Properties

Label 3744.2.a.bc.1.2
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: \(\Q(\zeta_{24})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) 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 1.2
Root \(-0.517638\) 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)
 
\(f(q)\) \(=\) \(q-1.03528 q^{5} +3.86370 q^{7} +O(q^{10})\) \(q-1.03528 q^{5} +3.86370 q^{7} +1.46410 q^{11} -1.00000 q^{13} -5.65685 q^{17} -1.79315 q^{19} -6.92820 q^{23} -3.92820 q^{25} -2.07055 q^{29} +3.86370 q^{31} -4.00000 q^{35} -2.00000 q^{37} +1.03528 q^{41} -5.65685 q^{43} -2.53590 q^{47} +7.92820 q^{49} -5.65685 q^{53} -1.51575 q^{55} -9.46410 q^{59} -8.92820 q^{61} +1.03528 q^{65} +1.79315 q^{67} -2.53590 q^{71} +11.8564 q^{73} +5.65685 q^{77} -6.53590 q^{83} +5.85641 q^{85} +14.4195 q^{89} -3.86370 q^{91} +1.85641 q^{95} +6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{11} - 4 q^{13} + 12 q^{25} - 16 q^{35} - 8 q^{37} - 24 q^{47} + 4 q^{49} - 24 q^{59} - 8 q^{61} - 24 q^{71} - 8 q^{73} - 40 q^{83} - 32 q^{85} - 48 q^{95} + 24 q^{97}+O(q^{100}) \) 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
\(5\) −1.03528 −0.462990 −0.231495 0.972836i \(-0.574362\pi\)
−0.231495 + 0.972836i \(0.574362\pi\)
\(6\) 0 0
\(7\) 3.86370 1.46034 0.730171 0.683264i \(-0.239440\pi\)
0.730171 + 0.683264i \(0.239440\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.46410 0.441443 0.220722 0.975337i \(-0.429159\pi\)
0.220722 + 0.975337i \(0.429159\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.65685 −1.37199 −0.685994 0.727607i \(-0.740633\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) 0 0
\(19\) −1.79315 −0.411377 −0.205689 0.978618i \(-0.565943\pi\)
−0.205689 + 0.978618i \(0.565943\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.92820 −1.44463 −0.722315 0.691564i \(-0.756922\pi\)
−0.722315 + 0.691564i \(0.756922\pi\)
\(24\) 0 0
\(25\) −3.92820 −0.785641
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.07055 −0.384492 −0.192246 0.981347i \(-0.561577\pi\)
−0.192246 + 0.981347i \(0.561577\pi\)
\(30\) 0 0
\(31\) 3.86370 0.693942 0.346971 0.937876i \(-0.387210\pi\)
0.346971 + 0.937876i \(0.387210\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.03528 0.161683 0.0808415 0.996727i \(-0.474239\pi\)
0.0808415 + 0.996727i \(0.474239\pi\)
\(42\) 0 0
\(43\) −5.65685 −0.862662 −0.431331 0.902194i \(-0.641956\pi\)
−0.431331 + 0.902194i \(0.641956\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.53590 −0.369899 −0.184949 0.982748i \(-0.559212\pi\)
−0.184949 + 0.982748i \(0.559212\pi\)
\(48\) 0 0
\(49\) 7.92820 1.13260
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.65685 −0.777029 −0.388514 0.921443i \(-0.627012\pi\)
−0.388514 + 0.921443i \(0.627012\pi\)
\(54\) 0 0
\(55\) −1.51575 −0.204384
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −9.46410 −1.23212 −0.616061 0.787699i \(-0.711272\pi\)
−0.616061 + 0.787699i \(0.711272\pi\)
\(60\) 0 0
\(61\) −8.92820 −1.14314 −0.571570 0.820554i \(-0.693665\pi\)
−0.571570 + 0.820554i \(0.693665\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.03528 0.128410
\(66\) 0 0
\(67\) 1.79315 0.219068 0.109534 0.993983i \(-0.465064\pi\)
0.109534 + 0.993983i \(0.465064\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.53590 −0.300956 −0.150478 0.988613i \(-0.548081\pi\)
−0.150478 + 0.988613i \(0.548081\pi\)
\(72\) 0 0
\(73\) 11.8564 1.38769 0.693844 0.720126i \(-0.255916\pi\)
0.693844 + 0.720126i \(0.255916\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.65685 0.644658
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −6.53590 −0.717408 −0.358704 0.933451i \(-0.616781\pi\)
−0.358704 + 0.933451i \(0.616781\pi\)
\(84\) 0 0
\(85\) 5.85641 0.635216
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 14.4195 1.52847 0.764234 0.644939i \(-0.223117\pi\)
