Properties

Label 2156.2.a.f.1.1
Level $2156$
Weight $2$
Character 2156.1
Self dual yes
Analytic conductor $17.216$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2156,2,Mod(1,2156)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2156, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2156.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2156 = 2^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2156.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: \(17.2157466758\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2156.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.82843 q^{3} -1.41421 q^{5} +5.00000 q^{9} +O(q^{10})\) \(q-2.82843 q^{3} -1.41421 q^{5} +5.00000 q^{9} -1.00000 q^{11} +1.41421 q^{13} +4.00000 q^{15} -7.07107 q^{17} -2.82843 q^{19} +4.00000 q^{23} -3.00000 q^{25} -5.65685 q^{27} +5.65685 q^{31} +2.82843 q^{33} -8.00000 q^{37} -4.00000 q^{39} -9.89949 q^{41} +4.00000 q^{43} -7.07107 q^{45} +20.0000 q^{51} -6.00000 q^{53} +1.41421 q^{55} +8.00000 q^{57} -8.48528 q^{59} +1.41421 q^{61} -2.00000 q^{65} -8.00000 q^{67} -11.3137 q^{69} +8.00000 q^{71} +1.41421 q^{73} +8.48528 q^{75} +16.0000 q^{79} +1.00000 q^{81} +2.82843 q^{83} +10.0000 q^{85} +15.5563 q^{89} -16.0000 q^{93} +4.00000 q^{95} -9.89949 q^{97} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 10 q^{9} - 2 q^{11} + 8 q^{15} + 8 q^{23} - 6 q^{25} - 16 q^{37} - 8 q^{39} + 8 q^{43} + 40 q^{51} - 12 q^{53} + 16 q^{57} - 4 q^{65} - 16 q^{67} + 16 q^{71} + 32 q^{79} + 2 q^{81} + 20 q^{85} - 32 q^{93} + 8 q^{95} - 10 q^{99}+O(q^{100}) \) 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
\(3\) −2.82843 −1.63299 −0.816497 0.577350i \(-0.804087\pi\)
−0.816497 + 0.577350i \(0.804087\pi\)
\(4\) 0 0
\(5\) −1.41421 −0.632456 −0.316228 0.948683i \(-0.602416\pi\)
−0.316228 + 0.948683i \(0.602416\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 5.00000 1.66667
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 1.41421 0.392232 0.196116 0.980581i \(-0.437167\pi\)
0.196116 + 0.980581i \(0.437167\pi\)
\(14\) 0 0
\(15\) 4.00000 1.03280
\(16\) 0 0
\(17\) −7.07107 −1.71499 −0.857493 0.514496i \(-0.827979\pi\)
−0.857493 + 0.514496i \(0.827979\pi\)
\(18\) 0 0
\(19\) −2.82843 −0.648886 −0.324443 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) −5.65685 −1.08866
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 5.65685 1.01600 0.508001 0.861357i \(-0.330385\pi\)
0.508001 + 0.861357i \(0.330385\pi\)
\(32\) 0 0
\(33\) 2.82843 0.492366
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −9.89949 −1.54604 −0.773021 0.634381i \(-0.781255\pi\)
−0.773021 + 0.634381i \(0.781255\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) −7.07107 −1.05409
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 20.0000 2.80056
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 1.41421 0.190693
\(56\) 0 0
\(57\) 8.00000 1.05963
\(58\) 0 0
\(59\) −8.48528 −1.10469 −0.552345 0.833616i \(-0.686267\pi\)
−0.552345 + 0.833616i \(0.686267\pi\)
\(60\) 0 0
\(61\) 1.41421 0.181071 0.0905357 0.995893i \(-0.471142\pi\)
0.0905357 + 0.995893i \(0.471142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) −11.3137 −1.36201
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 1.41421 0.165521 0.0827606 0.996569i \(-0.473626\pi\)
0.0827606 + 0.996569i \(0.473626\pi\)
\(74\) 0 0
\(75\) 8.48528 0.979796
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 16.0000 1.80014 0.900070 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 2.82843 0.310460 0.155230 0.987878i \(-0.450388\pi\)
