Properties

Label 3800.2.a.s.1.2
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 1.2
Root \(-0.347296\) 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)
 
\(f(q)\) \(=\) \(q-0.347296 q^{3} +3.41147 q^{7} -2.87939 q^{9} +O(q^{10})\) \(q-0.347296 q^{3} +3.41147 q^{7} -2.87939 q^{9} +2.41147 q^{11} -6.29086 q^{13} +2.34730 q^{17} +1.00000 q^{19} -1.18479 q^{21} -2.49020 q^{23} +2.04189 q^{27} -8.17024 q^{29} -2.77332 q^{31} -0.837496 q^{33} +0.977711 q^{37} +2.18479 q^{39} -3.49020 q^{41} +2.75877 q^{43} -6.29086 q^{47} +4.63816 q^{49} -0.815207 q^{51} -2.38919 q^{53} -0.347296 q^{57} +3.67499 q^{59} -12.7442 q^{61} -9.82295 q^{63} -2.41147 q^{67} +0.864837 q^{69} +4.51754 q^{71} -1.81521 q^{73} +8.22668 q^{77} -5.04189 q^{79} +7.92902 q^{81} +8.07192 q^{83} +2.83750 q^{87} -2.94356 q^{89} -21.4611 q^{91} +0.963163 q^{93} +3.09833 q^{97} -6.94356 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{9} - 3 q^{11} - 3 q^{13} + 6 q^{17} + 3 q^{19} - 6 q^{23} + 3 q^{27} - 3 q^{29} - 15 q^{31} + 9 q^{37} + 3 q^{39} - 9 q^{41} - 3 q^{43} - 3 q^{47} - 3 q^{49} - 6 q^{51} - 3 q^{53} + 6 q^{59} - 9 q^{61} - 9 q^{63} + 3 q^{67} - 21 q^{69} - 9 q^{71} - 9 q^{73} + 18 q^{77} - 12 q^{79} - 9 q^{81} - 9 q^{83} + 6 q^{87} + 6 q^{89} - 27 q^{91} - 9 q^{93} + 21 q^{97} - 6 q^{99}+O(q^{100}) \) 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
\(3\) −0.347296 −0.200512 −0.100256 0.994962i \(-0.531966\pi\)
−0.100256 + 0.994962i \(0.531966\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.41147 1.28942 0.644708 0.764429i \(-0.276979\pi\)
0.644708 + 0.764429i \(0.276979\pi\)
\(8\) 0 0
\(9\) −2.87939 −0.959795
\(10\) 0 0
\(11\) 2.41147 0.727087 0.363543 0.931577i \(-0.381567\pi\)
0.363543 + 0.931577i \(0.381567\pi\)
\(12\) 0 0
\(13\) −6.29086 −1.74477 −0.872385 0.488819i \(-0.837428\pi\)
−0.872385 + 0.488819i \(0.837428\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.34730 0.569303 0.284651 0.958631i \(-0.408122\pi\)
0.284651 + 0.958631i \(0.408122\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −1.18479 −0.258543
\(22\) 0 0
\(23\) −2.49020 −0.519243 −0.259621 0.965711i \(-0.583598\pi\)
−0.259621 + 0.965711i \(0.583598\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2.04189 0.392962
\(28\) 0 0
\(29\) −8.17024 −1.51718 −0.758588 0.651570i \(-0.774111\pi\)
−0.758588 + 0.651570i \(0.774111\pi\)
\(30\) 0 0
\(31\) −2.77332 −0.498103 −0.249051 0.968490i \(-0.580119\pi\)
−0.249051 + 0.968490i \(0.580119\pi\)
\(32\) 0 0
\(33\) −0.837496 −0.145789
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.977711 0.160735 0.0803674 0.996765i \(-0.474391\pi\)
0.0803674 + 0.996765i \(0.474391\pi\)
\(38\) 0 0
\(39\) 2.18479 0.349847
\(40\) 0 0
\(41\) −3.49020 −0.545078 −0.272539 0.962145i \(-0.587863\pi\)
−0.272539 + 0.962145i \(0.587863\pi\)
\(42\) 0 0
\(43\) 2.75877 0.420709 0.210354 0.977625i \(-0.432538\pi\)
0.210354 + 0.977625i \(0.432538\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.29086 −0.917616 −0.458808 0.888535i \(-0.651723\pi\)
−0.458808 + 0.888535i \(0.651723\pi\)
\(48\) 0 0
\(49\) 4.63816 0.662594
\(50\) 0 0
\(51\) −0.815207 −0.114152
\(52\) 0 0
\(53\) −2.38919 −0.328180 −0.164090 0.986445i \(-0.552469\pi\)
−0.164090 + 0.986445i \(0.552469\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.347296 −0.0460005
\(58\) 0 0
\(59\) 3.67499 0.478443 0.239222 0.970965i \(-0.423108\pi\)
0.239222 + 0.970965i \(0.423108\pi\)
\(60\) 0 0
\(61\) −12.7442 −1.63173 −0.815865 0.578242i \(-0.803739\pi\)
−0.815865 + 0.578242i \(0.803739\pi\)
\(62\) 0 0
\(63\) −9.82295 −1.23758
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −2.41147 −0.294608 −0.147304 0.989091i \(-0.547060\pi\)
−0.147304 + 0.989091i \(0.547060\pi\)
\(68\) 0 0
\(69\) 0.864837 0.104114
\(70\) 0 0
\(71\) 4.51754 0.536133 0.268067 0.963400i \(-0.413615\pi\)
0.268067 + 0.963400i \(0.413615\pi\)
\(72\) 0 0
\(73\) −1.81521 −0.212454 −0.106227 0.994342i \(-0.533877\pi\)
−0.106227 + 0.994342i \(0.533877\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.22668 0.937517
\(78\) 0 0
