Properties

Label 1900.3.e.c.1101.2
Level $1900$
Weight $3$
Character 1900.1101
Analytic conductor $51.771$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1900,3,Mod(1101,1900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1900.1101");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1900.e (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(51.7712502285\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{6}, \sqrt{-14})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1101.2
Root \(1.22474 + 1.87083i\) of defining polynomial
Character \(\chi\) \(=\) 1900.1101
Dual form 1900.3.e.c.1101.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.74166i q^{3} +9.79796 q^{7} -5.00000 q^{9} +O(q^{10})\) \(q-3.74166i q^{3} +9.79796 q^{7} -5.00000 q^{9} +4.00000 q^{11} -11.2250i q^{13} +19.5959 q^{17} +(5.00000 + 18.3303i) q^{19} -36.6606i q^{21} +9.79796 q^{23} -14.9666i q^{27} +36.6606i q^{29} -36.6606i q^{31} -14.9666i q^{33} +33.6749i q^{37} -42.0000 q^{39} +36.6606i q^{41} +68.5857 q^{43} -9.79796 q^{47} +47.0000 q^{49} -73.3212i q^{51} +56.1249i q^{53} +(68.5857 - 18.7083i) q^{57} -73.3212i q^{59} +100.000 q^{61} -48.9898 q^{63} +11.2250i q^{67} -36.6606i q^{69} -36.6606i q^{71} +19.5959 q^{73} +39.1918 q^{77} +109.982i q^{79} -101.000 q^{81} +29.3939 q^{83} +137.171 q^{87} -146.642i q^{89} -109.982i q^{91} -137.171 q^{93} +123.475i q^{97} -20.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 20 q^{9} + 16 q^{11} + 20 q^{19} - 168 q^{39} + 188 q^{49} + 400 q^{61} - 404 q^{81} - 80 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 3.74166i 1.24722i −0.781736 0.623610i \(-0.785666\pi\)
0.781736 0.623610i \(-0.214334\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 9.79796 1.39971 0.699854 0.714286i \(-0.253248\pi\)
0.699854 + 0.714286i \(0.253248\pi\)
\(8\) 0 0
\(9\) −5.00000 −0.555556
\(10\) 0 0
\(11\) 4.00000 0.363636 0.181818 0.983332i \(-0.441802\pi\)
0.181818 + 0.983332i \(0.441802\pi\)
\(12\) 0 0
\(13\) 11.2250i 0.863459i −0.902003 0.431730i \(-0.857903\pi\)
0.902003 0.431730i \(-0.142097\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 19.5959 1.15270 0.576351 0.817203i \(-0.304476\pi\)
0.576351 + 0.817203i \(0.304476\pi\)
\(18\) 0 0
\(19\) 5.00000 + 18.3303i 0.263158 + 0.964753i
\(20\) 0 0
\(21\) 36.6606i 1.74574i
\(22\) 0 0
\(23\) 9.79796 0.425998 0.212999 0.977052i \(-0.431677\pi\)
0.212999 + 0.977052i \(0.431677\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 14.9666i 0.554320i
\(28\) 0 0
\(29\) 36.6606i 1.26416i 0.774904 + 0.632079i \(0.217798\pi\)
−0.774904 + 0.632079i \(0.782202\pi\)
\(30\) 0 0
\(31\) 36.6606i 1.18260i −0.806452 0.591300i \(-0.798615\pi\)
0.806452 0.591300i \(-0.201385\pi\)
\(32\) 0 0
\(33\) 14.9666i 0.453534i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 33.6749i 0.910133i 0.890457 + 0.455066i \(0.150384\pi\)
−0.890457 + 0.455066i \(0.849616\pi\)
\(38\) 0 0
\(39\) −42.0000 −1.07692
\(40\) 0 0
\(41\) 36.6606i 0.894161i 0.894494 + 0.447081i \(0.147536\pi\)
−0.894494 + 0.447081i \(0.852464\pi\)
\(42\) 0 0
\(43\) 68.5857 1.59502 0.797508 0.603308i \(-0.206151\pi\)
0.797508 + 0.603308i \(0.206151\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −9.79796 −0.208467 −0.104234 0.994553i \(-0.533239\pi\)
−0.104234 + 0.994553i \(0.533239\pi\)
\(48\) 0 0
\(49\) 47.0000 0.959184
\(50\) 0 0
\(51\) 73.3212i 1.43767i
\(52\) 0 0
\(53\) 56.1249i 1.05896i 0.848323 + 0.529480i \(0.177613\pi\)
−0.848323 + 0.529480i \(0.822387\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 68.5857 18.7083i 1.20326 0.328216i
\(58\) 0 0
\(59\) 73.3212i 1.24273i −0.783520 0.621366i \(-0.786578\pi\)
0.783520 0.621366i \(-0.213422\pi\)
\(60\) 0 0
\(61\) 100.000 1.63934 0.819672 0.572833i \(-0.194156\pi\)
0.819672 + 0.572833i \(0.194156\pi\)
\(62\) 0 0
\(63\) −48.9898 −0.777616
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 11.2250i 0.167537i 0.996485 + 0.0837684i \(0.0266956\pi\)
−0.996485 + 0.0837684i \(0.973304\pi\)
\(68\) 0 0
\(69\) 36.6606i 0.531313i
\(70\) 0 0
\(71\) 36.6606i 0.516347i −0.966099 0.258173i \(-0.916879\pi\)
0.966099 0.258173i \(-0.0831205\pi\)
\(72\) 0 0
\(73\) 19.5959 0.268437 0.134219 0.990952i \(-0.457148\pi\)