0.155230 + 0.987878i \(0.450388\pi\)
\(84\) 0 0
\(85\) 10.0000 1.08465
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 15.5563 1.64897 0.824485 0.565884i \(-0.191465\pi\)
0.824485 + 0.565884i \(0.191465\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −16.0000 −1.65912
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) −9.89949 −1.00514 −0.502571 0.864536i \(-0.667612\pi\)
−0.502571 + 0.864536i \(0.667612\pi\)
\(98\) 0 0
\(99\) −5.00000 −0.502519
\(100\) 0 0
\(101\) 18.3848 1.82935 0.914677 0.404186i \(-0.132445\pi\)
0.914677 + 0.404186i \(0.132445\pi\)
\(102\) 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
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 0 0
\(109\) 16.0000 1.53252 0.766261 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 0 0
\(111\) 22.6274 2.14770
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) −5.65685 −0.527504
\(116\) 0 0
\(117\) 7.07107 0.653720
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 28.0000 2.52467
\(124\) 0 0
\(125\) 11.3137 1.01193
\(126\) 0 0
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) 0 0
\(129\) −11.3137 −0.996116
\(130\) 0 0
\(131\) −2.82843 −0.247121 −0.123560 0.992337i \(-0.539431\pi\)
−0.123560 + 0.992337i \(0.539431\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 8.00000 0.688530
\(136\) 0 0
\(137\) 16.0000 1.36697 0.683486 0.729964i \(-0.260463\pi\)
0.683486 + 0.729964i \(0.260463\pi\)
\(138\) 0 0
\(139\) 19.7990 1.67933 0.839664 0.543106i \(-0.182752\pi\)
0.839664 + 0.543106i \(0.182752\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1.41421 −0.118262
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −22.0000 −1.80231 −0.901155 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) 0 0
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 0 0
\(153\) −35.3553 −2.85831
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) 0 0
\(157\) 15.5563 1.24153 0.620766 0.783996i \(-0.286822\pi\)
0.620766 + 0.783996i \(0.286822\pi\)
\(158\) 0 0
\(159\) 16.9706 1.34585
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) −4.00000 −0.311400
\(166\) 0 0
\(167\) −5.65685 −0.437741 −0.218870 0.975754i \(-0.570237\pi\)
−0.218870 + 0.975754i \(0.570237\pi\)
\(168\) 0 0
\(169\) −11.0000 −0.846154
\(170\) 0 0
\(171\) −14.1421 −1.08148
\(172\) 0 0
\(173\) −1.41421 −0.107521 −0.0537603 0.998554i \(-0.517121\pi\)
−0.0537603 + 0.998554i \(0.517121\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 24.0000 1.80395
\(178\) 0 0
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 0 0
\(181\) −12.7279 −0.946059 −0.473029 0.881047i \(-0.656840\pi\)
−0.473029 + 0.881047i \(0.656840\pi\)
\(182\) 0 0
\(183\) −4.00000 −0.295689
\(184\) 0 0
\(185\) 11.3137 0.831800
\(186\) 0 0
\(187\) 7.07107 0.517088
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 0 0
\(195\) 5.65685 0.405096
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 16.9706 1.20301 0.601506 0.798869i \(-0.294568\pi\)
0.601506 + 0.798869i \(0.294568\pi\)
\(200\) 0 0
\(201\) 22.6274 1.59601
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 14.0000 0.977802
\(206\) 0 0
\(207\) 20.0000 1.39010
\(208\) 0 0
\(209\) 2.82843 0.195646
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 0 0
\(213\) −22.6274 −1.55041
\(214\) 0 0
\(215\) −5.65685 −0.385794
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) −10.0000 −0.672673
\(222\) 0 0
\(223\) 11.3137 0.757622 0.378811 0.925474i \(-0.376333\pi\)
0.378811 + 0.925474i \(0.376333\pi\)
\(224\) 0 0
\(225\) −15.0000 −1.00000
