Properties

Label 4840.2.a.s.1.3
Level $4840$
Weight $2$
Character 4840.1
Self dual yes
Analytic conductor $38.648$
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] = [4840,2,Mod(1,4840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4840, 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("4840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4840.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: \(38.6475945783\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.788.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 7x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.35386\) of defining polynomial
Character \(\chi\) \(=\) 4840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.35386 q^{3} -1.00000 q^{5} -3.81322 q^{7} +8.24835 q^{9} +O(q^{10})\) \(q+3.35386 q^{3} -1.00000 q^{5} -3.81322 q^{7} +8.24835 q^{9} +0.459363 q^{13} -3.35386 q^{15} -7.78899 q^{17} +1.54064 q^{19} -12.7890 q^{21} -8.24835 q^{23} +1.00000 q^{25} +17.6022 q^{27} -6.70771 q^{29} -7.54064 q^{31} +3.81322 q^{35} +4.70771 q^{37} +1.54064 q^{39} -1.45936 q^{41} +3.97577 q^{43} -8.24835 q^{45} +0.272582 q^{47} +7.54064 q^{49} -26.1231 q^{51} -2.62191 q^{53} +5.16707 q^{57} -8.24835 q^{59} +0.751651 q^{61} -31.4528 q^{63} -0.459363 q^{65} -4.06157 q^{67} -27.6638 q^{69} -10.4109 q^{71} +0.918726 q^{73} +3.35386 q^{75} +5.78899 q^{79} +34.2902 q^{81} -9.16707 q^{83} +7.78899 q^{85} -22.4967 q^{87} +12.7890 q^{89} -1.75165 q^{91} -25.2902 q^{93} -1.54064 q^{95} -0.459363 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9} + 2 q^{13} - q^{15} - 4 q^{17} + 4 q^{19} - 19 q^{21} - 6 q^{23} + 3 q^{25} + 25 q^{27} - 2 q^{29} - 22 q^{31} + 3 q^{35} - 4 q^{37} + 4 q^{39} - 5 q^{41} + q^{43} - 6 q^{45} - 7 q^{47} + 22 q^{49} - 24 q^{51} - 6 q^{53} - 2 q^{57} - 6 q^{59} + 21 q^{61} - 20 q^{63} - 2 q^{65} + 15 q^{67} - 28 q^{69} - 10 q^{71} + 4 q^{73} + q^{75} - 2 q^{79} + 31 q^{81} - 10 q^{83} + 4 q^{85} - 30 q^{87} + 19 q^{89} - 24 q^{91} - 4 q^{93} - 4 q^{95} - 2 q^{97}+O(q^{100}) \) 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
\(3\) 3.35386 1.93635 0.968175 0.250275i \(-0.0805211\pi\)
0.968175 + 0.250275i \(0.0805211\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.81322 −1.44126 −0.720631 0.693319i \(-0.756148\pi\)
−0.720631 + 0.693319i \(0.756148\pi\)
\(8\) 0 0
\(9\) 8.24835 2.74945
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 0.459363 0.127404 0.0637022 0.997969i \(-0.479709\pi\)
0.0637022 + 0.997969i \(0.479709\pi\)
\(14\) 0 0
\(15\) −3.35386 −0.865962
\(16\) 0 0
\(17\) −7.78899 −1.88911 −0.944553 0.328358i \(-0.893505\pi\)
−0.944553 + 0.328358i \(0.893505\pi\)
\(18\) 0 0
\(19\) 1.54064 0.353446 0.176723 0.984261i \(-0.443450\pi\)
0.176723 + 0.984261i \(0.443450\pi\)
\(20\) 0 0
\(21\) −12.7890 −2.79079
\(22\) 0 0
\(23\) −8.24835 −1.71990 −0.859950 0.510379i \(-0.829505\pi\)
−0.859950 + 0.510379i \(0.829505\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 17.6022 3.38755
\(28\) 0 0
\(29\) −6.70771 −1.24559 −0.622795 0.782385i \(-0.714003\pi\)
−0.622795 + 0.782385i \(0.714003\pi\)
\(30\) 0 0
\(31\) −7.54064 −1.35434 −0.677169 0.735827i \(-0.736793\pi\)
−0.677169 + 0.735827i \(0.736793\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.81322 0.644552
\(36\) 0 0
\(37\) 4.70771 0.773943 0.386972 0.922092i \(-0.373521\pi\)
0.386972 + 0.922092i \(0.373521\pi\)
\(38\) 0 0
\(39\) 1.54064 0.246699
\(40\) 0 0
\(41\) −1.45936 −0.227914 −0.113957 0.993486i \(-0.536353\pi\)
−0.113957 + 0.993486i \(0.536353\pi\)
\(42\) 0 0
\(43\) 3.97577 0.606299 0.303149 0.952943i \(-0.401962\pi\)
0.303149 + 0.952943i \(0.401962\pi\)
\(44\) 0 0
\(45\) −8.24835 −1.22959
\(46\) 0 0
\(47\) 0.272582 0.0397601 0.0198801 0.999802i \(-0.493672\pi\)
0.0198801 + 0.999802i \(0.493672\pi\)
\(48\) 0 0
\(49\) 7.54064 1.07723
\(50\) 0 0
\(51\) −26.1231 −3.65797
\(52\) 0 0
\(53\) −2.62191 −0.360147 −0.180074 0.983653i \(-0.557634\pi\)
−0.180074 + 0.983653i \(0.557634\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5.16707 0.684396
\(58\) 0 0
\(59\) −8.24835 −1.07384 −0.536922 0.843632i \(-0.680413\pi\)
−0.536922 + 0.843632i \(0.680413\pi\)
\(60\) 0 0
