Properties

Label 3800.2.a.t.1.3
Level $3800$
Weight $2$
Character 3800.1
Self dual yes
Analytic conductor $30.343$
Analytic rank $1$
Dimension $3$
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] = [3800,2,Mod(1,3800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3800, 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("3800.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.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: \(30.3431527681\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.53209\) of defining polynomial
Character \(\chi\) \(=\) 3800.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.53209 q^{3} +2.22668 q^{7} -0.652704 q^{9} -3.22668 q^{11} -1.57398 q^{13} -3.53209 q^{17} +1.00000 q^{19} +3.41147 q^{21} -4.47565 q^{23} -5.59627 q^{27} +1.92127 q^{29} -3.81521 q^{31} -4.94356 q^{33} -11.3550 q^{37} -2.41147 q^{39} +3.47565 q^{41} +1.69459 q^{43} -1.57398 q^{47} -2.04189 q^{49} -5.41147 q^{51} +7.12836 q^{53} +1.53209 q^{57} -7.88713 q^{59} -2.79561 q^{61} -1.45336 q^{63} -3.22668 q^{67} -6.85710 q^{69} -4.38919 q^{71} +6.41147 q^{73} -7.18479 q^{77} -8.59627 q^{79} -6.61587 q^{81} +14.6236 q^{83} +2.94356 q^{87} +6.10607 q^{89} -3.50475 q^{91} -5.84524 q^{93} -15.7023 q^{97} +2.10607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{9} - 3 q^{11} + 3 q^{13} - 6 q^{17} + 3 q^{19} + 6 q^{23} - 3 q^{27} - 3 q^{29} - 15 q^{31} - 9 q^{37} + 3 q^{39} - 9 q^{41} + 3 q^{43} + 3 q^{47} - 3 q^{49} - 6 q^{51} + 3 q^{53} + 6 q^{59}+ \cdots - 6 q^{99}+O(q^{100}) \) 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
\(3\) 1.53209 0.884552 0.442276 0.896879i \(-0.354171\pi\)
0.442276 + 0.896879i \(0.354171\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.22668 0.841607 0.420803 0.907152i \(-0.361748\pi\)
0.420803 + 0.907152i \(0.361748\pi\)
\(8\) 0 0
\(9\) −0.652704 −0.217568
\(10\) 0 0
\(11\) −3.22668 −0.972881 −0.486441 0.873714i \(-0.661705\pi\)
−0.486441 + 0.873714i \(0.661705\pi\)
\(12\) 0 0
\(13\) −1.57398 −0.436543 −0.218271 0.975888i \(-0.570042\pi\)
−0.218271 + 0.975888i \(0.570042\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.53209 −0.856657 −0.428329 0.903623i \(-0.640897\pi\)
−0.428329 + 0.903623i \(0.640897\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 3.41147 0.744445
\(22\) 0 0
\(23\) −4.47565 −0.933238 −0.466619 0.884458i \(-0.654528\pi\)
−0.466619 + 0.884458i \(0.654528\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.59627 −1.07700
\(28\) 0 0
\(29\) 1.92127 0.356772 0.178386 0.983961i \(-0.442912\pi\)
0.178386 + 0.983961i \(0.442912\pi\)
\(30\) 0 0
\(31\) −3.81521 −0.685231 −0.342616 0.939476i \(-0.611313\pi\)
−0.342616 + 0.939476i \(0.611313\pi\)
\(32\) 0 0
\(33\) −4.94356 −0.860564
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −11.3550 −1.86676 −0.933378 0.358894i \(-0.883154\pi\)
−0.933378 + 0.358894i \(0.883154\pi\)
\(38\) 0 0
\(39\) −2.41147 −0.386145
\(40\) 0 0
\(41\) 3.47565 0.542806 0.271403 0.962466i \(-0.412512\pi\)
0.271403 + 0.962466i \(0.412512\pi\)
\(42\) 0 0
\(43\) 1.69459 0.258423 0.129211 0.991617i \(-0.458755\pi\)
0.129211 + 0.991617i \(0.458755\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.57398 −0.229588 −0.114794 0.993389i \(-0.536621\pi\)
−0.114794 + 0.993389i \(0.536621\pi\)
\(48\) 0 0
\(49\) −2.04189 −0.291698
\(50\) 0 0
\(51\) −5.41147 −0.757758
\(52\) 0 0
\(53\) 7.12836 0.979155 0.489577 0.871960i \(-0.337151\pi\)
0.489577 + 0.871960i \(0.337151\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.53209 0.202930
\(58\) 0 0
\(59\) −7.88713 −1.02682 −0.513408 0.858145i \(-0.671617\pi\)
−0.513408 + 0.858145i \(0.671617\pi\)
\(60\) 0 0
\(61\) −2.79561 −0.357941 −0.178970 0.983854i \(-0.557277\pi\)
−0.178970 + 0.983854i \(0.557277\pi\)
\(62\) 0 0
\(63\) −1.45336 −0.183107
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.22668 −0.394202 −0.197101 0.980383i \(-0.563153\pi\)
−0.197101 + 0.980383i \(0.563153\pi\)
\(68\) 0 0
\(69\) −6.85710 −0.825497
\(70\) 0 0
\(71\) −4.38919 −0.520900 −0.260450 0.965487i \(-0.583871\pi\)
−0.260450 + 0.965487i \(0.583871\pi\)