\(226\) 0 0
\(227\) 2.82843 0.187729 0.0938647 0.995585i \(-0.470078\pi\)
0.0938647 + 0.995585i \(0.470078\pi\)
\(228\) 0 0
\(229\) −12.7279 −0.841085 −0.420542 0.907273i \(-0.638160\pi\)
−0.420542 + 0.907273i \(0.638160\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −45.2548 −2.93962
\(238\) 0 0
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 0 0
\(241\) 26.8701 1.73085 0.865426 0.501036i \(-0.167048\pi\)
0.865426 + 0.501036i \(0.167048\pi\)
\(242\) 0 0
\(243\) 14.1421 0.907218
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) 0 0
\(249\) −8.00000 −0.506979
\(250\) 0 0
\(251\) −14.1421 −0.892644 −0.446322 0.894873i \(-0.647266\pi\)
−0.446322 + 0.894873i \(0.647266\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 0 0
\(255\) −28.2843 −1.77123
\(256\) 0 0
\(257\) 1.41421 0.0882162 0.0441081 0.999027i \(-0.485955\pi\)
0.0441081 + 0.999027i \(0.485955\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 8.48528 0.521247
\(266\) 0 0
\(267\) −44.0000 −2.69276
\(268\) 0 0
\(269\) 7.07107 0.431131 0.215565 0.976489i \(-0.430841\pi\)
0.215565 + 0.976489i \(0.430841\pi\)
\(270\) 0 0
\(271\) −16.9706 −1.03089 −0.515444 0.856923i \(-0.672373\pi\)
−0.515444 + 0.856923i \(0.672373\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.00000 0.180907
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 0 0
\(279\) 28.2843 1.69334
\(280\) 0 0
\(281\) 24.0000 1.43172 0.715860 0.698244i \(-0.246035\pi\)
0.715860 + 0.698244i \(0.246035\pi\)
\(282\) 0 0
\(283\) −8.48528 −0.504398 −0.252199 0.967675i \(-0.581154\pi\)
−0.252199 + 0.967675i \(0.581154\pi\)
\(284\) 0 0
\(285\) −11.3137 −0.670166
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 33.0000 1.94118
\(290\) 0 0
\(291\) 28.0000 1.64139
\(292\) 0 0
\(293\) 29.6985 1.73500 0.867502 0.497434i \(-0.165724\pi\)
0.867502 + 0.497434i \(0.165724\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) 0 0
\(297\) 5.65685 0.328244
\(298\) 0 0
\(299\) 5.65685 0.327144
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −52.0000 −2.98732
\(304\) 0 0
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) −14.1421 −0.807134 −0.403567 0.914950i \(-0.632230\pi\)
−0.403567 + 0.914950i \(0.632230\pi\)
\(308\) 0 0
\(309\) −48.0000 −2.73062
\(310\) 0 0
\(311\) 5.65685 0.320771 0.160385 0.987054i \(-0.448726\pi\)
0.160385 + 0.987054i \(0.448726\pi\)
\(312\) 0 0
\(313\) −26.8701 −1.51879 −0.759393 0.650633i \(-0.774504\pi\)
−0.759393 + 0.650633i \(0.774504\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 22.6274 1.26294
\(322\) 0 0
\(323\) 20.0000 1.11283
\(324\) 0 0
\(325\) −4.24264 −0.235339
\(326\) 0 0
\(327\) −45.2548 −2.50260
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 0 0
\(333\) −40.0000 −2.19199
\(334\) 0 0
\(335\) 11.3137 0.618134
\(336\) 0 0
\(337\) 16.0000 0.871576 0.435788 0.900049i \(-0.356470\pi\)
0.435788 + 0.900049i \(0.356470\pi\)
\(338\) 0 0
\(339\) −16.9706 −0.921714
\(340\) 0 0
\(341\) −5.65685 −0.306336
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 16.0000 0.861411
\(346\) 0 0
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) 9.89949 0.529908 0.264954 0.964261i \(-0.414643\pi\)
0.264954 + 0.964261i \(0.414643\pi\)
\(350\) 0 0
\(351\) −8.00000 −0.427008
\(352\) 0 0
\(353\) −9.89949 −0.526897 −0.263448 0.964673i \(-0.584860\pi\)
−0.263448 + 0.964673i \(0.584860\pi\)
\(354\) 0 0
\(355\) −11.3137 −0.600469
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 20.0000 1.05556 0.527780 0.849381i \(-0.323025\pi\)
