Properties

Label 8640.2.a.cr.1.2
Level $8640$
Weight $2$
Character 8640.1
Self dual yes
Analytic conductor $68.991$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8640,2,Mod(1,8640)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8640, 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("8640.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8640 = 2^{6} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8640.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: \(68.9907473464\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 135)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 8640.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{5} +4.60555 q^{7} +O(q^{10})\) \(q-1.00000 q^{5} +4.60555 q^{7} +2.60555 q^{11} +0.605551 q^{13} -5.60555 q^{17} +3.60555 q^{19} -3.00000 q^{23} +1.00000 q^{25} -8.60555 q^{29} +1.60555 q^{31} -4.60555 q^{35} -2.00000 q^{37} +2.60555 q^{41} +6.60555 q^{43} +5.21110 q^{47} +14.2111 q^{49} +5.60555 q^{53} -2.60555 q^{55} +8.60555 q^{59} -10.2111 q^{61} -0.605551 q^{65} +15.2111 q^{67} +14.6056 q^{71} +5.39445 q^{73} +12.0000 q^{77} -4.39445 q^{79} -3.00000 q^{83} +5.60555 q^{85} +7.81665 q^{89} +2.78890 q^{91} -3.60555 q^{95} +8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + 2 q^{7} - 2 q^{11} - 6 q^{13} - 4 q^{17} - 6 q^{23} + 2 q^{25} - 10 q^{29} - 4 q^{31} - 2 q^{35} - 4 q^{37} - 2 q^{41} + 6 q^{43} - 4 q^{47} + 14 q^{49} + 4 q^{53} + 2 q^{55} + 10 q^{59} - 6 q^{61} + 6 q^{65} + 16 q^{67} + 22 q^{71} + 18 q^{73} + 24 q^{77} - 16 q^{79} - 6 q^{83} + 4 q^{85} - 6 q^{89} + 20 q^{91} + 16 q^{97}+O(q^{100}) \) 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
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.60555 1.74073 0.870367 0.492403i \(-0.163881\pi\)
0.870367 + 0.492403i \(0.163881\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.60555 0.785603 0.392802 0.919623i \(-0.371506\pi\)
0.392802 + 0.919623i \(0.371506\pi\)
\(12\) 0 0
\(13\) 0.605551 0.167950 0.0839749 0.996468i \(-0.473238\pi\)
0.0839749 + 0.996468i \(0.473238\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.60555 −1.35955 −0.679773 0.733423i \(-0.737922\pi\)
−0.679773 + 0.733423i \(0.737922\pi\)
\(18\) 0 0
\(19\) 3.60555 0.827170 0.413585 0.910465i \(-0.364276\pi\)
0.413585 + 0.910465i \(0.364276\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −8.60555 −1.59801 −0.799005 0.601324i \(-0.794640\pi\)
−0.799005 + 0.601324i \(0.794640\pi\)
\(30\) 0 0
\(31\) 1.60555 0.288366 0.144183 0.989551i \(-0.453945\pi\)
0.144183 + 0.989551i \(0.453945\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.60555 −0.778480
\(36\) 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
\(41\) 2.60555 0.406919 0.203459 0.979083i \(-0.434782\pi\)
0.203459 + 0.979083i \(0.434782\pi\)
\(42\) 0 0
\(43\) 6.60555 1.00734 0.503669 0.863897i \(-0.331983\pi\)
0.503669 + 0.863897i \(0.331983\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.21110 0.760117 0.380059 0.924962i \(-0.375904\pi\)
0.380059 + 0.924962i \(0.375904\pi\)
\(48\) 0 0
\(49\) 14.2111 2.03016
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.60555 0.769982 0.384991 0.922920i \(-0.374205\pi\)
0.384991 + 0.922920i \(0.374205\pi\)
\(54\) 0 0
\(55\) −2.60555 −0.351332
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.60555 1.12035 0.560174 0.828375i \(-0.310734\pi\)
0.560174 + 0.828375i \(0.310734\pi\)
\(60\) 0 0
\(61\) −10.2111 −1.30740 −0.653699 0.756755i \(-0.726784\pi\)
−0.653699 + 0.756755i \(0.726784\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.605551 −0.0751094
\(66\) 0 0
\(67\) 15.2111 1.85833 0.929166 0.369663i \(-0.120527\pi\)
0.929166 + 0.369663i \(0.120527\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 14.6056 1.73336 0.866680 0.498864i \(-0.166249\pi\)
0.866680 + 0.498864i \(0.166249\pi\)
\(72\) 0 0
\(73\) 5.39445 0.631372 0.315686 0.948864i \(-0.397765\pi\)