0.134219 + 0.990952i \(0.457148\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 39.1918 0.508985
\(78\) 0 0
\(79\) 109.982i 1.39217i 0.717957 + 0.696087i \(0.245077\pi\)
−0.717957 + 0.696087i \(0.754923\pi\)
\(80\) 0 0
\(81\) −101.000 −1.24691
\(82\) 0 0
\(83\) 29.3939 0.354143 0.177072 0.984198i \(-0.443338\pi\)
0.177072 + 0.984198i \(0.443338\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 137.171 1.57668
\(88\) 0 0
\(89\) 146.642i 1.64767i −0.566831 0.823834i \(-0.691831\pi\)
0.566831 0.823834i \(-0.308169\pi\)
\(90\) 0 0
\(91\) 109.982i 1.20859i
\(92\) 0 0
\(93\) −137.171 −1.47496
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 123.475i 1.27293i 0.771304 + 0.636467i \(0.219605\pi\)
−0.771304 + 0.636467i \(0.780395\pi\)
\(98\) 0 0
\(99\) −20.0000 −0.202020
\(100\) 0 0
\(101\) 4.00000 0.0396040 0.0198020 0.999804i \(-0.493696\pi\)
0.0198020 + 0.999804i \(0.493696\pi\)
\(102\) 0 0
\(103\) 78.5748i 0.762862i 0.924397 + 0.381431i \(0.124569\pi\)
−0.924397 + 0.381431i \(0.875431\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 101.025i 0.944157i 0.881557 + 0.472078i \(0.156496\pi\)
−0.881557 + 0.472078i \(0.843504\pi\)
\(108\) 0 0
\(109\) 73.3212i 0.672672i −0.941742 0.336336i \(-0.890812\pi\)
0.941742 0.336336i \(-0.109188\pi\)
\(110\) 0 0
\(111\) 126.000 1.13514
\(112\) 0 0
\(113\) 11.2250i 0.0993360i −0.998766 0.0496680i \(-0.984184\pi\)
0.998766 0.0496680i \(-0.0158163\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 56.1249i 0.479700i
\(118\) 0 0
\(119\) 192.000 1.61345
\(120\) 0 0
\(121\) −105.000 −0.867769
\(122\) 0 0
\(123\) 137.171 1.11521
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 235.724i 1.85610i −0.372458 0.928049i \(-0.621485\pi\)
0.372458 0.928049i \(-0.378515\pi\)
\(128\) 0 0
\(129\) 256.624i 1.98934i
\(130\) 0 0
\(131\) 26.0000 0.198473 0.0992366 0.995064i \(-0.468360\pi\)
0.0992366 + 0.995064i \(0.468360\pi\)
\(132\) 0 0
\(133\) 48.9898 + 179.600i 0.368344 + 1.35037i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −254.747 −1.85947 −0.929733 0.368234i \(-0.879963\pi\)
−0.929733 + 0.368234i \(0.879963\pi\)
\(138\) 0 0
\(139\) −148.000 −1.06475 −0.532374 0.846509i \(-0.678700\pi\)
−0.532374 + 0.846509i \(0.678700\pi\)
\(140\) 0 0
\(141\) 36.6606i 0.260004i
\(142\) 0 0
\(143\) 44.8999i 0.313985i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 175.858i 1.19631i
\(148\) 0 0
\(149\) −52.0000 −0.348993 −0.174497 0.984658i \(-0.555830\pi\)
−0.174497 + 0.984658i \(0.555830\pi\)
\(150\) 0 0
\(151\) 73.3212i 0.485571i −0.970080 0.242785i \(-0.921939\pi\)
0.970080 0.242785i \(-0.0780611\pi\)
\(152\) 0 0
\(153\) −97.9796 −0.640389
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −117.576 −0.748889 −0.374444 0.927249i \(-0.622167\pi\)
−0.374444 + 0.927249i \(0.622167\pi\)
\(158\) 0 0
\(159\) 210.000 1.32075
\(160\) 0 0
\(161\) 96.0000 0.596273
\(162\) 0 0
\(163\) −9.79796 −0.0601102 −0.0300551 0.999548i \(-0.509568\pi\)
−0.0300551 + 0.999548i \(0.509568\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 33.6749i 0.201646i 0.994904 + 0.100823i \(0.0321476\pi\)
−0.994904 + 0.100823i \(0.967852\pi\)
\(168\) 0 0
\(169\) 43.0000 0.254438
\(170\) 0 0
\(171\) −25.0000 91.6515i −0.146199 0.535974i
\(172\) 0 0
\(173\) 280.624i 1.62211i −0.584973 0.811053i \(-0.698895\pi\)
0.584973 0.811053i \(-0.301105\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −274.343 −1.54996
\(178\) 0 0
\(179\) 146.642i 0.819231i 0.912258 + 0.409616i \(0.134337\pi\)
−0.912258 + 0.409616i \(0.865663\pi\)
\(180\) 0 0
\(181\) 256.624i 1.41781i −0.705302 0.708907i \(-0.749189\pi\)
0.705302 0.708907i \(-0.250811\pi\)
\(182\) 0 0
\(183\) 374.166i 2.04462i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 78.3837 0.419164
\(188\) 0 0
\(189\) 146.642i 0.775886i
\(190\) 0 0
\(191\) 158.000 0.827225 0.413613 0.910453i \(-0.364267\pi\)
0.413613 + 0.910453i \(0.364267\pi\)
\(192\) 0 0
\(193\) 213.274i 1.10505i −0.833497 0.552525i \(-0.813665\pi\)
0.833497 0.552525i \(-0.186335\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −176.363 −0.895245 −0.447622 0.894223i \(-0.647729\pi\)
−0.447622 + 0.894223i \(0.647729\pi\)
\(198\) 0 0
\(199\) −302.000 −1.51759 −0.758794 0.651331i \(-0.774211\pi\)