\(79\) −5.04189 −0.567257 −0.283628 0.958934i \(-0.591538\pi\)
−0.283628 + 0.958934i \(0.591538\pi\)
\(80\) 0 0
\(81\) 7.92902 0.881002
\(82\) 0 0
\(83\) 8.07192 0.886008 0.443004 0.896520i \(-0.353913\pi\)
0.443004 + 0.896520i \(0.353913\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.83750 0.304212
\(88\) 0 0
\(89\) −2.94356 −0.312017 −0.156009 0.987756i \(-0.549863\pi\)
−0.156009 + 0.987756i \(0.549863\pi\)
\(90\) 0 0
\(91\) −21.4611 −2.24973
\(92\) 0 0
\(93\) 0.963163 0.0998754
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 3.09833 0.314587 0.157294 0.987552i \(-0.449723\pi\)
0.157294 + 0.987552i \(0.449723\pi\)
\(98\) 0 0
\(99\) −6.94356 −0.697854
\(100\) 0 0
\(101\) −6.02229 −0.599240 −0.299620 0.954059i \(-0.596860\pi\)
−0.299620 + 0.954059i \(0.596860\pi\)
\(102\) 0 0
\(103\) 7.25402 0.714760 0.357380 0.933959i \(-0.383670\pi\)
0.357380 + 0.933959i \(0.383670\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −8.94356 −0.864607 −0.432303 0.901728i \(-0.642299\pi\)
−0.432303 + 0.901728i \(0.642299\pi\)
\(108\) 0 0
\(109\) −9.81521 −0.940126 −0.470063 0.882633i \(-0.655769\pi\)
−0.470063 + 0.882633i \(0.655769\pi\)
\(110\) 0 0
\(111\) −0.339556 −0.0322292
\(112\) 0 0
\(113\) 7.57667 0.712753 0.356376 0.934342i \(-0.384012\pi\)
0.356376 + 0.934342i \(0.384012\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 18.1138 1.67462
\(118\) 0 0
\(119\) 8.00774 0.734068
\(120\) 0 0
\(121\) −5.18479 −0.471345
\(122\) 0 0
\(123\) 1.21213 0.109294
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −8.33275 −0.739412 −0.369706 0.929149i \(-0.620542\pi\)
−0.369706 + 0.929149i \(0.620542\pi\)
\(128\) 0 0
\(129\) −0.958111 −0.0843570
\(130\) 0 0
\(131\) −13.0915 −1.14381 −0.571906 0.820319i \(-0.693796\pi\)
−0.571906 + 0.820319i \(0.693796\pi\)
\(132\) 0 0
\(133\) 3.41147 0.295812
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 15.3327 1.30997 0.654983 0.755644i \(-0.272676\pi\)
0.654983 + 0.755644i \(0.272676\pi\)
\(138\) 0 0
\(139\) −7.77837 −0.659753 −0.329876 0.944024i \(-0.607007\pi\)
−0.329876 + 0.944024i \(0.607007\pi\)
\(140\) 0 0
\(141\) 2.18479 0.183993
\(142\) 0 0
\(143\) −15.1702 −1.26860
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −1.61081 −0.132858
\(148\) 0 0
\(149\) −17.1361 −1.40384 −0.701922 0.712254i \(-0.747674\pi\)
−0.701922 + 0.712254i \(0.747674\pi\)
\(150\) 0 0
\(151\) −21.4638 −1.74670 −0.873349 0.487094i \(-0.838057\pi\)
−0.873349 + 0.487094i \(0.838057\pi\)
\(152\) 0 0
\(153\) −6.75877 −0.546414
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 19.5253 1.55829 0.779144 0.626846i \(-0.215654\pi\)
0.779144 + 0.626846i \(0.215654\pi\)
\(158\) 0 0
\(159\) 0.829755 0.0658039
\(160\) 0 0
\(161\) −8.49525 −0.669520
\(162\) 0 0
\(163\) 3.59627 0.281681 0.140841 0.990032i \(-0.455019\pi\)
0.140841 + 0.990032i \(0.455019\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 13.1361 1.01650 0.508251 0.861209i \(-0.330292\pi\)
0.508251 + 0.861209i \(0.330292\pi\)
\(168\) 0 0
\(169\) 26.5749 2.04422
\(170\) 0 0
\(171\) −2.87939 −0.220192
\(172\) 0 0
\(173\) −16.4757 −1.25262 −0.626310 0.779574i \(-0.715436\pi\)
−0.626310 + 0.779574i \(0.715436\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1.27631 −0.0959334
\(178\) 0 0
\(179\) −23.1438 −1.72985 −0.864926 0.501900i \(-0.832635\pi\)
−0.864926 + 0.501900i \(0.832635\pi\)
\(180\) 0 0
\(181\) 24.1634 1.79605 0.898027 0.439940i \(-0.145000\pi\)
0.898027 + 0.439940i \(0.145000\pi\)
\(182\) 0 0
\(183\) 4.42602 0.327181
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 5.66044 0.413933
\(188\) 0 0
\(189\) 6.96585 0.506691
\(190\) 0 0
\(191\) 2.62361 0.189838 0.0949188 0.995485i \(-0.469741\pi\)
0.0949188 + 0.995485i \(0.469741\pi\)
\(192\) 0 0
\(193\) 12.5544 0.903684 0.451842 0.892098i \(-0.350767\pi\)
0.451842 + 0.892098i \(0.350767\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 16.9094 1.20475 0.602373 0.798215i \(-0.294222\pi\)
0.602373 + 0.798215i \(0.294222\pi\)
\(198\) 0 0
\(199\) 19.1607 1.35827 0.679135 0.734014i \(-0.262355\pi\)