\(61\) 0.751651 0.0962391 0.0481195 0.998842i \(-0.484677\pi\)
0.0481195 + 0.998842i \(0.484677\pi\)
\(62\) 0 0
\(63\) −31.4528 −3.96267
\(64\) 0 0
\(65\) −0.459363 −0.0569770
\(66\) 0 0
\(67\) −4.06157 −0.496199 −0.248100 0.968735i \(-0.579806\pi\)
−0.248100 + 0.968735i \(0.579806\pi\)
\(68\) 0 0
\(69\) −27.6638 −3.33033
\(70\) 0 0
\(71\) −10.4109 −1.23555 −0.617773 0.786356i \(-0.711965\pi\)
−0.617773 + 0.786356i \(0.711965\pi\)
\(72\) 0 0
\(73\) 0.918726 0.107529 0.0537644 0.998554i \(-0.482878\pi\)
0.0537644 + 0.998554i \(0.482878\pi\)
\(74\) 0 0
\(75\) 3.35386 0.387270
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 5.78899 0.651312 0.325656 0.945488i \(-0.394415\pi\)
0.325656 + 0.945488i \(0.394415\pi\)
\(80\) 0 0
\(81\) 34.2902 3.81002
\(82\) 0 0
\(83\) −9.16707 −1.00622 −0.503109 0.864223i \(-0.667810\pi\)
−0.503109 + 0.864223i \(0.667810\pi\)
\(84\) 0 0
\(85\) 7.78899 0.844834
\(86\) 0 0
\(87\) −22.4967 −2.41190
\(88\) 0 0
\(89\) 12.7890 1.35563 0.677815 0.735233i \(-0.262927\pi\)
0.677815 + 0.735233i \(0.262927\pi\)
\(90\) 0 0
\(91\) −1.75165 −0.183623
\(92\) 0 0
\(93\) −25.2902 −2.62247
\(94\) 0 0
\(95\) −1.54064 −0.158066
\(96\) 0 0
\(97\) −0.459363 −0.0466412 −0.0233206 0.999728i \(-0.507424\pi\)
−0.0233206 + 0.999728i \(0.507424\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −9.54516 −0.949779 −0.474890 0.880045i \(-0.657512\pi\)
−0.474890 + 0.880045i \(0.657512\pi\)
\(102\) 0 0
\(103\) 6.41090 0.631684 0.315842 0.948812i \(-0.397713\pi\)
0.315842 + 0.948812i \(0.397713\pi\)
\(104\) 0 0
\(105\) 12.7890 1.24808
\(106\) 0 0
\(107\) 0.435130 0.0420656 0.0210328 0.999779i \(-0.493305\pi\)
0.0210328 + 0.999779i \(0.493305\pi\)
\(108\) 0 0
\(109\) 8.66377 0.829839 0.414919 0.909858i \(-0.363810\pi\)
0.414919 + 0.909858i \(0.363810\pi\)
\(110\) 0 0
\(111\) 15.7890 1.49862
\(112\) 0 0
\(113\) −17.2529 −1.62301 −0.811507 0.584343i \(-0.801352\pi\)
−0.811507 + 0.584343i \(0.801352\pi\)
\(114\) 0 0
\(115\) 8.24835 0.769162
\(116\) 0 0
\(117\) 3.78899 0.350292
\(118\) 0 0
\(119\) 29.7011 2.72270
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −4.89449 −0.441322
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 3.14284 0.278882 0.139441 0.990230i \(-0.455469\pi\)
0.139441 + 0.990230i \(0.455469\pi\)
\(128\) 0 0
\(129\) 13.3341 1.17401
\(130\) 0 0
\(131\) 16.7935 1.46726 0.733628 0.679551i \(-0.237826\pi\)
0.733628 + 0.679551i \(0.237826\pi\)
\(132\) 0 0
\(133\) −5.87479 −0.509409
\(134\) 0 0
\(135\) −17.6022 −1.51496
\(136\) 0 0
\(137\) 11.0045 0.940180 0.470090 0.882618i \(-0.344221\pi\)
0.470090 + 0.882618i \(0.344221\pi\)
\(138\) 0 0
\(139\) −20.9561 −1.77747 −0.888735 0.458421i \(-0.848415\pi\)
−0.888735 + 0.458421i \(0.848415\pi\)
\(140\) 0 0
\(141\) 0.914200 0.0769895
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 6.70771 0.557045
\(146\) 0 0
\(147\) 25.2902 2.08590
\(148\) 0 0
\(149\) 1.62191 0.132872 0.0664361 0.997791i \(-0.478837\pi\)
0.0664361 + 0.997791i \(0.478837\pi\)
\(150\) 0 0
\(151\) 7.00453 0.570020 0.285010 0.958525i \(-0.408003\pi\)
0.285010 + 0.958525i \(0.408003\pi\)
\(152\) 0 0
\(153\) −64.2463 −5.19400
\(154\) 0 0
\(155\) 7.54064 0.605679
\(156\) 0 0
\(157\) −10.3341 −0.824755 −0.412377 0.911013i \(-0.635301\pi\)
−0.412377 + 0.911013i \(0.635301\pi\)
\(158\) 0 0
\(159\) −8.79351 −0.697371
\(160\) 0 0
\(161\) 31.4528 2.47882
\(162\) 0 0
\(163\) −10.1868 −0.797890 −0.398945 0.916975i \(-0.630624\pi\)
−0.398945 + 0.916975i \(0.630624\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 22.8066 1.76483 0.882414 0.470473i \(-0.155917\pi\)
0.882414 + 0.470473i \(0.155917\pi\)
\(168\) 0 0
\(169\) −12.7890 −0.983768
\(170\) 0 0
\(171\) 12.7077 0.971783
\(172\) 0 0
\(173\) −17.8748 −1.35899 −0.679497 0.733678i \(-0.737802\pi\)
−0.679497 + 0.733678i \(0.737802\pi\)
\(174\) 0 0
\(175\) −3.81322 −0.288252
\(176\) 0 0
\(177\) −27.6638 −2.07934
\(178\) 0 0
\(179\) −6.91873 −0.517130 −0.258565 0.965994i \(-0.583250\pi\)
−0.258565 + 0.965994i \(0.583250\pi\)
\(180\) 0 0
\(181\) 14.9515 1.11134 0.555669 0.831403i \(-0.312462\pi\)
0.555669 + 0.831403i \(0.312462\pi\)
\(182\) 0 0
\(183\) 2.52093 0.186353
\(184\) 0 0
\(185\) −4.70771 −0.346118