0.315686 + 0.948864i \(0.397765\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 12.0000 1.36753
\(78\) 0 0
\(79\) −4.39445 −0.494414 −0.247207 0.968963i \(-0.579513\pi\)
−0.247207 + 0.968963i \(0.579513\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.00000 −0.329293 −0.164646 0.986353i \(-0.552648\pi\)
−0.164646 + 0.986353i \(0.552648\pi\)
\(84\) 0 0
\(85\) 5.60555 0.608007
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.81665 0.828564 0.414282 0.910149i \(-0.364033\pi\)
0.414282 + 0.910149i \(0.364033\pi\)
\(90\) 0 0
\(91\) 2.78890 0.292356
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.60555 −0.369922
\(96\) 0 0
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 7.00000 0.670478 0.335239 0.942133i \(-0.391183\pi\)
0.335239 + 0.942133i \(0.391183\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −0.788897 −0.0742132 −0.0371066 0.999311i \(-0.511814\pi\)
−0.0371066 + 0.999311i \(0.511814\pi\)
\(114\) 0 0
\(115\) 3.00000 0.279751
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −25.8167 −2.36661
\(120\) 0 0
\(121\) −4.21110 −0.382828
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −4.78890 −0.424946 −0.212473 0.977167i \(-0.568152\pi\)
−0.212473 + 0.977167i \(0.568152\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) 16.6056 1.43988
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −4.81665 −0.411515 −0.205757 0.978603i \(-0.565966\pi\)
−0.205757 + 0.978603i \(0.565966\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.57779 0.131942
\(144\) 0 0
\(145\) 8.60555 0.714652
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 13.0278 1.06728 0.533638 0.845713i \(-0.320825\pi\)
0.533638 + 0.845713i \(0.320825\pi\)
\(150\) 0 0
\(151\) −14.4222 −1.17366 −0.586831 0.809709i \(-0.699625\pi\)
−0.586831 + 0.809709i \(0.699625\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −1.60555 −0.128961
\(156\) 0 0
\(157\) −3.81665 −0.304602 −0.152301 0.988334i \(-0.548668\pi\)
−0.152301 + 0.988334i \(0.548668\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −13.8167 −1.08890
\(162\) 0 0
\(163\) −2.00000 −0.156652 −0.0783260 0.996928i \(-0.524958\pi\)
−0.0783260 + 0.996928i \(0.524958\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3.00000 −0.232147 −0.116073 0.993241i \(-0.537031\pi\)
−0.116073 + 0.993241i \(0.537031\pi\)
\(168\) 0 0
\(169\) −12.6333 −0.971793
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −10.8167 −0.822375 −0.411187 0.911551i \(-0.634886\pi\)
−0.411187 + 0.911551i \(0.634886\pi\)
\(174\) 0 0
\(175\) 4.60555 0.348147
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 6.78890 0.507426 0.253713 0.967280i \(-0.418348\pi\)
0.253713 + 0.967280i \(0.418348\pi\)
\(180\) 0 0
\(181\) 7.00000 0.520306 0.260153 0.965567i \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2.00000 0.147043
\(186\) 0 0
\(187\) −14.6056 −1.06806
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.4222 1.18827 0.594135 0.804366i \(-0.297495\pi\)
0.594135 + 0.804366i \(0.297495\pi\)
\(192\) 0 0
\(193\) 21.8167 1.57040 0.785199 0.619244i \(-0.212561\pi\)
0.785199 + 0.619244i \(0.212561\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.18335 0.0843099 0.0421550 0.999111i \(-0.486578\pi\)
0.0421550 + 0.999111i \(0.486578\pi\)
\(198\) 0 0
\(199\) 13.2111 0.936510 0.468255 0.883593i \(-0.344883\pi\)
0.468255 + 0.883593i \(0.344883\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −39.6333 −2.78171
\(204\) 0 0
\(205\) −2.60555 −0.181980
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 9.39445 0.649828
\(210\) 0 0
\(211\) −12.8167 −0.882335 −0.441167 0.897425i \(-0.645436\pi\)
−0.441167 + 0.897425i \(0.645436\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −6.60555 −0.450495
\(216\) 0 0
\(217\) 7.39445 0.501968
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −3.39445 −0.228335
\(222\) 0 0
