0.670124 0.742249i \(-0.266241\pi\)
0.670124 + 0.742249i \(0.266241\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −11.6125 −0.741895
\(246\) 0 0
\(247\) −5.17748 −0.329435
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 13.4453 0.848657 0.424329 0.905508i \(-0.360510\pi\)
0.424329 + 0.905508i \(0.360510\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −17.5078 −1.09211 −0.546054 0.837750i \(-0.683871\pi\)
−0.546054 + 0.837750i \(0.683871\pi\)
\(258\) 0 0
\(259\) −9.08080 −0.564254
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −10.3550 −0.638514 −0.319257 0.947668i \(-0.603433\pi\)
−0.319257 + 0.947668i \(0.603433\pi\)
\(264\) 0 0
\(265\) −30.8062 −1.89241
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −26.2094 −1.59801 −0.799007 0.601322i \(-0.794641\pi\)
−0.799007 + 0.601322i \(0.794641\pi\)
\(270\) 0 0
\(271\) −3.36131 −0.204185 −0.102093 0.994775i \(-0.532554\pi\)
−0.102093 + 0.994775i \(0.532554\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −11.9001 −0.717603
\(276\) 0 0
\(277\) −14.2094 −0.853758 −0.426879 0.904309i \(-0.640387\pi\)
−0.426879 + 0.904309i \(0.640387\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 11.4390 0.675221
\(288\) 0 0
\(289\) 27.9109 1.64182
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0.104686 0.00611584 0.00305792 0.999995i \(-0.499027\pi\)
0.00305792 + 0.999995i \(0.499027\pi\)
\(294\) 0 0
\(295\) −5.63861 −0.328292
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −28.7016 −1.65433
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 9.19375 0.526433
\(306\) 0 0
\(307\) −8.26778 −0.471867 −0.235934 0.971769i \(-0.575815\pi\)
−0.235934 + 0.971769i \(0.575815\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 10.8970 0.617910 0.308955 0.951077i \(-0.400021\pi\)
0.308955 + 0.951077i \(0.400021\pi\)
\(312\) 0 0
\(313\) −29.5078 −1.66788 −0.833940 0.551855i \(-0.813920\pi\)
−0.833940 + 0.551855i \(0.813920\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −0.806248 −0.0452834 −0.0226417 0.999744i \(-0.507208\pi\)
−0.0226417 + 0.999744i \(0.507208\pi\)
\(318\) 0 0
\(319\) −10.3550 −0.579766
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 34.6972 1.93060
\(324\) 0 0
\(325\) 2.29844 0.127494
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −11.2984 −0.622903
\(330\) 0 0
\(331\) −14.9904 −0.823948 −0.411974 0.911196i \(-0.635160\pi\)
−0.411974 + 0.911196i \(0.635160\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 33.6131 1.83648
\(336\) 0 0
\(337\) 16.1047 0.877278 0.438639 0.898663i \(-0.355461\pi\)
0.438639 + 0.898663i \(0.355461\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 45.6125 2.47006
\(342\) 0 0
\(343\) 9.08080 0.490317
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −26.1584 −1.40426 −0.702128 0.712051i \(-0.747766\pi\)
−0.702128 + 0.712051i \(0.747766\pi\)
\(348\) 0 0
\(349\) −10.9109 −0.584049 −0.292024 0.956411i \(-0.594329\pi\)
−0.292024 + 0.956411i \(0.594329\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −27.4031 −1.45852 −0.729261 0.684236i \(-0.760136\pi\)
−0.729261 + 0.684236i \(0.760136\pi\)
\(354\) 0 0
\(355\) 28.7067 1.52359
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 19.1647 1.01148 0.505738 0.862687i \(-0.331220\pi\)
0.505738 + 0.862687i \(0.331220\pi\)
\(360\) 0 0
\(361\) 7.80625 0.410855
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 16.2094 0.848437
\(366\) 0 0
\(367\) −28.5166 −1.48855 −0.744276 0.667872i \(-0.767205\pi\)
−0.744276 + 0.667872i \(0.767205\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −38.3295 −1.98997
\(372\) 0 0
\(373\) −22.2094 −1.14996 −0.574979 0.818168i \(-0.694990\pi\)