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −67.1211 −4.88234
\(190\) 0 0
\(191\) −5.32962 −0.385638 −0.192819 0.981234i \(-0.561763\pi\)
−0.192819 + 0.981234i \(0.561763\pi\)
\(192\) 0 0
\(193\) 5.29229 0.380947 0.190474 0.981692i \(-0.438998\pi\)
0.190474 + 0.981692i \(0.438998\pi\)
\(194\) 0 0
\(195\) −1.54064 −0.110327
\(196\) 0 0
\(197\) 12.1231 0.863738 0.431869 0.901936i \(-0.357854\pi\)
0.431869 + 0.901936i \(0.357854\pi\)
\(198\) 0 0
\(199\) 12.3341 0.874345 0.437172 0.899378i \(-0.355980\pi\)
0.437172 + 0.899378i \(0.355980\pi\)
\(200\) 0 0
\(201\) −13.6219 −0.960816
\(202\) 0 0
\(203\) 25.5780 1.79522
\(204\) 0 0
\(205\) 1.45936 0.101926
\(206\) 0 0
\(207\) −68.0353 −4.72878
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −4.95606 −0.341189 −0.170595 0.985341i \(-0.554569\pi\)
−0.170595 + 0.985341i \(0.554569\pi\)
\(212\) 0 0
\(213\) −34.9166 −2.39245
\(214\) 0 0
\(215\) −3.97577 −0.271145
\(216\) 0 0
\(217\) 28.7541 1.95196
\(218\) 0 0
\(219\) 3.08127 0.208213
\(220\) 0 0
\(221\) −3.57797 −0.240680
\(222\) 0 0
\(223\) −8.39780 −0.562358 −0.281179 0.959655i \(-0.590725\pi\)
−0.281179 + 0.959655i \(0.590725\pi\)
\(224\) 0 0
\(225\) 8.24835 0.549890
\(226\) 0 0
\(227\) −15.2681 −1.01338 −0.506688 0.862129i \(-0.669130\pi\)
−0.506688 + 0.862129i \(0.669130\pi\)
\(228\) 0 0
\(229\) 8.62644 0.570051 0.285026 0.958520i \(-0.407998\pi\)
0.285026 + 0.958520i \(0.407998\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −26.4109 −1.73024 −0.865118 0.501569i \(-0.832756\pi\)
−0.865118 + 0.501569i \(0.832756\pi\)
\(234\) 0 0
\(235\) −0.272582 −0.0177813
\(236\) 0 0
\(237\) 19.4154 1.26117
\(238\) 0 0
\(239\) −1.21554 −0.0786268 −0.0393134 0.999227i \(-0.512517\pi\)
−0.0393134 + 0.999227i \(0.512517\pi\)
\(240\) 0 0
\(241\) 14.1186 0.909460 0.454730 0.890629i \(-0.349736\pi\)
0.454730 + 0.890629i \(0.349736\pi\)
\(242\) 0 0
\(243\) 62.1978 3.98999
\(244\) 0 0
\(245\) −7.54064 −0.481754
\(246\) 0 0
\(247\) 0.707712 0.0450306
\(248\) 0 0
\(249\) −30.7450 −1.94839
\(250\) 0 0
\(251\) 9.16707 0.578621 0.289310 0.957235i \(-0.406574\pi\)
0.289310 + 0.957235i \(0.406574\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 26.1231 1.63589
\(256\) 0 0
\(257\) −10.4594 −0.652437 −0.326219 0.945294i \(-0.605775\pi\)
−0.326219 + 0.945294i \(0.605775\pi\)
\(258\) 0 0
\(259\) −17.9515 −1.11545
\(260\) 0 0
\(261\) −55.3275 −3.42469
\(262\) 0 0
\(263\) −10.6219 −0.654975 −0.327488 0.944855i \(-0.606202\pi\)
−0.327488 + 0.944855i \(0.606202\pi\)
\(264\) 0 0
\(265\) 2.62191 0.161063
\(266\) 0 0
\(267\) 42.8924 2.62497
\(268\) 0 0
\(269\) 10.0045 0.609987 0.304993 0.952354i \(-0.401346\pi\)
0.304993 + 0.952354i \(0.401346\pi\)
\(270\) 0 0
\(271\) −2.54516 −0.154608 −0.0773038 0.997008i \(-0.524631\pi\)
−0.0773038 + 0.997008i \(0.524631\pi\)
\(272\) 0 0
\(273\) −5.87479 −0.355558
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −17.2923 −1.03899 −0.519496 0.854473i \(-0.673880\pi\)
−0.519496 + 0.854473i \(0.673880\pi\)
\(278\) 0 0
\(279\) −62.1978 −3.72369
\(280\) 0 0
\(281\) −16.8308 −1.00404 −0.502022 0.864855i \(-0.667410\pi\)
−0.502022 + 0.864855i \(0.667410\pi\)
\(282\) 0 0
\(283\) 1.30539 0.0775974 0.0387987 0.999247i \(-0.487647\pi\)
0.0387987 + 0.999247i \(0.487647\pi\)
\(284\) 0 0
\(285\) −5.16707 −0.306071
\(286\) 0 0
\(287\) 5.56487 0.328484
\(288\) 0 0
\(289\) 43.6683 2.56872
\(290\) 0 0
\(291\) −1.54064 −0.0903137
\(292\) 0 0
\(293\) −3.91420 −0.228670 −0.114335 0.993442i \(-0.536474\pi\)
−0.114335 + 0.993442i \(0.536474\pi\)
\(294\) 0 0
\(295\) 8.24835 0.480237
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3.78899 −0.219123
\(300\) 0 0
\(301\) −15.1605 −0.873835
\(302\) 0 0
\(303\) −32.0131 −1.83910
\(304\) 0 0
\(305\) −0.751651 −0.0430394
\(306\) 0 0
\(307\) −21.1671 −1.20807 −0.604034 0.796958i \(-0.706441\pi\)
−0.604034 + 0.796958i \(0.706441\pi\)
\(308\) 0 0
\(309\) 21.5012 1.22316
\(310\) 0 0
\(311\) −5.96267 −0.338112 −0.169056 0.985606i \(-0.554072\pi\)
−0.169056 + 0.985606i \(0.554072\pi\)
\(312\) 0 0
\(313\) −28.3230 −1.60091 −0.800456 0.599392i \(-0.795409\pi\)
−0.800456 + 0.599392i \(0.795409\pi\)
\(314\) 0 0
\(315\) 31.4528 1.77216