0.764234 + 0.644939i \(0.223117\pi\)
\(90\) 0 0
\(91\) −3.86370 −0.405026
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.85641 0.190463
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −7.72741 −0.768906 −0.384453 0.923145i \(-0.625610\pi\)
−0.384453 + 0.923145i \(0.625610\pi\)
\(102\) 0 0
\(103\) 16.9706 1.67216 0.836080 0.548608i \(-0.184842\pi\)
0.836080 + 0.548608i \(0.184842\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −6.92820 −0.669775 −0.334887 0.942258i \(-0.608698\pi\)
−0.334887 + 0.942258i \(0.608698\pi\)
\(108\) 0 0
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 19.0411 1.79124 0.895619 0.444823i \(-0.146733\pi\)
0.895619 + 0.444823i \(0.146733\pi\)
\(114\) 0 0
\(115\) 7.17260 0.668849
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −21.8564 −2.00357
\(120\) 0 0
\(121\) −8.85641 −0.805128
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.24316 0.826733
\(126\) 0 0
\(127\) −11.8685 −1.05316 −0.526580 0.850126i \(-0.676526\pi\)
−0.526580 + 0.850126i \(0.676526\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −14.9282 −1.30428 −0.652142 0.758097i \(-0.726129\pi\)
−0.652142 + 0.758097i \(0.726129\pi\)
\(132\) 0 0
\(133\) −6.92820 −0.600751
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 5.17638 0.442248 0.221124 0.975246i \(-0.429027\pi\)
0.221124 + 0.975246i \(0.429027\pi\)
\(138\) 0 0
\(139\) −13.3843 −1.13524 −0.567619 0.823291i \(-0.692135\pi\)
−0.567619 + 0.823291i \(0.692135\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1.46410 −0.122434
\(144\) 0 0
\(145\) 2.14359 0.178016
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 20.6312 1.69017 0.845087 0.534629i \(-0.179549\pi\)
0.845087 + 0.534629i \(0.179549\pi\)
\(150\) 0 0
\(151\) −8.00481 −0.651422 −0.325711 0.945469i \(-0.605604\pi\)
−0.325711 + 0.945469i \(0.605604\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −3.07180 −0.245156 −0.122578 0.992459i \(-0.539116\pi\)
−0.122578 + 0.992459i \(0.539116\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −26.7685 −2.10966
\(162\) 0 0
\(163\) 21.3891 1.67532 0.837661 0.546191i \(-0.183923\pi\)
0.837661 + 0.546191i \(0.183923\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.53590 −0.196234 −0.0981169 0.995175i \(-0.531282\pi\)
−0.0981169 + 0.995175i \(0.531282\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −23.1822 −1.76251 −0.881256 0.472640i \(-0.843301\pi\)
−0.881256 + 0.472640i \(0.843301\pi\)
\(174\) 0 0
\(175\) −15.1774 −1.14730
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 9.85641 0.736702 0.368351 0.929687i \(-0.379922\pi\)
0.368351 + 0.929687i \(0.379922\pi\)
\(180\) 0 0
\(181\) −18.7846 −1.39625 −0.698125 0.715976i \(-0.745982\pi\)
−0.698125 + 0.715976i \(0.745982\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2.07055 0.152230
\(186\) 0 0
\(187\) −8.28221 −0.605655
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −20.0000 −1.44715 −0.723575 0.690246i \(-0.757502\pi\)
−0.723575 + 0.690246i \(0.757502\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.10583 0.221281 0.110641 0.993860i \(-0.464710\pi\)
0.110641 + 0.993860i \(0.464710\pi\)
\(198\) 0 0
\(199\) −3.58630 −0.254226 −0.127113 0.991888i \(-0.540571\pi\)
−0.127113 + 0.991888i \(0.540571\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −8.00000 −0.561490
\(204\) 0 0
\(205\) −1.07180 −0.0748575
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2.62536 −0.181600
\(210\) 0 0
\(211\) 4.14110 0.285085 0.142543 0.989789i \(-0.454472\pi\)
0.142543 + 0.989789i \(0.454472\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 5.85641 0.399404
\(216\) 0 0
\(217\) 14.9282 1.01339
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 5.65685 0.380521
\(222\) 0 0
\(223\) −15.7322 −1.05351 −0.526754 0.850018i \(-0.676591\pi\)