\(72\) 0 0
\(73\) 6.41147 0.750406 0.375203 0.926943i \(-0.377573\pi\)
0.375203 + 0.926943i \(0.377573\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −7.18479 −0.818783
\(78\) 0 0
\(79\) −8.59627 −0.967156 −0.483578 0.875301i \(-0.660663\pi\)
−0.483578 + 0.875301i \(0.660663\pi\)
\(80\) 0 0
\(81\) −6.61587 −0.735096
\(82\) 0 0
\(83\) 14.6236 1.60515 0.802575 0.596552i \(-0.203463\pi\)
0.802575 + 0.596552i \(0.203463\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.94356 0.315583
\(88\) 0 0
\(89\) 6.10607 0.647242 0.323621 0.946187i \(-0.395100\pi\)
0.323621 + 0.946187i \(0.395100\pi\)
\(90\) 0 0
\(91\) −3.50475 −0.367397
\(92\) 0 0
\(93\) −5.84524 −0.606123
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −15.7023 −1.59433 −0.797165 0.603761i \(-0.793668\pi\)
−0.797165 + 0.603761i \(0.793668\pi\)
\(98\) 0 0
\(99\) 2.10607 0.211668
\(100\) 0 0
\(101\) 4.35504 0.433342 0.216671 0.976245i \(-0.430480\pi\)
0.216671 + 0.976245i \(0.430480\pi\)
\(102\) 0 0
\(103\) −4.27126 −0.420860 −0.210430 0.977609i \(-0.567486\pi\)
−0.210430 + 0.977609i \(0.567486\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −0.106067 −0.0102539 −0.00512693 0.999987i \(-0.501632\pi\)
−0.00512693 + 0.999987i \(0.501632\pi\)
\(108\) 0 0
\(109\) −14.4115 −1.38037 −0.690184 0.723634i \(-0.742471\pi\)
−0.690184 + 0.723634i \(0.742471\pi\)
\(110\) 0 0
\(111\) −17.3969 −1.65124
\(112\) 0 0
\(113\) 16.5895 1.56061 0.780303 0.625402i \(-0.215065\pi\)
0.780303 + 0.625402i \(0.215065\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.02734 0.0949777
\(118\) 0 0
\(119\) −7.86484 −0.720968
\(120\) 0 0
\(121\) −0.588526 −0.0535024
\(122\) 0 0
\(123\) 5.32501 0.480140
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 4.02229 0.356920 0.178460 0.983947i \(-0.442888\pi\)
0.178460 + 0.983947i \(0.442888\pi\)
\(128\) 0 0
\(129\) 2.59627 0.228589
\(130\) 0 0
\(131\) −4.32770 −0.378113 −0.189056 0.981966i \(-0.560543\pi\)
−0.189056 + 0.981966i \(0.560543\pi\)
\(132\) 0 0
\(133\) 2.22668 0.193078
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −11.0223 −0.941698 −0.470849 0.882214i \(-0.656052\pi\)
−0.470849 + 0.882214i \(0.656052\pi\)
\(138\) 0 0
\(139\) −17.2567 −1.46370 −0.731848 0.681468i \(-0.761342\pi\)
−0.731848 + 0.681468i \(0.761342\pi\)
\(140\) 0 0
\(141\) −2.41147 −0.203083
\(142\) 0 0
\(143\) 5.07873 0.424704
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −3.12836 −0.258022
\(148\) 0 0
\(149\) 12.3824 1.01440 0.507202 0.861827i \(-0.330680\pi\)
0.507202 + 0.861827i \(0.330680\pi\)
\(150\) 0 0
\(151\) 20.8016 1.69281 0.846405 0.532540i \(-0.178762\pi\)
0.846405 + 0.532540i \(0.178762\pi\)
\(152\) 0 0
\(153\) 2.30541 0.186381
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 5.25402 0.419317 0.209658 0.977775i \(-0.432765\pi\)
0.209658 + 0.977775i \(0.432765\pi\)
\(158\) 0 0
\(159\) 10.9213 0.866113
\(160\) 0 0
\(161\) −9.96585 −0.785419
\(162\) 0 0
\(163\) 6.63816 0.519940 0.259970 0.965617i \(-0.416287\pi\)
0.259970 + 0.965617i \(0.416287\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 16.3824 1.26771 0.633853 0.773453i \(-0.281472\pi\)
0.633853 + 0.773453i \(0.281472\pi\)
\(168\) 0 0
\(169\) −10.5226 −0.809430
\(170\) 0 0
\(171\) −0.652704 −0.0499135
\(172\) 0 0
\(173\) 4.01455 0.305220 0.152610 0.988286i \(-0.451232\pi\)
0.152610 + 0.988286i \(0.451232\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −12.0838 −0.908272
\(178\) 0 0
\(179\) 22.2472 1.66283 0.831417 0.555648i \(-0.187530\pi\)
0.831417 + 0.555648i \(0.187530\pi\)
\(180\) 0 0
\(181\) −7.29591 −0.542301 −0.271150 0.962537i \(-0.587404\pi\)
−0.271150 + 0.962537i \(0.587404\pi\)
\(182\) 0 0
\(183\) −4.28312 −0.316617
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 11.3969 0.833426
\(188\) 0 0
\(189\) −12.4611 −0.906412
\(190\) 0 0
\(191\) −9.55169 −0.691136 −0.345568 0.938394i \(-0.612314\pi\)
−0.345568 + 0.938394i \(0.612314\pi\)
\(192\) 0 0
\(193\) 1.23442 0.0888557 0.0444278 0.999013i \(-0.485854\pi\)
0.0444278 + 0.999013i \(0.485854\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 11.5672 0.824127 0.412063 0.911155i \(-0.364808\pi\)