\(316\) 0 0
\(317\) −1.57797 −0.0886277 −0.0443139 0.999018i \(-0.514110\pi\)
−0.0443139 + 0.999018i \(0.514110\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 1.45936 0.0814537
\(322\) 0 0
\(323\) −12.0000 −0.667698
\(324\) 0 0
\(325\) 0.459363 0.0254809
\(326\) 0 0
\(327\) 29.0570 1.60686
\(328\) 0 0
\(329\) −1.03941 −0.0573047
\(330\) 0 0
\(331\) 17.9233 0.985151 0.492576 0.870270i \(-0.336056\pi\)
0.492576 + 0.870270i \(0.336056\pi\)
\(332\) 0 0
\(333\) 38.8308 2.12792
\(334\) 0 0
\(335\) 4.06157 0.221907
\(336\) 0 0
\(337\) 26.3715 1.43655 0.718273 0.695761i \(-0.244933\pi\)
0.718273 + 0.695761i \(0.244933\pi\)
\(338\) 0 0
\(339\) −57.8637 −3.14272
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −2.06157 −0.111314
\(344\) 0 0
\(345\) 27.6638 1.48937
\(346\) 0 0
\(347\) 8.80661 0.472764 0.236382 0.971660i \(-0.424038\pi\)
0.236382 + 0.971660i \(0.424038\pi\)
\(348\) 0 0
\(349\) 12.3736 0.662342 0.331171 0.943571i \(-0.392556\pi\)
0.331171 + 0.943571i \(0.392556\pi\)
\(350\) 0 0
\(351\) 8.08580 0.431588
\(352\) 0 0
\(353\) 13.5012 0.718598 0.359299 0.933223i \(-0.383016\pi\)
0.359299 + 0.933223i \(0.383016\pi\)
\(354\) 0 0
\(355\) 10.4109 0.552553
\(356\) 0 0
\(357\) 99.6132 5.27209
\(358\) 0 0
\(359\) 25.0792 1.32363 0.661815 0.749668i \(-0.269787\pi\)
0.661815 + 0.749668i \(0.269787\pi\)
\(360\) 0 0
\(361\) −16.6264 −0.875076
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −0.918726 −0.0480883
\(366\) 0 0
\(367\) −22.5977 −1.17959 −0.589795 0.807553i \(-0.700791\pi\)
−0.589795 + 0.807553i \(0.700791\pi\)
\(368\) 0 0
\(369\) −12.0373 −0.626639
\(370\) 0 0
\(371\) 9.99792 0.519066
\(372\) 0 0
\(373\) 17.5012 0.906179 0.453090 0.891465i \(-0.350322\pi\)
0.453090 + 0.891465i \(0.350322\pi\)
\(374\) 0 0
\(375\) −3.35386 −0.173192
\(376\) 0 0
\(377\) −3.08127 −0.158694
\(378\) 0 0
\(379\) −33.1165 −1.70108 −0.850541 0.525909i \(-0.823725\pi\)
−0.850541 + 0.525909i \(0.823725\pi\)
\(380\) 0 0
\(381\) 10.5406 0.540013
\(382\) 0 0
\(383\) 28.5340 1.45802 0.729010 0.684503i \(-0.239981\pi\)
0.729010 + 0.684503i \(0.239981\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 32.7935 1.66699
\(388\) 0 0
\(389\) 24.2812 1.23110 0.615552 0.788096i \(-0.288933\pi\)
0.615552 + 0.788096i \(0.288933\pi\)
\(390\) 0 0
\(391\) 64.2463 3.24907
\(392\) 0 0
\(393\) 56.3230 2.84112
\(394\) 0 0
\(395\) −5.78899 −0.291275
\(396\) 0 0
\(397\) −26.3341 −1.32167 −0.660837 0.750530i \(-0.729798\pi\)
−0.660837 + 0.750530i \(0.729798\pi\)
\(398\) 0 0
\(399\) −19.7032 −0.986393
\(400\) 0 0
\(401\) 17.8308 0.890430 0.445215 0.895424i \(-0.353127\pi\)
0.445215 + 0.895424i \(0.353127\pi\)
\(402\) 0 0
\(403\) −3.46389 −0.172549
\(404\) 0 0
\(405\) −34.2902 −1.70389
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 6.37809 0.315376 0.157688 0.987489i \(-0.449596\pi\)
0.157688 + 0.987489i \(0.449596\pi\)
\(410\) 0 0
\(411\) 36.9076 1.82052
\(412\) 0 0
\(413\) 31.4528 1.54769
\(414\) 0 0
\(415\) 9.16707 0.449994
\(416\) 0 0
\(417\) −70.2836 −3.44180
\(418\) 0 0
\(419\) 27.6638 1.35146 0.675732 0.737148i \(-0.263828\pi\)
0.675732 + 0.737148i \(0.263828\pi\)
\(420\) 0 0
\(421\) −7.78446 −0.379391 −0.189696 0.981843i \(-0.560750\pi\)
−0.189696 + 0.981843i \(0.560750\pi\)
\(422\) 0 0
\(423\) 2.24835 0.109318
\(424\) 0 0
\(425\) −7.78899 −0.377821
\(426\) 0 0
\(427\) −2.86621 −0.138706
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −21.3670 −1.02921 −0.514605 0.857427i \(-0.672061\pi\)
−0.514605 + 0.857427i \(0.672061\pi\)
\(432\) 0 0
\(433\) −16.3715 −0.786763 −0.393382 0.919375i \(-0.628695\pi\)
−0.393382 + 0.919375i \(0.628695\pi\)
\(434\) 0 0
\(435\) 22.4967 1.07863
\(436\) 0 0
\(437\) −12.7077 −0.607892
\(438\) 0 0
\(439\) −34.6198 −1.65231 −0.826157 0.563440i \(-0.809478\pi\)
−0.826157 + 0.563440i \(0.809478\pi\)
\(440\) 0 0
\(441\) 62.1978 2.96180
\(442\) 0 0
\(443\) −10.4725 −0.497562 −0.248781 0.968560i \(-0.580030\pi\)
−0.248781 + 0.968560i \(0.580030\pi\)
\(444\) 0 0
\(445\) −12.7890 −0.606256
\(446\) 0 0
\(447\) 5.43966 0.257287
\(448\) 0 0
\(449\) 39.5825 1.86801 0.934007 0.357255i \(-0.116287\pi\)
0.934007 + 0.357255i \(0.116287\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 23.4922 1.10376