\(223\) −10.0000 −0.669650 −0.334825 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 26.2111 1.73969 0.869846 0.493323i \(-0.164218\pi\)
0.869846 + 0.493323i \(0.164218\pi\)
\(228\) 0 0
\(229\) 6.21110 0.410441 0.205221 0.978716i \(-0.434209\pi\)
0.205221 + 0.978716i \(0.434209\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) −5.21110 −0.339935
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −0.788897 −0.0510295 −0.0255148 0.999674i \(-0.508122\pi\)
−0.0255148 + 0.999674i \(0.508122\pi\)
\(240\) 0 0
\(241\) 28.2111 1.81724 0.908618 0.417627i \(-0.137138\pi\)
0.908618 + 0.417627i \(0.137138\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −14.2111 −0.907914
\(246\) 0 0
\(247\) 2.18335 0.138923
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 15.6333 0.986766 0.493383 0.869812i \(-0.335760\pi\)
0.493383 + 0.869812i \(0.335760\pi\)
\(252\) 0 0
\(253\) −7.81665 −0.491429
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 22.8167 1.42326 0.711632 0.702553i \(-0.247956\pi\)
0.711632 + 0.702553i \(0.247956\pi\)
\(258\) 0 0
\(259\) −9.21110 −0.572350
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −17.2111 −1.06128 −0.530641 0.847597i \(-0.678049\pi\)
−0.530641 + 0.847597i \(0.678049\pi\)
\(264\) 0 0
\(265\) −5.60555 −0.344346
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −11.2111 −0.683553 −0.341776 0.939781i \(-0.611029\pi\)
−0.341776 + 0.939781i \(0.611029\pi\)
\(270\) 0 0
\(271\) −19.2389 −1.16868 −0.584339 0.811510i \(-0.698646\pi\)
−0.584339 + 0.811510i \(0.698646\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.60555 0.157121
\(276\) 0 0
\(277\) 29.0278 1.74411 0.872054 0.489409i \(-0.162787\pi\)
0.872054 + 0.489409i \(0.162787\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.81665 0.108372 0.0541862 0.998531i \(-0.482744\pi\)
0.0541862 + 0.998531i \(0.482744\pi\)
\(282\) 0 0
\(283\) −10.6056 −0.630435 −0.315217 0.949020i \(-0.602077\pi\)
−0.315217 + 0.949020i \(0.602077\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) 0 0
\(289\) 14.4222 0.848365
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 28.8167 1.68349 0.841743 0.539878i \(-0.181530\pi\)
0.841743 + 0.539878i \(0.181530\pi\)
\(294\) 0 0
\(295\) −8.60555 −0.501035
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.81665 −0.105060
\(300\) 0 0
\(301\) 30.4222 1.75351
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 10.2111 0.584686
\(306\) 0 0
\(307\) 20.4222 1.16556 0.582778 0.812631i \(-0.301966\pi\)
0.582778 + 0.812631i \(0.301966\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 13.8167 0.783471 0.391735 0.920078i \(-0.371875\pi\)
0.391735 + 0.920078i \(0.371875\pi\)
\(312\) 0 0
\(313\) 23.6333 1.33583 0.667917 0.744236i \(-0.267186\pi\)
0.667917 + 0.744236i \(0.267186\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0.394449 0.0221544 0.0110772 0.999939i \(-0.496474\pi\)
0.0110772 + 0.999939i \(0.496474\pi\)
\(318\) 0 0
\(319\) −22.4222 −1.25540
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −20.2111 −1.12458
\(324\) 0 0
\(325\) 0.605551 0.0335899
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) −14.7889 −0.812871 −0.406436 0.913679i \(-0.633228\pi\)
−0.406436 + 0.913679i \(0.633228\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −15.2111 −0.831071
\(336\) 0 0
\(337\) −0.605551 −0.0329865 −0.0164932 0.999864i \(-0.505250\pi\)
−0.0164932 + 0.999864i \(0.505250\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 4.18335 0.226541
\(342\) 0 0
\(343\) 33.2111 1.79323
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.57779 −0.0847005 −0.0423502 0.999103i \(-0.513485\pi\)
−0.0423502 + 0.999103i \(0.513485\pi\)
\(348\) 0 0
\(349\) −25.8444 −1.38342 −0.691710 0.722176i \(-0.743142\pi\)
−0.691710 + 0.722176i \(0.743142\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −21.6333 −1.15142 −0.575712 0.817652i \(-0.695275\pi\)
−0.575712 + 0.817652i \(0.695275\pi\)
\(354\) 0 0
