Properties

Label 5415.2.a.p.1.2
Level $5415$
Weight $2$
Character 5415.1
Self dual yes
Analytic conductor $43.239$
Analytic rank $1$
Dimension $2$
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] = [5415,2,Mod(1,5415)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5415, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("5415.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5415 = 3 \cdot 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5415.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: \(43.2389926945\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 285)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 5415.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} -1.00000 q^{5} +0.414214 q^{6} +0.585786 q^{7} -1.58579 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} -1.00000 q^{5} +0.414214 q^{6} +0.585786 q^{7} -1.58579 q^{8} +1.00000 q^{9} -0.414214 q^{10} +1.41421 q^{11} -1.82843 q^{12} -5.41421 q^{13} +0.242641 q^{14} -1.00000 q^{15} +3.00000 q^{16} -1.17157 q^{17} +0.414214 q^{18} +1.82843 q^{20} +0.585786 q^{21} +0.585786 q^{22} +7.65685 q^{23} -1.58579 q^{24} +1.00000 q^{25} -2.24264 q^{26} +1.00000 q^{27} -1.07107 q^{28} +9.07107 q^{29} -0.414214 q^{30} -6.48528 q^{31} +4.41421 q^{32} +1.41421 q^{33} -0.485281 q^{34} -0.585786 q^{35} -1.82843 q^{36} -11.0711 q^{37} -5.41421 q^{39} +1.58579 q^{40} +7.41421 q^{41} +0.242641 q^{42} +0.585786 q^{43} -2.58579 q^{44} -1.00000 q^{45} +3.17157 q^{46} +0.343146 q^{47} +3.00000 q^{48} -6.65685 q^{49} +0.414214 q^{50} -1.17157 q^{51} +9.89949 q^{52} -4.00000 q^{53} +0.414214 q^{54} -1.41421 q^{55} -0.928932 q^{56} +3.75736 q^{58} -8.48528 q^{59} +1.82843 q^{60} +5.65685 q^{61} -2.68629 q^{62} +0.585786 q^{63} -4.17157 q^{64} +5.41421 q^{65} +0.585786 q^{66} -12.0000 q^{67} +2.14214 q^{68} +7.65685 q^{69} -0.242641 q^{70} +4.48528 q^{71} -1.58579 q^{72} -2.00000 q^{73} -4.58579 q^{74} +1.00000 q^{75} +0.828427 q^{77} -2.24264 q^{78} +11.3137 q^{79} -3.00000 q^{80} +1.00000 q^{81} +3.07107 q^{82} -10.4853 q^{83} -1.07107 q^{84} +1.17157 q^{85} +0.242641 q^{86} +9.07107 q^{87} -2.24264 q^{88} -10.7279 q^{89} -0.414214 q^{90} -3.17157 q^{91} -14.0000 q^{92} -6.48528 q^{93} +0.142136 q^{94} +4.41421 q^{96} +4.24264 q^{97} -2.75736 q^{98} +1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 4 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 4 q^{7} - 6 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{12} - 8 q^{13} - 8 q^{14} - 2 q^{15} + 6 q^{16} - 8 q^{17} - 2 q^{18} - 2 q^{20} + 4 q^{21} + 4 q^{22} + 4 q^{23} - 6 q^{24} + 2 q^{25} + 4 q^{26} + 2 q^{27} + 12 q^{28} + 4 q^{29} + 2 q^{30} + 4 q^{31} + 6 q^{32} + 16 q^{34} - 4 q^{35} + 2 q^{36} - 8 q^{37} - 8 q^{39} + 6 q^{40} + 12 q^{41} - 8 q^{42} + 4 q^{43} - 8 q^{44} - 2 q^{45} + 12 q^{46} + 12 q^{47} + 6 q^{48} - 2 q^{49} - 2 q^{50} - 8 q^{51} - 8 q^{53} - 2 q^{54} - 16 q^{56} + 16 q^{58} - 2 q^{60} - 28 q^{62} + 4 q^{63} - 14 q^{64} + 8 q^{65} + 4 q^{66} - 24 q^{67} - 24 q^{68} + 4 q^{69} + 8 q^{70} - 8 q^{71} - 6 q^{72} - 4 q^{73} - 12 q^{74} + 2 q^{75} - 4 q^{77} + 4 q^{78} - 6 q^{80} + 2 q^{81} - 8 q^{82} - 4 q^{83} + 12 q^{84} + 8 q^{85} - 8 q^{86} + 4 q^{87} + 4 q^{88} + 4 q^{89} + 2 q^{90} - 12 q^{91} - 28 q^{92} + 4 q^{93} - 28 q^{94} + 6 q^{96} - 14 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.414214 0.292893 0.146447 0.989219i \(-0.453216\pi\)
0.146447 + 0.989219i \(0.453216\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.82843 −0.914214
\(5\) −1.00000 −0.447214
\(6\) 0.414214 0.169102
\(7\) 0.585786 0.221406 0.110703 0.993854i \(-0.464690\pi\)
0.110703 + 0.993854i \(0.464690\pi\)
\(8\) −1.58579 −0.560660
\(9\) 1.00000 0.333333
\(10\) −0.414214 −0.130986
\(11\) 1.41421 0.426401 0.213201 0.977008i \(-0.431611\pi\)
0.213201 + 0.977008i \(0.431611\pi\)
\(12\) −1.82843 −0.527821
\(13\) −5.41421 −1.50163 −0.750816 0.660511i \(-0.770340\pi\)
−0.750816 + 0.660511i \(0.770340\pi\)
\(14\) 0.242641 0.0648485
\(15\) −1.00000 −0.258199
\(16\) 3.00000 0.750000
\(17\) −1.17157 −0.284148 −0.142074 0.989856i \(-0.545377\pi\)
−0.142074 + 0.989856i \(0.545377\pi\)
\(18\) 0.414214 0.0976311
\(19\) 0 0
\(20\) 1.82843 0.408849
\(21\) 0.585786 0.127829
\(22\) 0.585786 0.124890
\(23\) 7.65685 1.59656 0.798282 0.602284i \(-0.205742\pi\)
0.798282 + 0.602284i \(0.205742\pi\)
\(24\) −1.58579 −0.323697
\(25\) 1.00000 0.200000
\(26\) −2.24264 −0.439818
\(27\) 1.00000 0.192450
\(28\) −1.07107 −0.202413
\(29\) 9.07107 1.68446 0.842228 0.539122i \(-0.181244\pi\)
0.842228 + 0.539122i \(0.181244\pi\)
\(30\) −0.414214 −0.0756247
\(31\) −6.48528 −1.16479 −0.582395 0.812906i \(-0.697884\pi\)
−0.582395 + 0.812906i \(0.697884\pi\)
\(32\) 4.41421 0.780330
\(33\) 1.41421 0.246183
\(34\) −0.485281 −0.0832251
\(35\) −0.585786 −0.0990160
\(36\) −1.82843 −0.304738
\(37\) −11.0711 −1.82007 −0.910036 0.414529i \(-0.863946\pi\)
−0.910036 + 0.414529i \(0.863946\pi\)
\(38\) 0 0
\(39\) −5.41421 −0.866968
\(40\) 1.58579 0.250735
\(41\) 7.41421 1.15791 0.578953 0.815361i \(-0.303462\pi\)
0.578953 + 0.815361i \(0.303462\pi\)
\(42\) 0.242641 0.0374403
\(43\) 0.585786 0.0893316 0.0446658 0.999002i \(-0.485778\pi\)
