Properties

Label 3744.2.a.bb.1.3
Level $3744$
Weight $2$
Character 3744.1
Self dual yes
Analytic conductor $29.896$
Analytic rank $1$
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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{20})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.90211\) 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)
 
\(f(q)\) \(=\) \(q+1.23607 q^{5} -2.35114 q^{7} +O(q^{10})\) \(q+1.23607 q^{5} -2.35114 q^{7} +6.15537 q^{11} +1.00000 q^{13} -6.47214 q^{17} -5.25731 q^{19} -3.47214 q^{25} -4.00000 q^{29} -2.35114 q^{31} -2.90617 q^{35} -10.9443 q^{37} -5.23607 q^{41} +7.60845 q^{43} -9.06154 q^{47} -1.47214 q^{49} +4.94427 q^{53} +7.60845 q^{55} +9.06154 q^{59} +4.47214 q^{61} +1.23607 q^{65} +14.6619 q^{67} +6.15537 q^{71} +6.00000 q^{73} -14.4721 q^{77} -15.2169 q^{79} -3.24920 q^{83} -8.00000 q^{85} -15.7082 q^{89} -2.35114 q^{91} -6.49839 q^{95} +10.9443 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{5} + 4 q^{13} - 8 q^{17} + 4 q^{25} - 16 q^{29} - 8 q^{37} - 12 q^{41} + 12 q^{49} - 16 q^{53} - 4 q^{65} + 24 q^{73} - 40 q^{77} - 32 q^{85} - 36 q^{89} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.23607 0.552786 0.276393 0.961045i \(-0.410861\pi\)
0.276393 + 0.961045i \(0.410861\pi\)
\(6\) 0 0
\(7\) −2.35114 −0.888648 −0.444324 0.895866i \(-0.646556\pi\)
−0.444324 + 0.895866i \(0.646556\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 6.15537 1.85591 0.927957 0.372689i \(-0.121564\pi\)
0.927957 + 0.372689i \(0.121564\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.47214 −1.56972 −0.784862 0.619671i \(-0.787266\pi\)
−0.784862 + 0.619671i \(0.787266\pi\)
\(18\) 0 0
\(19\) −5.25731 −1.20611 −0.603055 0.797700i \(-0.706050\pi\)
−0.603055 + 0.797700i \(0.706050\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −3.47214 −0.694427
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) −2.35114 −0.422277 −0.211139 0.977456i \(-0.567717\pi\)
−0.211139 + 0.977456i \(0.567717\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.90617 −0.491232
\(36\) 0 0
\(37\) −10.9443 −1.79923 −0.899614 0.436687i \(-0.856152\pi\)
−0.899614 + 0.436687i \(0.856152\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −5.23607 −0.817736 −0.408868 0.912593i \(-0.634076\pi\)
−0.408868 + 0.912593i \(0.634076\pi\)
\(42\) 0 0
\(43\) 7.60845 1.16028 0.580139 0.814517i \(-0.302998\pi\)
0.580139 + 0.814517i \(0.302998\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −9.06154 −1.32176 −0.660881 0.750491i \(-0.729817\pi\)
−0.660881 + 0.750491i \(0.729817\pi\)
\(48\) 0 0
\(49\) −1.47214 −0.210305
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.94427 0.679148 0.339574 0.940579i \(-0.389717\pi\)
0.339574 + 0.940579i \(0.389717\pi\)
\(54\) 0 0
\(55\) 7.60845 1.02592
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 9.06154 1.17971 0.589856 0.807509i \(-0.299185\pi\)
0.589856 + 0.807509i \(0.299185\pi\)
\(60\) 0 0
\(61\) 4.47214 0.572598 0.286299 0.958140i \(-0.407575\pi\)
0.286299 + 0.958140i \(0.407575\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.23607 0.153315
\(66\) 0 0
\(67\) 14.6619 1.79123 0.895617 0.444827i \(-0.146735\pi\)
0.895617 + 0.444827i \(0.146735\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.15537 0.730508 0.365254 0.930908i \(-0.380982\pi\)
0.365254 + 0.930908i \(0.380982\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −14.4721 −1.64925
\(78\) 0 0
\(79\) −15.2169 −1.71204 −0.856018 0.516946i \(-0.827069\pi\)
−0.856018 + 0.516946i \(0.827069\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.24920 −0.356646 −0.178323 0.983972i \(-0.557067\pi\)
−0.178323 + 0.983972i \(0.557067\pi\)
\(84\) 0 0
\(85\) −8.00000 −0.867722
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −15.7082 −1.66507 −0.832533 0.553975i \(-0.813110\pi\)