\(454\) 0 0
\(455\) 1.75165 0.0821187
\(456\) 0 0
\(457\) 34.5340 1.61543 0.807717 0.589570i \(-0.200703\pi\)
0.807717 + 0.589570i \(0.200703\pi\)
\(458\) 0 0
\(459\) −137.103 −6.39943
\(460\) 0 0
\(461\) 0.788986 0.0367467 0.0183734 0.999831i \(-0.494151\pi\)
0.0183734 + 0.999831i \(0.494151\pi\)
\(462\) 0 0
\(463\) −6.56034 −0.304885 −0.152443 0.988312i \(-0.548714\pi\)
−0.152443 + 0.988312i \(0.548714\pi\)
\(464\) 0 0
\(465\) 25.2902 1.17281
\(466\) 0 0
\(467\) 3.19131 0.147676 0.0738381 0.997270i \(-0.476475\pi\)
0.0738381 + 0.997270i \(0.476475\pi\)
\(468\) 0 0
\(469\) 15.4876 0.715153
\(470\) 0 0
\(471\) −34.6592 −1.59701
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 1.54064 0.0706893
\(476\) 0 0
\(477\) −21.6264 −0.990207
\(478\) 0 0
\(479\) 16.4482 0.751539 0.375769 0.926713i \(-0.377378\pi\)
0.375769 + 0.926713i \(0.377378\pi\)
\(480\) 0 0
\(481\) 2.16255 0.0986037
\(482\) 0 0
\(483\) 105.488 4.79987
\(484\) 0 0
\(485\) 0.459363 0.0208586
\(486\) 0 0
\(487\) −36.9166 −1.67285 −0.836426 0.548079i \(-0.815359\pi\)
−0.836426 + 0.548079i \(0.815359\pi\)
\(488\) 0 0
\(489\) −34.1650 −1.54499
\(490\) 0 0
\(491\) −23.0328 −1.03946 −0.519728 0.854332i \(-0.673967\pi\)
−0.519728 + 0.854332i \(0.673967\pi\)
\(492\) 0 0
\(493\) 52.2463 2.35305
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 39.6990 1.78074
\(498\) 0 0
\(499\) −27.0903 −1.21273 −0.606365 0.795187i \(-0.707373\pi\)
−0.606365 + 0.795187i \(0.707373\pi\)
\(500\) 0 0
\(501\) 76.4901 3.41733
\(502\) 0 0
\(503\) 30.0222 1.33862 0.669311 0.742983i \(-0.266589\pi\)
0.669311 + 0.742983i \(0.266589\pi\)
\(504\) 0 0
\(505\) 9.54516 0.424754
\(506\) 0 0
\(507\) −42.8924 −1.90492
\(508\) 0 0
\(509\) 19.1141 0.847217 0.423608 0.905845i \(-0.360763\pi\)
0.423608 + 0.905845i \(0.360763\pi\)
\(510\) 0 0
\(511\) −3.50330 −0.154977
\(512\) 0 0
\(513\) 27.1186 1.19732
\(514\) 0 0
\(515\) −6.41090 −0.282498
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −59.9495 −2.63149
\(520\) 0 0
\(521\) −5.28568 −0.231570 −0.115785 0.993274i \(-0.536938\pi\)
−0.115785 + 0.993274i \(0.536938\pi\)
\(522\) 0 0
\(523\) 24.0373 1.05108 0.525540 0.850769i \(-0.323863\pi\)
0.525540 + 0.850769i \(0.323863\pi\)
\(524\) 0 0
\(525\) −12.7890 −0.558157
\(526\) 0 0
\(527\) 58.7339 2.55849
\(528\) 0 0
\(529\) 45.0353 1.95805
\(530\) 0 0
\(531\) −68.0353 −2.95248
\(532\) 0 0
\(533\) −0.670377 −0.0290373
\(534\) 0 0
\(535\) −0.435130 −0.0188123
\(536\) 0 0
\(537\) −23.2044 −1.00134
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 24.2529 1.04271 0.521356 0.853339i \(-0.325426\pi\)
0.521356 + 0.853339i \(0.325426\pi\)
\(542\) 0 0
\(543\) 50.1453 2.15194
\(544\) 0 0
\(545\) −8.66377 −0.371115
\(546\) 0 0
\(547\) 37.8354 1.61772 0.808862 0.587999i \(-0.200084\pi\)
0.808862 + 0.587999i \(0.200084\pi\)
\(548\) 0 0
\(549\) 6.19988 0.264605
\(550\) 0 0
\(551\) −10.3341 −0.440250
\(552\) 0 0
\(553\) −22.0747 −0.938710
\(554\) 0 0
\(555\) −15.7890 −0.670205
\(556\) 0 0
\(557\) 6.54516 0.277327 0.138664 0.990340i \(-0.455719\pi\)
0.138664 + 0.990340i \(0.455719\pi\)
\(558\) 0 0
\(559\) 1.82632 0.0772451
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −33.1892 −1.39876 −0.699380 0.714750i \(-0.746540\pi\)
−0.699380 + 0.714750i \(0.746540\pi\)
\(564\) 0 0
\(565\) 17.2529 0.725834
\(566\) 0 0
\(567\) −130.756 −5.49124
\(568\) 0 0
\(569\) 12.0045 0.503256 0.251628 0.967824i \(-0.419034\pi\)
0.251628 + 0.967824i \(0.419034\pi\)
\(570\) 0 0
\(571\) −21.0419 −0.880574 −0.440287 0.897857i \(-0.645123\pi\)
−0.440287 + 0.897857i \(0.645123\pi\)
\(572\) 0 0
\(573\) −17.8748 −0.746730
\(574\) 0 0
\(575\) −8.24835 −0.343980
\(576\) 0 0
\(577\) 35.4922 1.47756 0.738779 0.673948i \(-0.235403\pi\)
0.738779 + 0.673948i \(0.235403\pi\)
\(578\) 0 0
\(579\) 17.7496 0.737647
\(580\) 0 0
\(581\) 34.9561 1.45022
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −3.78899 −0.156655
\(586\) 0 0
\(587\) −9.10551 −0.375825 −0.187912 0.982186i \(-0.560172\pi\)
−0.187912 + 0.982186i \(0.560172\pi\)
\(588\) 0 0
\(589\) −11.6174 −0.478686
\(590\) 0 0
\(591\) 40.6592 1.67250
\(592\) 0 0