0.0446658 + 0.999002i \(0.485778\pi\)
\(44\) −2.58579 −0.389822
\(45\) −1.00000 −0.149071
\(46\) 3.17157 0.467623
\(47\) 0.343146 0.0500530 0.0250265 0.999687i \(-0.492033\pi\)
0.0250265 + 0.999687i \(0.492033\pi\)
\(48\) 3.00000 0.433013
\(49\) −6.65685 −0.950979
\(50\) 0.414214 0.0585786
\(51\) −1.17157 −0.164053
\(52\) 9.89949 1.37281
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 0.414214 0.0563673
\(55\) −1.41421 −0.190693
\(56\) −0.928932 −0.124134
\(57\) 0 0
\(58\) 3.75736 0.493365
\(59\) −8.48528 −1.10469 −0.552345 0.833616i \(-0.686267\pi\)
−0.552345 + 0.833616i \(0.686267\pi\)
\(60\) 1.82843 0.236049
\(61\) 5.65685 0.724286 0.362143 0.932123i \(-0.382045\pi\)
0.362143 + 0.932123i \(0.382045\pi\)
\(62\) −2.68629 −0.341159
\(63\) 0.585786 0.0738022
\(64\) −4.17157 −0.521447
\(65\) 5.41421 0.671551
\(66\) 0.585786 0.0721053
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 2.14214 0.259772
\(69\) 7.65685 0.921777
\(70\) −0.242641 −0.0290011
\(71\) 4.48528 0.532305 0.266152 0.963931i \(-0.414248\pi\)
0.266152 + 0.963931i \(0.414248\pi\)
\(72\) −1.58579 −0.186887
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −4.58579 −0.533087
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 0.828427 0.0944080
\(78\) −2.24264 −0.253929
\(79\) 11.3137 1.27289 0.636446 0.771321i \(-0.280404\pi\)
0.636446 + 0.771321i \(0.280404\pi\)
\(80\) −3.00000 −0.335410
\(81\) 1.00000 0.111111
\(82\) 3.07107 0.339143
\(83\) −10.4853 −1.15091 −0.575455 0.817834i \(-0.695175\pi\)
−0.575455 + 0.817834i \(0.695175\pi\)
\(84\) −1.07107 −0.116863
\(85\) 1.17157 0.127075
\(86\) 0.242641 0.0261646
\(87\) 9.07107 0.972521
\(88\) −2.24264 −0.239066
\(89\) −10.7279 −1.13716 −0.568579 0.822629i \(-0.692507\pi\)
−0.568579 + 0.822629i \(0.692507\pi\)
\(90\) −0.414214 −0.0436619
\(91\) −3.17157 −0.332471
\(92\) −14.0000 −1.45960
\(93\) −6.48528 −0.672492
\(94\) 0.142136 0.0146602
\(95\) 0 0
\(96\) 4.41421 0.450524
\(97\) 4.24264 0.430775 0.215387 0.976529i \(-0.430899\pi\)
0.215387 + 0.976529i \(0.430899\pi\)
\(98\) −2.75736 −0.278535
\(99\) 1.41421 0.142134
\(100\) −1.82843 −0.182843
\(101\) −4.82843 −0.480446 −0.240223 0.970718i \(-0.577221\pi\)
−0.240223 + 0.970718i \(0.577221\pi\)
\(102\) −0.485281 −0.0480500
\(103\) 1.65685 0.163255 0.0816274 0.996663i \(-0.473988\pi\)
0.0816274 + 0.996663i \(0.473988\pi\)
\(104\) 8.58579 0.841906
\(105\) −0.585786 −0.0571669
\(106\) −1.65685 −0.160928
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) −1.82843 −0.175940
\(109\) −3.17157 −0.303782 −0.151891 0.988397i \(-0.548536\pi\)
−0.151891 + 0.988397i \(0.548536\pi\)
\(110\) −0.585786 −0.0558525
\(111\) −11.0711 −1.05082
\(112\) 1.75736 0.166055
\(113\) −12.4853 −1.17452 −0.587258 0.809400i \(-0.699793\pi\)
−0.587258 + 0.809400i \(0.699793\pi\)
\(114\) 0 0
\(115\) −7.65685 −0.714005
\(116\) −16.5858 −1.53995
\(117\) −5.41421 −0.500544
\(118\) −3.51472 −0.323556
\(119\) −0.686292 −0.0629122
\(120\) 1.58579 0.144762
\(121\) −9.00000 −0.818182
\(122\) 2.34315 0.212138
\(123\) 7.41421 0.668517
\(124\) 11.8579 1.06487
\(125\) −1.00000 −0.0894427
\(126\) 0.242641 0.0216162
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) −10.5563 −0.933058
\(129\) 0.585786 0.0515756
\(130\) 2.24264 0.196693
\(131\) 16.7279 1.46153 0.730763 0.682632i \(-0.239165\pi\)
0.730763 + 0.682632i \(0.239165\pi\)
\(132\) −2.58579 −0.225064
\(133\) 0 0
\(134\) −4.97056 −0.429391
\(135\) −1.00000 −0.0860663
\(136\) 1.85786 0.159311
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 3.17157 0.269982
\(139\) 1.17157 0.0993715 0.0496858 0.998765i \(-0.484178\pi\)
0.0496858 + 0.998765i \(0.484178\pi\)
\(140\) 1.07107 0.0905218
\(141\) 0.343146 0.0288981
\(142\) 1.85786 0.155909
\(143\) −7.65685 −0.640298
\(144\) 3.00000 0.250000
\(145\) −9.07107 −0.753311
\(146\) −0.828427 −0.0685611
\(147\) −6.65685 −0.549048
\(148\) 20.2426 1.66393
\(149\) −3.65685 −0.299581 −0.149791 0.988718i \(-0.547860\pi\)
−0.149791 + 0.988718i \(0.547860\pi\)
\(150\) 0.414214 0.0338204
\(151\) −17.7990 −1.44846 −0.724231 0.689558i \(-0.757805\pi\)
−0.724231 + 0.689558i \(0.757805\pi\)
\(152\) 0 0
\(153\) −1.17157 −0.0947161
\(154\) 0.343146 0.0276515
\(155\) 6.48528 0.520910
\(156\) 9.89949 0.792594
\(157\) −21.7990 −1.73975 −0.869874 0.493273i \(-0.835800\pi\)
−0.869874 + 0.493273i \(0.835800\pi\)
\(158\) 4.68629 0.372821
\(159\) −4.00000 −0.317221
\(160\) −4.41421 −0.348974
\(161\) 4.48528 0.353490
\(162\) 0.414214 0.0325437
\(163\) 7.89949 0.618736 0.309368 0.950942i \(-0.399882\pi\)
0.309368 + 0.950942i \(0.399882\pi\)
\(164\) −13.5563 −1.05857
\(165\) −1.41421 −0.110096
\(166\) −4.34315 −0.337093
\(167\) −10.0000 −0.773823 −0.386912 0.922117i \(-0.626458\pi\)
−0.386912 + 0.922117i \(0.626458\pi\)
\(168\) −0.928932 −0.0716687
\(169\) 16.3137 1.25490
\(170\) 0.485281 0.0372194
\(171\) 0 0
\(172\) −1.07107 −0.0816682
\(173\) 6.14214 0.466978 0.233489 0.972359i \(-0.424986\pi\)
0.233489 + 0.972359i \(0.424986\pi\)
\(174\) 3.75736 0.284845
\(175\) 0.585786 0.0442813
\(176\) 4.24264 0.319801
\(177\) −8.48528 −0.637793
\(178\) −4.44365 −0.333066
\(179\) −17.1716 −1.28346 −0.641732 0.766929i \(-0.721784\pi\)
−0.641732 + 0.766929i \(0.721784\pi\)