0.412063 + 0.911155i \(0.364808\pi\)
\(198\) 0 0
\(199\) 12.0104 0.851397 0.425698 0.904865i \(-0.360028\pi\)
0.425698 + 0.904865i \(0.360028\pi\)
\(200\) 0 0
\(201\) −4.94356 −0.348692
\(202\) 0 0
\(203\) 4.27807 0.300261
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.92127 0.203043
\(208\) 0 0
\(209\) −3.22668 −0.223194
\(210\) 0 0
\(211\) −11.3696 −0.782715 −0.391357 0.920239i \(-0.627994\pi\)
−0.391357 + 0.920239i \(0.627994\pi\)
\(212\) 0 0
\(213\) −6.72462 −0.460764
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −8.49525 −0.576695
\(218\) 0 0
\(219\) 9.82295 0.663773
\(220\) 0 0
\(221\) 5.55943 0.373968
\(222\) 0 0
\(223\) −13.7493 −0.920720 −0.460360 0.887732i \(-0.652280\pi\)
−0.460360 + 0.887732i \(0.652280\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −7.06149 −0.468688 −0.234344 0.972154i \(-0.575294\pi\)
−0.234344 + 0.972154i \(0.575294\pi\)
\(228\) 0 0
\(229\) −7.05644 −0.466302 −0.233151 0.972440i \(-0.574904\pi\)
−0.233151 + 0.972440i \(0.574904\pi\)
\(230\) 0 0
\(231\) −11.0077 −0.724256
\(232\) 0 0
\(233\) −11.9659 −0.783909 −0.391955 0.919985i \(-0.628201\pi\)
−0.391955 + 0.919985i \(0.628201\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −13.1702 −0.855499
\(238\) 0 0
\(239\) 2.75103 0.177949 0.0889747 0.996034i \(-0.471641\pi\)
0.0889747 + 0.996034i \(0.471641\pi\)
\(240\) 0 0
\(241\) −7.95811 −0.512627 −0.256313 0.966594i \(-0.582508\pi\)
−0.256313 + 0.966594i \(0.582508\pi\)
\(242\) 0 0
\(243\) 6.65270 0.426771
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1.57398 −0.100150
\(248\) 0 0
\(249\) 22.4047 1.41984
\(250\) 0 0
\(251\) −13.6827 −0.863646 −0.431823 0.901958i \(-0.642130\pi\)
−0.431823 + 0.901958i \(0.642130\pi\)
\(252\) 0 0
\(253\) 14.4415 0.907930
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −10.5517 −0.658196 −0.329098 0.944296i \(-0.606745\pi\)
−0.329098 + 0.944296i \(0.606745\pi\)
\(258\) 0 0
\(259\) −25.2841 −1.57107
\(260\) 0 0
\(261\) −1.25402 −0.0776221
\(262\) 0 0
\(263\) 9.40642 0.580025 0.290012 0.957023i \(-0.406341\pi\)
0.290012 + 0.957023i \(0.406341\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 9.35504 0.572519
\(268\) 0 0
\(269\) 9.02734 0.550407 0.275203 0.961386i \(-0.411255\pi\)
0.275203 + 0.961386i \(0.411255\pi\)
\(270\) 0 0
\(271\) −22.3354 −1.35678 −0.678391 0.734701i \(-0.737322\pi\)
−0.678391 + 0.734701i \(0.737322\pi\)
\(272\) 0 0
\(273\) −5.36959 −0.324982
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −17.2959 −1.03921 −0.519605 0.854406i \(-0.673921\pi\)
−0.519605 + 0.854406i \(0.673921\pi\)
\(278\) 0 0
\(279\) 2.49020 0.149084
\(280\) 0 0
\(281\) −10.4534 −0.623595 −0.311798 0.950149i \(-0.600931\pi\)
−0.311798 + 0.950149i \(0.600931\pi\)
\(282\) 0 0
\(283\) 23.3628 1.38877 0.694386 0.719602i \(-0.255676\pi\)
0.694386 + 0.719602i \(0.255676\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 7.73917 0.456829
\(288\) 0 0
\(289\) −4.52435 −0.266138
\(290\) 0 0
\(291\) −24.0574 −1.41027
\(292\) 0 0
\(293\) 27.9641 1.63368 0.816840 0.576864i \(-0.195724\pi\)
0.816840 + 0.576864i \(0.195724\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 18.0574 1.04779
\(298\) 0 0
\(299\) 7.04458 0.407398
\(300\) 0 0
\(301\) 3.77332 0.217490
\(302\) 0 0
\(303\) 6.67230 0.383314
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 22.8452 1.30385 0.651923 0.758285i \(-0.273962\pi\)
0.651923 + 0.758285i \(0.273962\pi\)
\(308\) 0 0
\(309\) −6.54395 −0.372272
\(310\) 0 0
\(311\) −11.8699 −0.673080 −0.336540 0.941669i \(-0.609257\pi\)
−0.336540 + 0.941669i \(0.609257\pi\)
\(312\) 0 0
\(313\) −25.5672 −1.44514 −0.722571 0.691297i \(-0.757040\pi\)
−0.722571 + 0.691297i \(0.757040\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −17.4020 −0.977392 −0.488696 0.872454i \(-0.662527\pi\)
−0.488696 + 0.872454i \(0.662527\pi\)
\(318\) 0 0
\(319\) −6.19934 −0.347096
\(320\) 0 0
\(321\) −0.162504 −0.00907008
\(322\) 0 0
\(323\) −3.53209 −0.196531
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −22.0797 −1.22101
\(328\) 0 0
\(329\) −3.50475 −0.193223
