Properties

Label 1900.3.e.g.1101.8
Level $1900$
Weight $3$
Character 1900.1101
Analytic conductor $51.771$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 89x^{12} + 3026x^{10} + 49092x^{8} + 390297x^{6} + 1440495x^{4} + 1994425x^{2} + 151875 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1101.8
Root \(0.284180i\) of defining polynomial
Character \(\chi\) \(=\) 1900.1101
Dual form 1900.3.e.g.1101.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.284180i q^{3} +2.19606 q^{7} +8.91924 q^{9} +O(q^{10})\) \(q+0.284180i q^{3} +2.19606 q^{7} +8.91924 q^{9} -2.14967 q^{11} +5.91448i q^{13} -21.3302 q^{17} +(-12.8601 - 13.9863i) q^{19} +0.624075i q^{21} -9.06199 q^{23} +5.09229i q^{27} +33.2891i q^{29} +44.1498i q^{31} -0.610894i q^{33} +50.0133i q^{37} -1.68078 q^{39} -61.1061i q^{41} -5.39704 q^{43} +9.02253 q^{47} -44.1773 q^{49} -6.06161i q^{51} +62.0535i q^{53} +(3.97463 - 3.65460i) q^{57} +37.5680i q^{59} +58.0153 q^{61} +19.5871 q^{63} -121.549i q^{67} -2.57524i q^{69} -103.884i q^{71} -120.816 q^{73} -4.72080 q^{77} +57.1212i q^{79} +78.8260 q^{81} -47.3469 q^{83} -9.46009 q^{87} -68.2779i q^{89} +12.9885i q^{91} -12.5465 q^{93} +22.7712i q^{97} -19.1735 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 4 q^{7} - 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 4 q^{7} - 52 q^{9} + 4 q^{11} + 6 q^{17} - 29 q^{19} + 28 q^{23} - 56 q^{39} - 22 q^{43} - 84 q^{47} + 138 q^{49} + 5 q^{57} - 50 q^{61} - 234 q^{63} + 204 q^{73} - 68 q^{77} + 66 q^{81} - 256 q^{83} - 18 q^{87} - 118 q^{93} + 92 q^{99}+O(q^{100}) \) 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
\(3\) 0.284180i 0.0947267i 0.998878 + 0.0473633i \(0.0150819\pi\)
−0.998878 + 0.0473633i \(0.984918\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.19606 0.313722 0.156861 0.987621i \(-0.449863\pi\)
0.156861 + 0.987621i \(0.449863\pi\)
\(8\) 0 0
\(9\) 8.91924 0.991027
\(10\) 0 0
\(11\) −2.14967 −0.195425 −0.0977124 0.995215i \(-0.531153\pi\)
−0.0977124 + 0.995215i \(0.531153\pi\)
\(12\) 0 0
\(13\) 5.91448i 0.454960i 0.973783 + 0.227480i \(0.0730486\pi\)
−0.973783 + 0.227480i \(0.926951\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −21.3302 −1.25472 −0.627358 0.778731i \(-0.715864\pi\)
−0.627358 + 0.778731i \(0.715864\pi\)
\(18\) 0 0
\(19\) −12.8601 13.9863i −0.676850 0.736121i
\(20\) 0 0
\(21\) 0.624075i 0.0297178i
\(22\) 0 0
\(23\) −9.06199 −0.394000 −0.197000 0.980404i \(-0.563120\pi\)
−0.197000 + 0.980404i \(0.563120\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.09229i 0.188603i
\(28\) 0 0
\(29\) 33.2891i 1.14790i 0.818891 + 0.573950i \(0.194589\pi\)
−0.818891 + 0.573950i \(0.805411\pi\)
\(30\) 0 0
\(31\) 44.1498i 1.42419i 0.702085 + 0.712093i \(0.252253\pi\)
−0.702085 + 0.712093i \(0.747747\pi\)
\(32\) 0 0
\(33\) 0.610894i 0.0185119i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 50.0133i 1.35171i 0.737035 + 0.675855i \(0.236225\pi\)
−0.737035 + 0.675855i \(0.763775\pi\)
\(38\) 0 0
\(39\) −1.68078 −0.0430968
\(40\) 0 0
\(41\) 61.1061i 1.49039i −0.666845 0.745196i \(-0.732356\pi\)
0.666845 0.745196i \(-0.267644\pi\)
\(42\) 0 0
\(43\) −5.39704 −0.125513 −0.0627563 0.998029i \(-0.519989\pi\)
−0.0627563 + 0.998029i \(0.519989\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.02253 0.191969 0.0959844 0.995383i \(-0.469400\pi\)
0.0959844 + 0.995383i \(0.469400\pi\)
\(48\) 0 0
\(49\) −44.1773 −0.901578
\(50\) 0 0
\(51\) 6.06161i 0.118855i
\(52\) 0 0
\(53\) 62.0535i 1.17082i 0.810737 + 0.585411i \(0.199067\pi\)
−0.810737 + 0.585411i \(0.800933\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.97463 3.65460i 0.0697303 0.0641157i
\(58\) 0 0
\(59\) 37.5680i 0.636746i 0.947966 + 0.318373i \(0.103136\pi\)
−0.947966 + 0.318373i \(0.896864\pi\)
\(60\) 0 0
\(61\) 58.0153 0.951071 0.475535 0.879697i \(-0.342254\pi\)
0.475535 + 0.879697i \(0.342254\pi\)
\(62\) 0 0
\(63\) 19.5871 0.310907
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 121.549i 1.81416i −0.420958 0.907080i \(-0.638306\pi\)
0.420958 0.907080i \(-0.361694\pi\)
\(68\) 0 0
\(69\) 2.57524i 0.0373223i
\(70\) 0 0
\(71\) 103.884i 1.46315i −0.681759 0.731576i \(-0.738785\pi\)
0.681759 0.731576i \(-0.261215\pi\)
\(72\) 0 0