−0.832533 + 0.553975i \(0.813110\pi\)
\(90\) 0 0
\(91\) −2.35114 −0.246467
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −6.49839 −0.666721
\(96\) 0 0
\(97\) 10.9443 1.11122 0.555611 0.831442i \(-0.312484\pi\)
0.555611 + 0.831442i \(0.312484\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −5.52786 −0.550043 −0.275022 0.961438i \(-0.588685\pi\)
−0.275022 + 0.961438i \(0.588685\pi\)
\(102\) 0 0
\(103\) 17.0130 1.67634 0.838171 0.545407i \(-0.183625\pi\)
0.838171 + 0.545407i \(0.183625\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −15.2169 −1.47107 −0.735537 0.677485i \(-0.763070\pi\)
−0.735537 + 0.677485i \(0.763070\pi\)
\(108\) 0 0
\(109\) −14.9443 −1.43140 −0.715701 0.698407i \(-0.753893\pi\)
−0.715701 + 0.698407i \(0.753893\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −4.00000 −0.376288 −0.188144 0.982141i \(-0.560247\pi\)
−0.188144 + 0.982141i \(0.560247\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 15.2169 1.39493
\(120\) 0 0
\(121\) 26.8885 2.44441
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −10.4721 −0.936656
\(126\) 0 0
\(127\) −4.70228 −0.417260 −0.208630 0.977995i \(-0.566900\pi\)
−0.208630 + 0.977995i \(0.566900\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −9.40456 −0.821681 −0.410840 0.911707i \(-0.634765\pi\)
−0.410840 + 0.911707i \(0.634765\pi\)
\(132\) 0 0
\(133\) 12.3607 1.07181
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −0.291796 −0.0249298 −0.0124649 0.999922i \(-0.503968\pi\)
−0.0124649 + 0.999922i \(0.503968\pi\)
\(138\) 0 0
\(139\) 2.90617 0.246498 0.123249 0.992376i \(-0.460669\pi\)
0.123249 + 0.992376i \(0.460669\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.15537 0.514738
\(144\) 0 0
\(145\) −4.94427 −0.410599
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −17.2361 −1.41203 −0.706017 0.708195i \(-0.749510\pi\)
−0.706017 + 0.708195i \(0.749510\pi\)
\(150\) 0 0
\(151\) −22.2703 −1.81233 −0.906167 0.422921i \(-0.861005\pi\)
−0.906167 + 0.422921i \(0.861005\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.90617 −0.233429
\(156\) 0 0
\(157\) 5.41641 0.432276 0.216138 0.976363i \(-0.430654\pi\)
0.216138 + 0.976363i \(0.430654\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −9.95959 −0.780096 −0.390048 0.920795i \(-0.627542\pi\)
−0.390048 + 0.920795i \(0.627542\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −9.06154 −0.701203 −0.350601 0.936525i \(-0.614023\pi\)
−0.350601 + 0.936525i \(0.614023\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −10.4721 −0.796182 −0.398091 0.917346i \(-0.630327\pi\)
−0.398091 + 0.917346i \(0.630327\pi\)
\(174\) 0 0
\(175\) 8.16348 0.617101
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −21.7153 −1.62308 −0.811539 0.584299i \(-0.801370\pi\)
−0.811539 + 0.584299i \(0.801370\pi\)
\(180\) 0 0
\(181\) −16.4721 −1.22436 −0.612182 0.790717i \(-0.709708\pi\)
−0.612182 + 0.790717i \(0.709708\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −13.5279 −0.994588
\(186\) 0 0
\(187\) −39.8384 −2.91327
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 6.49839 0.470207 0.235104 0.971970i \(-0.424457\pi\)
0.235104 + 0.971970i \(0.424457\pi\)
\(192\) 0 0
\(193\) −2.94427 −0.211933 −0.105967 0.994370i \(-0.533794\pi\)
−0.105967 + 0.994370i \(0.533794\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 14.1803 1.01031 0.505154 0.863029i \(-0.331436\pi\)
0.505154 + 0.863029i \(0.331436\pi\)
\(198\) 0 0
\(199\) 14.1068 1.00001 0.500004 0.866023i \(-0.333332\pi\)
0.500004 + 0.866023i \(0.333332\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 9.40456 0.660071
\(204\) 0 0
\(205\) −6.47214 −0.452034
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −32.3607 −2.23844