\(180\) 1.82843 0.136283
\(181\) 19.1716 1.42501 0.712506 0.701666i \(-0.247560\pi\)
0.712506 + 0.701666i \(0.247560\pi\)
\(182\) −1.31371 −0.0973786
\(183\) 5.65685 0.418167
\(184\) −12.1421 −0.895130
\(185\) 11.0711 0.813961
\(186\) −2.68629 −0.196968
\(187\) −1.65685 −0.121161
\(188\) −0.627417 −0.0457591
\(189\) 0.585786 0.0426097
\(190\) 0 0
\(191\) 1.89949 0.137443 0.0687213 0.997636i \(-0.478108\pi\)
0.0687213 + 0.997636i \(0.478108\pi\)
\(192\) −4.17157 −0.301057
\(193\) 15.0711 1.08484 0.542420 0.840108i \(-0.317508\pi\)
0.542420 + 0.840108i \(0.317508\pi\)
\(194\) 1.75736 0.126171
\(195\) 5.41421 0.387720
\(196\) 12.1716 0.869398
\(197\) −14.8284 −1.05648 −0.528241 0.849095i \(-0.677148\pi\)
−0.528241 + 0.849095i \(0.677148\pi\)
\(198\) 0.585786 0.0416300
\(199\) −16.4853 −1.16861 −0.584305 0.811534i \(-0.698633\pi\)
−0.584305 + 0.811534i \(0.698633\pi\)
\(200\) −1.58579 −0.112132
\(201\) −12.0000 −0.846415
\(202\) −2.00000 −0.140720
\(203\) 5.31371 0.372949
\(204\) 2.14214 0.149979
\(205\) −7.41421 −0.517831
\(206\) 0.686292 0.0478162
\(207\) 7.65685 0.532188
\(208\) −16.2426 −1.12622
\(209\) 0 0
\(210\) −0.242641 −0.0167438
\(211\) 15.3137 1.05424 0.527120 0.849791i \(-0.323272\pi\)
0.527120 + 0.849791i \(0.323272\pi\)
\(212\) 7.31371 0.502308
\(213\) 4.48528 0.307326
\(214\) −3.31371 −0.226520
\(215\) −0.585786 −0.0399503
\(216\) −1.58579 −0.107899
\(217\) −3.79899 −0.257892
\(218\) −1.31371 −0.0889756
\(219\) −2.00000 −0.135147
\(220\) 2.58579 0.174334
\(221\) 6.34315 0.426686
\(222\) −4.58579 −0.307778
\(223\) −6.34315 −0.424768 −0.212384 0.977186i \(-0.568123\pi\)
−0.212384 + 0.977186i \(0.568123\pi\)
\(224\) 2.58579 0.172770
\(225\) 1.00000 0.0666667
\(226\) −5.17157 −0.344008
\(227\) 18.9706 1.25912 0.629560 0.776952i \(-0.283235\pi\)
0.629560 + 0.776952i \(0.283235\pi\)
\(228\) 0 0
\(229\) −1.65685 −0.109488 −0.0547440 0.998500i \(-0.517434\pi\)
−0.0547440 + 0.998500i \(0.517434\pi\)
\(230\) −3.17157 −0.209127
\(231\) 0.828427 0.0545065
\(232\) −14.3848 −0.944407
\(233\) −8.34315 −0.546578 −0.273289 0.961932i \(-0.588111\pi\)
−0.273289 + 0.961932i \(0.588111\pi\)
\(234\) −2.24264 −0.146606
\(235\) −0.343146 −0.0223844
\(236\) 15.5147 1.00992
\(237\) 11.3137 0.734904
\(238\) −0.284271 −0.0184266
\(239\) 2.58579 0.167261 0.0836303 0.996497i \(-0.473349\pi\)
0.0836303 + 0.996497i \(0.473349\pi\)
\(240\) −3.00000 −0.193649
\(241\) 14.9706 0.964339 0.482169 0.876078i \(-0.339849\pi\)
0.482169 + 0.876078i \(0.339849\pi\)
\(242\) −3.72792 −0.239640
\(243\) 1.00000 0.0641500
\(244\) −10.3431 −0.662152
\(245\) 6.65685 0.425291
\(246\) 3.07107 0.195804
\(247\) 0 0
\(248\) 10.2843 0.653052
\(249\) −10.4853 −0.664478
\(250\) −0.414214 −0.0261972
\(251\) 12.9289 0.816067 0.408033 0.912967i \(-0.366215\pi\)
0.408033 + 0.912967i \(0.366215\pi\)
\(252\) −1.07107 −0.0674709
\(253\) 10.8284 0.680777
\(254\) −3.31371 −0.207921
\(255\) 1.17157 0.0733667
\(256\) 3.97056 0.248160
\(257\) −1.17157 −0.0730807 −0.0365404 0.999332i \(-0.511634\pi\)
−0.0365404 + 0.999332i \(0.511634\pi\)
\(258\) 0.242641 0.0151061
\(259\) −6.48528 −0.402976
\(260\) −9.89949 −0.613941
\(261\) 9.07107 0.561485
\(262\) 6.92893 0.428071
\(263\) −32.1421 −1.98197 −0.990984 0.133977i \(-0.957225\pi\)
−0.990984 + 0.133977i \(0.957225\pi\)
\(264\) −2.24264 −0.138025
\(265\) 4.00000 0.245718
\(266\) 0 0
\(267\) −10.7279 −0.656538
\(268\) 21.9411 1.34027
\(269\) −8.38478 −0.511229 −0.255614 0.966779i \(-0.582278\pi\)
−0.255614 + 0.966779i \(0.582278\pi\)
\(270\) −0.414214 −0.0252082
\(271\) −30.8284 −1.87269 −0.936347 0.351076i \(-0.885816\pi\)
−0.936347 + 0.351076i \(0.885816\pi\)
\(272\) −3.51472 −0.213111
\(273\) −3.17157 −0.191952
\(274\) −5.79899 −0.350330
\(275\) 1.41421 0.0852803
\(276\) −14.0000 −0.842701
\(277\) 10.9706 0.659157 0.329579 0.944128i \(-0.393093\pi\)
0.329579 + 0.944128i \(0.393093\pi\)
\(278\) 0.485281 0.0291052
\(279\) −6.48528 −0.388264
\(280\) 0.928932 0.0555143
\(281\) 14.7279 0.878594 0.439297 0.898342i \(-0.355228\pi\)
0.439297 + 0.898342i \(0.355228\pi\)
\(282\) 0.142136 0.00846405
\(283\) 6.24264 0.371086 0.185543 0.982636i \(-0.440596\pi\)
0.185543 + 0.982636i \(0.440596\pi\)
\(284\) −8.20101 −0.486640
\(285\) 0 0
\(286\) −3.17157 −0.187539
\(287\) 4.34315 0.256368
\(288\) 4.41421 0.260110
\(289\) −15.6274 −0.919260
\(290\) −3.75736 −0.220640
\(291\) 4.24264 0.248708
\(292\) 3.65685 0.214001
\(293\) 31.7990 1.85772 0.928858 0.370435i \(-0.120791\pi\)
0.928858 + 0.370435i \(0.120791\pi\)
\(294\) −2.75736 −0.160812
\(295\) 8.48528 0.494032
\(296\) 17.5563 1.02044
\(297\) 1.41421 0.0820610
\(298\) −1.51472 −0.0877453
\(299\) −41.4558 −2.39745
\(300\) −1.82843 −0.105564
\(301\) 0.343146 0.0197786
\(302\) −7.37258 −0.424244
\(303\) −4.82843 −0.277386
\(304\) 0 0
\(305\) −5.65685 −0.323911
\(306\) −0.485281 −0.0277417
\(307\) −7.79899 −0.445112 −0.222556 0.974920i \(-0.571440\pi\)
−0.222556 + 0.974920i \(0.571440\pi\)
\(308\) −1.51472 −0.0863091
\(309\) 1.65685 0.0942551
\(310\) 2.68629 0.152571
\(311\) −32.2426 −1.82831 −0.914156 0.405362i \(-0.867145\pi\)
−0.914156 + 0.405362i \(0.867145\pi\)
\(312\) 8.58579 0.486074