0.527780 + 0.849381i \(0.323025\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 0 0
\(363\) −2.82843 −0.148454
\(364\) 0 0
\(365\) −2.00000 −0.104685
\(366\) 0 0
\(367\) 33.9411 1.77171 0.885856 0.463960i \(-0.153572\pi\)
0.885856 + 0.463960i \(0.153572\pi\)
\(368\) 0 0
\(369\) −49.4975 −2.57674
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −38.0000 −1.96757 −0.983783 0.179364i \(-0.942596\pi\)
−0.983783 + 0.179364i \(0.942596\pi\)
\(374\) 0 0
\(375\) −32.0000 −1.65247
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 36.0000 1.84920 0.924598 0.380945i \(-0.124401\pi\)
0.924598 + 0.380945i \(0.124401\pi\)
\(380\) 0 0
\(381\) −33.9411 −1.73886
\(382\) 0 0
\(383\) −22.6274 −1.15621 −0.578103 0.815963i \(-0.696207\pi\)
−0.578103 + 0.815963i \(0.696207\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 20.0000 1.01666
\(388\) 0 0
\(389\) 8.00000 0.405616 0.202808 0.979219i \(-0.434993\pi\)
0.202808 + 0.979219i \(0.434993\pi\)
\(390\) 0 0
\(391\) −28.2843 −1.43040
\(392\) 0 0
\(393\) 8.00000 0.403547
\(394\) 0 0
\(395\) −22.6274 −1.13851
\(396\) 0 0
\(397\) 7.07107 0.354887 0.177443 0.984131i \(-0.443217\pi\)
0.177443 + 0.984131i \(0.443217\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −24.0000 −1.19850 −0.599251 0.800561i \(-0.704535\pi\)
−0.599251 + 0.800561i \(0.704535\pi\)
\(402\) 0 0
\(403\) 8.00000 0.398508
\(404\) 0 0
\(405\) −1.41421 −0.0702728
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) 24.0416 1.18878 0.594391 0.804176i \(-0.297393\pi\)
0.594391 + 0.804176i \(0.297393\pi\)
\(410\) 0 0
\(411\) −45.2548 −2.23226
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 0 0
\(417\) −56.0000 −2.74233
\(418\) 0 0
\(419\) 25.4558 1.24360 0.621800 0.783176i \(-0.286402\pi\)
0.621800 + 0.783176i \(0.286402\pi\)
\(420\) 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
\(425\) 21.2132 1.02899
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 4.00000 0.193122
\(430\) 0 0
\(431\) 4.00000 0.192673 0.0963366 0.995349i \(-0.469287\pi\)
0.0963366 + 0.995349i \(0.469287\pi\)
\(432\) 0 0
\(433\) −12.7279 −0.611665 −0.305832 0.952085i \(-0.598935\pi\)
−0.305832 + 0.952085i \(0.598935\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −11.3137 −0.541208
\(438\) 0 0
\(439\) 16.9706 0.809961 0.404980 0.914325i \(-0.367278\pi\)
0.404980 + 0.914325i \(0.367278\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) −22.0000 −1.04290
\(446\) 0 0
\(447\) 62.2254 2.94316
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 9.89949 0.466149
\(452\) 0 0
\(453\) −11.3137 −0.531564
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 14.0000 0.654892 0.327446 0.944870i \(-0.393812\pi\)
0.327446 + 0.944870i \(0.393812\pi\)
\(458\) 0 0
\(459\) 40.0000 1.86704
\(460\) 0 0
\(461\) −24.0416 −1.11973 −0.559865 0.828584i \(-0.689147\pi\)
−0.559865 + 0.828584i \(0.689147\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 0 0
\(465\) 22.6274 1.04932
\(466\) 0 0
\(467\) −8.48528 −0.392652 −0.196326 0.980539i \(-0.562901\pi\)
−0.196326 + 0.980539i \(0.562901\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −44.0000 −2.02741
\(472\) 0 0
\(473\) −4.00000 −0.183920
\(474\) 0 0
\(475\) 8.48528 0.389331
\(476\) 0 0
\(477\) −30.0000 −1.37361
\(478\) 0 0
\(479\) 33.9411 1.55081 0.775405 0.631464i \(-0.217546\pi\)
0.775405 + 0.631464i \(0.217546\pi\)
\(480\) 0 0
\(481\) −11.3137 −0.515861
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 14.0000 0.635707
\(486\) 0 0
\(487\) −20.0000 −0.906287 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) 0 0