\(73\) −120.816 −1.65501 −0.827507 0.561456i \(-0.810241\pi\)
−0.827507 + 0.561456i \(0.810241\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.72080 −0.0613091
\(78\) 0 0
\(79\) 57.1212i 0.723053i 0.932362 + 0.361526i \(0.117744\pi\)
−0.932362 + 0.361526i \(0.882256\pi\)
\(80\) 0 0
\(81\) 78.8260 0.973161
\(82\) 0 0
\(83\) −47.3469 −0.570445 −0.285222 0.958461i \(-0.592067\pi\)
−0.285222 + 0.958461i \(0.592067\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −9.46009 −0.108737
\(88\) 0 0
\(89\) 68.2779i 0.767168i −0.923506 0.383584i \(-0.874690\pi\)
0.923506 0.383584i \(-0.125310\pi\)
\(90\) 0 0
\(91\) 12.9885i 0.142731i
\(92\) 0 0
\(93\) −12.5465 −0.134908
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 22.7712i 0.234755i 0.993087 + 0.117378i \(0.0374488\pi\)
−0.993087 + 0.117378i \(0.962551\pi\)
\(98\) 0 0
\(99\) −19.1735 −0.193671
\(100\) 0 0
\(101\) 37.0879 0.367207 0.183603 0.983000i \(-0.441224\pi\)
0.183603 + 0.983000i \(0.441224\pi\)
\(102\) 0 0
\(103\) 167.221i 1.62350i 0.584003 + 0.811752i \(0.301486\pi\)
−0.584003 + 0.811752i \(0.698514\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 82.4568i 0.770624i 0.922786 + 0.385312i \(0.125906\pi\)
−0.922786 + 0.385312i \(0.874094\pi\)
\(108\) 0 0
\(109\) 137.915i 1.26528i 0.774448 + 0.632638i \(0.218028\pi\)
−0.774448 + 0.632638i \(0.781972\pi\)
\(110\) 0 0
\(111\) −14.2128 −0.128043
\(112\) 0 0
\(113\) 156.892i 1.38842i 0.719772 + 0.694211i \(0.244247\pi\)
−0.719772 + 0.694211i \(0.755753\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 52.7527i 0.450877i
\(118\) 0 0
\(119\) −46.8422 −0.393632
\(120\) 0 0
\(121\) −116.379 −0.961809
\(122\) 0 0
\(123\) 17.3651 0.141180
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 160.978i 1.26754i 0.773521 + 0.633771i \(0.218494\pi\)
−0.773521 + 0.633771i \(0.781506\pi\)
\(128\) 0 0
\(129\) 1.53373i 0.0118894i
\(130\) 0 0
\(131\) 200.942 1.53391 0.766954 0.641703i \(-0.221772\pi\)
0.766954 + 0.641703i \(0.221772\pi\)
\(132\) 0 0
\(133\) −28.2416 30.7147i −0.212343 0.230938i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −181.728 −1.32648 −0.663240 0.748406i \(-0.730819\pi\)
−0.663240 + 0.748406i \(0.730819\pi\)
\(138\) 0 0
\(139\) −49.6249 −0.357013 −0.178507 0.983939i \(-0.557127\pi\)
−0.178507 + 0.983939i \(0.557127\pi\)
\(140\) 0 0
\(141\) 2.56402i 0.0181846i
\(142\) 0 0
\(143\) 12.7142i 0.0889104i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 12.5543i 0.0854035i
\(148\) 0 0
\(149\) −197.358 −1.32455 −0.662276 0.749260i \(-0.730409\pi\)
−0.662276 + 0.749260i \(0.730409\pi\)
\(150\) 0 0
\(151\) 76.4718i 0.506436i −0.967409 0.253218i \(-0.918511\pi\)
0.967409 0.253218i \(-0.0814890\pi\)
\(152\) 0 0
\(153\) −190.249 −1.24346
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −163.139 −1.03910 −0.519550 0.854440i \(-0.673900\pi\)
−0.519550 + 0.854440i \(0.673900\pi\)
\(158\) 0 0
\(159\) −17.6344 −0.110908
\(160\) 0 0
\(161\) −19.9006 −0.123606
\(162\) 0 0
\(163\) −287.890 −1.76620 −0.883100 0.469185i \(-0.844548\pi\)
−0.883100 + 0.469185i \(0.844548\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 47.3144i 0.283320i 0.989915 + 0.141660i \(0.0452439\pi\)
−0.989915 + 0.141660i \(0.954756\pi\)
\(168\) 0 0
\(169\) 134.019 0.793012
\(170\) 0 0
\(171\) −114.703 124.747i −0.670776 0.729516i
\(172\) 0 0
\(173\) 64.2813i 0.371568i 0.982591 + 0.185784i \(0.0594825\pi\)
−0.982591 + 0.185784i \(0.940518\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −10.6761 −0.0603168
\(178\) 0 0
\(179\) 276.531i 1.54487i 0.635097 + 0.772433i \(0.280960\pi\)
−0.635097 + 0.772433i \(0.719040\pi\)
\(180\) 0 0
\(181\) 95.9988i 0.530380i −0.964196 0.265190i \(-0.914565\pi\)
0.964196 0.265190i \(-0.0854347\pi\)
\(182\) 0 0
\(183\) 16.4868i 0.0900918i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 45.8529 0.245203
\(188\) 0 0
\(189\) 11.1829i 0.0591690i
\(190\) 0 0
\(191\) 235.317 1.23203 0.616013 0.787736i \(-0.288747\pi\)
0.616013 + 0.787736i \(0.288747\pi\)
\(192\) 0 0
\(193\) 311.842i 1.61576i 0.589344 + 0.807882i \(0.299386\pi\)
−0.589344 + 0.807882i \(0.700614\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 212.226 1.07729 0.538644 0.842533i \(-0.318937\pi\)
0.538644 + 0.842533i \(0.318937\pi\)