\(355\) −14.6056 −0.775182
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 33.6333 1.77510 0.887549 0.460713i \(-0.152406\pi\)
0.887549 + 0.460713i \(0.152406\pi\)
\(360\) 0 0
\(361\) −6.00000 −0.315789
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −5.39445 −0.282358
\(366\) 0 0
\(367\) 4.60555 0.240408 0.120204 0.992749i \(-0.461645\pi\)
0.120204 + 0.992749i \(0.461645\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 25.8167 1.34033
\(372\) 0 0
\(373\) 10.7889 0.558628 0.279314 0.960200i \(-0.409893\pi\)
0.279314 + 0.960200i \(0.409893\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −5.21110 −0.268385
\(378\) 0 0
\(379\) −14.3944 −0.739393 −0.369697 0.929153i \(-0.620538\pi\)
−0.369697 + 0.929153i \(0.620538\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −18.6333 −0.952118 −0.476059 0.879413i \(-0.657935\pi\)
−0.476059 + 0.879413i \(0.657935\pi\)
\(384\) 0 0
\(385\) −12.0000 −0.611577
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −4.18335 −0.212104 −0.106052 0.994361i \(-0.533821\pi\)
−0.106052 + 0.994361i \(0.533821\pi\)
\(390\) 0 0
\(391\) 16.8167 0.850455
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 4.39445 0.221109
\(396\) 0 0
\(397\) 12.6056 0.632654 0.316327 0.948650i \(-0.397550\pi\)
0.316327 + 0.948650i \(0.397550\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) 0.972244 0.0484309
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −5.21110 −0.258305
\(408\) 0 0
\(409\) 5.00000 0.247234 0.123617 0.992330i \(-0.460551\pi\)
0.123617 + 0.992330i \(0.460551\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 39.6333 1.95023
\(414\) 0 0
\(415\) 3.00000 0.147264
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 13.0278 0.636448 0.318224 0.948016i \(-0.396914\pi\)
0.318224 + 0.948016i \(0.396914\pi\)
\(420\) 0 0
\(421\) 23.4222 1.14153 0.570764 0.821114i \(-0.306647\pi\)
0.570764 + 0.821114i \(0.306647\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −5.60555 −0.271909
\(426\) 0 0
\(427\) −47.0278 −2.27583
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 25.8167 1.24354 0.621772 0.783198i \(-0.286413\pi\)
0.621772 + 0.783198i \(0.286413\pi\)
\(432\) 0 0
\(433\) −28.2389 −1.35707 −0.678536 0.734567i \(-0.737385\pi\)
−0.678536 + 0.734567i \(0.737385\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −10.8167 −0.517431
\(438\) 0 0
\(439\) 20.3944 0.973374 0.486687 0.873576i \(-0.338205\pi\)
0.486687 + 0.873576i \(0.338205\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −18.6333 −0.885295 −0.442648 0.896696i \(-0.645961\pi\)
−0.442648 + 0.896696i \(0.645961\pi\)
\(444\) 0 0
\(445\) −7.81665 −0.370545
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −12.2389 −0.577587 −0.288794 0.957391i \(-0.593254\pi\)
−0.288794 + 0.957391i \(0.593254\pi\)
\(450\) 0 0
\(451\) 6.78890 0.319677
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −2.78890 −0.130746
\(456\) 0 0
\(457\) 1.21110 0.0566530 0.0283265 0.999599i \(-0.490982\pi\)
0.0283265 + 0.999599i \(0.490982\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 21.6333 1.00756 0.503782 0.863831i \(-0.331942\pi\)
0.503782 + 0.863831i \(0.331942\pi\)
\(462\) 0 0
\(463\) −15.2111 −0.706920 −0.353460 0.935450i \(-0.614995\pi\)
−0.353460 + 0.935450i \(0.614995\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2.21110 0.102318 0.0511588 0.998691i \(-0.483709\pi\)
0.0511588 + 0.998691i \(0.483709\pi\)
\(468\) 0 0
\(469\) 70.0555 3.23486
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 17.2111 0.791367
\(474\) 0 0
\(475\) 3.60555 0.165434
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −16.1833 −0.739436 −0.369718 0.929144i \(-0.620546\pi\)
−0.369718 + 0.929144i \(0.620546\pi\)
\(480\) 0 0
\(481\) −1.21110 −0.0552215
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −8.00000 −0.363261
\(486\) 0 0
\(487\) −8.18335 −0.370823 −0.185411 0.982661i \(-0.559362\pi\)