−0.526754 + 0.850018i \(0.676591\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −9.46410 −0.628154 −0.314077 0.949397i \(-0.601695\pi\)
−0.314077 + 0.949397i \(0.601695\pi\)
\(228\) 0 0
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −13.3843 −0.876832 −0.438416 0.898772i \(-0.644460\pi\)
−0.438416 + 0.898772i \(0.644460\pi\)
\(234\) 0 0
\(235\) 2.62536 0.171259
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −24.3923 −1.57781 −0.788904 0.614517i \(-0.789351\pi\)
−0.788904 + 0.614517i \(0.789351\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −8.20788 −0.524382
\(246\) 0 0
\(247\) 1.79315 0.114095
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −16.7846 −1.05944 −0.529718 0.848174i \(-0.677702\pi\)
−0.529718 + 0.848174i \(0.677702\pi\)
\(252\) 0 0
\(253\) −10.1436 −0.637722
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −17.5254 −1.09320 −0.546601 0.837393i \(-0.684079\pi\)
−0.546601 + 0.837393i \(0.684079\pi\)
\(258\) 0 0
\(259\) −7.72741 −0.480158
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 10.9282 0.673862 0.336931 0.941529i \(-0.390611\pi\)
0.336931 + 0.941529i \(0.390611\pi\)
\(264\) 0 0
\(265\) 5.85641 0.359756
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 13.3843 0.816053 0.408026 0.912970i \(-0.366217\pi\)
0.408026 + 0.912970i \(0.366217\pi\)
\(270\) 0 0
\(271\) −4.41851 −0.268405 −0.134203 0.990954i \(-0.542847\pi\)
−0.134203 + 0.990954i \(0.542847\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −5.75129 −0.346816
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −14.4195 −0.860197 −0.430099 0.902782i \(-0.641521\pi\)
−0.430099 + 0.902782i \(0.641521\pi\)
\(282\) 0 0
\(283\) 14.9000 0.885714 0.442857 0.896592i \(-0.353965\pi\)
0.442857 + 0.896592i \(0.353965\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.00000 0.236113
\(288\) 0 0
\(289\) 15.0000 0.882353
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 12.3490 0.721435 0.360718 0.932675i \(-0.382532\pi\)
0.360718 + 0.932675i \(0.382532\pi\)
\(294\) 0 0
\(295\) 9.79796 0.570459
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.92820 0.400668
\(300\) 0 0
\(301\) −21.8564 −1.25978
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 9.24316 0.529262
\(306\) 0 0
\(307\) 17.8028 1.01606 0.508029 0.861340i \(-0.330374\pi\)
0.508029 + 0.861340i \(0.330374\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −6.92820 −0.392862 −0.196431 0.980518i \(-0.562935\pi\)
−0.196431 + 0.980518i \(0.562935\pi\)
\(312\) 0 0
\(313\) 7.85641 0.444070 0.222035 0.975039i \(-0.428730\pi\)
0.222035 + 0.975039i \(0.428730\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −31.9449 −1.79420 −0.897102 0.441823i \(-0.854332\pi\)
−0.897102 + 0.441823i \(0.854332\pi\)
\(318\) 0 0
\(319\) −3.03150 −0.169731
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 10.1436 0.564405
\(324\) 0 0
\(325\) 3.92820 0.217898
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −9.79796 −0.540179
\(330\) 0 0
\(331\) 8.96575 0.492802 0.246401 0.969168i \(-0.420752\pi\)
0.246401 + 0.969168i \(0.420752\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.85641 −0.101426
\(336\) 0 0
\(337\) 27.8564 1.51744 0.758718 0.651420i \(-0.225826\pi\)
0.758718 + 0.651420i \(0.225826\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 5.65685 0.306336
\(342\) 0 0
\(343\) 3.58630 0.193642
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −24.0000 −1.28839 −0.644194 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) 0 0
\(349\) −18.0000 −0.963518 −0.481759 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −7.24693 −0.385715 −0.192858 0.981227i \(-0.561776\pi\)
−0.192858 + 0.981227i \(0.561776\pi\)
\(354\) 0 0
\(355\) 2.62536 0.139339
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 5.46410 0.288384 0.144192 0.989550i \(-0.453942\pi\)