\(210\) 0 0
\(211\) −5.81234 −0.400138 −0.200069 0.979782i \(-0.564117\pi\)
−0.200069 + 0.979782i \(0.564117\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 9.40456 0.641386
\(216\) 0 0
\(217\) 5.52786 0.375256
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −6.47214 −0.435363
\(222\) 0 0
\(223\) 22.2703 1.49133 0.745666 0.666320i \(-0.232132\pi\)
0.745666 + 0.666320i \(0.232132\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 27.8707 1.84984 0.924921 0.380161i \(-0.124131\pi\)
0.924921 + 0.380161i \(0.124131\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 7.41641 0.485865 0.242933 0.970043i \(-0.421891\pi\)
0.242933 + 0.970043i \(0.421891\pi\)
\(234\) 0 0
\(235\) −11.2007 −0.730652
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −27.8707 −1.80280 −0.901402 0.432984i \(-0.857461\pi\)
−0.901402 + 0.432984i \(0.857461\pi\)
\(240\) 0 0
\(241\) 7.88854 0.508146 0.254073 0.967185i \(-0.418230\pi\)
0.254073 + 0.967185i \(0.418230\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.81966 −0.116254
\(246\) 0 0
\(247\) −5.25731 −0.334515
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 18.1231 1.14392 0.571959 0.820282i \(-0.306184\pi\)
0.571959 + 0.820282i \(0.306184\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −13.5279 −0.843845 −0.421922 0.906632i \(-0.638645\pi\)
−0.421922 + 0.906632i \(0.638645\pi\)
\(258\) 0 0
\(259\) 25.7315 1.59888
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 12.3107 0.759112 0.379556 0.925169i \(-0.376077\pi\)
0.379556 + 0.925169i \(0.376077\pi\)
\(264\) 0 0
\(265\) 6.11146 0.375424
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 20.0000 1.21942 0.609711 0.792624i \(-0.291286\pi\)
0.609711 + 0.792624i \(0.291286\pi\)
\(270\) 0 0
\(271\) −2.35114 −0.142822 −0.0714108 0.997447i \(-0.522750\pi\)
−0.0714108 + 0.997447i \(0.522750\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −21.3723 −1.28880
\(276\) 0 0
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −0.291796 −0.0174071 −0.00870355 0.999962i \(-0.502770\pi\)
−0.00870355 + 0.999962i \(0.502770\pi\)
\(282\) 0 0
\(283\) 14.1068 0.838565 0.419282 0.907856i \(-0.362282\pi\)
0.419282 + 0.907856i \(0.362282\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.3107 0.726680
\(288\) 0 0
\(289\) 24.8885 1.46403
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −19.1246 −1.11727 −0.558636 0.829413i \(-0.688675\pi\)
−0.558636 + 0.829413i \(0.688675\pi\)
\(294\) 0 0
\(295\) 11.2007 0.652129
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −17.8885 −1.03108
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 5.52786 0.316525
\(306\) 0 0
\(307\) 9.95959 0.568424 0.284212 0.958761i \(-0.408268\pi\)
0.284212 + 0.958761i \(0.408268\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 23.8885 1.35026 0.675130 0.737699i \(-0.264087\pi\)
0.675130 + 0.737699i \(0.264087\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −21.5967 −1.21299 −0.606497 0.795086i \(-0.707426\pi\)
−0.606497 + 0.795086i \(0.707426\pi\)
\(318\) 0 0
\(319\) −24.6215 −1.37854
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 34.0260 1.89326
\(324\) 0 0
\(325\) −3.47214 −0.192599
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 21.3050 1.17458
\(330\) 0 0
\(331\) 5.25731 0.288968 0.144484 0.989507i \(-0.453848\pi\)
0.144484 + 0.989507i \(0.453848\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 18.1231 0.990169
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −14.4721 −0.783710
\(342\) 0 0
\(343\) 19.9192 1.07553
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 24.6215 1.32175 0.660875 0.750496i \(-0.270185\pi\)
0.660875 + 0.750496i \(0.270185\pi\)
\(348\) 0 0
\(349\) −7.88854 −0.422264 −0.211132 0.977458i \(-0.567715\pi\)