\(330\) 0 0
\(331\) −6.77063 −0.372147 −0.186074 0.982536i \(-0.559576\pi\)
−0.186074 + 0.982536i \(0.559576\pi\)
\(332\) 0 0
\(333\) 7.41147 0.406146
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −4.11112 −0.223947 −0.111973 0.993711i \(-0.535717\pi\)
−0.111973 + 0.993711i \(0.535717\pi\)
\(338\) 0 0
\(339\) 25.4165 1.38044
\(340\) 0 0
\(341\) 12.3105 0.666649
\(342\) 0 0
\(343\) −20.1334 −1.08710
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 11.0247 0.591834 0.295917 0.955214i \(-0.404375\pi\)
0.295917 + 0.955214i \(0.404375\pi\)
\(348\) 0 0
\(349\) −22.8307 −1.22210 −0.611049 0.791592i \(-0.709252\pi\)
−0.611049 + 0.791592i \(0.709252\pi\)
\(350\) 0 0
\(351\) 8.80840 0.470158
\(352\) 0 0
\(353\) −14.6732 −0.780978 −0.390489 0.920608i \(-0.627694\pi\)
−0.390489 + 0.920608i \(0.627694\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −12.0496 −0.637734
\(358\) 0 0
\(359\) 3.60307 0.190163 0.0950815 0.995469i \(-0.469689\pi\)
0.0950815 + 0.995469i \(0.469689\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −0.901674 −0.0473256
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 19.8007 1.03359 0.516793 0.856110i \(-0.327126\pi\)
0.516793 + 0.856110i \(0.327126\pi\)
\(368\) 0 0
\(369\) −2.26857 −0.118097
\(370\) 0 0
\(371\) 15.8726 0.824063
\(372\) 0 0
\(373\) 17.6631 0.914562 0.457281 0.889322i \(-0.348823\pi\)
0.457281 + 0.889322i \(0.348823\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −3.02404 −0.155746
\(378\) 0 0
\(379\) 6.05644 0.311098 0.155549 0.987828i \(-0.450285\pi\)
0.155549 + 0.987828i \(0.450285\pi\)
\(380\) 0 0
\(381\) 6.16250 0.315715
\(382\) 0 0
\(383\) 20.8895 1.06740 0.533702 0.845673i \(-0.320801\pi\)
0.533702 + 0.845673i \(0.320801\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1.10607 −0.0562245
\(388\) 0 0
\(389\) −2.30541 −0.116889 −0.0584444 0.998291i \(-0.518614\pi\)
−0.0584444 + 0.998291i \(0.518614\pi\)
\(390\) 0 0
\(391\) 15.8084 0.799465
\(392\) 0 0
\(393\) −6.63041 −0.334460
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 22.2354 1.11596 0.557980 0.829854i \(-0.311576\pi\)
0.557980 + 0.829854i \(0.311576\pi\)
\(398\) 0 0
\(399\) 3.41147 0.170787
\(400\) 0 0
\(401\) −14.1584 −0.707036 −0.353518 0.935428i \(-0.615015\pi\)
−0.353518 + 0.935428i \(0.615015\pi\)
\(402\) 0 0
\(403\) 6.00505 0.299133
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 36.6391 1.81613
\(408\) 0 0
\(409\) −4.22432 −0.208879 −0.104440 0.994531i \(-0.533305\pi\)
−0.104440 + 0.994531i \(0.533305\pi\)
\(410\) 0 0
\(411\) −16.8871 −0.832980
\(412\) 0 0
\(413\) −17.5621 −0.864175
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −26.4388 −1.29471
\(418\) 0 0
\(419\) −11.9632 −0.584439 −0.292219 0.956351i \(-0.594394\pi\)
−0.292219 + 0.956351i \(0.594394\pi\)
\(420\) 0 0
\(421\) −10.2395 −0.499041 −0.249521 0.968369i \(-0.580273\pi\)
−0.249521 + 0.968369i \(0.580273\pi\)
\(422\) 0 0
\(423\) 1.02734 0.0499510
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −6.22493 −0.301245
\(428\) 0 0
\(429\) 7.78106 0.375673
\(430\) 0 0
\(431\) 22.2517 1.07182 0.535912 0.844274i \(-0.319968\pi\)
0.535912 + 0.844274i \(0.319968\pi\)
\(432\) 0 0
\(433\) −24.9581 −1.19941 −0.599705 0.800221i \(-0.704715\pi\)
−0.599705 + 0.800221i \(0.704715\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −4.47565 −0.214099
\(438\) 0 0
\(439\) 16.0942 0.768135 0.384067 0.923305i \(-0.374523\pi\)
0.384067 + 0.923305i \(0.374523\pi\)
\(440\) 0 0
\(441\) 1.33275 0.0634642
\(442\) 0 0
\(443\) −17.7270 −0.842235 −0.421117 0.907006i \(-0.638362\pi\)
−0.421117 + 0.907006i \(0.638362\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 18.9709 0.897293
\(448\) 0 0
\(449\) −37.6049 −1.77469 −0.887343 0.461109i \(-0.847452\pi\)
−0.887343 + 0.461109i \(0.847452\pi\)
\(450\) 0 0
\(451\) −11.2148 −0.528085
\(452\) 0 0
\(453\) 31.8699 1.49738
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 20.1949 0.944677 0.472339 0.881417i \(-0.343410\pi\)
0.472339 + 0.881417i \(0.343410\pi\)
\(458\) 0 0
\(459\) 19.7665 0.922622
\(460\) 0 0
\(461\) 6.81521 0.317416 0.158708 0.987326i \(-0.449267\pi\)