\(489\) 11.3137 0.511624
\(490\) 0 0
\(491\) −32.0000 −1.44414 −0.722070 0.691820i \(-0.756809\pi\)
−0.722070 + 0.691820i \(0.756809\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 7.07107 0.317821
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 24.0000 1.07439 0.537194 0.843459i \(-0.319484\pi\)
0.537194 + 0.843459i \(0.319484\pi\)
\(500\) 0 0
\(501\) 16.0000 0.714827
\(502\) 0 0
\(503\) 5.65685 0.252227 0.126113 0.992016i \(-0.459750\pi\)
0.126113 + 0.992016i \(0.459750\pi\)
\(504\) 0 0
\(505\) −26.0000 −1.15698
\(506\) 0 0
\(507\) 31.1127 1.38176
\(508\) 0 0
\(509\) 29.6985 1.31636 0.658181 0.752860i \(-0.271326\pi\)
0.658181 + 0.752860i \(0.271326\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 16.0000 0.706417
\(514\) 0 0
\(515\) −24.0000 −1.05757
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 4.00000 0.175581
\(520\) 0 0
\(521\) 7.07107 0.309789 0.154895 0.987931i \(-0.450496\pi\)
0.154895 + 0.987931i \(0.450496\pi\)
\(522\) 0 0
\(523\) −42.4264 −1.85518 −0.927589 0.373603i \(-0.878122\pi\)
−0.927589 + 0.373603i \(0.878122\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −40.0000 −1.74243
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −42.4264 −1.84115
\(532\) 0 0
\(533\) −14.0000 −0.606407
\(534\) 0 0
\(535\) 11.3137 0.489134
\(536\) 0 0
\(537\) 45.2548 1.95289
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) 0 0
\(543\) 36.0000 1.54491
\(544\) 0 0
\(545\) −22.6274 −0.969252
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 0 0
\(549\) 7.07107 0.301786
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −32.0000 −1.35832
\(556\) 0 0
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) 5.65685 0.239259
\(560\) 0 0
\(561\) −20.0000 −0.844401
\(562\) 0 0
\(563\) −36.7696 −1.54965 −0.774826 0.632175i \(-0.782163\pi\)
−0.774826 + 0.632175i \(0.782163\pi\)
\(564\) 0 0
\(565\) −8.48528 −0.356978
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 44.0000 1.84134 0.920671 0.390339i \(-0.127642\pi\)
0.920671 + 0.390339i \(0.127642\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −12.0000 −0.500435
\(576\) 0 0
\(577\) 21.2132 0.883117 0.441559 0.897232i \(-0.354426\pi\)
0.441559 + 0.897232i \(0.354426\pi\)
\(578\) 0 0
\(579\) −16.9706 −0.705273
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 6.00000 0.248495
\(584\) 0 0
\(585\) −10.0000 −0.413449
\(586\) 0 0
\(587\) 14.1421 0.583708 0.291854 0.956463i \(-0.405728\pi\)
0.291854 + 0.956463i \(0.405728\pi\)
\(588\) 0 0
\(589\) −16.0000 −0.659269
\(590\) 0 0
\(591\) −16.9706 −0.698076
\(592\) 0 0
\(593\) −7.07107 −0.290374 −0.145187 0.989404i \(-0.546378\pi\)
−0.145187 + 0.989404i \(0.546378\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −48.0000 −1.96451
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 35.3553 1.44217 0.721087 0.692844i \(-0.243643\pi\)
0.721087 + 0.692844i \(0.243643\pi\)
\(602\) 0 0
\(603\) −40.0000 −1.62893
\(604\) 0 0
\(605\) −1.41421 −0.0574960
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 32.0000 1.29247 0.646234 0.763139i \(-0.276343\pi\)
0.646234 + 0.763139i \(0.276343\pi\)
\(614\) 0 0
\(615\) −39.5980 −1.59674
\(616\) 0 0
\(617\) −24.0000 −0.966204 −0.483102 0.875564i \(-0.660490\pi\)
−0.483102 + 0.875564i \(0.660490\pi\)
\(618\) 0 0
\(619\) −8.48528 −0.341052 −0.170526 0.985353i \(-0.554547\pi\)
−0.170526 + 0.985353i \(0.554547\pi\)
\(620\) 0 0
\(621\) −22.6274 −0.908007
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) −8.00000 −0.319489
\(628\) 0 0
\(629\) 56.5685 2.25554
\(630\) 0 0