−0.758794 + 0.651331i \(0.774211\pi\)
\(200\) 0 0
\(201\) 42.0000 0.208955
\(202\) 0 0
\(203\) 359.199i 1.76945i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −48.9898 −0.236666
\(208\) 0 0
\(209\) 20.0000 + 73.3212i 0.0956938 + 0.350819i
\(210\) 0 0
\(211\) 183.303i 0.868735i −0.900736 0.434367i \(-0.856972\pi\)
0.900736 0.434367i \(-0.143028\pi\)
\(212\) 0 0
\(213\) −137.171 −0.643997
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 359.199i 1.65530i
\(218\) 0 0
\(219\) 73.3212i 0.334800i
\(220\) 0 0
\(221\) 219.964i 0.995311i
\(222\) 0 0
\(223\) 213.274i 0.956388i −0.878254 0.478194i \(-0.841292\pi\)
0.878254 0.478194i \(-0.158708\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 280.624i 1.23623i 0.786088 + 0.618115i \(0.212103\pi\)
−0.786088 + 0.618115i \(0.787897\pi\)
\(228\) 0 0
\(229\) 236.000 1.03057 0.515284 0.857020i \(-0.327687\pi\)
0.515284 + 0.857020i \(0.327687\pi\)
\(230\) 0 0
\(231\) 146.642i 0.634816i
\(232\) 0 0
\(233\) −215.555 −0.925129 −0.462565 0.886586i \(-0.653071\pi\)
−0.462565 + 0.886586i \(0.653071\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 411.514 1.73635
\(238\) 0 0
\(239\) −146.000 −0.610879 −0.305439 0.952212i \(-0.598803\pi\)
−0.305439 + 0.952212i \(0.598803\pi\)
\(240\) 0 0
\(241\) 183.303i 0.760593i −0.924865 0.380297i \(-0.875822\pi\)
0.924865 0.380297i \(-0.124178\pi\)
\(242\) 0 0
\(243\) 243.208i 1.00085i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 205.757 56.1249i 0.833025 0.227226i
\(248\) 0 0
\(249\) 109.982i 0.441694i
\(250\) 0 0
\(251\) −250.000 −0.996016 −0.498008 0.867172i \(-0.665935\pi\)
−0.498008 + 0.867172i \(0.665935\pi\)
\(252\) 0 0
\(253\) 39.1918 0.154908
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 145.925i 0.567800i −0.958854 0.283900i \(-0.908372\pi\)
0.958854 0.283900i \(-0.0916284\pi\)
\(258\) 0 0
\(259\) 329.945i 1.27392i
\(260\) 0 0
\(261\) 183.303i 0.702310i
\(262\) 0 0
\(263\) 88.1816 0.335291 0.167646 0.985847i \(-0.446384\pi\)
0.167646 + 0.985847i \(0.446384\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −548.686 −2.05500
\(268\) 0 0
\(269\) 73.3212i 0.272570i −0.990670 0.136285i \(-0.956484\pi\)
0.990670 0.136285i \(-0.0435162\pi\)
\(270\) 0 0
\(271\) −68.0000 −0.250923 −0.125461 0.992099i \(-0.540041\pi\)
−0.125461 + 0.992099i \(0.540041\pi\)
\(272\) 0 0
\(273\) −411.514 −1.50738
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −156.767 −0.565947 −0.282974 0.959128i \(-0.591321\pi\)
−0.282974 + 0.959128i \(0.591321\pi\)
\(278\) 0 0
\(279\) 183.303i 0.657000i
\(280\) 0 0
\(281\) 109.982i 0.391394i −0.980664 0.195697i \(-0.937303\pi\)
0.980664 0.195697i \(-0.0626970\pi\)
\(282\) 0 0
\(283\) 480.100 1.69647 0.848233 0.529623i \(-0.177667\pi\)
0.848233 + 0.529623i \(0.177667\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 359.199i 1.25156i
\(288\) 0 0
\(289\) 95.0000 0.328720
\(290\) 0 0
\(291\) 462.000 1.58763
\(292\) 0 0
\(293\) 505.124i 1.72397i 0.506932 + 0.861986i \(0.330779\pi\)
−0.506932 + 0.861986i \(0.669221\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 59.8665i 0.201571i
\(298\) 0 0
\(299\) 109.982i 0.367832i
\(300\) 0 0
\(301\) 672.000 2.23256
\(302\) 0 0
\(303\) 14.9666i 0.0493948i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 101.025i 0.329071i 0.986371 + 0.164535i \(0.0526125\pi\)
−0.986371 + 0.164535i \(0.947388\pi\)
\(308\) 0 0
\(309\) 294.000 0.951456
\(310\) 0 0
\(311\) −500.000 −1.60772 −0.803859 0.594821i \(-0.797223\pi\)
−0.803859 + 0.594821i \(0.797223\pi\)
\(312\) 0 0
\(313\) −215.555 −0.688674 −0.344337 0.938846i \(-0.611896\pi\)
−0.344337 + 0.938846i \(0.611896\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 280.624i 0.885250i 0.896707 + 0.442625i \(0.145953\pi\)
−0.896707 + 0.442625i \(0.854047\pi\)
\(318\) 0 0
\(319\) 146.642i 0.459694i
\(320\) 0 0
\(321\) 378.000 1.17757
\(322\) 0 0
\(323\) 97.9796 + 359.199i 0.303342 + 1.11207i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −274.343 −0.838969
\(328\) 0 0
\(329\) −96.0000 −0.291793
\(330\) 0 0
\(331\) 403.267i 1.21833i −0.793044 0.609164i \(-0.791505\pi\)
0.793044 0.609164i \(-0.208495\pi\)
\(332\) 0 0
\(333\) 168.375i 0.505629i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 101.025i 0.299777i 0.988703 + 0.149888i \(0.0478914\pi\)