\(198\) 0 0
\(199\) −63.8764 −0.320987 −0.160493 0.987037i \(-0.551309\pi\)
−0.160493 + 0.987037i \(0.551309\pi\)
\(200\) 0 0
\(201\) 34.5417 0.171849
\(202\) 0 0
\(203\) 73.1046i 0.360121i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −80.8261 −0.390464
\(208\) 0 0
\(209\) 27.6451 + 30.0660i 0.132273 + 0.143856i
\(210\) 0 0
\(211\) 153.634i 0.728125i −0.931375 0.364063i \(-0.881389\pi\)
0.931375 0.364063i \(-0.118611\pi\)
\(212\) 0 0
\(213\) 29.5217 0.138600
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 96.9553i 0.446799i
\(218\) 0 0
\(219\) 34.3335i 0.156774i
\(220\) 0 0
\(221\) 126.157i 0.570845i
\(222\) 0 0
\(223\) 64.1517i 0.287676i 0.989601 + 0.143838i \(0.0459444\pi\)
−0.989601 + 0.143838i \(0.954056\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 185.129i 0.815546i −0.913083 0.407773i \(-0.866306\pi\)
0.913083 0.407773i \(-0.133694\pi\)
\(228\) 0 0
\(229\) −244.498 −1.06768 −0.533838 0.845587i \(-0.679251\pi\)
−0.533838 + 0.845587i \(0.679251\pi\)
\(230\) 0 0
\(231\) 1.34156i 0.00580761i
\(232\) 0 0
\(233\) −76.4331 −0.328039 −0.164019 0.986457i \(-0.552446\pi\)
−0.164019 + 0.986457i \(0.552446\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −16.2327 −0.0684924
\(238\) 0 0
\(239\) −221.151 −0.925317 −0.462659 0.886537i \(-0.653104\pi\)
−0.462659 + 0.886537i \(0.653104\pi\)
\(240\) 0 0
\(241\) 14.4423i 0.0599266i 0.999551 + 0.0299633i \(0.00953904\pi\)
−0.999551 + 0.0299633i \(0.990461\pi\)
\(242\) 0 0
\(243\) 68.2314i 0.280788i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 82.7217 76.0610i 0.334906 0.307939i
\(248\) 0 0
\(249\) 13.4550i 0.0540363i
\(250\) 0 0
\(251\) 120.711 0.480920 0.240460 0.970659i \(-0.422702\pi\)
0.240460 + 0.970659i \(0.422702\pi\)
\(252\) 0 0
\(253\) 19.4803 0.0769973
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 396.891i 1.54432i 0.635426 + 0.772161i \(0.280824\pi\)
−0.635426 + 0.772161i \(0.719176\pi\)
\(258\) 0 0
\(259\) 109.832i 0.424061i
\(260\) 0 0
\(261\) 296.913i 1.13760i
\(262\) 0 0
\(263\) −435.465 −1.65576 −0.827880 0.560905i \(-0.810453\pi\)
−0.827880 + 0.560905i \(0.810453\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 19.4032 0.0726713
\(268\) 0 0
\(269\) 59.6306i 0.221675i −0.993839 0.110838i \(-0.964647\pi\)
0.993839 0.110838i \(-0.0353533\pi\)
\(270\) 0 0
\(271\) 419.450 1.54779 0.773893 0.633317i \(-0.218307\pi\)
0.773893 + 0.633317i \(0.218307\pi\)
\(272\) 0 0
\(273\) −3.69108 −0.0135204
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −262.373 −0.947196 −0.473598 0.880741i \(-0.657045\pi\)
−0.473598 + 0.880741i \(0.657045\pi\)
\(278\) 0 0
\(279\) 393.783i 1.41141i
\(280\) 0 0
\(281\) 223.815i 0.796495i −0.917278 0.398247i \(-0.869619\pi\)
0.917278 0.398247i \(-0.130381\pi\)
\(282\) 0 0
\(283\) 240.958 0.851441 0.425721 0.904855i \(-0.360021\pi\)
0.425721 + 0.904855i \(0.360021\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 134.192i 0.467569i
\(288\) 0 0
\(289\) 165.976 0.574312
\(290\) 0 0
\(291\) −6.47113 −0.0222376
\(292\) 0 0
\(293\) 418.683i 1.42895i −0.699660 0.714476i \(-0.746665\pi\)
0.699660 0.714476i \(-0.253335\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 10.9468i 0.0368578i
\(298\) 0 0
\(299\) 53.5969i 0.179254i
\(300\) 0 0
\(301\) −11.8522 −0.0393761
\(302\) 0 0
\(303\) 10.5396i 0.0347843i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 31.3393i 0.102082i −0.998697 0.0510412i \(-0.983746\pi\)
0.998697 0.0510412i \(-0.0162540\pi\)
\(308\) 0 0
\(309\) −47.5208 −0.153789
\(310\) 0 0
\(311\) −475.395 −1.52860 −0.764301 0.644860i \(-0.776916\pi\)
−0.764301 + 0.644860i \(0.776916\pi\)
\(312\) 0 0
\(313\) 97.9348 0.312891 0.156445 0.987687i \(-0.449996\pi\)
0.156445 + 0.987687i \(0.449996\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 491.774i 1.55134i −0.631140 0.775669i \(-0.717413\pi\)
0.631140 0.775669i \(-0.282587\pi\)
\(318\) 0 0
\(319\) 71.5606i 0.224328i
\(320\) 0 0
\(321\) −23.4326 −0.0729987
\(322\) 0 0
\(323\) 274.309 + 298.330i 0.849254 + 0.923623i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −39.1927 −0.119855
\(328\) 0 0
\(329\) 19.8140 0.0602249
\(330\) 0 0
\(331\) 329.980i 0.996919i −0.866913 0.498460i \(-0.833899\pi\)