0.144192 + 0.989550i \(0.453942\pi\)
\(360\) 0 0
\(361\) −15.7846 −0.830769
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −12.2747 −0.642485
\(366\) 0 0
\(367\) 6.21166 0.324246 0.162123 0.986771i \(-0.448166\pi\)
0.162123 + 0.986771i \(0.448166\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −21.8564 −1.13473
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.07055 0.106639
\(378\) 0 0
\(379\) 9.52056 0.489038 0.244519 0.969644i \(-0.421370\pi\)
0.244519 + 0.969644i \(0.421370\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 2.53590 0.129578 0.0647892 0.997899i \(-0.479363\pi\)
0.0647892 + 0.997899i \(0.479363\pi\)
\(384\) 0 0
\(385\) −5.85641 −0.298470
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 15.4548 0.783590 0.391795 0.920053i \(-0.371854\pi\)
0.391795 + 0.920053i \(0.371854\pi\)
\(390\) 0 0
\(391\) 39.1918 1.98202
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −31.8564 −1.59883 −0.799414 0.600781i \(-0.794856\pi\)
−0.799414 + 0.600781i \(0.794856\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 21.5921 1.07826 0.539130 0.842223i \(-0.318753\pi\)
0.539130 + 0.842223i \(0.318753\pi\)
\(402\) 0 0
\(403\) −3.86370 −0.192465
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2.92820 −0.145146
\(408\) 0 0
\(409\) 0.143594 0.00710024 0.00355012 0.999994i \(-0.498870\pi\)
0.00355012 + 0.999994i \(0.498870\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −36.5665 −1.79932
\(414\) 0 0
\(415\) 6.76646 0.332152
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 28.7846 1.40622 0.703110 0.711081i \(-0.251794\pi\)
0.703110 + 0.711081i \(0.251794\pi\)
\(420\) 0 0
\(421\) 37.7128 1.83801 0.919005 0.394246i \(-0.128994\pi\)
0.919005 + 0.394246i \(0.128994\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 22.2213 1.07789
\(426\) 0 0
\(427\) −34.4959 −1.66937
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0.392305 0.0188967 0.00944833 0.999955i \(-0.496992\pi\)
0.00944833 + 0.999955i \(0.496992\pi\)
\(432\) 0 0
\(433\) 16.9282 0.813518 0.406759 0.913536i \(-0.366659\pi\)
0.406759 + 0.913536i \(0.366659\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 12.4233 0.594288
\(438\) 0 0
\(439\) −40.7076 −1.94287 −0.971434 0.237312i \(-0.923734\pi\)
−0.971434 + 0.237312i \(0.923734\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 34.6410 1.64584 0.822922 0.568154i \(-0.192342\pi\)
0.822922 + 0.568154i \(0.192342\pi\)
\(444\) 0 0
\(445\) −14.9282 −0.707665
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 29.8744 1.40986 0.704929 0.709278i \(-0.250979\pi\)
0.704929 + 0.709278i \(0.250979\pi\)
\(450\) 0 0
\(451\) 1.51575 0.0713739
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) −2.00000 −0.0935561 −0.0467780 0.998905i \(-0.514895\pi\)
−0.0467780 + 0.998905i \(0.514895\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 41.1881 1.91832 0.959160 0.282866i \(-0.0912850\pi\)
0.959160 + 0.282866i \(0.0912850\pi\)
\(462\) 0 0
\(463\) −27.0459 −1.25693 −0.628465 0.777838i \(-0.716317\pi\)
−0.628465 + 0.777838i \(0.716317\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 34.9282 1.61628 0.808142 0.588987i \(-0.200473\pi\)
0.808142 + 0.588987i \(0.200473\pi\)
\(468\) 0 0
\(469\) 6.92820 0.319915
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −8.28221 −0.380816
\(474\) 0 0
\(475\) 7.04386 0.323195
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 18.5359 0.846927 0.423463 0.905913i \(-0.360814\pi\)
0.423463 + 0.905913i \(0.360814\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −6.21166 −0.282057
\(486\) 0 0
\(487\) 19.3185 0.875406 0.437703 0.899120i \(-0.355792\pi\)
0.437703 + 0.899120i \(0.355792\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 16.7846 0.757479 0.378739 0.925503i \(-0.376358\pi\)