\(313\) −9.51472 −0.537804 −0.268902 0.963168i \(-0.586661\pi\)
−0.268902 + 0.963168i \(0.586661\pi\)
\(314\) −9.02944 −0.509561
\(315\) −0.585786 −0.0330053
\(316\) −20.6863 −1.16369
\(317\) −11.3137 −0.635441 −0.317721 0.948184i \(-0.602917\pi\)
−0.317721 + 0.948184i \(0.602917\pi\)
\(318\) −1.65685 −0.0929118
\(319\) 12.8284 0.718254
\(320\) 4.17157 0.233198
\(321\) −8.00000 −0.446516
\(322\) 1.85786 0.103535
\(323\) 0 0
\(324\) −1.82843 −0.101579
\(325\) −5.41421 −0.300327
\(326\) 3.27208 0.181224
\(327\) −3.17157 −0.175388
\(328\) −11.7574 −0.649192
\(329\) 0.201010 0.0110820
\(330\) −0.585786 −0.0322465
\(331\) −7.17157 −0.394185 −0.197093 0.980385i \(-0.563150\pi\)
−0.197093 + 0.980385i \(0.563150\pi\)
\(332\) 19.1716 1.05218
\(333\) −11.0711 −0.606691
\(334\) −4.14214 −0.226648
\(335\) 12.0000 0.655630
\(336\) 1.75736 0.0958718
\(337\) −28.2426 −1.53847 −0.769237 0.638963i \(-0.779364\pi\)
−0.769237 + 0.638963i \(0.779364\pi\)
\(338\) 6.75736 0.367552
\(339\) −12.4853 −0.678107
\(340\) −2.14214 −0.116174
\(341\) −9.17157 −0.496669
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) −0.928932 −0.0500847
\(345\) −7.65685 −0.412231
\(346\) 2.54416 0.136775
\(347\) 10.4853 0.562879 0.281440 0.959579i \(-0.409188\pi\)
0.281440 + 0.959579i \(0.409188\pi\)
\(348\) −16.5858 −0.889091
\(349\) 29.3137 1.56913 0.784563 0.620049i \(-0.212887\pi\)
0.784563 + 0.620049i \(0.212887\pi\)
\(350\) 0.242641 0.0129697
\(351\) −5.41421 −0.288989
\(352\) 6.24264 0.332734
\(353\) 3.65685 0.194635 0.0973174 0.995253i \(-0.468974\pi\)
0.0973174 + 0.995253i \(0.468974\pi\)
\(354\) −3.51472 −0.186805
\(355\) −4.48528 −0.238054
\(356\) 19.6152 1.03960
\(357\) −0.686292 −0.0363224
\(358\) −7.11270 −0.375918
\(359\) 9.89949 0.522475 0.261238 0.965275i \(-0.415869\pi\)
0.261238 + 0.965275i \(0.415869\pi\)
\(360\) 1.58579 0.0835783
\(361\) 0 0
\(362\) 7.94113 0.417376
\(363\) −9.00000 −0.472377
\(364\) 5.79899 0.303950
\(365\) 2.00000 0.104685
\(366\) 2.34315 0.122478
\(367\) −19.4142 −1.01341 −0.506707 0.862118i \(-0.669137\pi\)
−0.506707 + 0.862118i \(0.669137\pi\)
\(368\) 22.9706 1.19742
\(369\) 7.41421 0.385969
\(370\) 4.58579 0.238404
\(371\) −2.34315 −0.121650
\(372\) 11.8579 0.614802
\(373\) 9.89949 0.512576 0.256288 0.966600i \(-0.417500\pi\)
0.256288 + 0.966600i \(0.417500\pi\)
\(374\) −0.686292 −0.0354873
\(375\) −1.00000 −0.0516398
\(376\) −0.544156 −0.0280627
\(377\) −49.1127 −2.52943
\(378\) 0.242641 0.0124801
\(379\) −24.1421 −1.24010 −0.620049 0.784563i \(-0.712887\pi\)
−0.620049 + 0.784563i \(0.712887\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 0.786797 0.0402560
\(383\) −12.0000 −0.613171 −0.306586 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) −10.5563 −0.538701
\(385\) −0.828427 −0.0422206
\(386\) 6.24264 0.317742
\(387\) 0.585786 0.0297772
\(388\) −7.75736 −0.393820
\(389\) 14.9706 0.759038 0.379519 0.925184i \(-0.376090\pi\)
0.379519 + 0.925184i \(0.376090\pi\)
\(390\) 2.24264 0.113561
\(391\) −8.97056 −0.453661
\(392\) 10.5563 0.533176
\(393\) 16.7279 0.843812
\(394\) −6.14214 −0.309436
\(395\) −11.3137 −0.569254
\(396\) −2.58579 −0.129941
\(397\) 35.6569 1.78957 0.894783 0.446501i \(-0.147330\pi\)
0.894783 + 0.446501i \(0.147330\pi\)
\(398\) −6.82843 −0.342278
\(399\) 0 0
\(400\) 3.00000 0.150000
\(401\) −30.0416 −1.50021 −0.750104 0.661320i \(-0.769996\pi\)
−0.750104 + 0.661320i \(0.769996\pi\)
\(402\) −4.97056 −0.247909
\(403\) 35.1127 1.74909
\(404\) 8.82843 0.439231
\(405\) −1.00000 −0.0496904
\(406\) 2.20101 0.109234
\(407\) −15.6569 −0.776081
\(408\) 1.85786 0.0919780
\(409\) −9.51472 −0.470473 −0.235236 0.971938i \(-0.575586\pi\)
−0.235236 + 0.971938i \(0.575586\pi\)
\(410\) −3.07107 −0.151669
\(411\) −14.0000 −0.690569
\(412\) −3.02944 −0.149250
\(413\) −4.97056 −0.244585
\(414\) 3.17157 0.155874
\(415\) 10.4853 0.514702
\(416\) −23.8995 −1.17177
\(417\) 1.17157 0.0573722
\(418\) 0 0
\(419\) −34.8701 −1.70351 −0.851757 0.523937i \(-0.824463\pi\)
−0.851757 + 0.523937i \(0.824463\pi\)
\(420\) 1.07107 0.0522628
\(421\) 14.6863 0.715766 0.357883 0.933766i \(-0.383499\pi\)
0.357883 + 0.933766i \(0.383499\pi\)
\(422\) 6.34315 0.308780
\(423\) 0.343146 0.0166843
\(424\) 6.34315 0.308050
\(425\) −1.17157 −0.0568296
\(426\) 1.85786 0.0900138
\(427\) 3.31371 0.160362
\(428\) 14.6274 0.707043
\(429\) −7.65685 −0.369676
\(430\) −0.242641 −0.0117012
\(431\) −3.51472 −0.169298 −0.0846490 0.996411i \(-0.526977\pi\)
−0.0846490 + 0.996411i \(0.526977\pi\)
\(432\) 3.00000 0.144338
\(433\) −0.928932 −0.0446416 −0.0223208 0.999751i \(-0.507106\pi\)
−0.0223208 + 0.999751i \(0.507106\pi\)
\(434\) −1.57359 −0.0755349
\(435\) −9.07107 −0.434924
\(436\) 5.79899 0.277721
\(437\) 0 0
\(438\) −0.828427 −0.0395838
\(439\) −0.970563 −0.0463224 −0.0231612 0.999732i \(-0.507373\pi\)
−0.0231612 + 0.999732i \(0.507373\pi\)
\(440\) 2.24264 0.106914
\(441\) −6.65685 −0.316993
\(442\) 2.62742 0.124973
\(443\) −1.31371 −0.0624162 −0.0312081 0.999513i \(-0.509935\pi\)
−0.0312081 + 0.999513i \(0.509935\pi\)
\(444\) 20.2426 0.960673
\(445\) 10.7279 0.508552
\(446\) −2.62742 −0.124412
\(447\) −3.65685 −0.172963
\(448\) −2.44365 −0.115452
\(449\) −7.89949 −0.372800 −0.186400 0.982474i \(-0.559682\pi\)