−0.185411 + 0.982661i \(0.559362\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −12.7889 −0.577155 −0.288577 0.957457i \(-0.593182\pi\)
−0.288577 + 0.957457i \(0.593182\pi\)
\(492\) 0 0
\(493\) 48.2389 2.17257
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 67.2666 3.01732
\(498\) 0 0
\(499\) 27.6056 1.23579 0.617897 0.786259i \(-0.287985\pi\)
0.617897 + 0.786259i \(0.287985\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 31.4222 1.40105 0.700523 0.713629i \(-0.252950\pi\)
0.700523 + 0.713629i \(0.252950\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −32.8444 −1.45580 −0.727901 0.685682i \(-0.759504\pi\)
−0.727901 + 0.685682i \(0.759504\pi\)
\(510\) 0 0
\(511\) 24.8444 1.09905
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 4.00000 0.176261
\(516\) 0 0
\(517\) 13.5778 0.597151
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −21.3944 −0.937308 −0.468654 0.883382i \(-0.655261\pi\)
−0.468654 + 0.883382i \(0.655261\pi\)
\(522\) 0 0
\(523\) −23.3944 −1.02297 −0.511484 0.859293i \(-0.670904\pi\)
−0.511484 + 0.859293i \(0.670904\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −9.00000 −0.392046
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1.57779 0.0683419
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 37.0278 1.59490
\(540\) 0 0
\(541\) 11.5778 0.497768 0.248884 0.968533i \(-0.419936\pi\)
0.248884 + 0.968533i \(0.419936\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −7.00000 −0.299847
\(546\) 0 0
\(547\) 6.60555 0.282433 0.141216 0.989979i \(-0.454899\pi\)
0.141216 + 0.989979i \(0.454899\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −31.0278 −1.32183
\(552\) 0 0
\(553\) −20.2389 −0.860644
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 33.6333 1.42509 0.712544 0.701627i \(-0.247543\pi\)
0.712544 + 0.701627i \(0.247543\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) 0.788897 0.0331892
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 37.8167 1.58536 0.792678 0.609640i \(-0.208686\pi\)
0.792678 + 0.609640i \(0.208686\pi\)
\(570\) 0 0
\(571\) 36.4500 1.52538 0.762692 0.646762i \(-0.223877\pi\)
0.762692 + 0.646762i \(0.223877\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −3.00000 −0.125109
\(576\) 0 0
\(577\) 27.8167 1.15802 0.579011 0.815320i \(-0.303439\pi\)
0.579011 + 0.815320i \(0.303439\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −13.8167 −0.573211
\(582\) 0 0
\(583\) 14.6056 0.604900
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −21.0000 −0.866763 −0.433381 0.901211i \(-0.642680\pi\)
−0.433381 + 0.901211i \(0.642680\pi\)
\(588\) 0 0
\(589\) 5.78890 0.238527
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 21.2389 0.872175 0.436088 0.899904i \(-0.356364\pi\)
0.436088 + 0.899904i \(0.356364\pi\)
\(594\) 0 0
\(595\) 25.8167 1.05838
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −15.3944 −0.629000 −0.314500 0.949257i \(-0.601837\pi\)
−0.314500 + 0.949257i \(0.601837\pi\)
\(600\) 0 0
\(601\) 32.6333 1.33114 0.665570 0.746335i \(-0.268188\pi\)
0.665570 + 0.746335i \(0.268188\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 4.21110 0.171206
\(606\) 0 0
\(607\) 17.3944 0.706019 0.353009 0.935620i \(-0.385158\pi\)
0.353009 + 0.935620i \(0.385158\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3.15559 0.127661
\(612\) 0 0
\(613\) −28.8444 −1.16501 −0.582507 0.812825i \(-0.697928\pi\)
−0.582507 + 0.812825i \(0.697928\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −26.4500 −1.06484 −0.532418 0.846482i \(-0.678716\pi\)
−0.532418 + 0.846482i \(0.678716\pi\)
\(618\) 0 0
\(619\) 7.63331 0.306809 0.153404 0.988164i \(-0.450976\pi\)
0.153404 + 0.988164i \(0.450976\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 36.0000 1.44231
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 11.2111 0.447016
\(630\) 0 0
\(631\) 30.0278 1.19539 0.597693 0.801725i \(-0.296084\pi\)