−0.988703 + 0.149888i \(0.952109\pi\)
\(338\) 0 0
\(339\) −42.0000 −0.123894
\(340\) 0 0
\(341\) 146.642i 0.430036i
\(342\) 0 0
\(343\) −19.5959 −0.0571310
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 617.271 1.77888 0.889440 0.457052i \(-0.151095\pi\)
0.889440 + 0.457052i \(0.151095\pi\)
\(348\) 0 0
\(349\) −230.000 −0.659026 −0.329513 0.944151i \(-0.606885\pi\)
−0.329513 + 0.944151i \(0.606885\pi\)
\(350\) 0 0
\(351\) −168.000 −0.478632
\(352\) 0 0
\(353\) 548.686 1.55435 0.777175 0.629284i \(-0.216652\pi\)
0.777175 + 0.629284i \(0.216652\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 718.398i 2.01232i
\(358\) 0 0
\(359\) −604.000 −1.68245 −0.841226 0.540684i \(-0.818165\pi\)
−0.841226 + 0.540684i \(0.818165\pi\)
\(360\) 0 0
\(361\) −311.000 + 183.303i −0.861496 + 0.507765i
\(362\) 0 0
\(363\) 392.874i 1.08230i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −342.929 −0.934410 −0.467205 0.884149i \(-0.654739\pi\)
−0.467205 + 0.884149i \(0.654739\pi\)
\(368\) 0 0
\(369\) 183.303i 0.496756i
\(370\) 0 0
\(371\) 549.909i 1.48223i
\(372\) 0 0
\(373\) 123.475i 0.331031i −0.986207 0.165516i \(-0.947071\pi\)
0.986207 0.165516i \(-0.0529288\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 411.514 1.09155
\(378\) 0 0
\(379\) 329.945i 0.870568i 0.900293 + 0.435284i \(0.143352\pi\)
−0.900293 + 0.435284i \(0.856648\pi\)
\(380\) 0 0
\(381\) −882.000 −2.31496
\(382\) 0 0
\(383\) 11.2250i 0.0293080i −0.999893 0.0146540i \(-0.995335\pi\)
0.999893 0.0146540i \(-0.00466468\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −342.929 −0.886120
\(388\) 0 0
\(389\) −122.000 −0.313625 −0.156812 0.987628i \(-0.550122\pi\)
−0.156812 + 0.987628i \(0.550122\pi\)
\(390\) 0 0
\(391\) 192.000 0.491049
\(392\) 0 0
\(393\) 97.2831i 0.247540i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 470.302 1.18464 0.592320 0.805703i \(-0.298212\pi\)
0.592320 + 0.805703i \(0.298212\pi\)
\(398\) 0 0
\(399\) 672.000 183.303i 1.68421 0.459406i
\(400\) 0 0
\(401\) 219.964i 0.548538i 0.961653 + 0.274269i \(0.0884358\pi\)
−0.961653 + 0.274269i \(0.911564\pi\)
\(402\) 0 0
\(403\) −411.514 −1.02113
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 134.700i 0.330957i
\(408\) 0 0
\(409\) 329.945i 0.806713i −0.915043 0.403356i \(-0.867844\pi\)
0.915043 0.403356i \(-0.132156\pi\)
\(410\) 0 0
\(411\) 953.176i 2.31916i
\(412\) 0 0
\(413\) 718.398i 1.73946i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 553.765i 1.32797i
\(418\) 0 0
\(419\) 358.000 0.854415 0.427208 0.904154i \(-0.359497\pi\)
0.427208 + 0.904154i \(0.359497\pi\)
\(420\) 0 0
\(421\) 73.3212i 0.174160i −0.996201 0.0870798i \(-0.972246\pi\)
0.996201 0.0870798i \(-0.0277535\pi\)
\(422\) 0 0
\(423\) 48.9898 0.115815
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 979.796 2.29460
\(428\) 0 0
\(429\) −168.000 −0.391608
\(430\) 0 0
\(431\) 733.212i 1.70119i 0.525823 + 0.850594i \(0.323757\pi\)
−0.525823 + 0.850594i \(0.676243\pi\)
\(432\) 0 0
\(433\) 190.825i 0.440703i −0.975420 0.220352i \(-0.929280\pi\)
0.975420 0.220352i \(-0.0707205\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 48.9898 + 179.600i 0.112105 + 0.410983i
\(438\) 0 0
\(439\) 183.303i 0.417547i 0.977964 + 0.208773i \(0.0669471\pi\)
−0.977964 + 0.208773i \(0.933053\pi\)
\(440\) 0 0
\(441\) −235.000 −0.532880
\(442\) 0 0
\(443\) −264.545 −0.597167 −0.298583 0.954384i \(-0.596514\pi\)
−0.298583 + 0.954384i \(0.596514\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 194.566i 0.435271i
\(448\) 0 0
\(449\) 476.588i 1.06144i −0.847546 0.530721i \(-0.821921\pi\)
0.847546 0.530721i \(-0.178079\pi\)
\(450\) 0 0
\(451\) 146.642i 0.325149i
\(452\) 0 0
\(453\) −274.343 −0.605613
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 470.302 1.02911 0.514554 0.857458i \(-0.327958\pi\)
0.514554 + 0.857458i \(0.327958\pi\)
\(458\) 0 0
\(459\) 293.285i 0.638965i
\(460\) 0 0
\(461\) −22.0000 −0.0477223 −0.0238612 0.999715i \(-0.507596\pi\)
−0.0238612 + 0.999715i \(0.507596\pi\)
\(462\) 0 0
\(463\) 578.080 1.24855 0.624276 0.781204i \(-0.285394\pi\)
0.624276 + 0.781204i \(0.285394\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 421.312 0.902168 0.451084 0.892482i \(-0.351038\pi\)