−0.211132 + 0.977458i \(0.567715\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 18.7639 0.998703 0.499352 0.866399i \(-0.333572\pi\)
0.499352 + 0.866399i \(0.333572\pi\)
\(354\) 0 0
\(355\) 7.60845 0.403815
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −12.6538 −0.667840 −0.333920 0.942601i \(-0.608372\pi\)
−0.333920 + 0.942601i \(0.608372\pi\)
\(360\) 0 0
\(361\) 8.63932 0.454701
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 7.41641 0.388193
\(366\) 0 0
\(367\) 18.1231 0.946017 0.473008 0.881058i \(-0.343168\pi\)
0.473008 + 0.881058i \(0.343168\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −11.6247 −0.603523
\(372\) 0 0
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) 9.95959 0.511590 0.255795 0.966731i \(-0.417663\pi\)
0.255795 + 0.966731i \(0.417663\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −11.9677 −0.611521 −0.305761 0.952108i \(-0.598911\pi\)
−0.305761 + 0.952108i \(0.598911\pi\)
\(384\) 0 0
\(385\) −17.8885 −0.911685
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −14.4721 −0.733766 −0.366883 0.930267i \(-0.619575\pi\)
−0.366883 + 0.930267i \(0.619575\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −18.8091 −0.946390
\(396\) 0 0
\(397\) 24.8328 1.24632 0.623162 0.782093i \(-0.285848\pi\)
0.623162 + 0.782093i \(0.285848\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −31.7082 −1.58343 −0.791716 0.610889i \(-0.790812\pi\)
−0.791716 + 0.610889i \(0.790812\pi\)
\(402\) 0 0
\(403\) −2.35114 −0.117119
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −67.3660 −3.33921
\(408\) 0 0
\(409\) −11.8885 −0.587851 −0.293925 0.955828i \(-0.594962\pi\)
−0.293925 + 0.955828i \(0.594962\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −21.3050 −1.04835
\(414\) 0 0
\(415\) −4.01623 −0.197149
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 21.0292 1.02735 0.513673 0.857986i \(-0.328284\pi\)
0.513673 + 0.857986i \(0.328284\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 22.4721 1.09006
\(426\) 0 0
\(427\) −10.5146 −0.508838
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 12.6538 0.609510 0.304755 0.952431i \(-0.401425\pi\)
0.304755 + 0.952431i \(0.401425\pi\)
\(432\) 0 0
\(433\) −9.41641 −0.452524 −0.226262 0.974067i \(-0.572651\pi\)
−0.226262 + 0.974067i \(0.572651\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −26.4176 −1.26084 −0.630421 0.776253i \(-0.717118\pi\)
−0.630421 + 0.776253i \(0.717118\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −15.2169 −0.722977 −0.361488 0.932377i \(-0.617731\pi\)
−0.361488 + 0.932377i \(0.617731\pi\)
\(444\) 0 0
\(445\) −19.4164 −0.920426
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 26.1803 1.23553 0.617763 0.786364i \(-0.288039\pi\)
0.617763 + 0.786364i \(0.288039\pi\)
\(450\) 0 0
\(451\) −32.2299 −1.51765
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −2.90617 −0.136243
\(456\) 0 0
\(457\) 26.9443 1.26040 0.630200 0.776433i \(-0.282973\pi\)
0.630200 + 0.776433i \(0.282973\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 24.0689 1.12100 0.560500 0.828155i \(-0.310609\pi\)
0.560500 + 0.828155i \(0.310609\pi\)
\(462\) 0 0
\(463\) −11.7557 −0.546334 −0.273167 0.961967i \(-0.588071\pi\)
−0.273167 + 0.961967i \(0.588071\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −6.49839 −0.300710 −0.150355 0.988632i \(-0.548042\pi\)
−0.150355 + 0.988632i \(0.548042\pi\)
\(468\) 0 0
\(469\) −34.4721 −1.59178
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 46.8328 2.15338
\(474\) 0 0
\(475\) 18.2541 0.837556
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −15.5599 −0.710951 −0.355476 0.934686i \(-0.615681\pi\)
−0.355476 + 0.934686i \(0.615681\pi\)
\(480\) 0 0
\(481\) −10.9443 −0.499016
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 13.5279 0.614269