−0.186400 + 0.982474i \(0.559682\pi\)
\(450\) 0.414214 0.0195262
\(451\) 10.4853 0.493733
\(452\) 22.8284 1.07376
\(453\) −17.7990 −0.836269
\(454\) 7.85786 0.368788
\(455\) 3.17157 0.148686
\(456\) 0 0
\(457\) 28.8284 1.34854 0.674268 0.738486i \(-0.264459\pi\)
0.674268 + 0.738486i \(0.264459\pi\)
\(458\) −0.686292 −0.0320683
\(459\) −1.17157 −0.0546843
\(460\) 14.0000 0.652753
\(461\) 10.6863 0.497710 0.248855 0.968541i \(-0.419946\pi\)
0.248855 + 0.968541i \(0.419946\pi\)
\(462\) 0.343146 0.0159646
\(463\) −8.10051 −0.376462 −0.188231 0.982125i \(-0.560275\pi\)
−0.188231 + 0.982125i \(0.560275\pi\)
\(464\) 27.2132 1.26334
\(465\) 6.48528 0.300748
\(466\) −3.45584 −0.160089
\(467\) 16.3431 0.756271 0.378135 0.925750i \(-0.376565\pi\)
0.378135 + 0.925750i \(0.376565\pi\)
\(468\) 9.89949 0.457604
\(469\) −7.02944 −0.324589
\(470\) −0.142136 −0.00655623
\(471\) −21.7990 −1.00444
\(472\) 13.4558 0.619355
\(473\) 0.828427 0.0380911
\(474\) 4.68629 0.215248
\(475\) 0 0
\(476\) 1.25483 0.0575152
\(477\) −4.00000 −0.183147
\(478\) 1.07107 0.0489895
\(479\) −38.1838 −1.74466 −0.872330 0.488917i \(-0.837392\pi\)
−0.872330 + 0.488917i \(0.837392\pi\)
\(480\) −4.41421 −0.201480
\(481\) 59.9411 2.73308
\(482\) 6.20101 0.282448
\(483\) 4.48528 0.204087
\(484\) 16.4558 0.747993
\(485\) −4.24264 −0.192648
\(486\) 0.414214 0.0187891
\(487\) 2.82843 0.128168 0.0640841 0.997944i \(-0.479587\pi\)
0.0640841 + 0.997944i \(0.479587\pi\)
\(488\) −8.97056 −0.406078
\(489\) 7.89949 0.357228
\(490\) 2.75736 0.124565
\(491\) 21.8995 0.988310 0.494155 0.869374i \(-0.335477\pi\)
0.494155 + 0.869374i \(0.335477\pi\)
\(492\) −13.5563 −0.611167
\(493\) −10.6274 −0.478635
\(494\) 0 0
\(495\) −1.41421 −0.0635642
\(496\) −19.4558 −0.873593
\(497\) 2.62742 0.117856
\(498\) −4.34315 −0.194621
\(499\) −23.7990 −1.06539 −0.532695 0.846308i \(-0.678821\pi\)
−0.532695 + 0.846308i \(0.678821\pi\)
\(500\) 1.82843 0.0817697
\(501\) −10.0000 −0.446767
\(502\) 5.35534 0.239020
\(503\) 0.828427 0.0369377 0.0184689 0.999829i \(-0.494121\pi\)
0.0184689 + 0.999829i \(0.494121\pi\)
\(504\) −0.928932 −0.0413779
\(505\) 4.82843 0.214862
\(506\) 4.48528 0.199395
\(507\) 16.3137 0.724517
\(508\) 14.6274 0.648987
\(509\) −32.3848 −1.43543 −0.717715 0.696337i \(-0.754812\pi\)
−0.717715 + 0.696337i \(0.754812\pi\)
\(510\) 0.485281 0.0214886
\(511\) −1.17157 −0.0518273
\(512\) 22.7574 1.00574
\(513\) 0 0
\(514\) −0.485281 −0.0214048
\(515\) −1.65685 −0.0730097
\(516\) −1.07107 −0.0471511
\(517\) 0.485281 0.0213427
\(518\) −2.68629 −0.118029
\(519\) 6.14214 0.269610
\(520\) −8.58579 −0.376512
\(521\) 24.3848 1.06832 0.534158 0.845385i \(-0.320629\pi\)
0.534158 + 0.845385i \(0.320629\pi\)
\(522\) 3.75736 0.164455
\(523\) 15.7990 0.690842 0.345421 0.938448i \(-0.387736\pi\)
0.345421 + 0.938448i \(0.387736\pi\)
\(524\) −30.5858 −1.33615
\(525\) 0.585786 0.0255658
\(526\) −13.3137 −0.580505
\(527\) 7.59798 0.330973
\(528\) 4.24264 0.184637
\(529\) 35.6274 1.54902
\(530\) 1.65685 0.0719691
\(531\) −8.48528 −0.368230
\(532\) 0 0
\(533\) −40.1421 −1.73875
\(534\) −4.44365 −0.192296
\(535\) 8.00000 0.345870
\(536\) 19.0294 0.821946
\(537\) −17.1716 −0.741008
\(538\) −3.47309 −0.149735
\(539\) −9.41421 −0.405499
\(540\) 1.82843 0.0786830
\(541\) −32.6274 −1.40276 −0.701381 0.712786i \(-0.747433\pi\)
−0.701381 + 0.712786i \(0.747433\pi\)
\(542\) −12.7696 −0.548499
\(543\) 19.1716 0.822731
\(544\) −5.17157 −0.221729
\(545\) 3.17157 0.135855
\(546\) −1.31371 −0.0562215
\(547\) 34.1421 1.45981 0.729906 0.683547i \(-0.239564\pi\)
0.729906 + 0.683547i \(0.239564\pi\)
\(548\) 25.5980 1.09349
\(549\) 5.65685 0.241429
\(550\) 0.585786 0.0249780
\(551\) 0 0
\(552\) −12.1421 −0.516804
\(553\) 6.62742 0.281826
\(554\) 4.54416 0.193063
\(555\) 11.0711 0.469941
\(556\) −2.14214 −0.0908468
\(557\) 10.0000 0.423714 0.211857 0.977301i \(-0.432049\pi\)
0.211857 + 0.977301i \(0.432049\pi\)
\(558\) −2.68629 −0.113720
\(559\) −3.17157 −0.134143
\(560\) −1.75736 −0.0742620
\(561\) −1.65685 −0.0699524
\(562\) 6.10051 0.257334
\(563\) −42.2843 −1.78207 −0.891035 0.453935i \(-0.850020\pi\)
−0.891035 + 0.453935i \(0.850020\pi\)
\(564\) −0.627417 −0.0264190
\(565\) 12.4853 0.525260
\(566\) 2.58579 0.108689
\(567\) 0.585786 0.0246007
\(568\) −7.11270 −0.298442
\(569\) 6.72792 0.282049 0.141025 0.990006i \(-0.454960\pi\)
0.141025 + 0.990006i \(0.454960\pi\)
\(570\) 0 0
\(571\) 19.7990 0.828562 0.414281 0.910149i \(-0.364033\pi\)
0.414281 + 0.910149i \(0.364033\pi\)
\(572\) 14.0000 0.585369
\(573\) 1.89949 0.0793525
\(574\) 1.79899 0.0750884
\(575\) 7.65685 0.319313
\(576\) −4.17157 −0.173816
\(577\) −37.7990 −1.57359 −0.786796 0.617213i \(-0.788262\pi\)
−0.786796 + 0.617213i \(0.788262\pi\)
\(578\) −6.47309 −0.269245
\(579\) 15.0711 0.626332
\(580\) 16.5858 0.688687
\(581\) −6.14214 −0.254819
\(582\) 1.75736 0.0728449
\(583\) −5.65685 −0.234283
\(584\) 3.17157 0.131241
\(585\) 5.41421 0.223850
\(586\) 13.1716 0.544113
\(587\) 11.6569 0.481130 0.240565 0.970633i \(-0.422667\pi\)
0.240565 + 0.970633i \(0.422667\pi\)
\(588\) 12.1716 0.501947
\(589\) 0 0
\(590\) 3.51472 0.144699
\(591\) −14.8284 −0.609960
\(592\) −33.2132 −1.36505