0.679135 + 0.734014i \(0.262355\pi\)
\(200\) 0 0
\(201\) 0.837496 0.0590724
\(202\) 0 0
\(203\) −27.8726 −1.95627
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 7.17024 0.498366
\(208\) 0 0
\(209\) 2.41147 0.166805
\(210\) 0 0
\(211\) −13.4534 −0.926168 −0.463084 0.886314i \(-0.653257\pi\)
−0.463084 + 0.886314i \(0.653257\pi\)
\(212\) 0 0
\(213\) −1.56893 −0.107501
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −9.46110 −0.642262
\(218\) 0 0
\(219\) 0.630415 0.0425995
\(220\) 0 0
\(221\) −14.7665 −0.993303
\(222\) 0 0
\(223\) −28.9864 −1.94107 −0.970536 0.240956i \(-0.922539\pi\)
−0.970536 + 0.240956i \(0.922539\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −10.6091 −0.704148 −0.352074 0.935972i \(-0.614523\pi\)
−0.352074 + 0.935972i \(0.614523\pi\)
\(228\) 0 0
\(229\) −12.8375 −0.848326 −0.424163 0.905586i \(-0.639432\pi\)
−0.424163 + 0.905586i \(0.639432\pi\)
\(230\) 0 0
\(231\) −2.85710 −0.187983
\(232\) 0 0
\(233\) 10.4953 0.687567 0.343783 0.939049i \(-0.388291\pi\)
0.343783 + 0.939049i \(0.388291\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.75103 0.113742
\(238\) 0 0
\(239\) 4.07873 0.263831 0.131915 0.991261i \(-0.457887\pi\)
0.131915 + 0.991261i \(0.457887\pi\)
\(240\) 0 0
\(241\) −14.6382 −0.942927 −0.471463 0.881886i \(-0.656274\pi\)
−0.471463 + 0.881886i \(0.656274\pi\)
\(242\) 0 0
\(243\) −8.87939 −0.569613
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −6.29086 −0.400278
\(248\) 0 0
\(249\) −2.80335 −0.177655
\(250\) 0 0
\(251\) −12.0692 −0.761803 −0.380902 0.924616i \(-0.624386\pi\)
−0.380902 + 0.924616i \(0.624386\pi\)
\(252\) 0 0
\(253\) −6.00505 −0.377534
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1.62361 −0.101278 −0.0506389 0.998717i \(-0.516126\pi\)
−0.0506389 + 0.998717i \(0.516126\pi\)
\(258\) 0 0
\(259\) 3.33544 0.207254
\(260\) 0 0
\(261\) 23.5253 1.45618
\(262\) 0 0
\(263\) −28.2618 −1.74269 −0.871347 0.490666i \(-0.836753\pi\)
−0.871347 + 0.490666i \(0.836753\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 1.02229 0.0625631
\(268\) 0 0
\(269\) −10.1138 −0.616650 −0.308325 0.951281i \(-0.599768\pi\)
−0.308325 + 0.951281i \(0.599768\pi\)
\(270\) 0 0
\(271\) −22.9486 −1.39403 −0.697015 0.717057i \(-0.745489\pi\)
−0.697015 + 0.717057i \(0.745489\pi\)
\(272\) 0 0
\(273\) 7.45336 0.451098
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −14.1634 −0.850998 −0.425499 0.904959i \(-0.639901\pi\)
−0.425499 + 0.904959i \(0.639901\pi\)
\(278\) 0 0
\(279\) 7.98545 0.478077
\(280\) 0 0
\(281\) 0.822948 0.0490930 0.0245465 0.999699i \(-0.492186\pi\)
0.0245465 + 0.999699i \(0.492186\pi\)
\(282\) 0 0
\(283\) −4.83481 −0.287399 −0.143700 0.989621i \(-0.545900\pi\)
−0.143700 + 0.989621i \(0.545900\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −11.9067 −0.702832
\(288\) 0 0
\(289\) −11.4902 −0.675894
\(290\) 0 0
\(291\) −1.07604 −0.0630784
\(292\) 0 0
\(293\) 17.5699 1.02644 0.513221 0.858256i \(-0.328452\pi\)
0.513221 + 0.858256i \(0.328452\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 4.92396 0.285717
\(298\) 0 0
\(299\) 15.6655 0.905959
\(300\) 0 0
\(301\) 9.41147 0.542468
\(302\) 0 0
\(303\) 2.09152 0.120155
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −17.9632 −1.02521 −0.512606 0.858624i \(-0.671320\pi\)
−0.512606 + 0.858624i \(0.671320\pi\)
\(308\) 0 0
\(309\) −2.51930 −0.143318
\(310\) 0 0
\(311\) 27.4543 1.55679 0.778395 0.627775i \(-0.216034\pi\)
0.778395 + 0.627775i \(0.216034\pi\)
\(312\) 0 0
\(313\) −2.90941 −0.164450 −0.0822249 0.996614i \(-0.526203\pi\)
−0.0822249 + 0.996614i \(0.526203\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −23.1070 −1.29782 −0.648909 0.760866i \(-0.724775\pi\)
−0.648909 + 0.760866i \(0.724775\pi\)
\(318\) 0 0
\(319\) −19.7023 −1.10312
\(320\) 0 0
\(321\) 3.10607 0.173364
\(322\) 0 0
\(323\) 2.34730 0.130607
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 3.40879 0.188506
\(328\) 0 0
\(329\) −21.4611 −1.18319
\(330\) 0 0
\(331\) 2.89218 0.158969 0.0794843 0.996836i \(-0.474673\pi\)
0.0794843 + 0.996836i \(0.474673\pi\)
\(332\) 0 0