\(593\) 13.7032 0.562722 0.281361 0.959602i \(-0.409214\pi\)
0.281361 + 0.959602i \(0.409214\pi\)
\(594\) 0 0
\(595\) −29.7011 −1.21763
\(596\) 0 0
\(597\) 41.3670 1.69304
\(598\) 0 0
\(599\) −27.1277 −1.10841 −0.554203 0.832381i \(-0.686977\pi\)
−0.554203 + 0.832381i \(0.686977\pi\)
\(600\) 0 0
\(601\) −10.1141 −0.412562 −0.206281 0.978493i \(-0.566136\pi\)
−0.206281 + 0.978493i \(0.566136\pi\)
\(602\) 0 0
\(603\) −33.5012 −1.36428
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −8.63096 −0.350320 −0.175160 0.984540i \(-0.556044\pi\)
−0.175160 + 0.984540i \(0.556044\pi\)
\(608\) 0 0
\(609\) 85.7848 3.47618
\(610\) 0 0
\(611\) 0.125214 0.00506561
\(612\) 0 0
\(613\) −26.7824 −1.08173 −0.540865 0.841109i \(-0.681903\pi\)
−0.540865 + 0.841109i \(0.681903\pi\)
\(614\) 0 0
\(615\) 4.89449 0.197365
\(616\) 0 0
\(617\) 19.2417 0.774643 0.387322 0.921945i \(-0.373400\pi\)
0.387322 + 0.921945i \(0.373400\pi\)
\(618\) 0 0
\(619\) −17.4901 −0.702986 −0.351493 0.936190i \(-0.614326\pi\)
−0.351493 + 0.936190i \(0.614326\pi\)
\(620\) 0 0
\(621\) −145.189 −5.82624
\(622\) 0 0
\(623\) −48.7672 −1.95382
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −36.6683 −1.46206
\(630\) 0 0
\(631\) −13.5870 −0.540891 −0.270445 0.962735i \(-0.587171\pi\)
−0.270445 + 0.962735i \(0.587171\pi\)
\(632\) 0 0
\(633\) −16.6219 −0.660662
\(634\) 0 0
\(635\) −3.14284 −0.124720
\(636\) 0 0
\(637\) 3.46389 0.137244
\(638\) 0 0
\(639\) −85.8727 −3.39707
\(640\) 0 0
\(641\) −20.0091 −0.790310 −0.395155 0.918614i \(-0.629309\pi\)
−0.395155 + 0.918614i \(0.629309\pi\)
\(642\) 0 0
\(643\) 40.8551 1.61117 0.805584 0.592482i \(-0.201852\pi\)
0.805584 + 0.592482i \(0.201852\pi\)
\(644\) 0 0
\(645\) −13.3341 −0.525032
\(646\) 0 0
\(647\) −21.3427 −0.839069 −0.419535 0.907739i \(-0.637807\pi\)
−0.419535 + 0.907739i \(0.637807\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 96.4371 3.77967
\(652\) 0 0
\(653\) −23.5295 −0.920781 −0.460390 0.887717i \(-0.652291\pi\)
−0.460390 + 0.887717i \(0.652291\pi\)
\(654\) 0 0
\(655\) −16.7935 −0.656177
\(656\) 0 0
\(657\) 7.57797 0.295645
\(658\) 0 0
\(659\) −22.7824 −0.887476 −0.443738 0.896157i \(-0.646348\pi\)
−0.443738 + 0.896157i \(0.646348\pi\)
\(660\) 0 0
\(661\) −26.8838 −1.04566 −0.522830 0.852437i \(-0.675124\pi\)
−0.522830 + 0.852437i \(0.675124\pi\)
\(662\) 0 0
\(663\) −12.0000 −0.466041
\(664\) 0 0
\(665\) 5.87479 0.227814
\(666\) 0 0
\(667\) 55.3275 2.14229
\(668\) 0 0
\(669\) −28.1650 −1.08892
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 35.0307 1.35034 0.675168 0.737664i \(-0.264071\pi\)
0.675168 + 0.737664i \(0.264071\pi\)
\(674\) 0 0
\(675\) 17.6022 0.677509
\(676\) 0 0
\(677\) −29.7405 −1.14302 −0.571511 0.820595i \(-0.693642\pi\)
−0.571511 + 0.820595i \(0.693642\pi\)
\(678\) 0 0
\(679\) 1.75165 0.0672222
\(680\) 0 0
\(681\) −51.2069 −1.96225
\(682\) 0 0
\(683\) −14.6441 −0.560340 −0.280170 0.959950i \(-0.590391\pi\)
−0.280170 + 0.959950i \(0.590391\pi\)
\(684\) 0 0
\(685\) −11.0045 −0.420461
\(686\) 0 0
\(687\) 28.9318 1.10382
\(688\) 0 0
\(689\) −1.20441 −0.0458843
\(690\) 0 0
\(691\) 32.5825 1.23950 0.619748 0.784801i \(-0.287235\pi\)
0.619748 + 0.784801i \(0.287235\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 20.9561 0.794909
\(696\) 0 0
\(697\) 11.3670 0.430554
\(698\) 0 0
\(699\) −88.5783 −3.35034
\(700\) 0 0
\(701\) 31.5295 1.19085 0.595426 0.803410i \(-0.296983\pi\)
0.595426 + 0.803410i \(0.296983\pi\)
\(702\) 0 0
\(703\) 7.25287 0.273547
\(704\) 0 0
\(705\) −0.914200 −0.0344307
\(706\) 0 0
\(707\) 36.3978 1.36888
\(708\) 0 0
\(709\) 32.1650 1.20798 0.603991 0.796991i \(-0.293576\pi\)
0.603991 + 0.796991i \(0.293576\pi\)
\(710\) 0 0
\(711\) 47.7496 1.79075
\(712\) 0 0
\(713\) 62.1978 2.32933
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −4.07675 −0.152249
\(718\) 0 0
\(719\) 17.8263 0.664810 0.332405 0.943137i \(-0.392140\pi\)
0.332405 + 0.943137i \(0.392140\pi\)
\(720\) 0 0
\(721\) −24.4462 −0.910422
\(722\) 0 0
\(723\) 47.3518 1.76103
\(724\) 0 0
\(725\) −6.70771 −0.249118
\(726\) 0 0
\(727\) 0.511878 0.0189845 0.00949225 0.999955i \(-0.496978\pi\)
0.00949225 + 0.999955i \(0.496978\pi\)
\(728\) 0 0
\(729\) 105.732 3.91599