−0.574979 + 0.818168i \(0.694990\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.00000 0.103005
\(378\) 0 0
\(379\) 33.6940 1.73075 0.865373 0.501128i \(-0.167082\pi\)
0.865373 + 0.501128i \(0.167082\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −24.0712 −1.22998 −0.614991 0.788534i \(-0.710840\pi\)
−0.614991 + 0.788534i \(0.710840\pi\)
\(384\) 0 0
\(385\) −47.0156 −2.39614
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 7.19375 0.364738 0.182369 0.983230i \(-0.441624\pi\)
0.182369 + 0.983230i \(0.441624\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −8.34866 −0.420067
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −14.2094 −0.709582 −0.354791 0.934946i \(-0.615448\pi\)
−0.354791 + 0.934946i \(0.615448\pi\)
\(402\) 0 0
\(403\) −8.80980 −0.438847
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −13.9873 −0.693323
\(408\) 0 0
\(409\) 3.40312 0.168274 0.0841368 0.996454i \(-0.473187\pi\)
0.0841368 + 0.996454i \(0.473187\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −7.01562 −0.345216
\(414\) 0 0
\(415\) −4.17433 −0.204910
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 4.36446 0.213218 0.106609 0.994301i \(-0.466001\pi\)
0.106609 + 0.994301i \(0.466001\pi\)
\(420\) 0 0
\(421\) −0.104686 −0.00510210 −0.00255105 0.999997i \(-0.500812\pi\)
−0.00255105 + 0.999997i \(0.500812\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −15.4031 −0.747161
\(426\) 0 0
\(427\) 11.4390 0.553571
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 6.45162 0.310764 0.155382 0.987854i \(-0.450339\pi\)
0.155382 + 0.987854i \(0.450339\pi\)
\(432\) 0 0
\(433\) 24.1047 1.15840 0.579199 0.815186i \(-0.303366\pi\)
0.579199 + 0.815186i \(0.303366\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 30.5228 1.45678 0.728388 0.685165i \(-0.240270\pi\)
0.728388 + 0.685165i \(0.240270\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 39.6036 1.88163 0.940813 0.338926i \(-0.110064\pi\)
0.940813 + 0.338926i \(0.110064\pi\)
\(444\) 0 0
\(445\) −16.2094 −0.768398
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0.806248 0.0380492 0.0190246 0.999819i \(-0.493944\pi\)
0.0190246 + 0.999819i \(0.493944\pi\)
\(450\) 0 0
\(451\) 17.6196 0.829674
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 9.08080 0.425715
\(456\) 0 0
\(457\) −24.8062 −1.16039 −0.580194 0.814479i \(-0.697023\pi\)
−0.580194 + 0.814479i \(0.697023\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 5.29844 0.246773 0.123386 0.992359i \(-0.460625\pi\)
0.123386 + 0.992359i \(0.460625\pi\)
\(462\) 0 0
\(463\) −12.9841 −0.603424 −0.301712 0.953399i \(-0.597558\pi\)
−0.301712 + 0.953399i \(0.597558\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −6.18062 −0.286005 −0.143002 0.989722i \(-0.545676\pi\)
−0.143002 + 0.989722i \(0.545676\pi\)
\(468\) 0 0
\(469\) 41.8219 1.93115
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −44.2094 −2.03275
\(474\) 0 0
\(475\) −11.9001 −0.546014
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −2.81930 −0.128817 −0.0644086 0.997924i \(-0.520516\pi\)
−0.0644086 + 0.997924i \(0.520516\pi\)
\(480\) 0 0
\(481\) 2.70156 0.123181
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 45.4031 2.06165
\(486\) 0 0
\(487\) 19.1647 0.868438 0.434219 0.900807i \(-0.357024\pi\)
0.434219 + 0.900807i \(0.357024\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −14.7194 −0.664278 −0.332139 0.943231i \(-0.607770\pi\)
−0.332139 + 0.943231i \(0.607770\pi\)
\(492\) 0 0
\(493\) −13.4031 −0.603646
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 35.7172 1.60213
\(498\) 0 0
\(499\) 18.0807 0.809404 0.404702 0.914449i \(-0.367375\pi\)