\(333\) −2.81521 −0.154272
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −28.3901 −1.54651 −0.773254 0.634096i \(-0.781372\pi\)
−0.773254 + 0.634096i \(0.781372\pi\)
\(338\) 0 0
\(339\) −2.63135 −0.142915
\(340\) 0 0
\(341\) −6.68779 −0.362164
\(342\) 0 0
\(343\) −8.05737 −0.435057
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 23.4175 1.25712 0.628558 0.777763i \(-0.283646\pi\)
0.628558 + 0.777763i \(0.283646\pi\)
\(348\) 0 0
\(349\) −5.48751 −0.293740 −0.146870 0.989156i \(-0.546920\pi\)
−0.146870 + 0.989156i \(0.546920\pi\)
\(350\) 0 0
\(351\) −12.8452 −0.685628
\(352\) 0 0
\(353\) −22.8530 −1.21634 −0.608171 0.793806i \(-0.708096\pi\)
−0.608171 + 0.793806i \(0.708096\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −2.78106 −0.147189
\(358\) 0 0
\(359\) 20.6604 1.09042 0.545208 0.838301i \(-0.316451\pi\)
0.545208 + 0.838301i \(0.316451\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 1.80066 0.0945101
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −6.29767 −0.328736 −0.164368 0.986399i \(-0.552558\pi\)
−0.164368 + 0.986399i \(0.552558\pi\)
\(368\) 0 0
\(369\) 10.0496 0.523163
\(370\) 0 0
\(371\) −8.15064 −0.423160
\(372\) 0 0
\(373\) −27.0401 −1.40009 −0.700043 0.714101i \(-0.746836\pi\)
−0.700043 + 0.714101i \(0.746836\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 51.3979 2.64712
\(378\) 0 0
\(379\) 11.8375 0.608051 0.304026 0.952664i \(-0.401669\pi\)
0.304026 + 0.952664i \(0.401669\pi\)
\(380\) 0 0
\(381\) 2.89393 0.148261
\(382\) 0 0
\(383\) 29.4252 1.50356 0.751779 0.659415i \(-0.229196\pi\)
0.751779 + 0.659415i \(0.229196\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −7.94356 −0.403794
\(388\) 0 0
\(389\) −6.75877 −0.342683 −0.171342 0.985212i \(-0.554810\pi\)
−0.171342 + 0.985212i \(0.554810\pi\)
\(390\) 0 0
\(391\) −5.84524 −0.295606
\(392\) 0 0
\(393\) 4.54664 0.229347
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 20.3158 1.01962 0.509811 0.860286i \(-0.329715\pi\)
0.509811 + 0.860286i \(0.329715\pi\)
\(398\) 0 0
\(399\) −1.18479 −0.0593138
\(400\) 0 0
\(401\) −5.57903 −0.278603 −0.139302 0.990250i \(-0.544486\pi\)
−0.139302 + 0.990250i \(0.544486\pi\)
\(402\) 0 0
\(403\) 17.4466 0.869075
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.35773 0.116868
\(408\) 0 0
\(409\) −37.3387 −1.84628 −0.923141 0.384462i \(-0.874387\pi\)
−0.923141 + 0.384462i \(0.874387\pi\)
\(410\) 0 0
\(411\) −5.32501 −0.262663
\(412\) 0 0
\(413\) 12.5371 0.616912
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 2.70140 0.132288
\(418\) 0 0
\(419\) 4.80840 0.234906 0.117453 0.993078i \(-0.462527\pi\)
0.117453 + 0.993078i \(0.462527\pi\)
\(420\) 0 0
\(421\) 27.0009 1.31594 0.657972 0.753042i \(-0.271414\pi\)
0.657972 + 0.753042i \(0.271414\pi\)
\(422\) 0 0
\(423\) 18.1138 0.880723
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −43.4766 −2.10398
\(428\) 0 0
\(429\) 5.26857 0.254369
\(430\) 0 0
\(431\) 36.2249 1.74489 0.872447 0.488709i \(-0.162532\pi\)
0.872447 + 0.488709i \(0.162532\pi\)
\(432\) 0 0
\(433\) 31.6382 1.52043 0.760216 0.649670i \(-0.225093\pi\)
0.760216 + 0.649670i \(0.225093\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2.49020 −0.119122
\(438\) 0 0
\(439\) 9.88444 0.471758 0.235879 0.971782i \(-0.424203\pi\)
0.235879 + 0.971782i \(0.424203\pi\)
\(440\) 0 0
\(441\) −13.3550 −0.635954
\(442\) 0 0
\(443\) −29.3191 −1.39299 −0.696497 0.717560i \(-0.745259\pi\)
−0.696497 + 0.717560i \(0.745259\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 5.95130 0.281487
\(448\) 0 0
\(449\) 2.86247 0.135088 0.0675442 0.997716i \(-0.478484\pi\)
0.0675442 + 0.997716i \(0.478484\pi\)
\(450\) 0 0
\(451\) −8.41653 −0.396319
\(452\) 0 0
\(453\) 7.45430 0.350233
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 25.6664 1.20062 0.600312 0.799766i \(-0.295043\pi\)
0.600312 + 0.799766i \(0.295043\pi\)
\(458\) 0 0
\(459\) 4.79292 0.223714
\(460\) 0 0
\(461\) 5.77332 0.268890 0.134445 0.990921i \(-0.457075\pi\)
0.134445 + 0.990921i \(0.457075\pi\)
\(462\) 0 0
\(463\) 31.1097 1.44579 0.722895 0.690958i \(-0.242811\pi\)