\(486\) 0 0
\(487\) −32.7849 −1.48563 −0.742814 0.669498i \(-0.766509\pi\)
−0.742814 + 0.669498i \(0.766509\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −36.9322 −1.66673 −0.833363 0.552725i \(-0.813588\pi\)
−0.833363 + 0.552725i \(0.813588\pi\)
\(492\) 0 0
\(493\) 25.8885 1.16596
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −14.4721 −0.649164
\(498\) 0 0
\(499\) 11.0697 0.495546 0.247773 0.968818i \(-0.420301\pi\)
0.247773 + 0.968818i \(0.420301\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 18.1231 0.808068 0.404034 0.914744i \(-0.367608\pi\)
0.404034 + 0.914744i \(0.367608\pi\)
\(504\) 0 0
\(505\) −6.83282 −0.304056
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 12.2918 0.544824 0.272412 0.962181i \(-0.412179\pi\)
0.272412 + 0.962181i \(0.412179\pi\)
\(510\) 0 0
\(511\) −14.1068 −0.624050
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 21.0292 0.926659
\(516\) 0 0
\(517\) −55.7771 −2.45307
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 2.11146 0.0925046 0.0462523 0.998930i \(-0.485272\pi\)
0.0462523 + 0.998930i \(0.485272\pi\)
\(522\) 0 0
\(523\) 2.90617 0.127078 0.0635390 0.997979i \(-0.479761\pi\)
0.0635390 + 0.997979i \(0.479761\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 15.2169 0.662859
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −5.23607 −0.226799
\(534\) 0 0
\(535\) −18.8091 −0.813190
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −9.06154 −0.390308
\(540\) 0 0
\(541\) 10.9443 0.470531 0.235266 0.971931i \(-0.424404\pi\)
0.235266 + 0.971931i \(0.424404\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −18.4721 −0.791259
\(546\) 0 0
\(547\) −2.90617 −0.124259 −0.0621294 0.998068i \(-0.519789\pi\)
−0.0621294 + 0.998068i \(0.519789\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 21.0292 0.895876
\(552\) 0 0
\(553\) 35.7771 1.52140
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 21.5967 0.915084 0.457542 0.889188i \(-0.348730\pi\)
0.457542 + 0.889188i \(0.348730\pi\)
\(558\) 0 0
\(559\) 7.60845 0.321803
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −2.90617 −0.122480 −0.0612402 0.998123i \(-0.519506\pi\)
−0.0612402 + 0.998123i \(0.519506\pi\)
\(564\) 0 0
\(565\) −4.94427 −0.208007
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −5.52786 −0.231740 −0.115870 0.993264i \(-0.536966\pi\)
−0.115870 + 0.993264i \(0.536966\pi\)
\(570\) 0 0
\(571\) −44.5407 −1.86397 −0.931984 0.362499i \(-0.881924\pi\)
−0.931984 + 0.362499i \(0.881924\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 30.9443 1.28823 0.644113 0.764930i \(-0.277226\pi\)
0.644113 + 0.764930i \(0.277226\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 7.63932 0.316932
\(582\) 0 0
\(583\) 30.4338 1.26044
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −12.6538 −0.522277 −0.261138 0.965301i \(-0.584098\pi\)
−0.261138 + 0.965301i \(0.584098\pi\)
\(588\) 0 0
\(589\) 12.3607 0.509313
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 32.2918 1.32607 0.663033 0.748591i \(-0.269269\pi\)
0.663033 + 0.748591i \(0.269269\pi\)
\(594\) 0 0
\(595\) 18.8091 0.771099
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −6.49839 −0.265517 −0.132759 0.991148i \(-0.542383\pi\)
−0.132759 + 0.991148i \(0.542383\pi\)
\(600\) 0 0
\(601\) 7.88854 0.321780 0.160890 0.986972i \(-0.448563\pi\)
0.160890 + 0.986972i \(0.448563\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 33.2361 1.35124
\(606\) 0 0
\(607\) 8.71851 0.353873 0.176937 0.984222i \(-0.443381\pi\)
0.176937 + 0.984222i \(0.443381\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −9.06154 −0.366591
\(612\) 0 0
\(613\) −14.0000 −0.565455 −0.282727 0.959200i \(-0.591239\pi\)