0.451084 + 0.892482i \(0.351038\pi\)
\(468\) 0 0
\(469\) 109.982i 0.234503i
\(470\) 0 0
\(471\) 439.927i 0.934028i
\(472\) 0 0
\(473\) 274.343 0.580006
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 280.624i 0.588311i
\(478\) 0 0
\(479\) 500.000 1.04384 0.521921 0.852994i \(-0.325216\pi\)
0.521921 + 0.852994i \(0.325216\pi\)
\(480\) 0 0
\(481\) 378.000 0.785863
\(482\) 0 0
\(483\) 359.199i 0.743683i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 11.2250i 0.0230492i 0.999934 + 0.0115246i \(0.00366848\pi\)
−0.999934 + 0.0115246i \(0.996332\pi\)
\(488\) 0 0
\(489\) 36.6606i 0.0749706i
\(490\) 0 0
\(491\) 230.000 0.468432 0.234216 0.972185i \(-0.424748\pi\)
0.234216 + 0.972185i \(0.424748\pi\)
\(492\) 0 0
\(493\) 718.398i 1.45720i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 359.199i 0.722735i
\(498\) 0 0
\(499\) 620.000 1.24248 0.621242 0.783618i \(-0.286628\pi\)
0.621242 + 0.783618i \(0.286628\pi\)
\(500\) 0 0
\(501\) 126.000 0.251497
\(502\) 0 0
\(503\) 656.463 1.30510 0.652548 0.757747i \(-0.273700\pi\)
0.652548 + 0.757747i \(0.273700\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 160.891i 0.317340i
\(508\) 0 0
\(509\) 769.873i 1.51252i 0.654271 + 0.756260i \(0.272976\pi\)
−0.654271 + 0.756260i \(0.727024\pi\)
\(510\) 0 0
\(511\) 192.000 0.375734
\(512\) 0 0
\(513\) 274.343 74.8331i 0.534781 0.145874i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −39.1918 −0.0758063
\(518\) 0 0
\(519\) −1050.00 −2.02312
\(520\) 0 0
\(521\) 659.891i 1.26659i 0.773912 + 0.633293i \(0.218297\pi\)
−0.773912 + 0.633293i \(0.781703\pi\)
\(522\) 0 0
\(523\) 841.873i 1.60970i −0.593479 0.804850i \(-0.702246\pi\)
0.593479 0.804850i \(-0.297754\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 718.398i 1.36318i
\(528\) 0 0
\(529\) −433.000 −0.818526
\(530\) 0 0
\(531\) 366.606i 0.690407i
\(532\) 0 0
\(533\) 411.514 0.772072
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 548.686 1.02176
\(538\) 0 0
\(539\) 188.000 0.348794
\(540\) 0 0
\(541\) −620.000 −1.14603 −0.573013 0.819546i \(-0.694226\pi\)
−0.573013 + 0.819546i \(0.694226\pi\)
\(542\) 0 0
\(543\) −960.200 −1.76832
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 931.673i 1.70324i 0.524159 + 0.851620i \(0.324380\pi\)
−0.524159 + 0.851620i \(0.675620\pi\)
\(548\) 0 0
\(549\) −500.000 −0.910747
\(550\) 0 0
\(551\) −672.000 + 183.303i −1.21960 + 0.332673i
\(552\) 0 0
\(553\) 1077.60i 1.94864i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −548.686 −0.985073 −0.492537 0.870292i \(-0.663930\pi\)
−0.492537 + 0.870292i \(0.663930\pi\)
\(558\) 0 0
\(559\) 769.873i 1.37723i
\(560\) 0 0
\(561\) 293.285i 0.522789i
\(562\) 0 0
\(563\) 325.524i 0.578196i 0.957300 + 0.289098i \(0.0933553\pi\)
−0.957300 + 0.289098i \(0.906645\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −989.594 −1.74532
\(568\) 0 0
\(569\) 36.6606i 0.0644299i −0.999481 0.0322149i \(-0.989744\pi\)
0.999481 0.0322149i \(-0.0102561\pi\)
\(570\) 0 0
\(571\) −332.000 −0.581436 −0.290718 0.956809i \(-0.593894\pi\)
−0.290718 + 0.956809i \(0.593894\pi\)
\(572\) 0 0
\(573\) 591.182i 1.03173i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −979.796 −1.69809 −0.849043 0.528323i \(-0.822821\pi\)
−0.849043 + 0.528323i \(0.822821\pi\)
\(578\) 0 0
\(579\) −798.000 −1.37824
\(580\) 0 0
\(581\) 288.000 0.495697
\(582\) 0 0
\(583\) 224.499i 0.385076i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 852.422 1.45217 0.726084 0.687606i \(-0.241338\pi\)
0.726084 + 0.687606i \(0.241338\pi\)
\(588\) 0 0
\(589\) 672.000 183.303i 1.14092 0.311211i
\(590\) 0 0
\(591\) 659.891i 1.11657i
\(592\) 0 0
\(593\) 489.898 0.826135 0.413067 0.910700i \(-0.364457\pi\)
0.413067 + 0.910700i \(0.364457\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1129.98i 1.89276i
\(598\) 0 0
\(599\) 916.515i 1.53008i 0.643986 + 0.765038i \(0.277280\pi\)
−0.643986 + 0.765038i \(0.722720\pi\)
\(600\) 0 0
\(601\) 843.194i 1.40298i −0.712677 0.701492i \(-0.752517\pi\)
0.712677 0.701492i \(-0.247483\pi\)
\(602\) 0 0
\(603\) 56.1249i 0.0930761i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 976.573i 1.60885i −0.594054 0.804426i \(-0.702473\pi\)
0.594054 0.804426i \(-0.297527\pi\)