0.722895 + 0.690958i \(0.242811\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 37.8408 1.75106 0.875532 0.483161i \(-0.160511\pi\)
0.875532 + 0.483161i \(0.160511\pi\)
\(468\) 0 0
\(469\) −8.22668 −0.379873
\(470\) 0 0
\(471\) −6.78106 −0.312455
\(472\) 0 0
\(473\) 6.65270 0.305892
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 6.87939 0.314985
\(478\) 0 0
\(479\) 5.10338 0.233179 0.116590 0.993180i \(-0.462804\pi\)
0.116590 + 0.993180i \(0.462804\pi\)
\(480\) 0 0
\(481\) −6.15064 −0.280445
\(482\) 0 0
\(483\) 2.95037 0.134246
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −18.8503 −0.854188 −0.427094 0.904207i \(-0.640463\pi\)
−0.427094 + 0.904207i \(0.640463\pi\)
\(488\) 0 0
\(489\) −1.24897 −0.0564804
\(490\) 0 0
\(491\) 0.593578 0.0267878 0.0133939 0.999910i \(-0.495736\pi\)
0.0133939 + 0.999910i \(0.495736\pi\)
\(492\) 0 0
\(493\) −19.1780 −0.863733
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 15.4115 0.691299
\(498\) 0 0
\(499\) −29.7965 −1.33388 −0.666938 0.745113i \(-0.732395\pi\)
−0.666938 + 0.745113i \(0.732395\pi\)
\(500\) 0 0
\(501\) −4.56212 −0.203820
\(502\) 0 0
\(503\) 21.8357 0.973608 0.486804 0.873511i \(-0.338163\pi\)
0.486804 + 0.873511i \(0.338163\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −9.22937 −0.409891
\(508\) 0 0
\(509\) 12.4861 0.553436 0.276718 0.960951i \(-0.410753\pi\)
0.276718 + 0.960951i \(0.410753\pi\)
\(510\) 0 0
\(511\) −6.19253 −0.273942
\(512\) 0 0
\(513\) 2.04189 0.0901516
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −15.1702 −0.667187
\(518\) 0 0
\(519\) 5.72193 0.251165
\(520\) 0 0
\(521\) 3.37908 0.148040 0.0740201 0.997257i \(-0.476417\pi\)
0.0740201 + 0.997257i \(0.476417\pi\)
\(522\) 0 0
\(523\) 15.7419 0.688343 0.344172 0.938907i \(-0.388160\pi\)
0.344172 + 0.938907i \(0.388160\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −6.50980 −0.283571
\(528\) 0 0
\(529\) −16.7989 −0.730387
\(530\) 0 0
\(531\) −10.5817 −0.459207
\(532\) 0 0
\(533\) 21.9564 0.951035
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 8.03777 0.346855
\(538\) 0 0
\(539\) 11.1848 0.481763
\(540\) 0 0
\(541\) −1.31820 −0.0566739 −0.0283369 0.999598i \(-0.509021\pi\)
−0.0283369 + 0.999598i \(0.509021\pi\)
\(542\) 0 0
\(543\) −8.39187 −0.360130
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −9.68954 −0.414295 −0.207147 0.978310i \(-0.566418\pi\)
−0.207147 + 0.978310i \(0.566418\pi\)
\(548\) 0 0
\(549\) 36.6955 1.56613
\(550\) 0 0
\(551\) −8.17024 −0.348064
\(552\) 0 0
\(553\) −17.2003 −0.731430
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −14.4757 −0.613353 −0.306677 0.951814i \(-0.599217\pi\)
−0.306677 + 0.951814i \(0.599217\pi\)
\(558\) 0 0
\(559\) −17.3550 −0.734040
\(560\) 0 0
\(561\) −1.96585 −0.0829983
\(562\) 0 0
\(563\) 25.8898 1.09113 0.545563 0.838070i \(-0.316316\pi\)
0.545563 + 0.838070i \(0.316316\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 27.0496 1.13598
\(568\) 0 0
\(569\) 34.8384 1.46050 0.730251 0.683178i \(-0.239403\pi\)
0.730251 + 0.683178i \(0.239403\pi\)
\(570\) 0 0
\(571\) 24.8557 1.04018 0.520089 0.854112i \(-0.325899\pi\)
0.520089 + 0.854112i \(0.325899\pi\)
\(572\) 0 0
\(573\) −0.911169 −0.0380646
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 36.4620 1.51793 0.758967 0.651129i \(-0.225704\pi\)
0.758967 + 0.651129i \(0.225704\pi\)
\(578\) 0 0
\(579\) −4.36009 −0.181199
\(580\) 0 0
\(581\) 27.5371 1.14243
\(582\) 0 0
\(583\) −5.76146 −0.238615
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −30.9249 −1.27641 −0.638204 0.769868i \(-0.720322\pi\)
−0.638204 + 0.769868i \(0.720322\pi\)
\(588\) 0 0
\(589\) −2.77332 −0.114273
\(590\) 0 0
\(591\) −5.87258 −0.241566
\(592\) 0 0
\(593\) 30.2909 1.24390 0.621948 0.783058i \(-0.286341\pi\)
0.621948 + 0.783058i \(0.286341\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −6.65446 −0.272349
\(598\) 0 0
\(599\) 25.3236 1.03469 0.517347 0.855776i \(-0.326920\pi\)
0.517347 + 0.855776i \(0.326920\pi\)
\(600\) 0 0
\(601\) −34.2894 −1.39869 −0.699347 0.714782i \(-0.746526\pi\)