0.158708 + 0.987326i \(0.449267\pi\)
\(462\) 0 0
\(463\) 33.7083 1.56656 0.783279 0.621670i \(-0.213546\pi\)
0.783279 + 0.621670i \(0.213546\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −27.5648 −1.27555 −0.637774 0.770224i \(-0.720144\pi\)
−0.637774 + 0.770224i \(0.720144\pi\)
\(468\) 0 0
\(469\) −7.18479 −0.331763
\(470\) 0 0
\(471\) 8.04963 0.370907
\(472\) 0 0
\(473\) −5.46791 −0.251415
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −4.65270 −0.213033
\(478\) 0 0
\(479\) 26.1438 1.19454 0.597271 0.802039i \(-0.296252\pi\)
0.597271 + 0.802039i \(0.296252\pi\)
\(480\) 0 0
\(481\) 17.8726 0.814919
\(482\) 0 0
\(483\) −15.2686 −0.694744
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 5.63310 0.255260 0.127630 0.991822i \(-0.459263\pi\)
0.127630 + 0.991822i \(0.459263\pi\)
\(488\) 0 0
\(489\) 10.1702 0.459914
\(490\) 0 0
\(491\) 14.6682 0.661966 0.330983 0.943637i \(-0.392620\pi\)
0.330983 + 0.943637i \(0.392620\pi\)
\(492\) 0 0
\(493\) −6.78611 −0.305631
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −9.77332 −0.438393
\(498\) 0 0
\(499\) 16.7793 0.751145 0.375572 0.926793i \(-0.377446\pi\)
0.375572 + 0.926793i \(0.377446\pi\)
\(500\) 0 0
\(501\) 25.0993 1.12135
\(502\) 0 0
\(503\) −3.12330 −0.139261 −0.0696306 0.997573i \(-0.522182\pi\)
−0.0696306 + 0.997573i \(0.522182\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −16.1215 −0.715983
\(508\) 0 0
\(509\) −40.1566 −1.77991 −0.889956 0.456047i \(-0.849265\pi\)
−0.889956 + 0.456047i \(0.849265\pi\)
\(510\) 0 0
\(511\) 14.2763 0.631547
\(512\) 0 0
\(513\) −5.59627 −0.247081
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 5.07873 0.223362
\(518\) 0 0
\(519\) 6.15064 0.269983
\(520\) 0 0
\(521\) −8.75465 −0.383548 −0.191774 0.981439i \(-0.561424\pi\)
−0.191774 + 0.981439i \(0.561424\pi\)
\(522\) 0 0
\(523\) 26.9522 1.17854 0.589270 0.807937i \(-0.299416\pi\)
0.589270 + 0.807937i \(0.299416\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 13.4757 0.587009
\(528\) 0 0
\(529\) −2.96854 −0.129067
\(530\) 0 0
\(531\) 5.14796 0.223402
\(532\) 0 0
\(533\) −5.47060 −0.236958
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 34.0847 1.47086
\(538\) 0 0
\(539\) 6.58853 0.283788
\(540\) 0 0
\(541\) 8.48751 0.364907 0.182453 0.983215i \(-0.441596\pi\)
0.182453 + 0.983215i \(0.441596\pi\)
\(542\) 0 0
\(543\) −11.1780 −0.479693
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 3.62267 0.154894 0.0774472 0.996996i \(-0.475323\pi\)
0.0774472 + 0.996996i \(0.475323\pi\)
\(548\) 0 0
\(549\) 1.82470 0.0778764
\(550\) 0 0
\(551\) 1.92127 0.0818490
\(552\) 0 0
\(553\) −19.1411 −0.813964
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 2.01455 0.0853591 0.0426796 0.999089i \(-0.486411\pi\)
0.0426796 + 0.999089i \(0.486411\pi\)
\(558\) 0 0
\(559\) −2.66725 −0.112813
\(560\) 0 0
\(561\) 17.4611 0.737208
\(562\) 0 0
\(563\) 16.5185 0.696171 0.348085 0.937463i \(-0.386832\pi\)
0.348085 + 0.937463i \(0.386832\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −14.7314 −0.618662
\(568\) 0 0
\(569\) −8.18304 −0.343051 −0.171525 0.985180i \(-0.554870\pi\)
−0.171525 + 0.985180i \(0.554870\pi\)
\(570\) 0 0
\(571\) −36.9796 −1.54755 −0.773774 0.633462i \(-0.781633\pi\)
−0.773774 + 0.633462i \(0.781633\pi\)
\(572\) 0 0
\(573\) −14.6340 −0.611346
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 18.7347 0.779937 0.389968 0.920828i \(-0.372486\pi\)
0.389968 + 0.920828i \(0.372486\pi\)
\(578\) 0 0
\(579\) 1.89124 0.0785975
\(580\) 0 0
\(581\) 32.5621 1.35090
\(582\) 0 0
\(583\) −23.0009 −0.952601
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −29.2968 −1.20921 −0.604605 0.796525i \(-0.706669\pi\)
−0.604605 + 0.796525i \(0.706669\pi\)
\(588\) 0 0
\(589\) −3.81521 −0.157203
\(590\) 0 0
\(591\) 17.7219 0.728983
\(592\) 0 0
\(593\) −22.4260 −0.920926 −0.460463 0.887679i \(-0.652317\pi\)
−0.460463 + 0.887679i \(0.652317\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 18.4010 0.753105
\(598\) 0 0
\(599\) −33.1002 −1.35244 −0.676219 0.736701i \(-0.736383\pi\)
−0.676219 + 0.736701i \(0.736383\pi\)