0.404702 + 0.914449i \(0.367375\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −6.72263 −0.299747 −0.149874 0.988705i \(-0.547887\pi\)
−0.149874 + 0.988705i \(0.547887\pi\)
\(504\) 0 0
\(505\) 20.0000 0.889988
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −14.0000 −0.620539 −0.310270 0.950649i \(-0.600419\pi\)
−0.310270 + 0.950649i \(0.600419\pi\)
\(510\) 0 0
\(511\) 20.1679 0.892175
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −19.6259 −0.864820
\(516\) 0 0
\(517\) −17.4031 −0.765389
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 13.5078 0.591788 0.295894 0.955221i \(-0.404382\pi\)
0.295894 + 0.955221i \(0.404382\pi\)
\(522\) 0 0
\(523\) −23.8002 −1.04071 −0.520355 0.853950i \(-0.674200\pi\)
−0.520355 + 0.853950i \(0.674200\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 59.0394 2.57180
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −3.40312 −0.147406
\(534\) 0 0
\(535\) −8.34866 −0.360944
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −22.2551 −0.958593
\(540\) 0 0
\(541\) −14.7016 −0.632070 −0.316035 0.948748i \(-0.602352\pi\)
−0.316035 + 0.948748i \(0.602352\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −28.9109 −1.23841
\(546\) 0 0
\(547\) −12.7131 −0.543574 −0.271787 0.962357i \(-0.587615\pi\)
−0.271787 + 0.962357i \(0.587615\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −10.3550 −0.441136
\(552\) 0 0
\(553\) −10.3875 −0.441722
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1.29844 −0.0550166 −0.0275083 0.999622i \(-0.508757\pi\)
−0.0275083 + 0.999622i \(0.508757\pi\)
\(558\) 0 0
\(559\) 8.53879 0.361152
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 40.1457 1.69194 0.845969 0.533232i \(-0.179023\pi\)
0.845969 + 0.533232i \(0.179023\pi\)
\(564\) 0 0
\(565\) 27.0156 1.13656
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −29.2984 −1.22825 −0.614127 0.789207i \(-0.710492\pi\)
−0.614127 + 0.789207i \(0.710492\pi\)
\(570\) 0 0
\(571\) −8.53879 −0.357337 −0.178669 0.983909i \(-0.557179\pi\)
−0.178669 + 0.983909i \(0.557179\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −32.8062 −1.36574 −0.682871 0.730539i \(-0.739269\pi\)
−0.682871 + 0.730539i \(0.739269\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −5.19375 −0.215473
\(582\) 0 0
\(583\) −59.0394 −2.44516
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −11.9001 −0.491170 −0.245585 0.969375i \(-0.578980\pi\)
−0.245585 + 0.969375i \(0.578980\pi\)
\(588\) 0 0
\(589\) 45.6125 1.87943
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 30.0000 1.23195 0.615976 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 0 0
\(595\) −60.8556 −2.49483
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −7.80665 −0.318971 −0.159486 0.987200i \(-0.550984\pi\)
−0.159486 + 0.987200i \(0.550984\pi\)
\(600\) 0 0
\(601\) 30.7016 1.25234 0.626171 0.779685i \(-0.284621\pi\)
0.626171 + 0.779685i \(0.284621\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −42.7016 −1.73607
\(606\) 0 0
\(607\) −41.9618 −1.70318 −0.851589 0.524211i \(-0.824360\pi\)
−0.851589 + 0.524211i \(0.824360\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3.36131 0.135984
\(612\) 0 0
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −19.4031 −0.781140 −0.390570 0.920573i \(-0.627722\pi\)
−0.390570 + 0.920573i \(0.627722\pi\)
\(618\) 0 0
\(619\) 7.72577 0.310525 0.155263 0.987873i \(-0.450378\pi\)
0.155263 + 0.987873i \(0.450378\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −20.1679 −0.808009
\(624\) 0 0
\(625\) −31.2094 −1.24837
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −18.1047 −0.721881
\(630\) 0 0
\(631\) 41.6908 1.65968 0.829842 0.557998i \(-0.188430\pi\)