−0.282727 + 0.959200i \(0.591239\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −36.0689 −1.45208 −0.726039 0.687653i \(-0.758641\pi\)
−0.726039 + 0.687653i \(0.758641\pi\)
\(618\) 0 0
\(619\) 28.7687 1.15631 0.578156 0.815926i \(-0.303772\pi\)
0.578156 + 0.815926i \(0.303772\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 36.9322 1.47966
\(624\) 0 0
\(625\) 4.41641 0.176656
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 70.8328 2.82429
\(630\) 0 0
\(631\) 13.9758 0.556369 0.278184 0.960528i \(-0.410267\pi\)
0.278184 + 0.960528i \(0.410267\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −5.81234 −0.230656
\(636\) 0 0
\(637\) −1.47214 −0.0583282
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −24.3607 −0.962189 −0.481095 0.876669i \(-0.659761\pi\)
−0.481095 + 0.876669i \(0.659761\pi\)
\(642\) 0 0
\(643\) −6.36737 −0.251105 −0.125552 0.992087i \(-0.540070\pi\)
−0.125552 + 0.992087i \(0.540070\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 55.7771 2.18944
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −32.9443 −1.28921 −0.644604 0.764516i \(-0.722978\pi\)
−0.644604 + 0.764516i \(0.722978\pi\)
\(654\) 0 0
\(655\) −11.6247 −0.454214
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 5.81234 0.226417 0.113208 0.993571i \(-0.463887\pi\)
0.113208 + 0.993571i \(0.463887\pi\)
\(660\) 0 0
\(661\) 45.7771 1.78052 0.890261 0.455450i \(-0.150522\pi\)
0.890261 + 0.455450i \(0.150522\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 15.2786 0.592480
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 27.5276 1.06269
\(672\) 0 0
\(673\) 17.4164 0.671353 0.335677 0.941977i \(-0.391035\pi\)
0.335677 + 0.941977i \(0.391035\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 27.7771 1.06756 0.533780 0.845623i \(-0.320771\pi\)
0.533780 + 0.845623i \(0.320771\pi\)
\(678\) 0 0
\(679\) −25.7315 −0.987485
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −9.06154 −0.346730 −0.173365 0.984858i \(-0.555464\pi\)
−0.173365 + 0.984858i \(0.555464\pi\)
\(684\) 0 0
\(685\) −0.360680 −0.0137809
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 4.94427 0.188362
\(690\) 0 0
\(691\) −15.7719 −0.599993 −0.299996 0.953940i \(-0.596985\pi\)
−0.299996 + 0.953940i \(0.596985\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3.59222 0.136261
\(696\) 0 0
\(697\) 33.8885 1.28362
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −6.47214 −0.244449 −0.122225 0.992502i \(-0.539003\pi\)
−0.122225 + 0.992502i \(0.539003\pi\)
\(702\) 0 0
\(703\) 57.5374 2.17007
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 12.9968 0.488795
\(708\) 0 0
\(709\) 25.7771 0.968079 0.484039 0.875046i \(-0.339169\pi\)
0.484039 + 0.875046i \(0.339169\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 7.60845 0.284540
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −5.81234 −0.216764 −0.108382 0.994109i \(-0.534567\pi\)
−0.108382 + 0.994109i \(0.534567\pi\)
\(720\) 0 0
\(721\) −40.0000 −1.48968
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 13.8885 0.515808
\(726\) 0 0
\(727\) 1.11006 0.0411698 0.0205849 0.999788i \(-0.493447\pi\)
0.0205849 + 0.999788i \(0.493447\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −49.2429 −1.82132
\(732\) 0 0
\(733\) −22.9443 −0.847466 −0.423733 0.905787i \(-0.639281\pi\)
−0.423733 + 0.905787i \(0.639281\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 90.2492 3.32437
\(738\) 0 0
\(739\) −8.84953 −0.325535 −0.162768 0.986664i \(-0.552042\pi\)
−0.162768 + 0.986664i \(0.552042\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −5.46931 −0.200650 −0.100325 0.994955i \(-0.531988\pi\)
−0.100325 + 0.994955i \(0.531988\pi\)
\(744\) 0 0
\(745\) −21.3050 −0.780553
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 35.7771 1.30727
\(750\) 0 0