\(631\) 48.0000 1.91085 0.955425 0.295234i \(-0.0953977\pi\)
0.955425 + 0.295234i \(0.0953977\pi\)
\(632\) 0 0
\(633\) 22.6274 0.899359
\(634\) 0 0
\(635\) −16.9706 −0.673456
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 40.0000 1.58238
\(640\) 0 0
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 0 0
\(643\) −42.4264 −1.67313 −0.836567 0.547865i \(-0.815441\pi\)
−0.836567 + 0.547865i \(0.815441\pi\)
\(644\) 0 0
\(645\) 16.0000 0.629999
\(646\) 0 0
\(647\) −28.2843 −1.11197 −0.555985 0.831193i \(-0.687659\pi\)
−0.555985 + 0.831193i \(0.687659\pi\)
\(648\) 0 0
\(649\) 8.48528 0.333076
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 24.0000 0.939193 0.469596 0.882881i \(-0.344399\pi\)
0.469596 + 0.882881i \(0.344399\pi\)
\(654\) 0 0
\(655\) 4.00000 0.156293
\(656\) 0 0
\(657\) 7.07107 0.275869
\(658\) 0 0
\(659\) −28.0000 −1.09073 −0.545363 0.838200i \(-0.683608\pi\)
−0.545363 + 0.838200i \(0.683608\pi\)
\(660\) 0 0
\(661\) 21.2132 0.825098 0.412549 0.910935i \(-0.364639\pi\)
0.412549 + 0.910935i \(0.364639\pi\)
\(662\) 0 0
\(663\) 28.2843 1.09847
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −32.0000 −1.23719
\(670\) 0 0
\(671\) −1.41421 −0.0545951
\(672\) 0 0
\(673\) 8.00000 0.308377 0.154189 0.988041i \(-0.450724\pi\)
0.154189 + 0.988041i \(0.450724\pi\)
\(674\) 0 0
\(675\) 16.9706 0.653197
\(676\) 0 0
\(677\) 49.4975 1.90234 0.951171 0.308664i \(-0.0998818\pi\)
0.951171 + 0.308664i \(0.0998818\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −8.00000 −0.306561
\(682\) 0 0
\(683\) 48.0000 1.83667 0.918334 0.395805i \(-0.129534\pi\)
0.918334 + 0.395805i \(0.129534\pi\)
\(684\) 0 0
\(685\) −22.6274 −0.864549
\(686\) 0 0
\(687\) 36.0000 1.37349
\(688\) 0 0
\(689\) −8.48528 −0.323263
\(690\) 0 0
\(691\) 2.82843 0.107598 0.0537992 0.998552i \(-0.482867\pi\)
0.0537992 + 0.998552i \(0.482867\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −28.0000 −1.06210
\(696\) 0 0
\(697\) 70.0000 2.65144
\(698\) 0 0
\(699\) 67.8823 2.56754
\(700\) 0 0
\(701\) 32.0000 1.20862 0.604312 0.796748i \(-0.293448\pi\)
0.604312 + 0.796748i \(0.293448\pi\)
\(702\) 0 0
\(703\) 22.6274 0.853409
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 40.0000 1.50223 0.751116 0.660171i \(-0.229516\pi\)
0.751116 + 0.660171i \(0.229516\pi\)
\(710\) 0 0
\(711\) 80.0000 3.00023
\(712\) 0 0
\(713\) 22.6274 0.847403
\(714\) 0 0
\(715\) 2.00000 0.0747958
\(716\) 0 0
\(717\) −56.5685 −2.11259
\(718\) 0 0
\(719\) 11.3137 0.421930 0.210965 0.977494i \(-0.432339\pi\)
0.210965 + 0.977494i \(0.432339\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −76.0000 −2.82647
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 11.3137 0.419602 0.209801 0.977744i \(-0.432718\pi\)
0.209801 + 0.977744i \(0.432718\pi\)
\(728\) 0 0
\(729\) −43.0000 −1.59259
\(730\) 0 0
\(731\) −28.2843 −1.04613
\(732\) 0 0
\(733\) 21.2132 0.783528 0.391764 0.920066i \(-0.371865\pi\)
0.391764 + 0.920066i \(0.371865\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) 36.0000 1.32428 0.662141 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) 0 0
\(741\) 11.3137 0.415619
\(742\) 0 0
\(743\) 12.0000 0.440237 0.220119 0.975473i \(-0.429356\pi\)
0.220119 + 0.975473i \(0.429356\pi\)
\(744\) 0 0
\(745\) 31.1127 1.13988
\(746\) 0 0
\(747\) 14.1421 0.517434
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −28.0000 −1.02173 −0.510867 0.859660i \(-0.670676\pi\)
−0.510867 + 0.859660i \(0.670676\pi\)
\(752\) 0 0
\(753\) 40.0000 1.45768
\(754\) 0 0
\(755\) −5.65685 −0.205874