0.829842 + 0.557998i \(0.188430\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −36.3232 −1.44144
\(636\) 0 0
\(637\) 4.29844 0.170310
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −7.19375 −0.284136 −0.142068 0.989857i \(-0.545375\pi\)
−0.142068 + 0.989857i \(0.545375\pi\)
\(642\) 0 0
\(643\) 28.9777 1.14277 0.571384 0.820683i \(-0.306407\pi\)
0.571384 + 0.820683i \(0.306407\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 10.8970 0.428404 0.214202 0.976789i \(-0.431285\pi\)
0.214202 + 0.976789i \(0.431285\pi\)
\(648\) 0 0
\(649\) −10.8062 −0.424182
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 17.0156 0.665873 0.332936 0.942949i \(-0.391961\pi\)
0.332936 + 0.942949i \(0.391961\pi\)
\(654\) 0 0
\(655\) 51.0426 1.99440
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 23.8002 0.927125 0.463562 0.886064i \(-0.346571\pi\)
0.463562 + 0.886064i \(0.346571\pi\)
\(660\) 0 0
\(661\) 3.19375 0.124223 0.0621113 0.998069i \(-0.480217\pi\)
0.0621113 + 0.998069i \(0.480217\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −47.0156 −1.82319
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 17.6196 0.680197
\(672\) 0 0
\(673\) 28.3141 1.09143 0.545713 0.837972i \(-0.316259\pi\)
0.545713 + 0.837972i \(0.316259\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −4.59688 −0.176672 −0.0883361 0.996091i \(-0.528155\pi\)
−0.0883361 + 0.996091i \(0.528155\pi\)
\(678\) 0 0
\(679\) 56.4911 2.16793
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −0.461133 −0.0176448 −0.00882238 0.999961i \(-0.502808\pi\)
−0.00882238 + 0.999961i \(0.502808\pi\)
\(684\) 0 0
\(685\) −49.1938 −1.87960
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 11.4031 0.434424
\(690\) 0 0
\(691\) −35.7003 −1.35810 −0.679052 0.734090i \(-0.737609\pi\)
−0.679052 + 0.734090i \(0.737609\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 24.5323 0.930565
\(696\) 0 0
\(697\) 22.8062 0.863848
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −4.59688 −0.173622 −0.0868108 0.996225i \(-0.527668\pi\)
−0.0868108 + 0.996225i \(0.527668\pi\)
\(702\) 0 0
\(703\) −13.9873 −0.527540
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 24.8842 0.935868
\(708\) 0 0
\(709\) −28.8062 −1.08184 −0.540921 0.841074i \(-0.681924\pi\)
−0.540921 + 0.841074i \(0.681924\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 13.9873 0.523094
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 18.1616 0.677313 0.338657 0.940910i \(-0.390027\pi\)
0.338657 + 0.940910i \(0.390027\pi\)
\(720\) 0 0
\(721\) −24.4187 −0.909402
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 4.59688 0.170724
\(726\) 0 0
\(727\) −45.0521 −1.67089 −0.835445 0.549574i \(-0.814790\pi\)
−0.835445 + 0.549574i \(0.814790\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −57.2232 −2.11648
\(732\) 0 0
\(733\) 6.91093 0.255261 0.127631 0.991822i \(-0.459263\pi\)
0.127631 + 0.991822i \(0.459263\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 64.4187 2.37289
\(738\) 0 0
\(739\) 16.0744 0.591308 0.295654 0.955295i \(-0.404463\pi\)
0.295654 + 0.955295i \(0.404463\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −6.99364 −0.256572 −0.128286 0.991737i \(-0.540947\pi\)
−0.128286 + 0.991737i \(0.540947\pi\)
\(744\) 0 0
\(745\) −34.5969 −1.26753
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −10.3875 −0.379551
\(750\) 0 0
\(751\) 29.0586 1.06036 0.530181 0.847884i \(-0.322124\pi\)
0.530181 + 0.847884i \(0.322124\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 18.8937 0.687614
\(756\) 0 0
\(757\) 26.2094 0.952596 0.476298 0.879284i \(-0.341978\pi\)
0.476298 + 0.879284i \(0.341978\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 27.6125 1.00095 0.500476 0.865750i \(-0.333158\pi\)