0.378739 + 0.925503i \(0.376358\pi\)
\(492\) 0 0
\(493\) 11.7128 0.527519
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −9.79796 −0.439499
\(498\) 0 0
\(499\) −33.2576 −1.48881 −0.744407 0.667726i \(-0.767268\pi\)
−0.744407 + 0.667726i \(0.767268\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −10.9282 −0.487264 −0.243632 0.969868i \(-0.578339\pi\)
−0.243632 + 0.969868i \(0.578339\pi\)
\(504\) 0 0
\(505\) 8.00000 0.355995
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 18.5606 0.822686 0.411343 0.911481i \(-0.365060\pi\)
0.411343 + 0.911481i \(0.365060\pi\)
\(510\) 0 0
\(511\) 45.8096 2.02650
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −17.5692 −0.774192
\(516\) 0 0
\(517\) −3.71281 −0.163289
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −0.554803 −0.0243064 −0.0121532 0.999926i \(-0.503869\pi\)
−0.0121532 + 0.999926i \(0.503869\pi\)
\(522\) 0 0
\(523\) 17.5254 0.766331 0.383165 0.923680i \(-0.374834\pi\)
0.383165 + 0.923680i \(0.374834\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −21.8564 −0.952080
\(528\) 0 0
\(529\) 25.0000 1.08696
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1.03528 −0.0448428
\(534\) 0 0
\(535\) 7.17260 0.310099
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 11.6077 0.499979
\(540\) 0 0
\(541\) −27.8564 −1.19764 −0.598820 0.800883i \(-0.704364\pi\)
−0.598820 + 0.800883i \(0.704364\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 6.21166 0.266078
\(546\) 0 0
\(547\) −40.1528 −1.71681 −0.858405 0.512973i \(-0.828544\pi\)
−0.858405 + 0.512973i \(0.828544\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 3.71281 0.158171
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 27.8038 1.17808 0.589042 0.808102i \(-0.299505\pi\)
0.589042 + 0.808102i \(0.299505\pi\)
\(558\) 0 0
\(559\) 5.65685 0.239259
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −1.85641 −0.0782382 −0.0391191 0.999235i \(-0.512455\pi\)
−0.0391191 + 0.999235i \(0.512455\pi\)
\(564\) 0 0
\(565\) −19.7128 −0.829324
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 25.8076 1.08191 0.540955 0.841052i \(-0.318063\pi\)
0.540955 + 0.841052i \(0.318063\pi\)
\(570\) 0 0
\(571\) −45.8096 −1.91707 −0.958537 0.284969i \(-0.908017\pi\)
−0.958537 + 0.284969i \(0.908017\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 27.2154 1.13496
\(576\) 0 0
\(577\) −11.8564 −0.493589 −0.246794 0.969068i \(-0.579377\pi\)
−0.246794 + 0.969068i \(0.579377\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −25.2528 −1.04766
\(582\) 0 0
\(583\) −8.28221 −0.343014
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −28.3923 −1.17188 −0.585938 0.810356i \(-0.699274\pi\)
−0.585938 + 0.810356i \(0.699274\pi\)
\(588\) 0 0
\(589\) −6.92820 −0.285472
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −8.20788 −0.337057 −0.168529 0.985697i \(-0.553902\pi\)
−0.168529 + 0.985697i \(0.553902\pi\)
\(594\) 0 0
\(595\) 22.6274 0.927634
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −2.92820 −0.119643 −0.0598216 0.998209i \(-0.519053\pi\)
−0.0598216 + 0.998209i \(0.519053\pi\)
\(600\) 0 0
\(601\) −8.14359 −0.332184 −0.166092 0.986110i \(-0.553115\pi\)
−0.166092 + 0.986110i \(0.553115\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 9.16883 0.372766
\(606\) 0 0
\(607\) 36.0117 1.46167 0.730834 0.682555i \(-0.239131\pi\)
0.730834 + 0.682555i \(0.239131\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.53590 0.102591
\(612\) 0 0
\(613\) 11.8564 0.478876 0.239438 0.970912i \(-0.423037\pi\)
0.239438 + 0.970912i \(0.423037\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −18.5606 −0.747223 −0.373612 0.927585i \(-0.621881\pi\)
−0.373612 + 0.927585i \(0.621881\pi\)
\(618\) 0 0
\(619\) 32.1480 1.29214 0.646068 0.763280i \(-0.276412\pi\)