0.866913 0.498460i \(-0.166101\pi\)
\(332\) 0 0
\(333\) 446.080i 1.33958i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 548.629i 1.62798i 0.580879 + 0.813990i \(0.302709\pi\)
−0.580879 + 0.813990i \(0.697291\pi\)
\(338\) 0 0
\(339\) −44.5855 −0.131521
\(340\) 0 0
\(341\) 94.9076i 0.278321i
\(342\) 0 0
\(343\) −204.623 −0.596567
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 225.865 0.650907 0.325454 0.945558i \(-0.394483\pi\)
0.325454 + 0.945558i \(0.394483\pi\)
\(348\) 0 0
\(349\) 236.046 0.676350 0.338175 0.941083i \(-0.390190\pi\)
0.338175 + 0.941083i \(0.390190\pi\)
\(350\) 0 0
\(351\) −30.1182 −0.0858069
\(352\) 0 0
\(353\) −205.291 −0.581560 −0.290780 0.956790i \(-0.593915\pi\)
−0.290780 + 0.956790i \(0.593915\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 13.3116i 0.0372875i
\(358\) 0 0
\(359\) −132.769 −0.369830 −0.184915 0.982755i \(-0.559201\pi\)
−0.184915 + 0.982755i \(0.559201\pi\)
\(360\) 0 0
\(361\) −30.2334 + 359.732i −0.0837490 + 0.996487i
\(362\) 0 0
\(363\) 33.0726i 0.0911090i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 314.585 0.857181 0.428590 0.903499i \(-0.359010\pi\)
0.428590 + 0.903499i \(0.359010\pi\)
\(368\) 0 0
\(369\) 545.020i 1.47702i
\(370\) 0 0
\(371\) 136.273i 0.367313i
\(372\) 0 0
\(373\) 79.5881i 0.213373i 0.994293 + 0.106687i \(0.0340241\pi\)
−0.994293 + 0.106687i \(0.965976\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −196.887 −0.522248
\(378\) 0 0
\(379\) 212.412i 0.560454i −0.959934 0.280227i \(-0.909590\pi\)
0.959934 0.280227i \(-0.0904098\pi\)
\(380\) 0 0
\(381\) −45.7467 −0.120070
\(382\) 0 0
\(383\) 709.304i 1.85197i −0.377563 0.925984i \(-0.623238\pi\)
0.377563 0.925984i \(-0.376762\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −48.1375 −0.124386
\(388\) 0 0
\(389\) 604.145 1.55307 0.776536 0.630073i \(-0.216975\pi\)
0.776536 + 0.630073i \(0.216975\pi\)
\(390\) 0 0
\(391\) 193.294 0.494358
\(392\) 0 0
\(393\) 57.1036i 0.145302i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −374.253 −0.942703 −0.471352 0.881945i \(-0.656234\pi\)
−0.471352 + 0.881945i \(0.656234\pi\)
\(398\) 0 0
\(399\) 8.72850 8.02569i 0.0218759 0.0201145i
\(400\) 0 0
\(401\) 298.430i 0.744215i 0.928190 + 0.372108i \(0.121365\pi\)
−0.928190 + 0.372108i \(0.878635\pi\)
\(402\) 0 0
\(403\) −261.123 −0.647948
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 107.512i 0.264158i
\(408\) 0 0
\(409\) 290.042i 0.709150i −0.935028 0.354575i \(-0.884626\pi\)
0.935028 0.354575i \(-0.115374\pi\)
\(410\) 0 0
\(411\) 51.6434i 0.125653i
\(412\) 0 0
\(413\) 82.5014i 0.199761i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 14.1024i 0.0338187i
\(418\) 0 0
\(419\) −44.1734 −0.105426 −0.0527129 0.998610i \(-0.516787\pi\)
−0.0527129 + 0.998610i \(0.516787\pi\)
\(420\) 0 0
\(421\) 326.685i 0.775975i 0.921665 + 0.387987i \(0.126830\pi\)
−0.921665 + 0.387987i \(0.873170\pi\)
\(422\) 0 0
\(423\) 80.4742 0.190246
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 127.405 0.298372
\(428\) 0 0
\(429\) 3.61312 0.00842219
\(430\) 0 0
\(431\) 152.204i 0.353141i −0.984288 0.176571i \(-0.943500\pi\)
0.984288 0.176571i \(-0.0565004\pi\)
\(432\) 0 0
\(433\) 223.747i 0.516736i −0.966047 0.258368i \(-0.916815\pi\)
0.966047 0.258368i \(-0.0831848\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 116.539 + 126.744i 0.266679 + 0.290031i
\(438\) 0 0
\(439\) 253.627i 0.577737i 0.957369 + 0.288869i \(0.0932791\pi\)
−0.957369 + 0.288869i \(0.906721\pi\)
\(440\) 0 0
\(441\) −394.028 −0.893488
\(442\) 0 0
\(443\) 471.418 1.06415 0.532075 0.846697i \(-0.321413\pi\)
0.532075 + 0.846697i \(0.321413\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 56.0852i 0.125470i
\(448\) 0 0
\(449\) 546.839i 1.21790i 0.793207 + 0.608952i \(0.208410\pi\)
−0.793207 + 0.608952i \(0.791590\pi\)
\(450\) 0 0
\(451\) 131.358i 0.291260i
\(452\) 0 0
\(453\) 21.7318 0.0479730
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −166.467 −0.364261 −0.182130 0.983274i \(-0.558299\pi\)
−0.182130 + 0.983274i \(0.558299\pi\)
\(458\) 0 0
\(459\) 108.619i 0.236644i
\(460\) 0 0
\(461\) −258.722 −0.561220 −0.280610 0.959822i \(-0.590537\pi\)
−0.280610 + 0.959822i \(0.590537\pi\)
\(462\) 0 0