−0.699347 + 0.714782i \(0.746526\pi\)
\(602\) 0 0
\(603\) 6.94356 0.282764
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −42.2550 −1.71508 −0.857538 0.514421i \(-0.828007\pi\)
−0.857538 + 0.514421i \(0.828007\pi\)
\(608\) 0 0
\(609\) 9.68004 0.392255
\(610\) 0 0
\(611\) 39.5749 1.60103
\(612\) 0 0
\(613\) −23.6759 −0.956262 −0.478131 0.878289i \(-0.658686\pi\)
−0.478131 + 0.878289i \(0.658686\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 21.8425 0.879348 0.439674 0.898157i \(-0.355094\pi\)
0.439674 + 0.898157i \(0.355094\pi\)
\(618\) 0 0
\(619\) 38.2841 1.53877 0.769383 0.638788i \(-0.220564\pi\)
0.769383 + 0.638788i \(0.220564\pi\)
\(620\) 0 0
\(621\) −5.08471 −0.204042
\(622\) 0 0
\(623\) −10.0419 −0.402320
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −0.837496 −0.0334464
\(628\) 0 0
\(629\) 2.29498 0.0915068
\(630\) 0 0
\(631\) 7.28817 0.290138 0.145069 0.989422i \(-0.453660\pi\)
0.145069 + 0.989422i \(0.453660\pi\)
\(632\) 0 0
\(633\) 4.67230 0.185707
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −29.1780 −1.15607
\(638\) 0 0
\(639\) −13.0077 −0.514578
\(640\) 0 0
\(641\) 14.2026 0.560970 0.280485 0.959858i \(-0.409505\pi\)
0.280485 + 0.959858i \(0.409505\pi\)
\(642\) 0 0
\(643\) −0.867526 −0.0342119 −0.0171059 0.999854i \(-0.505445\pi\)
−0.0171059 + 0.999854i \(0.505445\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −46.5503 −1.83008 −0.915040 0.403364i \(-0.867841\pi\)
−0.915040 + 0.403364i \(0.867841\pi\)
\(648\) 0 0
\(649\) 8.86215 0.347870
\(650\) 0 0
\(651\) 3.28581 0.128781
\(652\) 0 0
\(653\) −27.9050 −1.09201 −0.546003 0.837783i \(-0.683851\pi\)
−0.546003 + 0.837783i \(0.683851\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 5.22668 0.203912
\(658\) 0 0
\(659\) 49.5340 1.92957 0.964784 0.263042i \(-0.0847257\pi\)
0.964784 + 0.263042i \(0.0847257\pi\)
\(660\) 0 0
\(661\) 40.0164 1.55646 0.778229 0.627980i \(-0.216118\pi\)
0.778229 + 0.627980i \(0.216118\pi\)
\(662\) 0 0
\(663\) 5.12836 0.199169
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 20.3455 0.787783
\(668\) 0 0
\(669\) 10.0669 0.389208
\(670\) 0 0
\(671\) −30.7324 −1.18641
\(672\) 0 0
\(673\) 26.5449 1.02323 0.511615 0.859215i \(-0.329047\pi\)
0.511615 + 0.859215i \(0.329047\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −22.5398 −0.866276 −0.433138 0.901328i \(-0.642594\pi\)
−0.433138 + 0.901328i \(0.642594\pi\)
\(678\) 0 0
\(679\) 10.5699 0.405634
\(680\) 0 0
\(681\) 3.68449 0.141190
\(682\) 0 0
\(683\) −44.2154 −1.69186 −0.845928 0.533297i \(-0.820953\pi\)
−0.845928 + 0.533297i \(0.820953\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 4.45842 0.170099
\(688\) 0 0
\(689\) 15.0300 0.572599
\(690\) 0 0
\(691\) 31.0743 1.18212 0.591061 0.806627i \(-0.298709\pi\)
0.591061 + 0.806627i \(0.298709\pi\)
\(692\) 0 0
\(693\) −23.6878 −0.899825
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −8.19253 −0.310314
\(698\) 0 0
\(699\) −3.64496 −0.137865
\(700\) 0 0
\(701\) 44.8185 1.69277 0.846386 0.532570i \(-0.178774\pi\)
0.846386 + 0.532570i \(0.178774\pi\)
\(702\) 0 0
\(703\) 0.977711 0.0368751
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −20.5449 −0.772670
\(708\) 0 0
\(709\) −40.1334 −1.50724 −0.753621 0.657309i \(-0.771695\pi\)
−0.753621 + 0.657309i \(0.771695\pi\)
\(710\) 0 0
\(711\) 14.5175 0.544450
\(712\) 0 0
\(713\) 6.90612 0.258636
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −1.41653 −0.0529012
\(718\) 0 0
\(719\) 20.8571 0.777838 0.388919 0.921272i \(-0.372849\pi\)
0.388919 + 0.921272i \(0.372849\pi\)
\(720\) 0 0
\(721\) 24.7469 0.921623
\(722\) 0 0
\(723\) 5.08378 0.189068
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 11.2540 0.417389 0.208694 0.977981i \(-0.433079\pi\)
0.208694 + 0.977981i \(0.433079\pi\)
\(728\) 0 0
\(729\) −20.7033 −0.766788
\(730\) 0 0
\(731\) 6.47565 0.239511
\(732\) 0 0
\(733\) 11.3952 0.420890 0.210445 0.977606i \(-0.432509\pi\)
0.210445 + 0.977606i \(0.432509\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −5.81521 −0.214206
\(738\) 0 0