0.597693 + 0.801725i \(0.296084\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 4.78890 0.190042
\(636\) 0 0
\(637\) 8.60555 0.340964
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 0 0
\(643\) 23.0278 0.908126 0.454063 0.890970i \(-0.349974\pi\)
0.454063 + 0.890970i \(0.349974\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 38.2111 1.50223 0.751117 0.660169i \(-0.229516\pi\)
0.751117 + 0.660169i \(0.229516\pi\)
\(648\) 0 0
\(649\) 22.4222 0.880149
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −27.2389 −1.06594 −0.532969 0.846134i \(-0.678924\pi\)
−0.532969 + 0.846134i \(0.678924\pi\)
\(654\) 0 0
\(655\) 6.00000 0.234439
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 20.6056 0.802678 0.401339 0.915930i \(-0.368545\pi\)
0.401339 + 0.915930i \(0.368545\pi\)
\(660\) 0 0
\(661\) −22.8444 −0.888545 −0.444272 0.895892i \(-0.646538\pi\)
−0.444272 + 0.895892i \(0.646538\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −16.6056 −0.643936
\(666\) 0 0
\(667\) 25.8167 0.999625
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −26.6056 −1.02710
\(672\) 0 0
\(673\) −11.0278 −0.425089 −0.212544 0.977151i \(-0.568175\pi\)
−0.212544 + 0.977151i \(0.568175\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 21.6333 0.831436 0.415718 0.909494i \(-0.363530\pi\)
0.415718 + 0.909494i \(0.363530\pi\)
\(678\) 0 0
\(679\) 36.8444 1.41396
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 9.00000 0.344375 0.172188 0.985064i \(-0.444916\pi\)
0.172188 + 0.985064i \(0.444916\pi\)
\(684\) 0 0
\(685\) 4.81665 0.184035
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 3.39445 0.129318
\(690\) 0 0
\(691\) −2.39445 −0.0910891 −0.0455446 0.998962i \(-0.514502\pi\)
−0.0455446 + 0.998962i \(0.514502\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) −14.6056 −0.553225
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −40.4222 −1.52673 −0.763363 0.645970i \(-0.776453\pi\)
−0.763363 + 0.645970i \(0.776453\pi\)
\(702\) 0 0
\(703\) −7.21110 −0.271972
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −55.2666 −2.07851
\(708\) 0 0
\(709\) −34.8444 −1.30861 −0.654305 0.756231i \(-0.727039\pi\)
−0.654305 + 0.756231i \(0.727039\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −4.81665 −0.180385
\(714\) 0 0
\(715\) −1.57779 −0.0590062
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −49.2666 −1.83733 −0.918667 0.395032i \(-0.870733\pi\)
−0.918667 + 0.395032i \(0.870733\pi\)
\(720\) 0 0
\(721\) −18.4222 −0.686079
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −8.60555 −0.319602
\(726\) 0 0
\(727\) −7.63331 −0.283104 −0.141552 0.989931i \(-0.545209\pi\)
−0.141552 + 0.989931i \(0.545209\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −37.0278 −1.36952
\(732\) 0 0
\(733\) −32.0000 −1.18195 −0.590973 0.806691i \(-0.701256\pi\)
−0.590973 + 0.806691i \(0.701256\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 39.6333 1.45991
\(738\) 0 0
\(739\) −30.0278 −1.10459 −0.552294 0.833649i \(-0.686248\pi\)
−0.552294 + 0.833649i \(0.686248\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −34.4222 −1.26283 −0.631414 0.775446i \(-0.717525\pi\)
−0.631414 + 0.775446i \(0.717525\pi\)
\(744\) 0 0
\(745\) −13.0278 −0.477300
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 6.02776 0.219956 0.109978 0.993934i \(-0.464922\pi\)
0.109978 + 0.993934i \(0.464922\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 14.4222 0.524878
\(756\) 0 0
\(757\) −31.2111 −1.13439 −0.567193 0.823585i \(-0.691971\pi\)
−0.567193 + 0.823585i \(0.691971\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −53.4500 −1.93756 −0.968780 0.247923i \(-0.920252\pi\)
−0.968780 + 0.247923i \(0.920252\pi\)
\(762\) 0 0
\(763\) 32.2389 1.16713
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 5.21110 0.188162
\(768\) 0 0
\(769\) 41.0000 1.47850 0.739249 0.673432i \(-0.235181\pi\)