\(600\) 0 0
\(601\) 43.5613 1.77690 0.888451 0.458971i \(-0.151782\pi\)
0.888451 + 0.458971i \(0.151782\pi\)
\(602\) 0 0
\(603\) 2.10607 0.0857657
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 2.03178 0.0824676 0.0412338 0.999150i \(-0.486871\pi\)
0.0412338 + 0.999150i \(0.486871\pi\)
\(608\) 0 0
\(609\) 6.55438 0.265597
\(610\) 0 0
\(611\) 2.47741 0.100225
\(612\) 0 0
\(613\) −25.1266 −1.01485 −0.507427 0.861695i \(-0.669403\pi\)
−0.507427 + 0.861695i \(0.669403\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −24.4979 −0.986250 −0.493125 0.869958i \(-0.664145\pi\)
−0.493125 + 0.869958i \(0.664145\pi\)
\(618\) 0 0
\(619\) 9.05138 0.363806 0.181903 0.983316i \(-0.441774\pi\)
0.181903 + 0.983316i \(0.441774\pi\)
\(620\) 0 0
\(621\) 25.0469 1.00510
\(622\) 0 0
\(623\) 13.5963 0.544723
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −4.94356 −0.197427
\(628\) 0 0
\(629\) 40.1070 1.59917
\(630\) 0 0
\(631\) 23.7324 0.944770 0.472385 0.881392i \(-0.343393\pi\)
0.472385 + 0.881392i \(0.343393\pi\)
\(632\) 0 0
\(633\) −17.4192 −0.692352
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 3.21389 0.127339
\(638\) 0 0
\(639\) 2.86484 0.113331
\(640\) 0 0
\(641\) 10.6067 0.418939 0.209470 0.977815i \(-0.432826\pi\)
0.209470 + 0.977815i \(0.432826\pi\)
\(642\) 0 0
\(643\) −31.1634 −1.22897 −0.614483 0.788930i \(-0.710635\pi\)
−0.614483 + 0.788930i \(0.710635\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −12.9154 −0.507757 −0.253878 0.967236i \(-0.581706\pi\)
−0.253878 + 0.967236i \(0.581706\pi\)
\(648\) 0 0
\(649\) 25.4492 0.998970
\(650\) 0 0
\(651\) −13.0155 −0.510117
\(652\) 0 0
\(653\) 10.8060 0.422873 0.211436 0.977392i \(-0.432186\pi\)
0.211436 + 0.977392i \(0.432186\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −4.18479 −0.163264
\(658\) 0 0
\(659\) −28.3583 −1.10468 −0.552342 0.833618i \(-0.686266\pi\)
−0.552342 + 0.833618i \(0.686266\pi\)
\(660\) 0 0
\(661\) −28.9691 −1.12677 −0.563385 0.826195i \(-0.690501\pi\)
−0.563385 + 0.826195i \(0.690501\pi\)
\(662\) 0 0
\(663\) 8.51754 0.330794
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −8.59896 −0.332953
\(668\) 0 0
\(669\) −21.0651 −0.814424
\(670\) 0 0
\(671\) 9.02053 0.348234
\(672\) 0 0
\(673\) −15.6973 −0.605086 −0.302543 0.953136i \(-0.597836\pi\)
−0.302543 + 0.953136i \(0.597836\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 3.25578 0.125130 0.0625648 0.998041i \(-0.480072\pi\)
0.0625648 + 0.998041i \(0.480072\pi\)
\(678\) 0 0
\(679\) −34.9641 −1.34180
\(680\) 0 0
\(681\) −10.8188 −0.414578
\(682\) 0 0
\(683\) 33.1834 1.26973 0.634863 0.772625i \(-0.281057\pi\)
0.634863 + 0.772625i \(0.281057\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −10.8111 −0.412469
\(688\) 0 0
\(689\) −11.2199 −0.427443
\(690\) 0 0
\(691\) 41.1242 1.56444 0.782220 0.623002i \(-0.214087\pi\)
0.782220 + 0.623002i \(0.214087\pi\)
\(692\) 0 0
\(693\) 4.68954 0.178141
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −12.2763 −0.464998
\(698\) 0 0
\(699\) −18.3327 −0.693408
\(700\) 0 0
\(701\) 44.9198 1.69660 0.848300 0.529517i \(-0.177627\pi\)
0.848300 + 0.529517i \(0.177627\pi\)
\(702\) 0 0
\(703\) −11.3550 −0.428263
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9.69728 0.364704
\(708\) 0 0
\(709\) −34.9240 −1.31160 −0.655798 0.754936i \(-0.727668\pi\)
−0.655798 + 0.754936i \(0.727668\pi\)
\(710\) 0 0
\(711\) 5.61081 0.210422
\(712\) 0 0
\(713\) 17.0755 0.639484
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 4.21482 0.157405
\(718\) 0 0
\(719\) 29.0077 1.08181 0.540903 0.841085i \(-0.318083\pi\)
0.540903 + 0.841085i \(0.318083\pi\)
\(720\) 0 0
\(721\) −9.51073 −0.354198
\(722\) 0 0
\(723\) −12.1925 −0.453445
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −8.27126 −0.306764 −0.153382 0.988167i \(-0.549017\pi\)
−0.153382 + 0.988167i \(0.549017\pi\)
\(728\) 0 0
\(729\) 30.0401 1.11260
\(730\) 0 0
\(731\) −5.98545 −0.221380
\(732\) 0 0
\(733\) 12.6696 0.467963 0.233981 0.972241i \(-0.424824\pi\)
0.233981 + 0.972241i \(0.424824\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 10.4115 0.383512