0.646068 + 0.763280i \(0.276412\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 55.7128 2.23209
\(624\) 0 0
\(625\) 10.0718 0.402872
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 11.3137 0.451107
\(630\) 0 0
\(631\) −41.9459 −1.66984 −0.834921 0.550370i \(-0.814487\pi\)
−0.834921 + 0.550370i \(0.814487\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 12.2872 0.487602
\(636\) 0 0
\(637\) −7.92820 −0.314127
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −12.8295 −0.506733 −0.253367 0.967370i \(-0.581538\pi\)
−0.253367 + 0.967370i \(0.581538\pi\)
\(642\) 0 0
\(643\) 1.79315 0.0707150 0.0353575 0.999375i \(-0.488743\pi\)
0.0353575 + 0.999375i \(0.488743\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 45.8564 1.80280 0.901401 0.432986i \(-0.142540\pi\)
0.901401 + 0.432986i \(0.142540\pi\)
\(648\) 0 0
\(649\) −13.8564 −0.543912
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −16.4158 −0.642398 −0.321199 0.947012i \(-0.604086\pi\)
−0.321199 + 0.947012i \(0.604086\pi\)
\(654\) 0 0
\(655\) 15.4548 0.603870
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −10.1436 −0.395138 −0.197569 0.980289i \(-0.563305\pi\)
−0.197569 + 0.980289i \(0.563305\pi\)
\(660\) 0 0
\(661\) 11.8564 0.461161 0.230580 0.973053i \(-0.425938\pi\)
0.230580 + 0.973053i \(0.425938\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 7.17260 0.278142
\(666\) 0 0
\(667\) 14.3452 0.555449
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −13.0718 −0.504631
\(672\) 0 0
\(673\) −15.0718 −0.580975 −0.290488 0.956879i \(-0.593817\pi\)
−0.290488 + 0.956879i \(0.593817\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 16.9706 0.652232 0.326116 0.945330i \(-0.394260\pi\)
0.326116 + 0.945330i \(0.394260\pi\)
\(678\) 0 0
\(679\) 23.1822 0.889652
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 45.1769 1.72865 0.864323 0.502937i \(-0.167747\pi\)
0.864323 + 0.502937i \(0.167747\pi\)
\(684\) 0 0
\(685\) −5.35898 −0.204756
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 5.65685 0.215509
\(690\) 0 0
\(691\) −29.1165 −1.10764 −0.553821 0.832635i \(-0.686831\pi\)
−0.553821 + 0.832635i \(0.686831\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 13.8564 0.525603
\(696\) 0 0
\(697\) −5.85641 −0.221827
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −19.5959 −0.740128 −0.370064 0.929006i \(-0.620664\pi\)
−0.370064 + 0.929006i \(0.620664\pi\)
\(702\) 0 0
\(703\) 3.58630 0.135260
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −29.8564 −1.12287
\(708\) 0 0
\(709\) −17.7128 −0.665219 −0.332609 0.943065i \(-0.607929\pi\)
−0.332609 + 0.943065i \(0.607929\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −26.7685 −1.00249
\(714\) 0 0
\(715\) 1.51575 0.0566858
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 44.7846 1.67018 0.835092 0.550110i \(-0.185414\pi\)
0.835092 + 0.550110i \(0.185414\pi\)
\(720\) 0 0
\(721\) 65.5692 2.44193
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 8.13355 0.302072
\(726\) 0 0
\(727\) −7.72741 −0.286594 −0.143297 0.989680i \(-0.545770\pi\)
−0.143297 + 0.989680i \(0.545770\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 32.0000 1.18356
\(732\) 0 0
\(733\) −17.7128 −0.654238 −0.327119 0.944983i \(-0.606078\pi\)
−0.327119 + 0.944983i \(0.606078\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2.62536 0.0967062
\(738\) 0 0
\(739\) −14.2165 −0.522961 −0.261481 0.965209i \(-0.584211\pi\)
−0.261481 + 0.965209i \(0.584211\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −30.2487 −1.10972 −0.554859 0.831945i \(-0.687228\pi\)
−0.554859 + 0.831945i \(0.687228\pi\)
\(744\) 0 0
\(745\) −21.3590 −0.782533
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −26.7685 −0.978100
\(750\) 0 0
\(751\) −6.76646 −0.246912 −0.123456 0.992350i \(-0.539398\pi\)