\(593\) −29.3137 −1.20377 −0.601885 0.798583i \(-0.705583\pi\)
−0.601885 + 0.798583i \(0.705583\pi\)
\(594\) 0.585786 0.0240351
\(595\) 0.686292 0.0281352
\(596\) 6.68629 0.273881
\(597\) −16.4853 −0.674698
\(598\) −17.1716 −0.702198
\(599\) 21.9411 0.896490 0.448245 0.893911i \(-0.352049\pi\)
0.448245 + 0.893911i \(0.352049\pi\)
\(600\) −1.58579 −0.0647395
\(601\) 28.8284 1.17594 0.587968 0.808884i \(-0.299928\pi\)
0.587968 + 0.808884i \(0.299928\pi\)
\(602\) 0.142136 0.00579302
\(603\) −12.0000 −0.488678
\(604\) 32.5442 1.32420
\(605\) 9.00000 0.365902
\(606\) −2.00000 −0.0812444
\(607\) 2.14214 0.0869466 0.0434733 0.999055i \(-0.486158\pi\)
0.0434733 + 0.999055i \(0.486158\pi\)
\(608\) 0 0
\(609\) 5.31371 0.215322
\(610\) −2.34315 −0.0948712
\(611\) −1.85786 −0.0751611
\(612\) 2.14214 0.0865907
\(613\) 25.1127 1.01429 0.507146 0.861860i \(-0.330700\pi\)
0.507146 + 0.861860i \(0.330700\pi\)
\(614\) −3.23045 −0.130370
\(615\) −7.41421 −0.298970
\(616\) −1.31371 −0.0529308
\(617\) 10.1421 0.408307 0.204154 0.978939i \(-0.434556\pi\)
0.204154 + 0.978939i \(0.434556\pi\)
\(618\) 0.686292 0.0276067
\(619\) −43.7990 −1.76043 −0.880215 0.474575i \(-0.842602\pi\)
−0.880215 + 0.474575i \(0.842602\pi\)
\(620\) −11.8579 −0.476223
\(621\) 7.65685 0.307259
\(622\) −13.3553 −0.535500
\(623\) −6.28427 −0.251774
\(624\) −16.2426 −0.650226
\(625\) 1.00000 0.0400000
\(626\) −3.94113 −0.157519
\(627\) 0 0
\(628\) 39.8579 1.59050
\(629\) 12.9706 0.517170
\(630\) −0.242641 −0.00966704
\(631\) 22.6274 0.900783 0.450392 0.892831i \(-0.351284\pi\)
0.450392 + 0.892831i \(0.351284\pi\)
\(632\) −17.9411 −0.713660
\(633\) 15.3137 0.608665
\(634\) −4.68629 −0.186116
\(635\) 8.00000 0.317470
\(636\) 7.31371 0.290007
\(637\) 36.0416 1.42802
\(638\) 5.31371 0.210372
\(639\) 4.48528 0.177435
\(640\) 10.5563 0.417276
\(641\) 8.58579 0.339118 0.169559 0.985520i \(-0.445766\pi\)
0.169559 + 0.985520i \(0.445766\pi\)
\(642\) −3.31371 −0.130782
\(643\) 6.04163 0.238259 0.119129 0.992879i \(-0.461990\pi\)
0.119129 + 0.992879i \(0.461990\pi\)
\(644\) −8.20101 −0.323165
\(645\) −0.585786 −0.0230653
\(646\) 0 0
\(647\) 45.1127 1.77356 0.886782 0.462189i \(-0.152936\pi\)
0.886782 + 0.462189i \(0.152936\pi\)
\(648\) −1.58579 −0.0622956
\(649\) −12.0000 −0.471041
\(650\) −2.24264 −0.0879636
\(651\) −3.79899 −0.148894
\(652\) −14.4437 −0.565657
\(653\) 42.4264 1.66027 0.830137 0.557560i \(-0.188262\pi\)
0.830137 + 0.557560i \(0.188262\pi\)
\(654\) −1.31371 −0.0513701
\(655\) −16.7279 −0.653614
\(656\) 22.2426 0.868429
\(657\) −2.00000 −0.0780274
\(658\) 0.0832611 0.00324586
\(659\) 20.0000 0.779089 0.389545 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 2.58579 0.100652
\(661\) −18.4853 −0.718994 −0.359497 0.933146i \(-0.617052\pi\)
−0.359497 + 0.933146i \(0.617052\pi\)
\(662\) −2.97056 −0.115454
\(663\) 6.34315 0.246347
\(664\) 16.6274 0.645269
\(665\) 0 0
\(666\) −4.58579 −0.177696
\(667\) 69.4558 2.68934
\(668\) 18.2843 0.707440
\(669\) −6.34315 −0.245240
\(670\) 4.97056 0.192030
\(671\) 8.00000 0.308837
\(672\) 2.58579 0.0997489
\(673\) −21.8995 −0.844163 −0.422082 0.906558i \(-0.638700\pi\)
−0.422082 + 0.906558i \(0.638700\pi\)
\(674\) −11.6985 −0.450609
\(675\) 1.00000 0.0384900
\(676\) −29.8284 −1.14725
\(677\) 44.9706 1.72836 0.864180 0.503184i \(-0.167838\pi\)
0.864180 + 0.503184i \(0.167838\pi\)
\(678\) −5.17157 −0.198613
\(679\) 2.48528 0.0953763
\(680\) −1.85786 −0.0712458
\(681\) 18.9706 0.726954
\(682\) −3.79899 −0.145471
\(683\) −5.65685 −0.216454 −0.108227 0.994126i \(-0.534517\pi\)
−0.108227 + 0.994126i \(0.534517\pi\)
\(684\) 0 0
\(685\) 14.0000 0.534913
\(686\) −3.31371 −0.126518
\(687\) −1.65685 −0.0632129
\(688\) 1.75736 0.0669987
\(689\) 21.6569 0.825060
\(690\) −3.17157 −0.120740
\(691\) −23.1127 −0.879248 −0.439624 0.898182i \(-0.644888\pi\)
−0.439624 + 0.898182i \(0.644888\pi\)
\(692\) −11.2304 −0.426918
\(693\) 0.828427 0.0314693
\(694\) 4.34315 0.164864
\(695\) −1.17157 −0.0444403
\(696\) −14.3848 −0.545254
\(697\) −8.68629 −0.329017
\(698\) 12.1421 0.459587
\(699\) −8.34315 −0.315567
\(700\) −1.07107 −0.0404826
\(701\) −0.343146 −0.0129604 −0.00648022 0.999979i \(-0.502063\pi\)
−0.00648022 + 0.999979i \(0.502063\pi\)
\(702\) −2.24264 −0.0846430
\(703\) 0 0
\(704\) −5.89949 −0.222346
\(705\) −0.343146 −0.0129236
\(706\) 1.51472 0.0570072
\(707\) −2.82843 −0.106374
\(708\) 15.5147 0.583079
\(709\) −35.3137 −1.32623 −0.663117 0.748516i \(-0.730767\pi\)
−0.663117 + 0.748516i \(0.730767\pi\)
\(710\) −1.85786 −0.0697244
\(711\) 11.3137 0.424297
\(712\) 17.0122 0.637559
\(713\) −49.6569 −1.85966
\(714\) −0.284271 −0.0106386
\(715\) 7.65685 0.286350
\(716\) 31.3970 1.17336
\(717\) 2.58579 0.0965680
\(718\) 4.10051 0.153029
\(719\) −16.4437 −0.613245 −0.306622 0.951831i \(-0.599199\pi\)
−0.306622 + 0.951831i \(0.599199\pi\)
\(720\) −3.00000 −0.111803
\(721\) 0.970563 0.0361456
\(722\) 0 0
\(723\) 14.9706 0.556761
\(724\) −35.0538 −1.30277
\(725\) 9.07107 0.336891
\(726\) −3.72792 −0.138356
\(727\) −4.58579 −0.170077 −0.0850387 0.996378i \(-0.527101\pi\)
−0.0850387 + 0.996378i \(0.527101\pi\)
\(728\) 5.02944 0.186403
\(729\) 1.00000 0.0370370
\(730\) 0.828427 0.0306615