\(738\) 0 0
\(739\) 17.9733 0.661157 0.330579 0.943778i \(-0.392756\pi\)
0.330579 + 0.943778i \(0.392756\pi\)
\(740\) 0 0
\(741\) −2.41147 −0.0885877
\(742\) 0 0
\(743\) 5.47977 0.201033 0.100517 0.994935i \(-0.467950\pi\)
0.100517 + 0.994935i \(0.467950\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −9.54488 −0.349229
\(748\) 0 0
\(749\) −0.236177 −0.00862972
\(750\) 0 0
\(751\) 32.5803 1.18887 0.594436 0.804143i \(-0.297375\pi\)
0.594436 + 0.804143i \(0.297375\pi\)
\(752\) 0 0
\(753\) −20.9632 −0.763940
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −4.97359 −0.180768 −0.0903841 0.995907i \(-0.528809\pi\)
−0.0903841 + 0.995907i \(0.528809\pi\)
\(758\) 0 0
\(759\) 22.1257 0.803111
\(760\) 0 0
\(761\) 24.5262 0.889075 0.444537 0.895760i \(-0.353368\pi\)
0.444537 + 0.895760i \(0.353368\pi\)
\(762\) 0 0
\(763\) −32.0898 −1.16173
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 12.4142 0.448249
\(768\) 0 0
\(769\) −24.4688 −0.882369 −0.441185 0.897416i \(-0.645442\pi\)
−0.441185 + 0.897416i \(0.645442\pi\)
\(770\) 0 0
\(771\) −16.1661 −0.582209
\(772\) 0 0
\(773\) −4.43645 −0.159568 −0.0797840 0.996812i \(-0.525423\pi\)
−0.0797840 + 0.996812i \(0.525423\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −38.7374 −1.38970
\(778\) 0 0
\(779\) 3.47565 0.124528
\(780\) 0 0
\(781\) 14.1625 0.506774
\(782\) 0 0
\(783\) −10.7520 −0.384244
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −1.73143 −0.0617188 −0.0308594 0.999524i \(-0.509824\pi\)
−0.0308594 + 0.999524i \(0.509824\pi\)
\(788\) 0 0
\(789\) 14.4115 0.513062
\(790\) 0 0
\(791\) 36.9394 1.31342
\(792\) 0 0
\(793\) 4.40022 0.156257
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 48.3979 1.71434 0.857170 0.515033i \(-0.172221\pi\)
0.857170 + 0.515033i \(0.172221\pi\)
\(798\) 0 0
\(799\) 5.55943 0.196678
\(800\) 0 0
\(801\) −3.98545 −0.140819
\(802\) 0 0
\(803\) −20.6878 −0.730056
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 13.8307 0.486863
\(808\) 0 0
\(809\) 45.7861 1.60975 0.804877 0.593442i \(-0.202231\pi\)
0.804877 + 0.593442i \(0.202231\pi\)
\(810\) 0 0
\(811\) −40.0779 −1.40733 −0.703663 0.710534i \(-0.748453\pi\)
−0.703663 + 0.710534i \(0.748453\pi\)
\(812\) 0 0
\(813\) −34.2199 −1.20014
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1.69459 0.0592863
\(818\) 0 0
\(819\) 2.28756 0.0799339
\(820\) 0 0
\(821\) −51.9208 −1.81205 −0.906024 0.423227i \(-0.860897\pi\)
−0.906024 + 0.423227i \(0.860897\pi\)
\(822\) 0 0
\(823\) 18.0779 0.630156 0.315078 0.949066i \(-0.397969\pi\)
0.315078 + 0.949066i \(0.397969\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 46.5022 1.61704 0.808519 0.588469i \(-0.200269\pi\)
0.808519 + 0.588469i \(0.200269\pi\)
\(828\) 0 0
\(829\) −44.6946 −1.55231 −0.776154 0.630544i \(-0.782832\pi\)
−0.776154 + 0.630544i \(0.782832\pi\)
\(830\) 0 0
\(831\) −26.4989 −0.919236
\(832\) 0 0
\(833\) 7.21213 0.249886
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 21.3509 0.737996
\(838\) 0 0
\(839\) −6.37134 −0.219963 −0.109982 0.993934i \(-0.535079\pi\)
−0.109982 + 0.993934i \(0.535079\pi\)
\(840\) 0 0
\(841\) −25.3087 −0.872714
\(842\) 0 0
\(843\) −16.0155 −0.551602
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −1.31046 −0.0450279
\(848\) 0 0
\(849\) 35.7939 1.22844
\(850\) 0 0
\(851\) 50.8212 1.74213
\(852\) 0 0
\(853\) 37.9813 1.30046 0.650228 0.759739i \(-0.274673\pi\)
0.650228 + 0.759739i \(0.274673\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −47.2104 −1.61268 −0.806338 0.591455i \(-0.798554\pi\)
−0.806338 + 0.591455i \(0.798554\pi\)
\(858\) 0 0
\(859\) 15.6587 0.534268 0.267134 0.963659i \(-0.413923\pi\)
0.267134 + 0.963659i \(0.413923\pi\)
\(860\) 0 0
\(861\) 11.8571 0.404089
\(862\) 0 0
\(863\) −54.9026 −1.86891 −0.934453 0.356086i \(-0.884111\pi\)
−0.934453 + 0.356086i \(0.884111\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −6.93170 −0.235413
\(868\) 0 0
\(869\) 27.7374 0.940927
\(870\) 0 0
\(871\) 5.07873 0.172086
\(872\) 0 0
\(873\) 10.2490 0.346875