\(608\) 0 0
\(609\) 1344.00 2.20690
\(610\) 0 0
\(611\) 109.982i 0.180003i
\(612\) 0 0
\(613\) 1077.78 1.75820 0.879099 0.476639i \(-0.158145\pi\)
0.879099 + 0.476639i \(0.158145\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −685.857 −1.11160 −0.555800 0.831316i \(-0.687588\pi\)
−0.555800 + 0.831316i \(0.687588\pi\)
\(618\) 0 0
\(619\) −676.000 −1.09208 −0.546042 0.837758i \(-0.683866\pi\)
−0.546042 + 0.837758i \(0.683866\pi\)
\(620\) 0 0
\(621\) 146.642i 0.236139i
\(622\) 0 0
\(623\) 1436.80i 2.30625i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 274.343 74.8331i 0.437548 0.119351i
\(628\) 0 0
\(629\) 659.891i 1.04911i
\(630\) 0 0
\(631\) 124.000 0.196513 0.0982567 0.995161i \(-0.468673\pi\)
0.0982567 + 0.995161i \(0.468673\pi\)
\(632\) 0 0
\(633\) −685.857 −1.08350
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 527.574i 0.828216i
\(638\) 0 0
\(639\) 183.303i 0.286859i
\(640\) 0 0
\(641\) 623.230i 0.972278i 0.873881 + 0.486139i \(0.161595\pi\)
−0.873881 + 0.486139i \(0.838405\pi\)
\(642\) 0 0
\(643\) −244.949 −0.380947 −0.190474 0.981692i \(-0.561002\pi\)
−0.190474 + 0.981692i \(0.561002\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −342.929 −0.530029 −0.265014 0.964244i \(-0.585377\pi\)
−0.265014 + 0.964244i \(0.585377\pi\)
\(648\) 0 0
\(649\) 293.285i 0.451903i
\(650\) 0 0
\(651\) −1344.00 −2.06452
\(652\) 0 0
\(653\) −1038.58 −1.59048 −0.795240 0.606295i \(-0.792655\pi\)
−0.795240 + 0.606295i \(0.792655\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −97.9796 −0.149132
\(658\) 0 0
\(659\) 879.855i 1.33514i 0.744549 + 0.667568i \(0.232665\pi\)
−0.744549 + 0.667568i \(0.767335\pi\)
\(660\) 0 0
\(661\) 586.570i 0.887397i 0.896176 + 0.443699i \(0.146334\pi\)
−0.896176 + 0.443699i \(0.853666\pi\)
\(662\) 0 0
\(663\) −823.029 −1.24137
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 359.199i 0.538529i
\(668\) 0 0
\(669\) −798.000 −1.19283
\(670\) 0 0
\(671\) 400.000 0.596125
\(672\) 0 0
\(673\) 1156.17i 1.71794i 0.512028 + 0.858969i \(0.328894\pi\)
−0.512028 + 0.858969i \(0.671106\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 325.524i 0.480833i −0.970670 0.240417i \(-0.922716\pi\)
0.970670 0.240417i \(-0.0772841\pi\)
\(678\) 0 0
\(679\) 1209.80i 1.78174i
\(680\) 0 0
\(681\) 1050.00 1.54185
\(682\) 0 0
\(683\) 1133.72i 1.65992i 0.557826 + 0.829958i \(0.311636\pi\)
−0.557826 + 0.829958i \(0.688364\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 883.031i 1.28534i
\(688\) 0 0
\(689\) 630.000 0.914369
\(690\) 0 0
\(691\) −572.000 −0.827786 −0.413893 0.910326i \(-0.635831\pi\)
−0.413893 + 0.910326i \(0.635831\pi\)
\(692\) 0 0
\(693\) −195.959 −0.282769
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 718.398i 1.03070i
\(698\) 0 0
\(699\) 806.533i 1.15384i
\(700\) 0 0
\(701\) 820.000 1.16976 0.584879 0.811121i \(-0.301142\pi\)
0.584879 + 0.811121i \(0.301142\pi\)
\(702\) 0 0
\(703\) −617.271 + 168.375i −0.878053 + 0.239509i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 39.1918 0.0554340
\(708\) 0 0
\(709\) −278.000 −0.392102 −0.196051 0.980594i \(-0.562812\pi\)
−0.196051 + 0.980594i \(0.562812\pi\)
\(710\) 0 0
\(711\) 549.909i 0.773430i
\(712\) 0 0
\(713\) 359.199i 0.503786i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 546.282i 0.761900i
\(718\) 0 0
\(719\) −604.000 −0.840056 −0.420028 0.907511i \(-0.637980\pi\)
−0.420028 + 0.907511i \(0.637980\pi\)
\(720\) 0 0
\(721\) 769.873i 1.06778i
\(722\) 0 0
\(723\) −685.857 −0.948627
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −617.271 −0.849067 −0.424533 0.905412i \(-0.639562\pi\)
−0.424533 + 0.905412i \(0.639562\pi\)
\(728\) 0 0
\(729\) 1.00000 0.00137174
\(730\) 0 0
\(731\) 1344.00 1.83858
\(732\) 0 0
\(733\) 137.171 0.187137 0.0935685 0.995613i \(-0.470173\pi\)
0.0935685 + 0.995613i \(0.470173\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 44.8999i 0.0609225i
\(738\) 0 0
\(739\) 298.000 0.403248 0.201624 0.979463i \(-0.435378\pi\)
0.201624 + 0.979463i \(0.435378\pi\)
\(740\) 0 0
\(741\) −210.000 769.873i −0.283401 1.03896i
\(742\) 0 0
\(743\) 1313.32i 1.76759i 0.467871 + 0.883797i \(0.345021\pi\)
−0.467871 + 0.883797i \(0.654979\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −146.969 −0.196746