\(463\) 629.974 1.36064 0.680318 0.732917i \(-0.261842\pi\)
0.680318 + 0.732917i \(0.261842\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 151.843 0.325145 0.162573 0.986697i \(-0.448021\pi\)
0.162573 + 0.986697i \(0.448021\pi\)
\(468\) 0 0
\(469\) 266.928i 0.569142i
\(470\) 0 0
\(471\) 46.3608i 0.0984305i
\(472\) 0 0
\(473\) 11.6019 0.0245283
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 553.470i 1.16032i
\(478\) 0 0
\(479\) −15.2726 −0.0318843 −0.0159421 0.999873i \(-0.505075\pi\)
−0.0159421 + 0.999873i \(0.505075\pi\)
\(480\) 0 0
\(481\) −295.802 −0.614974
\(482\) 0 0
\(483\) 5.65536i 0.0117088i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 279.446i 0.573812i −0.957959 0.286906i \(-0.907373\pi\)
0.957959 0.286906i \(-0.0926267\pi\)
\(488\) 0 0
\(489\) 81.8127i 0.167306i
\(490\) 0 0
\(491\) 641.107 1.30572 0.652859 0.757480i \(-0.273569\pi\)
0.652859 + 0.757480i \(0.273569\pi\)
\(492\) 0 0
\(493\) 710.062i 1.44029i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 228.135i 0.459023i
\(498\) 0 0
\(499\) −102.056 −0.204521 −0.102261 0.994758i \(-0.532608\pi\)
−0.102261 + 0.994758i \(0.532608\pi\)
\(500\) 0 0
\(501\) −13.4458 −0.0268379
\(502\) 0 0
\(503\) 889.366 1.76812 0.884061 0.467371i \(-0.154799\pi\)
0.884061 + 0.467371i \(0.154799\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 38.0855i 0.0751193i
\(508\) 0 0
\(509\) 35.5445i 0.0698320i 0.999390 + 0.0349160i \(0.0111164\pi\)
−0.999390 + 0.0349160i \(0.988884\pi\)
\(510\) 0 0
\(511\) −265.319 −0.519214
\(512\) 0 0
\(513\) 71.2223 65.4876i 0.138835 0.127656i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −19.3955 −0.0375155
\(518\) 0 0
\(519\) −18.2674 −0.0351974
\(520\) 0 0
\(521\) 747.135i 1.43404i 0.697053 + 0.717020i \(0.254494\pi\)
−0.697053 + 0.717020i \(0.745506\pi\)
\(522\) 0 0
\(523\) 677.562i 1.29553i −0.761840 0.647765i \(-0.775704\pi\)
0.761840 0.647765i \(-0.224296\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 941.722i 1.78695i
\(528\) 0 0
\(529\) −446.880 −0.844764
\(530\) 0 0
\(531\) 335.078i 0.631032i
\(532\) 0 0
\(533\) 361.411 0.678069
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −78.5845 −0.146340
\(538\) 0 0
\(539\) 94.9668 0.176191
\(540\) 0 0
\(541\) −362.237 −0.669569 −0.334784 0.942295i \(-0.608663\pi\)
−0.334784 + 0.942295i \(0.608663\pi\)
\(542\) 0 0
\(543\) 27.2809 0.0502411
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 585.353i 1.07011i −0.844816 0.535057i \(-0.820290\pi\)
0.844816 0.535057i \(-0.179710\pi\)
\(548\) 0 0
\(549\) 517.453 0.942537
\(550\) 0 0
\(551\) 465.591 428.102i 0.844993 0.776955i
\(552\) 0 0
\(553\) 125.441i 0.226838i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1040.06 1.86725 0.933623 0.358257i \(-0.116629\pi\)
0.933623 + 0.358257i \(0.116629\pi\)
\(558\) 0 0
\(559\) 31.9207i 0.0571032i
\(560\) 0 0
\(561\) 13.0305i 0.0232272i
\(562\) 0 0
\(563\) 421.015i 0.747807i −0.927468 0.373903i \(-0.878019\pi\)
0.927468 0.373903i \(-0.121981\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 173.106 0.305302
\(568\) 0 0
\(569\) 24.7446i 0.0434879i 0.999764 + 0.0217440i \(0.00692187\pi\)
−0.999764 + 0.0217440i \(0.993078\pi\)
\(570\) 0 0
\(571\) −283.521 −0.496534 −0.248267 0.968692i \(-0.579861\pi\)
−0.248267 + 0.968692i \(0.579861\pi\)
\(572\) 0 0
\(573\) 66.8723i 0.116706i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −147.592 −0.255792 −0.127896 0.991788i \(-0.540822\pi\)
−0.127896 + 0.991788i \(0.540822\pi\)
\(578\) 0 0
\(579\) −88.6194 −0.153056
\(580\) 0 0
\(581\) −103.976 −0.178961
\(582\) 0 0
\(583\) 133.395i 0.228808i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 876.849 1.49378 0.746890 0.664948i \(-0.231546\pi\)
0.746890 + 0.664948i \(0.231546\pi\)
\(588\) 0 0
\(589\) 617.492 567.772i 1.04837 0.963960i
\(590\) 0 0
\(591\) 60.3103i 0.102048i
\(592\) 0 0
\(593\) −205.974 −0.347343 −0.173671 0.984804i \(-0.555563\pi\)
−0.173671 + 0.984804i \(0.555563\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 18.1524i 0.0304060i
\(598\) 0 0
\(599\) 631.341i 1.05399i 0.849868 + 0.526996i \(0.176682\pi\)
−0.849868 + 0.526996i \(0.823318\pi\)
\(600\) 0 0
\(601\) 139.903i 0.232784i 0.993203 + 0.116392i \(0.0371328\pi\)
−0.993203 + 0.116392i \(0.962867\pi\)