0.500476 + 0.865750i \(0.333158\pi\)
\(762\) 0 0
\(763\) −35.9713 −1.30225
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2.08717 0.0753632
\(768\) 0 0
\(769\) −26.2094 −0.945134 −0.472567 0.881295i \(-0.656673\pi\)
−0.472567 + 0.881295i \(0.656673\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 16.1047 0.579245 0.289623 0.957141i \(-0.406470\pi\)
0.289623 + 0.957141i \(0.406470\pi\)
\(774\) 0 0
\(775\) −20.2488 −0.727357
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 17.6196 0.631287
\(780\) 0 0
\(781\) 55.0156 1.96861
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 5.40312 0.192846
\(786\) 0 0
\(787\) 29.5197 1.05226 0.526132 0.850403i \(-0.323642\pi\)
0.526132 + 0.850403i \(0.323642\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 33.6131 1.19515
\(792\) 0 0
\(793\) −3.40312 −0.120848
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) −22.5261 −0.796915
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 31.0649 1.09625
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 7.89531 0.277584 0.138792 0.990322i \(-0.455678\pi\)
0.138792 + 0.990322i \(0.455678\pi\)
\(810\) 0 0
\(811\) −42.9650 −1.50870 −0.754352 0.656470i \(-0.772049\pi\)
−0.754352 + 0.656470i \(0.772049\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 32.1489 1.12613
\(816\) 0 0
\(817\) −44.2094 −1.54669
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 18.9109 0.659996 0.329998 0.943982i \(-0.392952\pi\)
0.329998 + 0.943982i \(0.392952\pi\)
\(822\) 0 0
\(823\) −51.2327 −1.78586 −0.892931 0.450194i \(-0.851355\pi\)
−0.892931 + 0.450194i \(0.851355\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −12.4421 −0.432655 −0.216327 0.976321i \(-0.569408\pi\)
−0.216327 + 0.976321i \(0.569408\pi\)
\(828\) 0 0
\(829\) 0.387503 0.0134585 0.00672927 0.999977i \(-0.497858\pi\)
0.00672927 + 0.999977i \(0.497858\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −28.8062 −0.998077
\(834\) 0 0
\(835\) 60.1234 2.08066
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 12.9841 0.448262 0.224131 0.974559i \(-0.428046\pi\)
0.224131 + 0.974559i \(0.428046\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −2.70156 −0.0929366
\(846\) 0 0
\(847\) −53.1298 −1.82556
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −3.89531 −0.133373 −0.0666865 0.997774i \(-0.521243\pi\)
−0.0666865 + 0.997774i \(0.521243\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −12.8062 −0.437453 −0.218727 0.975786i \(-0.570190\pi\)
−0.218727 + 0.975786i \(0.570190\pi\)
\(858\) 0 0
\(859\) 3.09031 0.105440 0.0527200 0.998609i \(-0.483211\pi\)
0.0527200 + 0.998609i \(0.483211\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −13.1743 −0.448457 −0.224228 0.974537i \(-0.571986\pi\)
−0.224228 + 0.974537i \(0.571986\pi\)
\(864\) 0 0
\(865\) −9.19375 −0.312597
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) −12.4421 −0.421585
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −24.5323 −0.829345
\(876\) 0 0
\(877\) 25.7172 0.868408 0.434204 0.900815i \(-0.357030\pi\)
0.434204 + 0.900815i \(0.357030\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 49.2984 1.66091 0.830453 0.557088i \(-0.188082\pi\)
0.830453 + 0.557088i \(0.188082\pi\)
\(882\) 0 0
\(883\) 11.0871 0.373110 0.186555 0.982444i \(-0.440268\pi\)
0.186555 + 0.982444i \(0.440268\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 23.8002 0.799133 0.399566 0.916704i \(-0.369161\pi\)
0.399566 + 0.916704i \(0.369161\pi\)
\(888\) 0 0
\(889\) −45.1938 −1.51575
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −17.4031 −0.582373
\(894\) 0 0
\(895\) 23.0681 0.771081