\(748\) 0 0
\(749\) 989.836i 1.32154i
\(750\) 0 0
\(751\) 659.891i 0.878683i 0.898320 + 0.439341i \(0.144788\pi\)
−0.898320 + 0.439341i \(0.855212\pi\)
\(752\) 0 0
\(753\) 935.414i 1.24225i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1058.18 −1.39786 −0.698930 0.715190i \(-0.746340\pi\)
−0.698930 + 0.715190i \(0.746340\pi\)
\(758\) 0 0
\(759\) 146.642i 0.193205i
\(760\) 0 0
\(761\) −788.000 −1.03548 −0.517740 0.855538i \(-0.673226\pi\)
−0.517740 + 0.855538i \(0.673226\pi\)
\(762\) 0 0
\(763\) 718.398i 0.941544i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −823.029 −1.07305
\(768\) 0 0
\(769\) 548.000 0.712614 0.356307 0.934369i \(-0.384036\pi\)
0.356307 + 0.934369i \(0.384036\pi\)
\(770\) 0 0
\(771\) −546.000 −0.708171
\(772\) 0 0
\(773\) 819.423i 1.06006i −0.847980 0.530028i \(-0.822181\pi\)
0.847980 0.530028i \(-0.177819\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 1234.54 1.58886
\(778\) 0 0
\(779\) −672.000 + 183.303i −0.862644 + 0.235306i
\(780\) 0 0
\(781\) 146.642i 0.187762i
\(782\) 0 0
\(783\) 548.686 0.700748
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 931.673i 1.18383i 0.806001 + 0.591914i \(0.201628\pi\)
−0.806001 + 0.591914i \(0.798372\pi\)
\(788\) 0 0
\(789\) 329.945i 0.418182i
\(790\) 0 0
\(791\) 109.982i 0.139041i
\(792\) 0 0
\(793\) 1122.50i 1.41551i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 774.523i 0.971798i −0.874015 0.485899i \(-0.838492\pi\)
0.874015 0.485899i \(-0.161508\pi\)
\(798\) 0 0
\(799\) −192.000 −0.240300
\(800\) 0 0
\(801\) 733.212i 0.915371i
\(802\) 0 0
\(803\) 78.3837 0.0976135
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −274.343 −0.339954
\(808\) 0 0
\(809\) −638.000 −0.788628 −0.394314 0.918976i \(-0.629018\pi\)
−0.394314 + 0.918976i \(0.629018\pi\)
\(810\) 0 0
\(811\) 73.3212i 0.0904084i 0.998978 + 0.0452042i \(0.0143938\pi\)
−0.998978 + 0.0452042i \(0.985606\pi\)
\(812\) 0 0
\(813\) 254.433i 0.312955i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 342.929 + 1257.20i 0.419741 + 1.53880i
\(818\) 0 0
\(819\) 549.909i 0.671440i
\(820\) 0 0
\(821\) −1318.00 −1.60536 −0.802680 0.596410i \(-0.796593\pi\)
−0.802680 + 0.596410i \(0.796593\pi\)
\(822\) 0 0
\(823\) −88.1816 −0.107147 −0.0535733 0.998564i \(-0.517061\pi\)
−0.0535733 + 0.998564i \(0.517061\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1133.72i 1.37089i −0.728127 0.685443i \(-0.759609\pi\)
0.728127 0.685443i \(-0.240391\pi\)
\(828\) 0 0
\(829\) 1466.42i 1.76891i −0.466628 0.884454i \(-0.654531\pi\)
0.466628 0.884454i \(-0.345469\pi\)
\(830\) 0 0
\(831\) 586.570i 0.705860i
\(832\) 0 0
\(833\) 921.008 1.10565
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −548.686 −0.655538
\(838\) 0 0
\(839\) 623.230i 0.742825i 0.928468 + 0.371413i \(0.121126\pi\)
−0.928468 + 0.371413i \(0.878874\pi\)
\(840\) 0 0
\(841\) −503.000 −0.598098
\(842\) 0 0
\(843\) −411.514 −0.488155
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −1028.79 −1.21462
\(848\) 0 0
\(849\) 1796.37i 2.11587i
\(850\) 0 0
\(851\) 329.945i 0.387715i
\(852\) 0 0
\(853\) 921.008 1.07973 0.539864 0.841752i \(-0.318476\pi\)
0.539864 + 0.841752i \(0.318476\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1133.72i 1.32290i −0.749991 0.661448i \(-0.769942\pi\)
0.749991 0.661448i \(-0.230058\pi\)
\(858\) 0 0
\(859\) −746.000 −0.868452 −0.434226 0.900804i \(-0.642978\pi\)
−0.434226 + 0.900804i \(0.642978\pi\)
\(860\) 0 0
\(861\) 1344.00 1.56098
\(862\) 0 0
\(863\) 235.724i 0.273145i 0.990630 + 0.136573i \(0.0436087\pi\)
−0.990630 + 0.136573i \(0.956391\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 355.457i 0.409986i
\(868\) 0 0
\(869\) 439.927i 0.506245i
\(870\) 0 0
\(871\) 126.000 0.144661
\(872\) 0 0
\(873\) 617.373i 0.707186i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 145.925i 0.166391i −0.996533 0.0831953i \(-0.973487\pi\)
0.996533 0.0831953i \(-0.0265125\pi\)
\(878\) 0 0
\(879\) 1890.00 2.15017
\(880\) 0 0
\(881\) −1028.00 −1.16686 −0.583428 0.812165i \(-0.698289\pi\)
−0.583428 + 0.812165i \(0.698289\pi\)
\(882\) 0 0
\(883\) 930.806 1.05414 0.527070 0.849822i \(-0.323290\pi\)
0.527070 + 0.849822i \(0.323290\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 168.375i 0.189825i −0.995486 0.0949124i \(-0.969743\pi\)