−0.123456 + 0.992350i \(0.539398\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 8.28719 0.301602
\(756\) 0 0
\(757\) 28.6410 1.04098 0.520488 0.853869i \(-0.325750\pi\)
0.520488 + 0.853869i \(0.325750\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −17.4510 −0.632600 −0.316300 0.948659i \(-0.602441\pi\)
−0.316300 + 0.948659i \(0.602441\pi\)
\(762\) 0 0
\(763\) −23.1822 −0.839253
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 9.46410 0.341729
\(768\) 0 0
\(769\) 21.7128 0.782984 0.391492 0.920181i \(-0.371959\pi\)
0.391492 + 0.920181i \(0.371959\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −44.2196 −1.59047 −0.795233 0.606303i \(-0.792652\pi\)
−0.795233 + 0.606303i \(0.792652\pi\)
\(774\) 0 0
\(775\) −15.1774 −0.545189
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.85641 −0.0665127
\(780\) 0 0
\(781\) −3.71281 −0.132855
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 3.18016 0.113505
\(786\) 0 0
\(787\) 29.1165 1.03789 0.518945 0.854808i \(-0.326325\pi\)
0.518945 + 0.854808i \(0.326325\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 73.5692 2.61582
\(792\) 0 0
\(793\) 8.92820 0.317050
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −12.4233 −0.440056 −0.220028 0.975494i \(-0.570615\pi\)
−0.220028 + 0.975494i \(0.570615\pi\)
\(798\) 0 0
\(799\) 14.3452 0.507497
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 17.3590 0.612585
\(804\) 0 0
\(805\) 27.7128 0.976748
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0.554803 0.0195058 0.00975291 0.999952i \(-0.496896\pi\)
0.00975291 + 0.999952i \(0.496896\pi\)
\(810\) 0 0
\(811\) −20.2795 −0.712108 −0.356054 0.934465i \(-0.615878\pi\)
−0.356054 + 0.934465i \(0.615878\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −22.1436 −0.775656
\(816\) 0 0
\(817\) 10.1436 0.354879
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −10.2784 −0.358720 −0.179360 0.983784i \(-0.557403\pi\)
−0.179360 + 0.983784i \(0.557403\pi\)
\(822\) 0 0
\(823\) −43.7391 −1.52465 −0.762324 0.647195i \(-0.775942\pi\)
−0.762324 + 0.647195i \(0.775942\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 51.0333 1.77460 0.887301 0.461190i \(-0.152577\pi\)
0.887301 + 0.461190i \(0.152577\pi\)
\(828\) 0 0
\(829\) 6.78461 0.235639 0.117820 0.993035i \(-0.462410\pi\)
0.117820 + 0.993035i \(0.462410\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −44.8487 −1.55392
\(834\) 0 0
\(835\) 2.62536 0.0908542
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −19.3205 −0.667018 −0.333509 0.942747i \(-0.608233\pi\)
−0.333509 + 0.942747i \(0.608233\pi\)
\(840\) 0 0
\(841\) −24.7128 −0.852166
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1.03528 −0.0356146
\(846\) 0 0
\(847\) −34.2185 −1.17576
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 13.8564 0.474991
\(852\) 0 0
\(853\) 41.7128 1.42822 0.714110 0.700034i \(-0.246832\pi\)
0.714110 + 0.700034i \(0.246832\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 24.6980 0.843667 0.421833 0.906673i \(-0.361387\pi\)
0.421833 + 0.906673i \(0.361387\pi\)
\(858\) 0 0
\(859\) 33.9411 1.15806 0.579028 0.815308i \(-0.303432\pi\)
0.579028 + 0.815308i \(0.303432\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −17.1769 −0.584709 −0.292354 0.956310i \(-0.594439\pi\)
−0.292354 + 0.956310i \(0.594439\pi\)
\(864\) 0 0
\(865\) 24.0000 0.816024
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −1.79315 −0.0607586
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 35.7128 1.20731
\(876\) 0 0
\(877\) −26.0000 −0.877958 −0.438979 0.898497i \(-0.644660\pi\)
−0.438979 + 0.898497i \(0.644660\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −16.4158 −0.553061 −0.276531 0.961005i \(-0.589185\pi\)
−0.276531 + 0.961005i \(0.589185\pi\)
\(882\) 0 0
\(883\) −11.8685 −0.399407 −0.199704 0.979856i \(-0.563998\pi\)