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −17.6196 −0.587646
\(900\) 0 0
\(901\) −76.4187 −2.54588
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −41.6125 −1.38325
\(906\) 0 0
\(907\) 1.27415 0.0423074 0.0211537 0.999776i \(-0.493266\pi\)
0.0211537 + 0.999776i \(0.493266\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −20.1679 −0.668192 −0.334096 0.942539i \(-0.608431\pi\)
−0.334096 + 0.942539i \(0.608431\pi\)
\(912\) 0 0
\(913\) −8.00000 −0.264761
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 63.5078 2.09721
\(918\) 0 0
\(919\) 3.09031 0.101940 0.0509700 0.998700i \(-0.483769\pi\)
0.0509700 + 0.998700i \(0.483769\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −10.6260 −0.349758
\(924\) 0 0
\(925\) 6.20937 0.204163
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 7.40312 0.242889 0.121444 0.992598i \(-0.461247\pi\)
0.121444 + 0.992598i \(0.461247\pi\)
\(930\) 0 0
\(931\) −22.2551 −0.729380
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −93.7366 −3.06551
\(936\) 0 0
\(937\) 7.19375 0.235010 0.117505 0.993072i \(-0.462510\pi\)
0.117505 + 0.993072i \(0.462510\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −36.1047 −1.17698 −0.588490 0.808505i \(-0.700277\pi\)
−0.588490 + 0.808505i \(0.700277\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −40.4167 −1.31336 −0.656682 0.754167i \(-0.728041\pi\)
−0.656682 + 0.754167i \(0.728041\pi\)
\(948\) 0 0
\(949\) −6.00000 −0.194768
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 8.31406 0.269319 0.134659 0.990892i \(-0.457006\pi\)
0.134659 + 0.990892i \(0.457006\pi\)
\(954\) 0 0
\(955\) 37.7875 1.22277
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −61.2074 −1.97649
\(960\) 0 0
\(961\) 46.6125 1.50363
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −23.2250 −0.747639
\(966\) 0 0
\(967\) −17.3486 −0.557893 −0.278946 0.960307i \(-0.589985\pi\)
−0.278946 + 0.960307i \(0.589985\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 33.4230 1.07260 0.536298 0.844029i \(-0.319822\pi\)
0.536298 + 0.844029i \(0.319822\pi\)
\(972\) 0 0
\(973\) 30.5234 0.978536
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) −31.0649 −0.992837
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 11.7100 0.373490 0.186745 0.982408i \(-0.440206\pi\)
0.186745 + 0.982408i \(0.440206\pi\)
\(984\) 0 0
\(985\) −61.3297 −1.95413
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0.542011 0.0172175 0.00860877 0.999963i \(-0.497260\pi\)
0.00860877 + 0.999963i \(0.497260\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 29.4388 0.933273
\(996\) 0 0
\(997\) −20.8062 −0.658941 −0.329470 0.944166i \(-0.606870\pi\)
−0.329470 + 0.944166i \(0.606870\pi\)
\(998\) 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.be.1.1 4
3.2 odd 2 416.2.a.f.1.3 yes 4
4.3 odd 2 inner 3744.2.a.be.1.2 4
8.3 odd 2 7488.2.a.da.1.4 4
8.5 even 2 7488.2.a.da.1.3 4
12.11 even 2 416.2.a.f.1.2 4
24.5 odd 2 832.2.a.p.1.2 4
24.11 even 2 832.2.a.p.1.3 4
39.38 odd 2 5408.2.a.bj.1.3 4
48.5 odd 4 3328.2.b.bb.1665.4 8
48.11 even 4 3328.2.b.bb.1665.6 8
48.29 odd 4 3328.2.b.bb.1665.5 8
48.35 even 4 3328.2.b.bb.1665.3 8
156.155 even 2 5408.2.a.bj.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
416.2.a.f.1.2 4 12.11 even 2
416.2.a.f.1.3 yes 4 3.2 odd 2
832.2.a.p.1.2 4 24.5 odd 2
832.2.a.p.1.3 4 24.11 even 2
3328.2.b.bb.1665.3 8 48.35 even 4
3328.2.b.bb.1665.4 8 48.5 odd 4
3328.2.b.bb.1665.5 8 48.29 odd 4
3328.2.b.bb.1665.6 8 48.11 even 4
3744.2.a.be.1.1 4 1.1 even 1 trivial
3744.2.a.be.1.2 4 4.3 odd 2 inner
5408.2.a.bj.1.2 4 156.155 even 2
5408.2.a.bj.1.3 4 39.38 odd 2
7488.2.a.da.1.3 4 8.5 even 2
7488.2.a.da.1.4 4 8.3 odd 2