\(751\) −11.2007 −0.408718 −0.204359 0.978896i \(-0.565511\pi\)
−0.204359 + 0.978896i \(0.565511\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −27.5276 −1.00183
\(756\) 0 0
\(757\) −18.5836 −0.675432 −0.337716 0.941248i \(-0.609654\pi\)
−0.337716 + 0.941248i \(0.609654\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 20.6525 0.748652 0.374326 0.927297i \(-0.377874\pi\)
0.374326 + 0.927297i \(0.377874\pi\)
\(762\) 0 0
\(763\) 35.1361 1.27201
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 9.06154 0.327193
\(768\) 0 0
\(769\) −36.8328 −1.32823 −0.664113 0.747633i \(-0.731190\pi\)
−0.664113 + 0.747633i \(0.731190\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 35.7082 1.28433 0.642167 0.766564i \(-0.278035\pi\)
0.642167 + 0.766564i \(0.278035\pi\)
\(774\) 0 0
\(775\) 8.16348 0.293241
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 27.5276 0.986280
\(780\) 0 0
\(781\) 37.8885 1.35576
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 6.69505 0.238957
\(786\) 0 0
\(787\) 46.2057 1.64706 0.823528 0.567275i \(-0.192002\pi\)
0.823528 + 0.567275i \(0.192002\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 9.40456 0.334388
\(792\) 0 0
\(793\) 4.47214 0.158810
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −11.4164 −0.404390 −0.202195 0.979345i \(-0.564807\pi\)
−0.202195 + 0.979345i \(0.564807\pi\)
\(798\) 0 0
\(799\) 58.6475 2.07480
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 36.9322 1.30331
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 4.00000 0.140633 0.0703163 0.997525i \(-0.477599\pi\)
0.0703163 + 0.997525i \(0.477599\pi\)
\(810\) 0 0
\(811\) 49.7980 1.74864 0.874322 0.485347i \(-0.161307\pi\)
0.874322 + 0.485347i \(0.161307\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −12.3107 −0.431226
\(816\) 0 0
\(817\) −40.0000 −1.39942
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 37.5967 1.31214 0.656068 0.754702i \(-0.272219\pi\)
0.656068 + 0.754702i \(0.272219\pi\)
\(822\) 0 0
\(823\) −22.8254 −0.795642 −0.397821 0.917463i \(-0.630233\pi\)
−0.397821 + 0.917463i \(0.630233\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 40.1814 1.39724 0.698622 0.715491i \(-0.253797\pi\)
0.698622 + 0.715491i \(0.253797\pi\)
\(828\) 0 0
\(829\) 0.472136 0.0163980 0.00819898 0.999966i \(-0.497390\pi\)
0.00819898 + 0.999966i \(0.497390\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 9.52786 0.330121
\(834\) 0 0
\(835\) −11.2007 −0.387615
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −36.5892 −1.26320 −0.631599 0.775295i \(-0.717601\pi\)
−0.631599 + 0.775295i \(0.717601\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.23607 0.0425220
\(846\) 0 0
\(847\) −63.2188 −2.17222
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −20.8328 −0.713302 −0.356651 0.934238i \(-0.616081\pi\)
−0.356651 + 0.934238i \(0.616081\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −49.3050 −1.68423 −0.842113 0.539302i \(-0.818688\pi\)
−0.842113 + 0.539302i \(0.818688\pi\)
\(858\) 0 0
\(859\) −15.2169 −0.519194 −0.259597 0.965717i \(-0.583590\pi\)
−0.259597 + 0.965717i \(0.583590\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −11.9677 −0.407385 −0.203693 0.979035i \(-0.565294\pi\)
−0.203693 + 0.979035i \(0.565294\pi\)
\(864\) 0 0
\(865\) −12.9443 −0.440118
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −93.6656 −3.17739
\(870\) 0 0
\(871\) 14.6619 0.496799
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 24.6215 0.832358
\(876\) 0 0
\(877\) 30.9443 1.04491 0.522457 0.852666i \(-0.325016\pi\)
0.522457 + 0.852666i \(0.325016\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 4.36068 0.146915 0.0734575 0.997298i \(-0.476597\pi\)
0.0734575 + 0.997298i \(0.476597\pi\)
\(882\) 0 0
\(883\) −48.1329 −1.61980 −0.809900 0.586568i \(-0.800479\pi\)