−0.199704 + 0.979856i \(0.563998\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 22.1436 0.743509 0.371755 0.928331i \(-0.378756\pi\)
0.371755 + 0.928331i \(0.378756\pi\)
\(888\) 0 0
\(889\) −45.8564 −1.53797
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 4.54725 0.152168
\(894\) 0 0
\(895\) −10.2041 −0.341086
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −8.00000 −0.266815
\(900\) 0 0
\(901\) 32.0000 1.06607
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 19.4473 0.646449
\(906\) 0 0
\(907\) 16.0096 0.531591 0.265795 0.964029i \(-0.414365\pi\)
0.265795 + 0.964029i \(0.414365\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −15.7128 −0.520589 −0.260294 0.965529i \(-0.583820\pi\)
−0.260294 + 0.965529i \(0.583820\pi\)
\(912\) 0 0
\(913\) −9.56922 −0.316695
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −57.6781 −1.90470
\(918\) 0 0
\(919\) −30.9096 −1.01961 −0.509807 0.860289i \(-0.670283\pi\)
−0.509807 + 0.860289i \(0.670283\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 2.53590 0.0834701
\(924\) 0 0
\(925\) 7.85641 0.258317
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 39.1175 1.28340 0.641702 0.766954i \(-0.278229\pi\)
0.641702 + 0.766954i \(0.278229\pi\)
\(930\) 0 0
\(931\) −14.2165 −0.465926
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 8.57437 0.280412
\(936\) 0 0
\(937\) −48.6410 −1.58903 −0.794516 0.607243i \(-0.792276\pi\)
−0.794516 + 0.607243i \(0.792276\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 9.31749 0.303741 0.151871 0.988400i \(-0.451470\pi\)
0.151871 + 0.988400i \(0.451470\pi\)
\(942\) 0 0
\(943\) −7.17260 −0.233572
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 34.2487 1.11293 0.556467 0.830870i \(-0.312157\pi\)
0.556467 + 0.830870i \(0.312157\pi\)
\(948\) 0 0
\(949\) −11.8564 −0.384875
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 51.4665 1.66716 0.833582 0.552396i \(-0.186286\pi\)
0.833582 + 0.552396i \(0.186286\pi\)
\(954\) 0 0
\(955\) 20.7055 0.670015
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 20.0000 0.645834
\(960\) 0 0
\(961\) −16.0718 −0.518445
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −10.3528 −0.333267
\(966\) 0 0
\(967\) 15.1774 0.488073 0.244036 0.969766i \(-0.421528\pi\)
0.244036 + 0.969766i \(0.421528\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −43.7128 −1.40281 −0.701405 0.712762i \(-0.747444\pi\)
−0.701405 + 0.712762i \(0.747444\pi\)
\(972\) 0 0
\(973\) −51.7128 −1.65784
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −13.4586 −0.430578 −0.215289 0.976550i \(-0.569069\pi\)
−0.215289 + 0.976550i \(0.569069\pi\)
\(978\) 0 0
\(979\) 21.1117 0.674732
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 32.3923 1.03315 0.516577 0.856241i \(-0.327206\pi\)
0.516577 + 0.856241i \(0.327206\pi\)
\(984\) 0 0
\(985\) −3.21539 −0.102451
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 39.1918 1.24623
\(990\) 0 0
\(991\) 13.3843 0.425165 0.212583 0.977143i \(-0.431813\pi\)
0.212583 + 0.977143i \(0.431813\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 3.71281 0.117704
\(996\) 0 0
\(997\) −13.7128 −0.434289 −0.217145 0.976139i \(-0.569674\pi\)
−0.217145 + 0.976139i \(0.569674\pi\)
\(998\) 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.bc.1.2 4
3.2 odd 2 3744.2.a.bd.1.3 yes 4
4.3 odd 2 3744.2.a.bd.1.2 yes 4
8.3 odd 2 7488.2.a.db.1.3 4
8.5 even 2 7488.2.a.dc.1.3 4
12.11 even 2 inner 3744.2.a.bc.1.3 yes 4
24.5 odd 2 7488.2.a.db.1.2 4
24.11 even 2 7488.2.a.dc.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3744.2.a.bc.1.2 4 1.1 even 1 trivial
3744.2.a.bc.1.3 yes 4 12.11 even 2 inner
3744.2.a.bd.1.2 yes 4 4.3 odd 2
3744.2.a.bd.1.3 yes 4 3.2 odd 2
7488.2.a.db.1.2 4 24.5 odd 2
7488.2.a.db.1.3 4 8.3 odd 2
7488.2.a.dc.1.2 4 24.11 even 2
7488.2.a.dc.1.3 4 8.5 even 2