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 7.74329 0.261472 0.130736 0.991417i \(-0.458266\pi\)
0.130736 + 0.991417i \(0.458266\pi\)
\(878\) 0 0
\(879\) 42.8435 1.44507
\(880\) 0 0
\(881\) −48.6005 −1.63739 −0.818696 0.574227i \(-0.805303\pi\)
−0.818696 + 0.574227i \(0.805303\pi\)
\(882\) 0 0
\(883\) −31.1625 −1.04870 −0.524351 0.851502i \(-0.675692\pi\)
−0.524351 + 0.851502i \(0.675692\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 4.52259 0.151854 0.0759269 0.997113i \(-0.475808\pi\)
0.0759269 + 0.997113i \(0.475808\pi\)
\(888\) 0 0
\(889\) 8.95636 0.300387
\(890\) 0 0
\(891\) 21.3473 0.715161
\(892\) 0 0
\(893\) −1.57398 −0.0526712
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 10.7929 0.360365
\(898\) 0 0
\(899\) −7.33006 −0.244471
\(900\) 0 0
\(901\) −25.1780 −0.838800
\(902\) 0 0
\(903\) 5.78106 0.192382
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 27.9777 0.928985 0.464492 0.885577i \(-0.346237\pi\)
0.464492 + 0.885577i \(0.346237\pi\)
\(908\) 0 0
\(909\) −2.84255 −0.0942814
\(910\) 0 0
\(911\) 35.5895 1.17913 0.589566 0.807720i \(-0.299299\pi\)
0.589566 + 0.807720i \(0.299299\pi\)
\(912\) 0 0
\(913\) −47.1857 −1.56162
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −9.63640 −0.318222
\(918\) 0 0
\(919\) 20.8854 0.688945 0.344472 0.938796i \(-0.388058\pi\)
0.344472 + 0.938796i \(0.388058\pi\)
\(920\) 0 0
\(921\) 35.0009 1.15332
\(922\) 0 0
\(923\) 6.90848 0.227395
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 2.78787 0.0915655
\(928\) 0 0
\(929\) 45.5885 1.49571 0.747856 0.663862i \(-0.231084\pi\)
0.747856 + 0.663862i \(0.231084\pi\)
\(930\) 0 0
\(931\) −2.04189 −0.0669202
\(932\) 0 0
\(933\) −18.1857 −0.595374
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 34.7638 1.13568 0.567842 0.823137i \(-0.307778\pi\)
0.567842 + 0.823137i \(0.307778\pi\)
\(938\) 0 0
\(939\) −39.1712 −1.27830
\(940\) 0 0
\(941\) 5.80066 0.189096 0.0945480 0.995520i \(-0.469859\pi\)
0.0945480 + 0.995520i \(0.469859\pi\)
\(942\) 0 0
\(943\) −15.5558 −0.506567
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 18.4466 0.599433 0.299716 0.954028i \(-0.403108\pi\)
0.299716 + 0.954028i \(0.403108\pi\)
\(948\) 0 0
\(949\) −10.0915 −0.327585
\(950\) 0 0
\(951\) −26.6614 −0.864554
\(952\) 0 0
\(953\) −15.6568 −0.507174 −0.253587 0.967313i \(-0.581610\pi\)
−0.253587 + 0.967313i \(0.581610\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −9.49794 −0.307025
\(958\) 0 0
\(959\) −24.5431 −0.792539
\(960\) 0 0
\(961\) −16.4442 −0.530458
\(962\) 0 0
\(963\) 0.0692302 0.00223091
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −13.9923 −0.449961 −0.224980 0.974363i \(-0.572232\pi\)
−0.224980 + 0.974363i \(0.572232\pi\)
\(968\) 0 0
\(969\) −5.41147 −0.173842
\(970\) 0 0
\(971\) 13.0487 0.418753 0.209376 0.977835i \(-0.432857\pi\)
0.209376 + 0.977835i \(0.432857\pi\)
\(972\) 0 0
\(973\) −38.4252 −1.23186
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −17.3327 −0.554524 −0.277262 0.960794i \(-0.589427\pi\)
−0.277262 + 0.960794i \(0.589427\pi\)
\(978\) 0 0
\(979\) −19.7023 −0.629689
\(980\) 0 0
\(981\) 9.40642 0.300324
\(982\) 0 0
\(983\) −25.3114 −0.807308 −0.403654 0.914912i \(-0.632260\pi\)
−0.403654 + 0.914912i \(0.632260\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −5.36959 −0.170916
\(988\) 0 0
\(989\) −7.58441 −0.241170
\(990\) 0 0
\(991\) 37.1762 1.18094 0.590471 0.807059i \(-0.298942\pi\)
0.590471 + 0.807059i \(0.298942\pi\)
\(992\) 0 0
\(993\) −10.3732 −0.329184
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 26.3865 0.835669 0.417834 0.908523i \(-0.362789\pi\)
0.417834 + 0.908523i \(0.362789\pi\)
\(998\) 0 0
\(999\) 63.5458 2.01050
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 3800.2.a.t.1.3 yes 3
4.3 odd 2 7600.2.a.bs.1.1 3
5.2 odd 4 3800.2.d.o.3649.2 6
5.3 odd 4 3800.2.d.o.3649.5 6
5.4 even 2 3800.2.a.s.1.1 3
20.19 odd 2 7600.2.a.br.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3800.2.a.s.1.1 3 5.4 even 2
3800.2.a.t.1.3 yes 3 1.1 even 1 trivial
3800.2.d.o.3649.2 6 5.2 odd 4
3800.2.d.o.3649.5 6 5.3 odd 4
7600.2.a.br.1.3 3 20.19 odd 2
7600.2.a.bs.1.1 3 4.3 odd 2