\(602\) 0 0
\(603\) 1084.12i 1.79788i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 18.9312i 0.0311881i −0.999878 0.0155940i \(-0.995036\pi\)
0.999878 0.0155940i \(-0.00496394\pi\)
\(608\) 0 0
\(609\) −20.7749 −0.0341131
\(610\) 0 0
\(611\) 53.3636i 0.0873381i
\(612\) 0 0
\(613\) 86.8754 0.141722 0.0708608 0.997486i \(-0.477425\pi\)
0.0708608 + 0.997486i \(0.477425\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −664.265 −1.07660 −0.538302 0.842752i \(-0.680934\pi\)
−0.538302 + 0.842752i \(0.680934\pi\)
\(618\) 0 0
\(619\) 1120.54 1.81024 0.905122 0.425151i \(-0.139779\pi\)
0.905122 + 0.425151i \(0.139779\pi\)
\(620\) 0 0
\(621\) 46.1463i 0.0743096i
\(622\) 0 0
\(623\) 149.942i 0.240678i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −8.54415 + 7.85618i −0.0136270 + 0.0125298i
\(628\) 0 0
\(629\) 1066.79i 1.69601i
\(630\) 0 0
\(631\) −940.484 −1.49047 −0.745233 0.666804i \(-0.767662\pi\)
−0.745233 + 0.666804i \(0.767662\pi\)
\(632\) 0 0
\(633\) 43.6598 0.0689729
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 261.286i 0.410182i
\(638\) 0 0
\(639\) 926.565i 1.45002i
\(640\) 0 0
\(641\) 585.344i 0.913174i −0.889679 0.456587i \(-0.849072\pi\)
0.889679 0.456587i \(-0.150928\pi\)
\(642\) 0 0
\(643\) 562.853 0.875354 0.437677 0.899132i \(-0.355801\pi\)
0.437677 + 0.899132i \(0.355801\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 723.636 1.11845 0.559224 0.829017i \(-0.311099\pi\)
0.559224 + 0.829017i \(0.311099\pi\)
\(648\) 0 0
\(649\) 80.7589i 0.124436i
\(650\) 0 0
\(651\) −27.5528 −0.0423238
\(652\) 0 0
\(653\) 211.465 0.323836 0.161918 0.986804i \(-0.448232\pi\)
0.161918 + 0.986804i \(0.448232\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −1077.59 −1.64016
\(658\) 0 0
\(659\) 692.365i 1.05063i −0.850908 0.525315i \(-0.823948\pi\)
0.850908 0.525315i \(-0.176052\pi\)
\(660\) 0 0
\(661\) 1121.07i 1.69602i 0.529978 + 0.848011i \(0.322200\pi\)
−0.529978 + 0.848011i \(0.677800\pi\)
\(662\) 0 0
\(663\) 35.8512 0.0540743
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 301.665i 0.452272i
\(668\) 0 0
\(669\) −18.2306 −0.0272506
\(670\) 0 0
\(671\) −124.714 −0.185863
\(672\) 0 0
\(673\) 536.583i 0.797301i 0.917103 + 0.398650i \(0.130521\pi\)
−0.917103 + 0.398650i \(0.869479\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 234.317i 0.346110i 0.984912 + 0.173055i \(0.0553639\pi\)
−0.984912 + 0.173055i \(0.944636\pi\)
\(678\) 0 0
\(679\) 50.0069i 0.0736478i
\(680\) 0 0
\(681\) 52.6099 0.0772539
\(682\) 0 0
\(683\) 545.339i 0.798447i 0.916854 + 0.399223i \(0.130720\pi\)
−0.916854 + 0.399223i \(0.869280\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 69.4814i 0.101137i
\(688\) 0 0
\(689\) −367.014 −0.532677
\(690\) 0 0
\(691\) −225.173 −0.325865 −0.162933 0.986637i \(-0.552095\pi\)
−0.162933 + 0.986637i \(0.552095\pi\)
\(692\) 0 0
\(693\) −42.1060 −0.0607590
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1303.40i 1.87002i
\(698\) 0 0
\(699\) 21.7207i 0.0310740i
\(700\) 0 0
\(701\) −517.342 −0.738005 −0.369003 0.929428i \(-0.620301\pi\)
−0.369003 + 0.929428i \(0.620301\pi\)
\(702\) 0 0
\(703\) 699.501 643.178i 0.995022 0.914905i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 81.4471 0.115201
\(708\) 0 0
\(709\) 272.183 0.383897 0.191948 0.981405i \(-0.438519\pi\)
0.191948 + 0.981405i \(0.438519\pi\)
\(710\) 0 0
\(711\) 509.478i 0.716565i
\(712\) 0 0
\(713\) 400.085i 0.561129i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 62.8466i 0.0876522i
\(718\) 0 0
\(719\) 547.539 0.761528 0.380764 0.924672i \(-0.375661\pi\)
0.380764 + 0.924672i \(0.375661\pi\)
\(720\) 0 0
\(721\) 367.226i 0.509329i
\(722\) 0 0
\(723\) −4.10421 −0.00567664
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 1211.49 1.66642 0.833210 0.552957i \(-0.186501\pi\)
0.833210 + 0.552957i \(0.186501\pi\)
\(728\) 0 0
\(729\) 690.044 0.946563
\(730\) 0 0
\(731\) 115.120 0.157483
\(732\) 0 0
\(733\) −3.28014 −0.00447495 −0.00223748 0.999997i \(-0.500712\pi\)
−0.00223748 + 0.999997i \(0.500712\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 261.290i 0.354532i
\(738\) 0 0
\(739\) −848.413 −1.14806 −0.574028 0.818836i \(-0.694620\pi\)
−0.574028 + 0.818836i \(0.694620\pi\)