\(731\) −0.686292 −0.0253834
\(732\) −10.3431 −0.382294
\(733\) −1.31371 −0.0485229 −0.0242615 0.999706i \(-0.507723\pi\)
−0.0242615 + 0.999706i \(0.507723\pi\)
\(734\) −8.04163 −0.296822
\(735\) 6.65685 0.245542
\(736\) 33.7990 1.24585
\(737\) −16.9706 −0.625119
\(738\) 3.07107 0.113048
\(739\) −9.65685 −0.355233 −0.177617 0.984100i \(-0.556839\pi\)
−0.177617 + 0.984100i \(0.556839\pi\)
\(740\) −20.2426 −0.744134
\(741\) 0 0
\(742\) −0.970563 −0.0356305
\(743\) 15.3137 0.561805 0.280903 0.959736i \(-0.409366\pi\)
0.280903 + 0.959736i \(0.409366\pi\)
\(744\) 10.2843 0.377040
\(745\) 3.65685 0.133977
\(746\) 4.10051 0.150130
\(747\) −10.4853 −0.383636
\(748\) 3.02944 0.110767
\(749\) −4.68629 −0.171233
\(750\) −0.414214 −0.0151249
\(751\) 37.1127 1.35426 0.677131 0.735863i \(-0.263223\pi\)
0.677131 + 0.735863i \(0.263223\pi\)
\(752\) 1.02944 0.0375397
\(753\) 12.9289 0.471156
\(754\) −20.3431 −0.740854
\(755\) 17.7990 0.647772
\(756\) −1.07107 −0.0389544
\(757\) 16.8284 0.611640 0.305820 0.952089i \(-0.401069\pi\)
0.305820 + 0.952089i \(0.401069\pi\)
\(758\) −10.0000 −0.363216
\(759\) 10.8284 0.393047
\(760\) 0 0
\(761\) −10.2843 −0.372805 −0.186402 0.982474i \(-0.559683\pi\)
−0.186402 + 0.982474i \(0.559683\pi\)
\(762\) −3.31371 −0.120043
\(763\) −1.85786 −0.0672592
\(764\) −3.47309 −0.125652
\(765\) 1.17157 0.0423583
\(766\) −4.97056 −0.179594
\(767\) 45.9411 1.65884
\(768\) 3.97056 0.143275
\(769\) −35.6569 −1.28582 −0.642910 0.765942i \(-0.722273\pi\)
−0.642910 + 0.765942i \(0.722273\pi\)
\(770\) −0.343146 −0.0123661
\(771\) −1.17157 −0.0421932
\(772\) −27.5563 −0.991775
\(773\) 32.9706 1.18587 0.592934 0.805251i \(-0.297969\pi\)
0.592934 + 0.805251i \(0.297969\pi\)
\(774\) 0.242641 0.00872154
\(775\) −6.48528 −0.232958
\(776\) −6.72792 −0.241518
\(777\) −6.48528 −0.232658
\(778\) 6.20101 0.222317
\(779\) 0 0
\(780\) −9.89949 −0.354459
\(781\) 6.34315 0.226976
\(782\) −3.71573 −0.132874
\(783\) 9.07107 0.324174
\(784\) −19.9706 −0.713234
\(785\) 21.7990 0.778039
\(786\) 6.92893 0.247147
\(787\) −13.4558 −0.479649 −0.239825 0.970816i \(-0.577090\pi\)
−0.239825 + 0.970816i \(0.577090\pi\)
\(788\) 27.1127 0.965850
\(789\) −32.1421 −1.14429
\(790\) −4.68629 −0.166731
\(791\) −7.31371 −0.260046
\(792\) −2.24264 −0.0796888
\(793\) −30.6274 −1.08761
\(794\) 14.7696 0.524152
\(795\) 4.00000 0.141865
\(796\) 30.1421 1.06836
\(797\) −10.8284 −0.383563 −0.191781 0.981438i \(-0.561426\pi\)
−0.191781 + 0.981438i \(0.561426\pi\)
\(798\) 0 0
\(799\) −0.402020 −0.0142225
\(800\) 4.41421 0.156066
\(801\) −10.7279 −0.379052
\(802\) −12.4437 −0.439401
\(803\) −2.82843 −0.0998130
\(804\) 21.9411 0.773804
\(805\) −4.48528 −0.158085
\(806\) 14.5442 0.512296
\(807\) −8.38478 −0.295158
\(808\) 7.65685 0.269367
\(809\) 49.3137 1.73378 0.866889 0.498502i \(-0.166116\pi\)
0.866889 + 0.498502i \(0.166116\pi\)
\(810\) −0.414214 −0.0145540
\(811\) 15.3137 0.537737 0.268869 0.963177i \(-0.413350\pi\)
0.268869 + 0.963177i \(0.413350\pi\)
\(812\) −9.71573 −0.340955
\(813\) −30.8284 −1.08120
\(814\) −6.48528 −0.227309
\(815\) −7.89949 −0.276707
\(816\) −3.51472 −0.123040
\(817\) 0 0
\(818\) −3.94113 −0.137798
\(819\) −3.17157 −0.110824
\(820\) 13.5563 0.473408
\(821\) 51.4558 1.79582 0.897911 0.440178i \(-0.145085\pi\)
0.897911 + 0.440178i \(0.145085\pi\)
\(822\) −5.79899 −0.202263
\(823\) −2.72792 −0.0950894 −0.0475447 0.998869i \(-0.515140\pi\)
−0.0475447 + 0.998869i \(0.515140\pi\)
\(824\) −2.62742 −0.0915304
\(825\) 1.41421 0.0492366
\(826\) −2.05887 −0.0716374
\(827\) −48.6274 −1.69094 −0.845470 0.534022i \(-0.820680\pi\)
−0.845470 + 0.534022i \(0.820680\pi\)
\(828\) −14.0000 −0.486534
\(829\) 38.4853 1.33665 0.668325 0.743870i \(-0.267012\pi\)
0.668325 + 0.743870i \(0.267012\pi\)
\(830\) 4.34315 0.150753
\(831\) 10.9706 0.380565
\(832\) 22.5858 0.783021
\(833\) 7.79899 0.270219
\(834\) 0.485281 0.0168039
\(835\) 10.0000 0.346064
\(836\) 0 0
\(837\) −6.48528 −0.224164
\(838\) −14.4437 −0.498948
\(839\) −27.1127 −0.936034 −0.468017 0.883719i \(-0.655031\pi\)
−0.468017 + 0.883719i \(0.655031\pi\)
\(840\) 0.928932 0.0320512
\(841\) 53.2843 1.83739
\(842\) 6.08326 0.209643
\(843\) 14.7279 0.507257
\(844\) −28.0000 −0.963800
\(845\) −16.3137 −0.561209
\(846\) 0.142136 0.00488672
\(847\) −5.27208 −0.181151
\(848\) −12.0000 −0.412082
\(849\) 6.24264 0.214247
\(850\) −0.485281 −0.0166450
\(851\) −84.7696 −2.90586
\(852\) −8.20101 −0.280962
\(853\) −9.51472 −0.325778 −0.162889 0.986644i \(-0.552081\pi\)
−0.162889 + 0.986644i \(0.552081\pi\)
\(854\) 1.37258 0.0469688
\(855\) 0 0
\(856\) 12.6863 0.433609
\(857\) −37.9411 −1.29604 −0.648022 0.761622i \(-0.724404\pi\)
−0.648022 + 0.761622i \(0.724404\pi\)
\(858\) −3.17157 −0.108276
\(859\) 25.9411 0.885100 0.442550 0.896744i \(-0.354074\pi\)
0.442550 + 0.896744i \(0.354074\pi\)
\(860\) 1.07107 0.0365231
\(861\) 4.34315 0.148014
\(862\) −1.45584 −0.0495862
\(863\) 31.3137 1.06593 0.532966 0.846137i \(-0.321078\pi\)
0.532966 + 0.846137i \(0.321078\pi\)
\(864\) 4.41421 0.150175
\(865\) −6.14214 −0.208839
\(866\) −0.384776 −0.0130752
\(867\) −15.6274 −0.530735
\(868\) 6.94618 0.235769
\(869\) 16.0000 0.542763
\(870\) −3.75736 −0.127386