\(756\) 0 0
\(757\) −8.00000 −0.290765 −0.145382 0.989376i \(-0.546441\pi\)
−0.145382 + 0.989376i \(0.546441\pi\)
\(758\) 0 0
\(759\) 11.3137 0.410662
\(760\) 0 0
\(761\) −12.7279 −0.461387 −0.230693 0.973026i \(-0.574099\pi\)
−0.230693 + 0.973026i \(0.574099\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 50.0000 1.80775
\(766\) 0 0
\(767\) −12.0000 −0.433295
\(768\) 0 0
\(769\) 4.24264 0.152994 0.0764968 0.997070i \(-0.475627\pi\)
0.0764968 + 0.997070i \(0.475627\pi\)
\(770\) 0 0
\(771\) −4.00000 −0.144056
\(772\) 0 0
\(773\) 9.89949 0.356060 0.178030 0.984025i \(-0.443028\pi\)
0.178030 + 0.984025i \(0.443028\pi\)
\(774\) 0 0
\(775\) −16.9706 −0.609601
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 28.0000 1.00320
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −22.0000 −0.785214
\(786\) 0 0
\(787\) 8.48528 0.302468 0.151234 0.988498i \(-0.451675\pi\)
0.151234 + 0.988498i \(0.451675\pi\)
\(788\) 0 0
\(789\) 67.8823 2.41667
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2.00000 0.0710221
\(794\) 0 0
\(795\) −24.0000 −0.851192
\(796\) 0 0
\(797\) −15.5563 −0.551034 −0.275517 0.961296i \(-0.588849\pi\)
−0.275517 + 0.961296i \(0.588849\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 77.7817 2.74828
\(802\) 0 0
\(803\) −1.41421 −0.0499065
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −20.0000 −0.704033
\(808\) 0 0
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) −8.48528 −0.297959 −0.148979 0.988840i \(-0.547599\pi\)
−0.148979 + 0.988840i \(0.547599\pi\)
\(812\) 0 0
\(813\) 48.0000 1.68343
\(814\) 0 0
\(815\) 5.65685 0.198151
\(816\) 0 0
\(817\) −11.3137 −0.395817
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −26.0000 −0.907406 −0.453703 0.891153i \(-0.649897\pi\)
−0.453703 + 0.891153i \(0.649897\pi\)
\(822\) 0 0
\(823\) −8.00000 −0.278862 −0.139431 0.990232i \(-0.544527\pi\)
−0.139431 + 0.990232i \(0.544527\pi\)
\(824\) 0 0
\(825\) −8.48528 −0.295420
\(826\) 0 0
\(827\) −8.00000 −0.278187 −0.139094 0.990279i \(-0.544419\pi\)
−0.139094 + 0.990279i \(0.544419\pi\)
\(828\) 0 0
\(829\) −15.5563 −0.540294 −0.270147 0.962819i \(-0.587072\pi\)
−0.270147 + 0.962819i \(0.587072\pi\)
\(830\) 0 0
\(831\) 28.2843 0.981170
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 8.00000 0.276851
\(836\) 0 0
\(837\) −32.0000 −1.10608
\(838\) 0 0
\(839\) −28.2843 −0.976481 −0.488241 0.872709i \(-0.662361\pi\)
−0.488241 + 0.872709i \(0.662361\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) −67.8823 −2.33799
\(844\) 0 0
\(845\) 15.5563 0.535155
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 24.0000 0.823678
\(850\) 0 0
\(851\) −32.0000 −1.09695
\(852\) 0 0
\(853\) −18.3848 −0.629483 −0.314741 0.949177i \(-0.601918\pi\)
−0.314741 + 0.949177i \(0.601918\pi\)
\(854\) 0 0
\(855\) 20.0000 0.683986
\(856\) 0 0
\(857\) 21.2132 0.724629 0.362315 0.932056i \(-0.381987\pi\)
0.362315 + 0.932056i \(0.381987\pi\)
\(858\) 0 0
\(859\) −8.48528 −0.289514 −0.144757 0.989467i \(-0.546240\pi\)
−0.144757 + 0.989467i \(0.546240\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −8.00000 −0.272323 −0.136162 0.990687i \(-0.543477\pi\)
−0.136162 + 0.990687i \(0.543477\pi\)
\(864\) 0 0
\(865\) 2.00000 0.0680020
\(866\) 0 0
\(867\) −93.3381 −3.16993
\(868\) 0 0
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) −11.3137 −0.383350
\(872\) 0 0
\(873\) −49.4975 −1.67524
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 32.0000 1.08056 0.540282 0.841484i \(-0.318318\pi\)
0.540282 + 0.841484i \(0.318318\pi\)