\(740\) 0 0
\(741\) 21.6150 + 23.5078i 0.0291701 + 0.0317245i
\(742\) 0 0
\(743\) 620.675i 0.835363i −0.908593 0.417682i \(-0.862843\pi\)
0.908593 0.417682i \(-0.137157\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −422.298 −0.565326
\(748\) 0 0
\(749\) 181.080i 0.241762i
\(750\) 0 0
\(751\) 88.8695i 0.118335i −0.998248 0.0591674i \(-0.981155\pi\)
0.998248 0.0591674i \(-0.0188446\pi\)
\(752\) 0 0
\(753\) 34.3036i 0.0455559i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −389.621 −0.514692 −0.257346 0.966319i \(-0.582848\pi\)
−0.257346 + 0.966319i \(0.582848\pi\)
\(758\) 0 0
\(759\) 5.53592i 0.00729370i
\(760\) 0 0
\(761\) 183.315 0.240887 0.120444 0.992720i \(-0.461568\pi\)
0.120444 + 0.992720i \(0.461568\pi\)
\(762\) 0 0
\(763\) 302.869i 0.396945i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −222.195 −0.289694
\(768\) 0 0
\(769\) 839.391 1.09154 0.545768 0.837936i \(-0.316238\pi\)
0.545768 + 0.837936i \(0.316238\pi\)
\(770\) 0 0
\(771\) −112.788 −0.146289
\(772\) 0 0
\(773\) 1149.34i 1.48685i −0.668819 0.743425i \(-0.733200\pi\)
0.668819 0.743425i \(-0.266800\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −31.2120 −0.0401699
\(778\) 0 0
\(779\) −854.649 + 785.833i −1.09711 + 1.00877i
\(780\) 0 0
\(781\) 223.316i 0.285936i
\(782\) 0 0
\(783\) −169.518 −0.216498
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 500.800i 0.636341i −0.948034 0.318170i \(-0.896932\pi\)
0.948034 0.318170i \(-0.103068\pi\)
\(788\) 0 0
\(789\) 123.750i 0.156845i
\(790\) 0 0
\(791\) 344.543i 0.435579i
\(792\) 0 0
\(793\) 343.130i 0.432699i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 481.090i 0.603626i 0.953367 + 0.301813i \(0.0975918\pi\)
−0.953367 + 0.301813i \(0.902408\pi\)
\(798\) 0 0
\(799\) −192.452 −0.240866
\(800\) 0 0
\(801\) 608.988i 0.760284i
\(802\) 0 0
\(803\) 259.715 0.323431
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 16.9458 0.0209985
\(808\) 0 0
\(809\) −16.0970 −0.0198974 −0.00994869 0.999951i \(-0.503167\pi\)
−0.00994869 + 0.999951i \(0.503167\pi\)
\(810\) 0 0
\(811\) 744.398i 0.917876i 0.888468 + 0.458938i \(0.151770\pi\)
−0.888468 + 0.458938i \(0.848230\pi\)
\(812\) 0 0
\(813\) 119.199i 0.146617i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 69.4067 + 75.4846i 0.0849531 + 0.0923925i
\(818\) 0 0
\(819\) 115.848i 0.141450i
\(820\) 0 0
\(821\) −817.427 −0.995648 −0.497824 0.867278i \(-0.665868\pi\)
−0.497824 + 0.867278i \(0.665868\pi\)
\(822\) 0 0
\(823\) −447.054 −0.543201 −0.271600 0.962410i \(-0.587553\pi\)
−0.271600 + 0.962410i \(0.587553\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1387.55i 1.67782i −0.544274 0.838908i \(-0.683195\pi\)
0.544274 0.838908i \(-0.316805\pi\)
\(828\) 0 0
\(829\) 1076.73i 1.29883i −0.760433 0.649416i \(-0.775013\pi\)
0.760433 0.649416i \(-0.224987\pi\)
\(830\) 0 0
\(831\) 74.5613i 0.0897247i
\(832\) 0 0
\(833\) 942.310 1.13122
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −224.823 −0.268606
\(838\) 0 0
\(839\) 606.394i 0.722758i 0.932419 + 0.361379i \(0.117694\pi\)
−0.932419 + 0.361379i \(0.882306\pi\)
\(840\) 0 0
\(841\) −267.163 −0.317673
\(842\) 0 0
\(843\) 63.6038 0.0754493
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −255.574 −0.301741
\(848\) 0 0
\(849\) 68.4754i 0.0806542i
\(850\) 0 0
\(851\) 453.220i 0.532573i
\(852\) 0 0
\(853\) 99.5618 0.116720 0.0583598 0.998296i \(-0.481413\pi\)
0.0583598 + 0.998296i \(0.481413\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1591.90i 1.85753i −0.370669 0.928765i \(-0.620872\pi\)
0.370669 0.928765i \(-0.379128\pi\)
\(858\) 0 0
\(859\) 974.586 1.13456 0.567279 0.823525i \(-0.307996\pi\)
0.567279 + 0.823525i \(0.307996\pi\)
\(860\) 0 0
\(861\) 38.1348 0.0442913
\(862\) 0 0
\(863\) 270.696i 0.313669i −0.987625 0.156834i \(-0.949871\pi\)
0.987625 0.156834i \(-0.0501289\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 47.1671i 0.0544027i
\(868\) 0 0
\(869\) 122.792i 0.141302i
\(870\) 0 0
\(871\) 718.897 0.825370
\(872\) 0 0
\(873\) 203.102i 0.232649i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 868.510i 0.990319i 0.868802 + 0.495160i \(0.164890\pi\)
−0.868802 + 0.495160i \(0.835110\pi\)
\(878\) 0 0
\(879\) 118.981 0.135360
\(880\) 0 0