0.739249 + 0.673432i \(0.235181\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −16.8167 −0.604853 −0.302426 0.953173i \(-0.597797\pi\)
−0.302426 + 0.953173i \(0.597797\pi\)
\(774\) 0 0
\(775\) 1.60555 0.0576731
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 9.39445 0.336591
\(780\) 0 0
\(781\) 38.0555 1.36173
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 3.81665 0.136222
\(786\) 0 0
\(787\) 2.97224 0.105949 0.0529745 0.998596i \(-0.483130\pi\)
0.0529745 + 0.998596i \(0.483130\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −3.63331 −0.129186
\(792\) 0 0
\(793\) −6.18335 −0.219577
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 37.6611 1.33402 0.667012 0.745047i \(-0.267573\pi\)
0.667012 + 0.745047i \(0.267573\pi\)
\(798\) 0 0
\(799\) −29.2111 −1.03341
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 14.0555 0.496008
\(804\) 0 0
\(805\) 13.8167 0.486973
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 50.6056 1.77920 0.889598 0.456744i \(-0.150984\pi\)
0.889598 + 0.456744i \(0.150984\pi\)
\(810\) 0 0
\(811\) −42.4222 −1.48965 −0.744823 0.667263i \(-0.767466\pi\)
−0.744823 + 0.667263i \(0.767466\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 2.00000 0.0700569
\(816\) 0 0
\(817\) 23.8167 0.833239
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −30.0000 −1.04701 −0.523504 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) 0 0
\(823\) 3.81665 0.133040 0.0665201 0.997785i \(-0.478810\pi\)
0.0665201 + 0.997785i \(0.478810\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 33.7889 1.17496 0.587478 0.809240i \(-0.300121\pi\)
0.587478 + 0.809240i \(0.300121\pi\)
\(828\) 0 0
\(829\) 27.2111 0.945081 0.472540 0.881309i \(-0.343337\pi\)
0.472540 + 0.881309i \(0.343337\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −79.6611 −2.76009
\(834\) 0 0
\(835\) 3.00000 0.103819
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 18.7889 0.648665 0.324332 0.945943i \(-0.394860\pi\)
0.324332 + 0.945943i \(0.394860\pi\)
\(840\) 0 0
\(841\) 45.0555 1.55364
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 12.6333 0.434599
\(846\) 0 0
\(847\) −19.3944 −0.666401
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 6.00000 0.205677
\(852\) 0 0
\(853\) −14.7889 −0.506362 −0.253181 0.967419i \(-0.581477\pi\)
−0.253181 + 0.967419i \(0.581477\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −45.2389 −1.54533 −0.772665 0.634814i \(-0.781077\pi\)
−0.772665 + 0.634814i \(0.781077\pi\)
\(858\) 0 0
\(859\) −6.02776 −0.205664 −0.102832 0.994699i \(-0.532790\pi\)
−0.102832 + 0.994699i \(0.532790\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 15.7889 0.537460 0.268730 0.963216i \(-0.413396\pi\)
0.268730 + 0.963216i \(0.413396\pi\)
\(864\) 0 0
\(865\) 10.8167 0.367777
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −11.4500 −0.388413
\(870\) 0 0
\(871\) 9.21110 0.312106
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −4.60555 −0.155696
\(876\) 0 0
\(877\) 56.6611 1.91331 0.956654 0.291227i \(-0.0940634\pi\)
0.956654 + 0.291227i \(0.0940634\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −32.6056 −1.09851 −0.549254 0.835655i \(-0.685088\pi\)
−0.549254 + 0.835655i \(0.685088\pi\)
\(882\) 0 0
\(883\) 5.81665 0.195746 0.0978730 0.995199i \(-0.468796\pi\)
0.0978730 + 0.995199i \(0.468796\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 12.6333 0.424185 0.212092 0.977250i \(-0.431972\pi\)
0.212092 + 0.977250i \(0.431972\pi\)
\(888\) 0 0
\(889\) −22.0555 −0.739718
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 18.7889 0.628746
\(894\) 0 0
\(895\) −6.78890 −0.226928
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −13.8167 −0.460811
\(900\) 0 0
\(901\) −31.4222 −1.04683
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −7.00000 −0.232688
\(906\) 0 0
\(907\) −30.4222 −1.01015 −0.505076 0.863075i \(-0.668536\pi\)