\(878\) 0 0
\(879\) −84.0000 −2.83325
\(880\) 0 0
\(881\) −12.7279 −0.428815 −0.214407 0.976744i \(-0.568782\pi\)
−0.214407 + 0.976744i \(0.568782\pi\)
\(882\) 0 0
\(883\) 24.0000 0.807664 0.403832 0.914833i \(-0.367678\pi\)
0.403832 + 0.914833i \(0.367678\pi\)
\(884\) 0 0
\(885\) −33.9411 −1.14092
\(886\) 0 0
\(887\) −5.65685 −0.189939 −0.0949693 0.995480i \(-0.530275\pi\)
−0.0949693 + 0.995480i \(0.530275\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 22.6274 0.756351
\(896\) 0 0
\(897\) −16.0000 −0.534224
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 42.4264 1.41343
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 18.0000 0.598340
\(906\) 0 0
\(907\) −32.0000 −1.06254 −0.531271 0.847202i \(-0.678286\pi\)
−0.531271 + 0.847202i \(0.678286\pi\)
\(908\) 0 0
\(909\) 91.9239 3.04892
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 0 0
\(913\) −2.82843 −0.0936073
\(914\) 0 0
\(915\) 5.65685 0.187010
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 0 0
\(921\) 40.0000 1.31804
\(922\) 0 0
\(923\) 11.3137 0.372395
\(924\) 0 0
\(925\) 24.0000 0.789115
\(926\) 0 0
\(927\) 84.8528 2.78693
\(928\) 0 0
\(929\) −1.41421 −0.0463988 −0.0231994 0.999731i \(-0.507385\pi\)
−0.0231994 + 0.999731i \(0.507385\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −16.0000 −0.523816
\(934\) 0 0
\(935\) −10.0000 −0.327035
\(936\) 0 0
\(937\) −21.2132 −0.693005 −0.346503 0.938049i \(-0.612631\pi\)
−0.346503 + 0.938049i \(0.612631\pi\)
\(938\) 0 0
\(939\) 76.0000 2.48017
\(940\) 0 0
\(941\) −43.8406 −1.42916 −0.714582 0.699552i \(-0.753383\pi\)
−0.714582 + 0.699552i \(0.753383\pi\)
\(942\) 0 0
\(943\) −39.5980 −1.28949
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) 0 0
\(949\) 2.00000 0.0649227
\(950\) 0 0
\(951\) 5.65685 0.183436
\(952\) 0 0
\(953\) 42.0000 1.36051 0.680257 0.732974i \(-0.261868\pi\)
0.680257 + 0.732974i \(0.261868\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) −40.0000 −1.28898
\(964\) 0 0
\(965\) −8.48528 −0.273151
\(966\) 0 0
\(967\) −44.0000 −1.41494 −0.707472 0.706741i \(-0.750165\pi\)
−0.707472 + 0.706741i \(0.750165\pi\)
\(968\) 0 0
\(969\) −56.5685 −1.81724
\(970\) 0 0
\(971\) 2.82843 0.0907685 0.0453843 0.998970i \(-0.485549\pi\)
0.0453843 + 0.998970i \(0.485549\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 12.0000 0.384308
\(976\) 0 0
\(977\) −32.0000 −1.02377 −0.511885 0.859054i \(-0.671053\pi\)
−0.511885 + 0.859054i \(0.671053\pi\)
\(978\) 0 0
\(979\) −15.5563 −0.497183
\(980\) 0 0
\(981\) 80.0000 2.55420
\(982\) 0 0
\(983\) −28.2843 −0.902128 −0.451064 0.892492i \(-0.648955\pi\)
−0.451064 + 0.892492i \(0.648955\pi\)
\(984\) 0 0
\(985\) −8.48528 −0.270364
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) −79.1960 −2.51321
\(994\) 0 0
\(995\) −24.0000 −0.760851
\(996\) 0 0
\(997\) 38.1838 1.20929 0.604646 0.796494i \(-0.293315\pi\)
0.604646 + 0.796494i \(0.293315\pi\)
\(998\) 0 0
\(999\) 45.2548 1.43180
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 2156.2.a.f.1.1 2
4.3 odd 2 8624.2.a.bx.1.2 2
7.2 even 3 2156.2.i.f.1145.2 4
7.3 odd 6 2156.2.i.f.177.1 4
7.4 even 3 2156.2.i.f.177.2 4
7.5 odd 6 2156.2.i.f.1145.1 4
7.6 odd 2 inner 2156.2.a.f.1.2 yes 2
28.27 even 2 8624.2.a.bx.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2156.2.a.f.1.1 2 1.1 even 1 trivial
2156.2.a.f.1.2 yes 2 7.6 odd 2 inner
2156.2.i.f.177.1 4 7.3 odd 6
2156.2.i.f.177.2 4 7.4 even 3
2156.2.i.f.1145.1 4 7.5 odd 6
2156.2.i.f.1145.2 4 7.2 even 3
8624.2.a.bx.1.1 2 28.27 even 2
8624.2.a.bx.1.2 2 4.3 odd 2