\(871\) 64.9706 2.20144
\(872\) 5.02944 0.170318
\(873\) 4.24264 0.143592
\(874\) 0 0
\(875\) −0.585786 −0.0198032
\(876\) 3.65685 0.123554
\(877\) −17.8995 −0.604423 −0.302211 0.953241i \(-0.597725\pi\)
−0.302211 + 0.953241i \(0.597725\pi\)
\(878\) −0.402020 −0.0135675
\(879\) 31.7990 1.07255
\(880\) −4.24264 −0.143019
\(881\) −47.4558 −1.59883 −0.799414 0.600781i \(-0.794857\pi\)
−0.799414 + 0.600781i \(0.794857\pi\)
\(882\) −2.75736 −0.0928451
\(883\) 46.5269 1.56576 0.782878 0.622176i \(-0.213751\pi\)
0.782878 + 0.622176i \(0.213751\pi\)
\(884\) −11.5980 −0.390082
\(885\) 8.48528 0.285230
\(886\) −0.544156 −0.0182813
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 17.5563 0.589153
\(889\) −4.68629 −0.157173
\(890\) 4.44365 0.148952
\(891\) 1.41421 0.0473779
\(892\) 11.5980 0.388329
\(893\) 0 0
\(894\) −1.51472 −0.0506598
\(895\) 17.1716 0.573982
\(896\) −6.18377 −0.206585
\(897\) −41.4558 −1.38417
\(898\) −3.27208 −0.109191
\(899\) −58.8284 −1.96204
\(900\) −1.82843 −0.0609476
\(901\) 4.68629 0.156123
\(902\) 4.34315 0.144611
\(903\) 0.343146 0.0114192
\(904\) 19.7990 0.658505
\(905\) −19.1716 −0.637285
\(906\) −7.37258 −0.244938
\(907\) −38.1421 −1.26649 −0.633244 0.773952i \(-0.718277\pi\)
−0.633244 + 0.773952i \(0.718277\pi\)
\(908\) −34.6863 −1.15111
\(909\) −4.82843 −0.160149
\(910\) 1.31371 0.0435490
\(911\) 23.3137 0.772418 0.386209 0.922411i \(-0.373784\pi\)
0.386209 + 0.922411i \(0.373784\pi\)
\(912\) 0 0
\(913\) −14.8284 −0.490749
\(914\) 11.9411 0.394977
\(915\) −5.65685 −0.187010
\(916\) 3.02944 0.100095
\(917\) 9.79899 0.323591
\(918\) −0.485281 −0.0160167
\(919\) −12.0000 −0.395843 −0.197922 0.980218i \(-0.563419\pi\)
−0.197922 + 0.980218i \(0.563419\pi\)
\(920\) 12.1421 0.400314
\(921\) −7.79899 −0.256985
\(922\) 4.42641 0.145776
\(923\) −24.2843 −0.799327
\(924\) −1.51472 −0.0498306
\(925\) −11.0711 −0.364014
\(926\) −3.35534 −0.110263
\(927\) 1.65685 0.0544182
\(928\) 40.0416 1.31443
\(929\) −51.4558 −1.68821 −0.844106 0.536177i \(-0.819868\pi\)
−0.844106 + 0.536177i \(0.819868\pi\)
\(930\) 2.68629 0.0880870
\(931\) 0 0
\(932\) 15.2548 0.499689
\(933\) −32.2426 −1.05558
\(934\) 6.76955 0.221507
\(935\) 1.65685 0.0541849
\(936\) 8.58579 0.280635
\(937\) −18.7696 −0.613175 −0.306587 0.951843i \(-0.599187\pi\)
−0.306587 + 0.951843i \(0.599187\pi\)
\(938\) −2.91169 −0.0950700
\(939\) −9.51472 −0.310501
\(940\) 0.627417 0.0204641
\(941\) 17.5563 0.572321 0.286160 0.958182i \(-0.407621\pi\)
0.286160 + 0.958182i \(0.407621\pi\)
\(942\) −9.02944 −0.294195
\(943\) 56.7696 1.84867
\(944\) −25.4558 −0.828517
\(945\) −0.585786 −0.0190556
\(946\) 0.343146 0.0111566
\(947\) 12.8284 0.416868 0.208434 0.978036i \(-0.433163\pi\)
0.208434 + 0.978036i \(0.433163\pi\)
\(948\) −20.6863 −0.671860
\(949\) 10.8284 0.351506
\(950\) 0 0
\(951\) −11.3137 −0.366872
\(952\) 1.08831 0.0352724
\(953\) 5.85786 0.189755 0.0948774 0.995489i \(-0.469754\pi\)
0.0948774 + 0.995489i \(0.469754\pi\)
\(954\) −1.65685 −0.0536426
\(955\) −1.89949 −0.0614662
\(956\) −4.72792 −0.152912
\(957\) 12.8284 0.414684
\(958\) −15.8162 −0.510999
\(959\) −8.20101 −0.264824
\(960\) 4.17157 0.134637
\(961\) 11.0589 0.356738
\(962\) 24.8284 0.800501
\(963\) −8.00000 −0.257796
\(964\) −27.3726 −0.881612
\(965\) −15.0711 −0.485155
\(966\) 1.85786 0.0597758
\(967\) −27.8995 −0.897187 −0.448594 0.893736i \(-0.648075\pi\)
−0.448594 + 0.893736i \(0.648075\pi\)
\(968\) 14.2721 0.458722
\(969\) 0 0
\(970\) −1.75736 −0.0564254
\(971\) −17.6569 −0.566635 −0.283318 0.959026i \(-0.591435\pi\)
−0.283318 + 0.959026i \(0.591435\pi\)
\(972\) −1.82843 −0.0586468
\(973\) 0.686292 0.0220015
\(974\) 1.17157 0.0375396
\(975\) −5.41421 −0.173394
\(976\) 16.9706 0.543214
\(977\) −39.5980 −1.26685 −0.633426 0.773803i \(-0.718352\pi\)
−0.633426 + 0.773803i \(0.718352\pi\)
\(978\) 3.27208 0.104630
\(979\) −15.1716 −0.484886
\(980\) −12.1716 −0.388807
\(981\) −3.17157 −0.101261
\(982\) 9.07107 0.289469
\(983\) −31.9411 −1.01876 −0.509382 0.860541i \(-0.670126\pi\)
−0.509382 + 0.860541i \(0.670126\pi\)
\(984\) −11.7574 −0.374811
\(985\) 14.8284 0.472473
\(986\) −4.40202 −0.140189
\(987\) 0.201010 0.00639822
\(988\) 0 0
\(989\) 4.48528 0.142624
\(990\) −0.585786 −0.0186175
\(991\) 61.6569 1.95859 0.979297 0.202427i \(-0.0648830\pi\)
0.979297 + 0.202427i \(0.0648830\pi\)
\(992\) −28.6274 −0.908921
\(993\) −7.17157 −0.227583
\(994\) 1.08831 0.0345192
\(995\) 16.4853 0.522619
\(996\) 19.1716 0.607475
\(997\) −50.4853 −1.59888 −0.799442 0.600743i \(-0.794872\pi\)
−0.799442 + 0.600743i \(0.794872\pi\)
\(998\) −9.85786 −0.312045
\(999\) −11.0711 −0.350273
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 5415.2.a.p.1.2 2
19.18 odd 2 285.2.a.f.1.1 2
57.56 even 2 855.2.a.e.1.2 2
76.75 even 2 4560.2.a.bj.1.2 2
95.18 even 4 1425.2.c.j.799.3 4
95.37 even 4 1425.2.c.j.799.2 4
95.94 odd 2 1425.2.a.l.1.2 2
285.284 even 2 4275.2.a.x.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.f.1.1 2 19.18 odd 2
855.2.a.e.1.2 2 57.56 even 2
1425.2.a.l.1.2 2 95.94 odd 2
1425.2.c.j.799.2 4 95.37 even 4
1425.2.c.j.799.3 4 95.18 even 4
4275.2.a.x.1.1 2 285.284 even 2
4560.2.a.bj.1.2 2 76.75 even 2
5415.2.a.p.1.2 2 1.1 even 1 trivial