0.995486 0.0949124i \(-0.0302571\pi\)
\(888\) 0 0
\(889\) 2309.62i 2.59800i
\(890\) 0 0
\(891\) −404.000 −0.453423
\(892\) 0 0
\(893\) −48.9898 179.600i −0.0548598 0.201119i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −411.514 −0.458767
\(898\) 0 0
\(899\) 1344.00 1.49499
\(900\) 0 0
\(901\) 1099.82i 1.22066i
\(902\) 0 0
\(903\) 2514.39i 2.78449i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 347.974i 0.383654i −0.981429 0.191827i \(-0.938559\pi\)
0.981429 0.191827i \(-0.0614412\pi\)
\(908\) 0 0
\(909\) −20.0000 −0.0220022
\(910\) 0 0
\(911\) 1246.46i 1.36823i −0.729373 0.684117i \(-0.760188\pi\)
0.729373 0.684117i \(-0.239812\pi\)
\(912\) 0 0
\(913\) 117.576 0.128779
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 254.747 0.277805
\(918\) 0 0
\(919\) 1150.00 1.25136 0.625680 0.780080i \(-0.284822\pi\)
0.625680 + 0.780080i \(0.284822\pi\)
\(920\) 0 0
\(921\) 378.000 0.410423
\(922\) 0 0
\(923\) −411.514 −0.445844
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 392.874i 0.423812i
\(928\) 0 0
\(929\) −1454.00 −1.56512 −0.782562 0.622573i \(-0.786087\pi\)
−0.782562 + 0.622573i \(0.786087\pi\)
\(930\) 0 0
\(931\) 235.000 + 861.524i 0.252417 + 0.925375i
\(932\) 0 0
\(933\) 1870.83i 2.00518i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 823.029 0.878366 0.439183 0.898398i \(-0.355268\pi\)
0.439183 + 0.898398i \(0.355268\pi\)
\(938\) 0 0
\(939\) 806.533i 0.858928i
\(940\) 0 0
\(941\) 329.945i 0.350633i −0.984512 0.175316i \(-0.943905\pi\)
0.984512 0.175316i \(-0.0560948\pi\)
\(942\) 0 0
\(943\) 359.199i 0.380911i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 754.443 0.796666 0.398333 0.917241i \(-0.369589\pi\)
0.398333 + 0.917241i \(0.369589\pi\)
\(948\) 0 0
\(949\) 219.964i 0.231785i
\(950\) 0 0
\(951\) 1050.00 1.10410
\(952\) 0 0
\(953\) 931.673i 0.977621i −0.872390 0.488810i \(-0.837431\pi\)
0.872390 0.488810i \(-0.162569\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 548.686 0.573339
\(958\) 0 0
\(959\) −2496.00 −2.60271
\(960\) 0 0
\(961\) −383.000 −0.398543
\(962\) 0 0
\(963\) 505.124i 0.524531i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −891.614 −0.922042 −0.461021 0.887389i \(-0.652517\pi\)
−0.461021 + 0.887389i \(0.652517\pi\)
\(968\) 0 0
\(969\) 1344.00 366.606i 1.38700 0.378334i
\(970\) 0 0
\(971\) 73.3212i 0.0755110i 0.999287 + 0.0377555i \(0.0120208\pi\)
−0.999287 + 0.0377555i \(0.987979\pi\)
\(972\) 0 0
\(973\) −1450.10 −1.49034
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 280.624i 0.287231i 0.989634 + 0.143615i \(0.0458728\pi\)
−0.989634 + 0.143615i \(0.954127\pi\)
\(978\) 0 0
\(979\) 586.570i 0.599152i
\(980\) 0 0
\(981\) 366.606i 0.373706i
\(982\) 0 0
\(983\) 460.224i 0.468183i −0.972215 0.234091i \(-0.924788\pi\)
0.972215 0.234091i \(-0.0752115\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 359.199i 0.363930i
\(988\) 0 0
\(989\) 672.000 0.679474
\(990\) 0 0
\(991\) 73.3212i 0.0739871i −0.999316 0.0369935i \(-0.988222\pi\)
0.999316 0.0369935i \(-0.0117781\pi\)
\(992\) 0 0
\(993\) −1508.89 −1.51952
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 529.090 0.530682 0.265341 0.964155i \(-0.414515\pi\)
0.265341 + 0.964155i \(0.414515\pi\)
\(998\) 0 0
\(999\) 504.000 0.504505
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 1900.3.e.c.1101.2 4
5.2 odd 4 380.3.g.a.189.2 yes 4
5.3 odd 4 380.3.g.a.189.3 yes 4
5.4 even 2 inner 1900.3.e.c.1101.3 4
15.2 even 4 3420.3.h.c.2089.3 4
15.8 even 4 3420.3.h.c.2089.2 4
19.18 odd 2 inner 1900.3.e.c.1101.4 4
95.18 even 4 380.3.g.a.189.1 4
95.37 even 4 380.3.g.a.189.4 yes 4
95.94 odd 2 inner 1900.3.e.c.1101.1 4
285.113 odd 4 3420.3.h.c.2089.1 4
285.227 odd 4 3420.3.h.c.2089.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.g.a.189.1 4 95.18 even 4
380.3.g.a.189.2 yes 4 5.2 odd 4
380.3.g.a.189.3 yes 4 5.3 odd 4
380.3.g.a.189.4 yes 4 95.37 even 4
1900.3.e.c.1101.1 4 95.94 odd 2 inner
1900.3.e.c.1101.2 4 1.1 even 1 trivial
1900.3.e.c.1101.3 4 5.4 even 2 inner
1900.3.e.c.1101.4 4 19.18 odd 2 inner
3420.3.h.c.2089.1 4 285.113 odd 4
3420.3.h.c.2089.2 4 15.8 even 4
3420.3.h.c.2089.3 4 15.2 even 4
3420.3.h.c.2089.4 4 285.227 odd 4