\(881\) 916.787 1.04062 0.520311 0.853977i \(-0.325816\pi\)
0.520311 + 0.853977i \(0.325816\pi\)
\(882\) 0 0
\(883\) −878.004 −0.994342 −0.497171 0.867653i \(-0.665628\pi\)
−0.497171 + 0.867653i \(0.665628\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1351.53i 1.52371i −0.647747 0.761856i \(-0.724289\pi\)
0.647747 0.761856i \(-0.275711\pi\)
\(888\) 0 0
\(889\) 353.516i 0.397656i
\(890\) 0 0
\(891\) −169.450 −0.190180
\(892\) 0 0
\(893\) −116.031 126.192i −0.129934 0.141312i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 15.2312 0.0169801
\(898\) 0 0
\(899\) −1469.71 −1.63482
\(900\) 0 0
\(901\) 1323.61i 1.46905i
\(902\) 0 0
\(903\) 3.36816i 0.00372996i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 872.623i 0.962098i 0.876694 + 0.481049i \(0.159744\pi\)
−0.876694 + 0.481049i \(0.840256\pi\)
\(908\) 0 0
\(909\) 330.796 0.363912
\(910\) 0 0
\(911\) 885.104i 0.971574i −0.874077 0.485787i \(-0.838533\pi\)
0.874077 0.485787i \(-0.161467\pi\)
\(912\) 0 0
\(913\) 101.780 0.111479
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 441.279 0.481221
\(918\) 0 0
\(919\) −446.590 −0.485952 −0.242976 0.970032i \(-0.578124\pi\)
−0.242976 + 0.970032i \(0.578124\pi\)
\(920\) 0 0
\(921\) 8.90600 0.00966992
\(922\) 0 0
\(923\) 614.419 0.665676
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 1491.48i 1.60894i
\(928\) 0 0
\(929\) 534.036 0.574850 0.287425 0.957803i \(-0.407201\pi\)
0.287425 + 0.957803i \(0.407201\pi\)
\(930\) 0 0
\(931\) 568.127 + 617.878i 0.610233 + 0.663671i
\(932\) 0 0
\(933\) 135.098i 0.144799i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1157.99 1.23585 0.617926 0.786236i \(-0.287973\pi\)
0.617926 + 0.786236i \(0.287973\pi\)
\(938\) 0 0
\(939\) 27.8311i 0.0296391i
\(940\) 0 0
\(941\) 868.034i 0.922459i 0.887281 + 0.461230i \(0.152592\pi\)
−0.887281 + 0.461230i \(0.847408\pi\)
\(942\) 0 0
\(943\) 553.743i 0.587214i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1754.37 1.85255 0.926276 0.376845i \(-0.122991\pi\)
0.926276 + 0.376845i \(0.122991\pi\)
\(948\) 0 0
\(949\) 714.563i 0.752965i
\(950\) 0 0
\(951\) 139.752 0.146953
\(952\) 0 0
\(953\) 541.232i 0.567924i −0.958836 0.283962i \(-0.908351\pi\)
0.958836 0.283962i \(-0.0916490\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 20.3361 0.0212498
\(958\) 0 0
\(959\) −399.084 −0.416146
\(960\) 0 0
\(961\) −988.203 −1.02831
\(962\) 0 0
\(963\) 735.452i 0.763709i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −632.276 −0.653853 −0.326927 0.945050i \(-0.606013\pi\)
−0.326927 + 0.945050i \(0.606013\pi\)
\(968\) 0 0
\(969\) −84.7795 + 77.9531i −0.0874917 + 0.0804470i
\(970\) 0 0
\(971\) 1326.62i 1.36624i −0.730305 0.683122i \(-0.760622\pi\)
0.730305 0.683122i \(-0.239378\pi\)
\(972\) 0 0
\(973\) −108.979 −0.112003
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 250.280i 0.256172i 0.991763 + 0.128086i \(0.0408833\pi\)
−0.991763 + 0.128086i \(0.959117\pi\)
\(978\) 0 0
\(979\) 146.775i 0.149924i
\(980\) 0 0
\(981\) 1230.10i 1.25392i
\(982\) 0 0
\(983\) 1291.69i 1.31403i 0.753879 + 0.657014i \(0.228181\pi\)
−0.753879 + 0.657014i \(0.771819\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 5.63074i 0.00570490i
\(988\) 0 0
\(989\) 48.9079 0.0494519
\(990\) 0 0
\(991\) 468.338i 0.472592i −0.971681 0.236296i \(-0.924067\pi\)
0.971681 0.236296i \(-0.0759335\pi\)
\(992\) 0 0
\(993\) 93.7738 0.0944348
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1592.68 −1.59747 −0.798735 0.601683i \(-0.794497\pi\)
−0.798735 + 0.601683i \(0.794497\pi\)
\(998\) 0 0
\(999\) −254.682 −0.254937
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.g.1101.8 yes 14
5.2 odd 4 1900.3.g.d.949.15 28
5.3 odd 4 1900.3.g.d.949.14 28
5.4 even 2 1900.3.e.h.1101.7 yes 14
19.18 odd 2 inner 1900.3.e.g.1101.7 14
95.18 even 4 1900.3.g.d.949.16 28
95.37 even 4 1900.3.g.d.949.13 28
95.94 odd 2 1900.3.e.h.1101.8 yes 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1900.3.e.g.1101.7 14 19.18 odd 2 inner
1900.3.e.g.1101.8 yes 14 1.1 even 1 trivial
1900.3.e.h.1101.7 yes 14 5.4 even 2
1900.3.e.h.1101.8 yes 14 95.94 odd 2
1900.3.g.d.949.13 28 95.37 even 4
1900.3.g.d.949.14 28 5.3 odd 4
1900.3.g.d.949.15 28 5.2 odd 4
1900.3.g.d.949.16 28 95.18 even 4