−0.809900 + 0.586568i \(0.800479\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 42.7445 1.43522 0.717611 0.696445i \(-0.245236\pi\)
0.717611 + 0.696445i \(0.245236\pi\)
\(888\) 0 0
\(889\) 11.0557 0.370797
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 47.6393 1.59419
\(894\) 0 0
\(895\) −26.8416 −0.897215
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 9.40456 0.313660
\(900\) 0 0
\(901\) −32.0000 −1.06607
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −20.3607 −0.676812
\(906\) 0 0
\(907\) 4.70228 0.156137 0.0780684 0.996948i \(-0.475125\pi\)
0.0780684 + 0.996948i \(0.475125\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 36.9322 1.22362 0.611809 0.791005i \(-0.290442\pi\)
0.611809 + 0.791005i \(0.290442\pi\)
\(912\) 0 0
\(913\) −20.0000 −0.661903
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 22.1115 0.730185
\(918\) 0 0
\(919\) −15.2169 −0.501959 −0.250980 0.967992i \(-0.580753\pi\)
−0.250980 + 0.967992i \(0.580753\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 6.15537 0.202606
\(924\) 0 0
\(925\) 38.0000 1.24943
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 13.8197 0.453408 0.226704 0.973964i \(-0.427205\pi\)
0.226704 + 0.973964i \(0.427205\pi\)
\(930\) 0 0
\(931\) 7.73948 0.253651
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −49.2429 −1.61042
\(936\) 0 0
\(937\) 27.5279 0.899296 0.449648 0.893206i \(-0.351549\pi\)
0.449648 + 0.893206i \(0.351549\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 11.7082 0.381677 0.190838 0.981621i \(-0.438879\pi\)
0.190838 + 0.981621i \(0.438879\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 5.46931 0.177729 0.0888644 0.996044i \(-0.471676\pi\)
0.0888644 + 0.996044i \(0.471676\pi\)
\(948\) 0 0
\(949\) 6.00000 0.194768
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 39.4164 1.27682 0.638411 0.769695i \(-0.279592\pi\)
0.638411 + 0.769695i \(0.279592\pi\)
\(954\) 0 0
\(955\) 8.03246 0.259924
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0.686054 0.0221538
\(960\) 0 0
\(961\) −25.4721 −0.821682
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −3.63932 −0.117154
\(966\) 0 0
\(967\) 41.0795 1.32103 0.660513 0.750815i \(-0.270339\pi\)
0.660513 + 0.750815i \(0.270339\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −30.4338 −0.976667 −0.488334 0.872657i \(-0.662395\pi\)
−0.488334 + 0.872657i \(0.662395\pi\)
\(972\) 0 0
\(973\) −6.83282 −0.219050
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 47.1246 1.50765 0.753825 0.657075i \(-0.228207\pi\)
0.753825 + 0.657075i \(0.228207\pi\)
\(978\) 0 0
\(979\) −96.6898 −3.09022
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −55.3983 −1.76693 −0.883466 0.468496i \(-0.844796\pi\)
−0.883466 + 0.468496i \(0.844796\pi\)
\(984\) 0 0
\(985\) 17.5279 0.558484
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −33.3400 −1.05908 −0.529540 0.848285i \(-0.677635\pi\)
−0.529540 + 0.848285i \(0.677635\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 17.4370 0.552791
\(996\) 0 0
\(997\) 4.11146 0.130211 0.0651056 0.997878i \(-0.479262\pi\)
0.0651056 + 0.997878i \(0.479262\pi\)
\(998\) 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.bb.1.3 4
3.2 odd 2 3744.2.a.bf.1.1 yes 4
4.3 odd 2 inner 3744.2.a.bb.1.4 yes 4
8.3 odd 2 7488.2.a.dd.1.2 4
8.5 even 2 7488.2.a.dd.1.1 4
12.11 even 2 3744.2.a.bf.1.2 yes 4
24.5 odd 2 7488.2.a.cz.1.3 4
24.11 even 2 7488.2.a.cz.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3744.2.a.bb.1.3 4 1.1 even 1 trivial
3744.2.a.bb.1.4 yes 4 4.3 odd 2 inner
3744.2.a.bf.1.1 yes 4 3.2 odd 2
3744.2.a.bf.1.2 yes 4 12.11 even 2
7488.2.a.cz.1.3 4 24.5 odd 2
7488.2.a.cz.1.4 4 24.11 even 2
7488.2.a.dd.1.1 4 8.5 even 2
7488.2.a.dd.1.2 4 8.3 odd 2