−0.505076 + 0.863075i \(0.668536\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 31.0278 1.02800 0.513998 0.857792i \(-0.328164\pi\)
0.513998 + 0.857792i \(0.328164\pi\)
\(912\) 0 0
\(913\) −7.81665 −0.258693
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −27.6333 −0.912532
\(918\) 0 0
\(919\) −2.42221 −0.0799012 −0.0399506 0.999202i \(-0.512720\pi\)
−0.0399506 + 0.999202i \(0.512720\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 8.84441 0.291117
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 26.6056 0.872900 0.436450 0.899729i \(-0.356236\pi\)
0.436450 + 0.899729i \(0.356236\pi\)
\(930\) 0 0
\(931\) 51.2389 1.67929
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 14.6056 0.477653
\(936\) 0 0
\(937\) 26.7889 0.875155 0.437578 0.899181i \(-0.355837\pi\)
0.437578 + 0.899181i \(0.355837\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −28.4222 −0.926537 −0.463269 0.886218i \(-0.653324\pi\)
−0.463269 + 0.886218i \(0.653324\pi\)
\(942\) 0 0
\(943\) −7.81665 −0.254545
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −39.0000 −1.26733 −0.633665 0.773608i \(-0.718450\pi\)
−0.633665 + 0.773608i \(0.718450\pi\)
\(948\) 0 0
\(949\) 3.26662 0.106039
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 26.8444 0.869576 0.434788 0.900533i \(-0.356823\pi\)
0.434788 + 0.900533i \(0.356823\pi\)
\(954\) 0 0
\(955\) −16.4222 −0.531410
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −22.1833 −0.716338
\(960\) 0 0
\(961\) −28.4222 −0.916845
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −21.8167 −0.702303
\(966\) 0 0
\(967\) 50.0000 1.60789 0.803946 0.594703i \(-0.202730\pi\)
0.803946 + 0.594703i \(0.202730\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 18.0000 0.577647 0.288824 0.957382i \(-0.406736\pi\)
0.288824 + 0.957382i \(0.406736\pi\)
\(972\) 0 0
\(973\) 18.4222 0.590589
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −0.788897 −0.0252391 −0.0126195 0.999920i \(-0.504017\pi\)
−0.0126195 + 0.999920i \(0.504017\pi\)
\(978\) 0 0
\(979\) 20.3667 0.650922
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0.633308 0.0201994 0.0100997 0.999949i \(-0.496785\pi\)
0.0100997 + 0.999949i \(0.496785\pi\)
\(984\) 0 0
\(985\) −1.18335 −0.0377045
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −19.8167 −0.630133
\(990\) 0 0
\(991\) −50.8167 −1.61424 −0.807122 0.590385i \(-0.798976\pi\)
−0.807122 + 0.590385i \(0.798976\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −13.2111 −0.418820
\(996\) 0 0
\(997\) −53.8722 −1.70615 −0.853074 0.521789i \(-0.825265\pi\)
−0.853074 + 0.521789i \(0.825265\pi\)
\(998\) 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 8640.2.a.cr.1.2 2
3.2 odd 2 8640.2.a.df.1.2 2
4.3 odd 2 8640.2.a.ck.1.1 2
8.3 odd 2 2160.2.a.ba.1.1 2
8.5 even 2 135.2.a.c.1.2 2
12.11 even 2 8640.2.a.cy.1.1 2
24.5 odd 2 135.2.a.d.1.1 yes 2
24.11 even 2 2160.2.a.y.1.1 2
40.13 odd 4 675.2.b.i.649.2 4
40.29 even 2 675.2.a.p.1.1 2
40.37 odd 4 675.2.b.i.649.3 4
56.13 odd 2 6615.2.a.p.1.2 2
72.5 odd 6 405.2.e.j.136.2 4
72.13 even 6 405.2.e.k.136.1 4
72.29 odd 6 405.2.e.j.271.2 4
72.61 even 6 405.2.e.k.271.1 4
120.29 odd 2 675.2.a.k.1.2 2
120.53 even 4 675.2.b.h.649.3 4
120.77 even 4 675.2.b.h.649.2 4
168.125 even 2 6615.2.a.v.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.2.a.c.1.2 2 8.5 even 2
135.2.a.d.1.1 yes 2 24.5 odd 2
405.2.e.j.136.2 4 72.5 odd 6
405.2.e.j.271.2 4 72.29 odd 6
405.2.e.k.136.1 4 72.13 even 6
405.2.e.k.271.1 4 72.61 even 6
675.2.a.k.1.2 2 120.29 odd 2
675.2.a.p.1.1 2 40.29 even 2
675.2.b.h.649.2 4 120.77 even 4
675.2.b.h.649.3 4 120.53 even 4
675.2.b.i.649.2 4 40.13 odd 4
675.2.b.i.649.3 4 40.37 odd 4
2160.2.a.y.1.1 2 24.11 even 2
2160.2.a.ba.1.1 2 8.3 odd 2
6615.2.a.p.1.2 2 56.13 odd 2
6615.2.a.v.1.1 2 168.125 even 2
8640.2.a.ck.1.1 2 4.3 odd 2
8640.2.a.cr.1.2 2 1.1 even 1 trivial
8640.2.a.cy.1.1 2 12.11 even 2
8640.2.a.df.1.2 2 3.2 odd 2