\(730\) 0 0
\(731\) −30.9672 −1.14536
\(732\) 0 0
\(733\) −50.1978 −1.85410 −0.927049 0.374940i \(-0.877663\pi\)
−0.927049 + 0.374940i \(0.877663\pi\)
\(734\) 0 0
\(735\) −25.2902 −0.932843
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 39.2044 1.44216 0.721079 0.692853i \(-0.243647\pi\)
0.721079 + 0.692853i \(0.243647\pi\)
\(740\) 0 0
\(741\) 2.37356 0.0871950
\(742\) 0 0
\(743\) 2.38666 0.0875582 0.0437791 0.999041i \(-0.486060\pi\)
0.0437791 + 0.999041i \(0.486060\pi\)
\(744\) 0 0
\(745\) −1.62191 −0.0594222
\(746\) 0 0
\(747\) −75.6132 −2.76654
\(748\) 0 0
\(749\) −1.65925 −0.0606275
\(750\) 0 0
\(751\) −8.25948 −0.301393 −0.150696 0.988580i \(-0.548152\pi\)
−0.150696 + 0.988580i \(0.548152\pi\)
\(752\) 0 0
\(753\) 30.7450 1.12041
\(754\) 0 0
\(755\) −7.00453 −0.254921
\(756\) 0 0
\(757\) 15.0792 0.548063 0.274031 0.961721i \(-0.411643\pi\)
0.274031 + 0.961721i \(0.411643\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 17.2529 0.625416 0.312708 0.949849i \(-0.398764\pi\)
0.312708 + 0.949849i \(0.398764\pi\)
\(762\) 0 0
\(763\) −33.0369 −1.19601
\(764\) 0 0
\(765\) 64.2463 2.32283
\(766\) 0 0
\(767\) −3.78899 −0.136812
\(768\) 0 0
\(769\) 42.1322 1.51933 0.759663 0.650317i \(-0.225364\pi\)
0.759663 + 0.650317i \(0.225364\pi\)
\(770\) 0 0
\(771\) −35.0792 −1.26335
\(772\) 0 0
\(773\) −46.8197 −1.68399 −0.841994 0.539487i \(-0.818618\pi\)
−0.841994 + 0.539487i \(0.818618\pi\)
\(774\) 0 0
\(775\) −7.54064 −0.270868
\(776\) 0 0
\(777\) −60.2069 −2.15991
\(778\) 0 0
\(779\) −2.24835 −0.0805554
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −118.071 −4.21950
\(784\) 0 0
\(785\) 10.3341 0.368842
\(786\) 0 0
\(787\) −11.0682 −0.394538 −0.197269 0.980349i \(-0.563207\pi\)
−0.197269 + 0.980349i \(0.563207\pi\)
\(788\) 0 0
\(789\) −35.6244 −1.26826
\(790\) 0 0
\(791\) 65.7890 2.33919
\(792\) 0 0
\(793\) 0.345281 0.0122613
\(794\) 0 0
\(795\) 8.79351 0.311874
\(796\) 0 0
\(797\) −3.41750 −0.121054 −0.0605271 0.998167i \(-0.519278\pi\)
−0.0605271 + 0.998167i \(0.519278\pi\)
\(798\) 0 0
\(799\) −2.12313 −0.0751111
\(800\) 0 0
\(801\) 105.488 3.72724
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −31.4528 −1.10856
\(806\) 0 0
\(807\) 33.5537 1.18115
\(808\) 0 0
\(809\) −44.5452 −1.56612 −0.783062 0.621943i \(-0.786343\pi\)
−0.783062 + 0.621943i \(0.786343\pi\)
\(810\) 0 0
\(811\) −3.63757 −0.127732 −0.0638662 0.997958i \(-0.520343\pi\)
−0.0638662 + 0.997958i \(0.520343\pi\)
\(812\) 0 0
\(813\) −8.53611 −0.299374
\(814\) 0 0
\(815\) 10.1868 0.356827
\(816\) 0 0
\(817\) 6.12521 0.214294
\(818\) 0 0
\(819\) −14.4482 −0.504862
\(820\) 0 0
\(821\) 8.96267 0.312799 0.156400 0.987694i \(-0.450011\pi\)
0.156400 + 0.987694i \(0.450011\pi\)
\(822\) 0 0
\(823\) −36.1756 −1.26100 −0.630502 0.776188i \(-0.717151\pi\)
−0.630502 + 0.776188i \(0.717151\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −43.7536 −1.52146 −0.760731 0.649067i \(-0.775159\pi\)
−0.760731 + 0.649067i \(0.775159\pi\)
\(828\) 0 0
\(829\) 19.6310 0.681812 0.340906 0.940097i \(-0.389266\pi\)
0.340906 + 0.940097i \(0.389266\pi\)
\(830\) 0 0
\(831\) −57.9958 −2.01185
\(832\) 0 0
\(833\) −58.7339 −2.03501
\(834\) 0 0
\(835\) −22.8066 −0.789255
\(836\) 0 0
\(837\) −132.732 −4.58788
\(838\) 0 0
\(839\) −4.00905 −0.138408 −0.0692039 0.997603i \(-0.522046\pi\)
−0.0692039 + 0.997603i \(0.522046\pi\)
\(840\) 0 0
\(841\) 15.9934 0.551496
\(842\) 0 0
\(843\) −56.4482 −1.94418
\(844\) 0 0
\(845\) 12.7890 0.439954
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 4.37809 0.150256
\(850\) 0 0
\(851\) −38.8308 −1.33110
\(852\) 0 0
\(853\) −19.2529 −0.659206 −0.329603 0.944120i \(-0.606915\pi\)
−0.329603 + 0.944120i \(0.606915\pi\)
\(854\) 0 0
\(855\) −12.7077 −0.434595
\(856\) 0 0
\(857\) 21.6174 0.738436 0.369218 0.929343i \(-0.379626\pi\)
0.369218 + 0.929343i \(0.379626\pi\)
\(858\) 0 0
\(859\) −37.0045 −1.26258 −0.631289 0.775548i \(-0.717474\pi\)
−0.631289 + 0.775548i \(0.717474\pi\)
\(860\) 0 0
\(861\) 18.6638 0.636060
\(862\) 0 0
\(863\) −28.8087 −0.980659 −0.490330 0.871537i \(-0.663124\pi\)
−0.490330 + 0.871537i \(0.663124\pi\)
\(864\) 0 0
\(865\) 17.8748 0.607761
\(866\) 0 0
\(867\) 146.457 4.97395