\(739\) −45.7015 −1.68116 −0.840579 0.541690i \(-0.817785\pi\)
−0.840579 + 0.541690i \(0.817785\pi\)
\(740\) 0 0
\(741\) 2.18479 0.0802604
\(742\) 0 0
\(743\) −3.82470 −0.140315 −0.0701574 0.997536i \(-0.522350\pi\)
−0.0701574 + 0.997536i \(0.522350\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −23.2422 −0.850386
\(748\) 0 0
\(749\) −30.5107 −1.11484
\(750\) 0 0
\(751\) 26.5550 0.969005 0.484503 0.874790i \(-0.339001\pi\)
0.484503 + 0.874790i \(0.339001\pi\)
\(752\) 0 0
\(753\) 4.19160 0.152750
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −4.64765 −0.168922 −0.0844609 0.996427i \(-0.526917\pi\)
−0.0844609 + 0.996427i \(0.526917\pi\)
\(758\) 0 0
\(759\) 2.08553 0.0757000
\(760\) 0 0
\(761\) −26.0327 −0.943685 −0.471843 0.881683i \(-0.656411\pi\)
−0.471843 + 0.881683i \(0.656411\pi\)
\(762\) 0 0
\(763\) −33.4843 −1.21221
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −23.1189 −0.834774
\(768\) 0 0
\(769\) 3.10876 0.112105 0.0560523 0.998428i \(-0.482149\pi\)
0.0560523 + 0.998428i \(0.482149\pi\)
\(770\) 0 0
\(771\) 0.563873 0.0203074
\(772\) 0 0
\(773\) 19.4516 0.699626 0.349813 0.936820i \(-0.386245\pi\)
0.349813 + 0.936820i \(0.386245\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −1.15839 −0.0415568
\(778\) 0 0
\(779\) −3.49020 −0.125049
\(780\) 0 0
\(781\) 10.8939 0.389816
\(782\) 0 0
\(783\) −16.6827 −0.596192
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 14.0496 0.500815 0.250408 0.968141i \(-0.419435\pi\)
0.250408 + 0.968141i \(0.419435\pi\)
\(788\) 0 0
\(789\) 9.81521 0.349431
\(790\) 0 0
\(791\) 25.8476 0.919035
\(792\) 0 0
\(793\) 80.1721 2.84700
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −2.57810 −0.0913209 −0.0456604 0.998957i \(-0.514539\pi\)
−0.0456604 + 0.998957i \(0.514539\pi\)
\(798\) 0 0
\(799\) −14.7665 −0.522402
\(800\) 0 0
\(801\) 8.47565 0.299472
\(802\) 0 0
\(803\) −4.37733 −0.154472
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 3.51249 0.123645
\(808\) 0 0
\(809\) 19.8220 0.696905 0.348452 0.937326i \(-0.386707\pi\)
0.348452 + 0.937326i \(0.386707\pi\)
\(810\) 0 0
\(811\) 22.6563 0.795571 0.397786 0.917478i \(-0.369779\pi\)
0.397786 + 0.917478i \(0.369779\pi\)
\(812\) 0 0
\(813\) 7.96997 0.279519
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 2.75877 0.0965172
\(818\) 0 0
\(819\) 61.7948 2.15928
\(820\) 0 0
\(821\) −23.0570 −0.804696 −0.402348 0.915487i \(-0.631806\pi\)
−0.402348 + 0.915487i \(0.631806\pi\)
\(822\) 0 0
\(823\) 44.6563 1.55662 0.778311 0.627879i \(-0.216077\pi\)
0.778311 + 0.627879i \(0.216077\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 52.4306 1.82319 0.911595 0.411090i \(-0.134852\pi\)
0.911595 + 0.411090i \(0.134852\pi\)
\(828\) 0 0
\(829\) −40.2412 −1.39764 −0.698818 0.715300i \(-0.746290\pi\)
−0.698818 + 0.715300i \(0.746290\pi\)
\(830\) 0 0
\(831\) 4.91891 0.170635
\(832\) 0 0
\(833\) 10.8871 0.377217
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −5.66281 −0.195735
\(838\) 0 0
\(839\) −52.5185 −1.81314 −0.906570 0.422056i \(-0.861308\pi\)
−0.906570 + 0.422056i \(0.861308\pi\)
\(840\) 0 0
\(841\) 37.7529 1.30182
\(842\) 0 0
\(843\) −0.285807 −0.00984371
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −17.6878 −0.607760
\(848\) 0 0
\(849\) 1.67911 0.0576269
\(850\) 0 0
\(851\) −2.43470 −0.0834603
\(852\) 0 0
\(853\) −20.2094 −0.691958 −0.345979 0.938242i \(-0.612453\pi\)
−0.345979 + 0.938242i \(0.612453\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −14.9522 −0.510759 −0.255379 0.966841i \(-0.582200\pi\)
−0.255379 + 0.966841i \(0.582200\pi\)
\(858\) 0 0
\(859\) −34.3286 −1.17128 −0.585639 0.810572i \(-0.699156\pi\)
−0.585639 + 0.810572i \(0.699156\pi\)
\(860\) 0 0
\(861\) 4.13516 0.140926
\(862\) 0 0
\(863\) 27.0392 0.920425 0.460213 0.887809i \(-0.347773\pi\)
0.460213 + 0.887809i \(0.347773\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 3.99050 0.135525
\(868\) 0 0
\(869\) −12.1584 −0.412445
\(870\) 0 0
\(871\) 15.1702 0.514024
\(872\) 0 0
\(873\) −8.92127 −0.301939
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −17.2216 −0.581533 −0.290767 0.956794i \(-0.593910\pi\)