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −1.86573 −0.0632180
\(872\) 0 0
\(873\) −3.78899 −0.128238
\(874\) 0 0
\(875\) 3.81322 0.128910
\(876\) 0 0
\(877\) −22.9166 −0.773840 −0.386920 0.922113i \(-0.626461\pi\)
−0.386920 + 0.922113i \(0.626461\pi\)
\(878\) 0 0
\(879\) −13.1277 −0.442785
\(880\) 0 0
\(881\) 8.76070 0.295156 0.147578 0.989050i \(-0.452852\pi\)
0.147578 + 0.989050i \(0.452852\pi\)
\(882\) 0 0
\(883\) −1.66377 −0.0559904 −0.0279952 0.999608i \(-0.508912\pi\)
−0.0279952 + 0.999608i \(0.508912\pi\)
\(884\) 0 0
\(885\) 27.6638 0.929908
\(886\) 0 0
\(887\) −28.4815 −0.956316 −0.478158 0.878274i \(-0.658695\pi\)
−0.478158 + 0.878274i \(0.658695\pi\)
\(888\) 0 0
\(889\) −11.9843 −0.401942
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0.419949 0.0140531
\(894\) 0 0
\(895\) 6.91873 0.231268
\(896\) 0 0
\(897\) −12.7077 −0.424298
\(898\) 0 0
\(899\) 50.5804 1.68695
\(900\) 0 0
\(901\) 20.4220 0.680356
\(902\) 0 0
\(903\) −50.8460 −1.69205
\(904\) 0 0
\(905\) −14.9515 −0.497006
\(906\) 0 0
\(907\) −7.22864 −0.240023 −0.120012 0.992772i \(-0.538293\pi\)
−0.120012 + 0.992772i \(0.538293\pi\)
\(908\) 0 0
\(909\) −78.7318 −2.61137
\(910\) 0 0
\(911\) 11.9515 0.395972 0.197986 0.980205i \(-0.436560\pi\)
0.197986 + 0.980205i \(0.436560\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −2.52093 −0.0833394
\(916\) 0 0
\(917\) −64.0373 −2.11470
\(918\) 0 0
\(919\) 8.58250 0.283110 0.141555 0.989930i \(-0.454790\pi\)
0.141555 + 0.989930i \(0.454790\pi\)
\(920\) 0 0
\(921\) −70.9913 −2.33924
\(922\) 0 0
\(923\) −4.78238 −0.157414
\(924\) 0 0
\(925\) 4.70771 0.154789
\(926\) 0 0
\(927\) 52.8793 1.73678
\(928\) 0 0
\(929\) −22.5936 −0.741273 −0.370636 0.928778i \(-0.620860\pi\)
−0.370636 + 0.928778i \(0.620860\pi\)
\(930\) 0 0
\(931\) 11.6174 0.380744
\(932\) 0 0
\(933\) −19.9979 −0.654703
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 31.5295 1.03002 0.515012 0.857183i \(-0.327787\pi\)
0.515012 + 0.857183i \(0.327787\pi\)
\(938\) 0 0
\(939\) −94.9913 −3.09992
\(940\) 0 0
\(941\) 59.5804 1.94227 0.971133 0.238538i \(-0.0766683\pi\)
0.971133 + 0.238538i \(0.0766683\pi\)
\(942\) 0 0
\(943\) 12.0373 0.391990
\(944\) 0 0
\(945\) 67.1211 2.18345
\(946\) 0 0
\(947\) 25.3781 0.824677 0.412339 0.911031i \(-0.364712\pi\)
0.412339 + 0.911031i \(0.364712\pi\)
\(948\) 0 0
\(949\) 0.422029 0.0136996
\(950\) 0 0
\(951\) −5.29229 −0.171614
\(952\) 0 0
\(953\) −3.13879 −0.101675 −0.0508377 0.998707i \(-0.516189\pi\)
−0.0508377 + 0.998707i \(0.516189\pi\)
\(954\) 0 0
\(955\) 5.32962 0.172463
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −41.9627 −1.35505
\(960\) 0 0
\(961\) 25.8612 0.834232
\(962\) 0 0
\(963\) 3.58910 0.115657
\(964\) 0 0
\(965\) −5.29229 −0.170365
\(966\) 0 0
\(967\) 1.66377 0.0535033 0.0267516 0.999642i \(-0.491484\pi\)
0.0267516 + 0.999642i \(0.491484\pi\)
\(968\) 0 0
\(969\) −40.2463 −1.29290
\(970\) 0 0
\(971\) 21.3013 0.683593 0.341796 0.939774i \(-0.388965\pi\)
0.341796 + 0.939774i \(0.388965\pi\)
\(972\) 0 0
\(973\) 79.9100 2.56180
\(974\) 0 0
\(975\) 1.54064 0.0493399
\(976\) 0 0
\(977\) −1.95153 −0.0624351 −0.0312176 0.999513i \(-0.509938\pi\)
−0.0312176 + 0.999513i \(0.509938\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 71.4618 2.28160
\(982\) 0 0
\(983\) 41.0065 1.30790 0.653952 0.756536i \(-0.273110\pi\)
0.653952 + 0.756536i \(0.273110\pi\)
\(984\) 0 0
\(985\) −12.1231 −0.386275
\(986\) 0 0
\(987\) −3.48604 −0.110962
\(988\) 0 0
\(989\) −32.7935 −1.04277
\(990\) 0 0
\(991\) 46.1120 1.46480 0.732398 0.680877i \(-0.238401\pi\)
0.732398 + 0.680877i \(0.238401\pi\)
\(992\) 0 0
\(993\) 60.1120 1.90760
\(994\) 0 0
\(995\) −12.3341 −0.391019
\(996\) 0 0
\(997\) 17.8354 0.564852 0.282426 0.959289i \(-0.408861\pi\)
0.282426 + 0.959289i \(0.408861\pi\)
\(998\) 0 0
\(999\) 82.8661 2.62177
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 4840.2.a.s.1.3 3
4.3 odd 2 9680.2.a.cd.1.1 3
11.10 odd 2 4840.2.a.v.1.3 yes 3
44.43 even 2 9680.2.a.by.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.s.1.3 3 1.1 even 1 trivial
4840.2.a.v.1.3 yes 3 11.10 odd 2
9680.2.a.by.1.1 3 44.43 even 2
9680.2.a.cd.1.1 3 4.3 odd 2