−0.290767 + 0.956794i \(0.593910\pi\)
\(878\) 0 0
\(879\) −6.10195 −0.205814
\(880\) 0 0
\(881\) 51.2312 1.72602 0.863012 0.505183i \(-0.168575\pi\)
0.863012 + 0.505183i \(0.168575\pi\)
\(882\) 0 0
\(883\) 27.8939 0.938706 0.469353 0.883011i \(-0.344487\pi\)
0.469353 + 0.883011i \(0.344487\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 32.5749 1.09376 0.546879 0.837211i \(-0.315816\pi\)
0.546879 + 0.837211i \(0.315816\pi\)
\(888\) 0 0
\(889\) −28.4270 −0.953409
\(890\) 0 0
\(891\) 19.1206 0.640565
\(892\) 0 0
\(893\) −6.29086 −0.210516
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −5.44057 −0.181655
\(898\) 0 0
\(899\) 22.6587 0.755710
\(900\) 0 0
\(901\) −5.60813 −0.186834
\(902\) 0 0
\(903\) −3.26857 −0.108771
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −23.6673 −0.785858 −0.392929 0.919569i \(-0.628538\pi\)
−0.392929 + 0.919569i \(0.628538\pi\)
\(908\) 0 0
\(909\) 17.3405 0.575148
\(910\) 0 0
\(911\) 11.4233 0.378472 0.189236 0.981932i \(-0.439399\pi\)
0.189236 + 0.981932i \(0.439399\pi\)
\(912\) 0 0
\(913\) 19.4652 0.644205
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −44.6614 −1.47485
\(918\) 0 0
\(919\) −34.7401 −1.14597 −0.572985 0.819566i \(-0.694215\pi\)
−0.572985 + 0.819566i \(0.694215\pi\)
\(920\) 0 0
\(921\) 6.23854 0.205567
\(922\) 0 0
\(923\) −28.4192 −0.935430
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −20.8871 −0.686023
\(928\) 0 0
\(929\) 50.1848 1.64651 0.823255 0.567672i \(-0.192156\pi\)
0.823255 + 0.567672i \(0.192156\pi\)
\(930\) 0 0
\(931\) 4.63816 0.152009
\(932\) 0 0
\(933\) −9.53478 −0.312155
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −4.48927 −0.146658 −0.0733290 0.997308i \(-0.523362\pi\)
−0.0733290 + 0.997308i \(0.523362\pi\)
\(938\) 0 0
\(939\) 1.01043 0.0329741
\(940\) 0 0
\(941\) −7.70233 −0.251089 −0.125544 0.992088i \(-0.540068\pi\)
−0.125544 + 0.992088i \(0.540068\pi\)
\(942\) 0 0
\(943\) 8.69129 0.283028
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 13.4415 0.436790 0.218395 0.975860i \(-0.429918\pi\)
0.218395 + 0.975860i \(0.429918\pi\)
\(948\) 0 0
\(949\) 11.4192 0.370683
\(950\) 0 0
\(951\) 8.02498 0.260228
\(952\) 0 0
\(953\) 23.1943 0.751337 0.375668 0.926754i \(-0.377413\pi\)
0.375668 + 0.926754i \(0.377413\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 6.84255 0.221188
\(958\) 0 0
\(959\) 52.3073 1.68909
\(960\) 0 0
\(961\) −23.3087 −0.751894
\(962\) 0 0
\(963\) 25.7520 0.829845
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 22.1429 0.712068 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(968\) 0 0
\(969\) −0.815207 −0.0261882
\(970\) 0 0
\(971\) 26.9804 0.865842 0.432921 0.901432i \(-0.357483\pi\)
0.432921 + 0.901432i \(0.357483\pi\)
\(972\) 0 0
\(973\) −26.5357 −0.850696
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 2.64496 0.0846199 0.0423099 0.999105i \(-0.486528\pi\)
0.0423099 + 0.999105i \(0.486528\pi\)
\(978\) 0 0
\(979\) −7.09833 −0.226863
\(980\) 0 0
\(981\) 28.2618 0.902329
\(982\) 0 0
\(983\) −22.4492 −0.716020 −0.358010 0.933718i \(-0.616545\pi\)
−0.358010 + 0.933718i \(0.616545\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 7.45336 0.237243
\(988\) 0 0
\(989\) −6.86989 −0.218450
\(990\) 0 0
\(991\) −26.4570 −0.840434 −0.420217 0.907424i \(-0.638046\pi\)
−0.420217 + 0.907424i \(0.638046\pi\)
\(992\) 0 0
\(993\) −1.00444 −0.0318750
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −2.17881 −0.0690035 −0.0345017 0.999405i \(-0.510984\pi\)
−0.0345017 + 0.999405i \(0.510984\pi\)
\(998\) 0 0
\(999\) 1.99638 0.0631626
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.s.1.2 3
4.3 odd 2 7600.2.a.br.1.2 3
5.2 odd 4 3800.2.d.o.3649.4 6
5.3 odd 4 3800.2.d.o.3649.3 6
5.4 even 2 3800.2.a.t.1.2 yes 3
20.19 odd 2 7600.2.a.bs.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3800.2.a.s.1.2 3 1.1 even 1 trivial
3800.2.a.t.1.2 yes 3 5.4 even 2
3800.2.d.o.3649.3 6 5.3 odd 4
3800.2.d.o.3649.4 6 5.2 odd 4
7600.2.a.br.1.2 3 4.3 odd 2
7600.2.a.bs.1.2 3 20.19 odd 2