Properties

Label 476.2.a.b.1.1
Level $476$
Weight $2$
Character 476.1
Self dual yes
Analytic conductor $3.801$
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] = [476,2,Mod(1,476)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(476, 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("476.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 476 = 2^{2} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 476.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: \(3.80087913621\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 476.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.61803 q^{3} +0.618034 q^{5} -1.00000 q^{7} -0.381966 q^{9} +O(q^{10})\) \(q-1.61803 q^{3} +0.618034 q^{5} -1.00000 q^{7} -0.381966 q^{9} -0.763932 q^{11} +1.23607 q^{13} -1.00000 q^{15} +1.00000 q^{17} -8.47214 q^{19} +1.61803 q^{21} -7.70820 q^{23} -4.61803 q^{25} +5.47214 q^{27} -5.70820 q^{29} +6.32624 q^{31} +1.23607 q^{33} -0.618034 q^{35} +0.472136 q^{37} -2.00000 q^{39} -0.0901699 q^{41} -12.0902 q^{43} -0.236068 q^{45} -8.47214 q^{47} +1.00000 q^{49} -1.61803 q^{51} +10.7984 q^{53} -0.472136 q^{55} +13.7082 q^{57} -7.32624 q^{61} +0.381966 q^{63} +0.763932 q^{65} -13.0902 q^{67} +12.4721 q^{69} +10.9443 q^{71} +7.14590 q^{73} +7.47214 q^{75} +0.763932 q^{77} +2.94427 q^{79} -7.70820 q^{81} +15.4164 q^{83} +0.618034 q^{85} +9.23607 q^{87} +2.00000 q^{89} -1.23607 q^{91} -10.2361 q^{93} -5.23607 q^{95} +15.0902 q^{97} +0.291796 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - q^{5} - 2 q^{7} - 3 q^{9} - 6 q^{11} - 2 q^{13} - 2 q^{15} + 2 q^{17} - 8 q^{19} + q^{21} - 2 q^{23} - 7 q^{25} + 2 q^{27} + 2 q^{29} - 3 q^{31} - 2 q^{33} + q^{35} - 8 q^{37} - 4 q^{39} + 11 q^{41} - 13 q^{43} + 4 q^{45} - 8 q^{47} + 2 q^{49} - q^{51} - 3 q^{53} + 8 q^{55} + 14 q^{57} + q^{61} + 3 q^{63} + 6 q^{65} - 15 q^{67} + 16 q^{69} + 4 q^{71} + 21 q^{73} + 6 q^{75} + 6 q^{77} - 12 q^{79} - 2 q^{81} + 4 q^{83} - q^{85} + 14 q^{87} + 4 q^{89} + 2 q^{91} - 16 q^{93} - 6 q^{95} + 19 q^{97} + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.61803 −0.934172 −0.467086 0.884212i \(-0.654696\pi\)
−0.467086 + 0.884212i \(0.654696\pi\)
\(4\) 0 0
\(5\) 0.618034 0.276393 0.138197 0.990405i \(-0.455869\pi\)
0.138197 + 0.990405i \(0.455869\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −0.381966 −0.127322
\(10\) 0 0
\(11\) −0.763932 −0.230334 −0.115167 0.993346i \(-0.536740\pi\)
−0.115167 + 0.993346i \(0.536740\pi\)
\(12\) 0 0
\(13\) 1.23607 0.342824 0.171412 0.985199i \(-0.445167\pi\)
0.171412 + 0.985199i \(0.445167\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 1.00000 0.242536
\(18\) 0 0
\(19\) −8.47214 −1.94364 −0.971821 0.235722i \(-0.924255\pi\)
−0.971821 + 0.235722i \(0.924255\pi\)
\(20\) 0 0
\(21\) 1.61803 0.353084
\(22\) 0 0
\(23\) −7.70820 −1.60727 −0.803636 0.595121i \(-0.797104\pi\)
−0.803636 + 0.595121i \(0.797104\pi\)
\(24\) 0 0
\(25\) −4.61803 −0.923607
\(26\) 0 0
\(27\) 5.47214 1.05311
\(28\) 0 0
\(29\) −5.70820 −1.05999 −0.529993 0.848002i \(-0.677806\pi\)
−0.529993 + 0.848002i \(0.677806\pi\)
\(30\) 0 0
\(31\) 6.32624 1.13623 0.568113 0.822951i \(-0.307674\pi\)
0.568113 + 0.822951i \(0.307674\pi\)
\(32\) 0 0
\(33\) 1.23607 0.215172
\(34\) 0 0
\(35\) −0.618034 −0.104467
\(36\) 0 0
\(37\) 0.472136 0.0776187 0.0388093 0.999247i \(-0.487644\pi\)
0.0388093 + 0.999247i \(0.487644\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −0.0901699 −0.0140822 −0.00704109 0.999975i \(-0.502241\pi\)
−0.00704109 + 0.999975i \(0.502241\pi\)
\(42\) 0 0
\(43\) −12.0902 −1.84373 −0.921867 0.387507i \(-0.873336\pi\)
−0.921867 + 0.387507i \(0.873336\pi\)
\(44\) 0 0
\(45\) −0.236068 −0.0351909
\(46\) 0 0
\(47\) −8.47214 −1.23579 −0.617894 0.786261i \(-0.712014\pi\)
−0.617894 + 0.786261i \(0.712014\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −1.61803 −0.226570
\(52\) 0 0
\(53\) 10.7984 1.48327 0.741635 0.670803i \(-0.234050\pi\)
0.741635 + 0.670803i \(0.234050\pi\)
\(54\) 0 0
\(55\) −0.472136 −0.0636628
\(56\) 0 0
\(57\) 13.7082 1.81570
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −7.32624 −0.938029 −0.469014 0.883191i \(-0.655391\pi\)
−0.469014 + 0.883191i \(0.655391\pi\)
\(62\) 0 0
\(63\) 0.381966 0.0481232
\(64\) 0 0
\(65\) 0.763932 0.0947541
\(66\) 0 0
\(67\) −13.0902 −1.59922 −0.799609 0.600520i \(-0.794960\pi\)
−0.799609 + 0.600520i \(0.794960\pi\)
\(68\) 0 0
\(69\) 12.4721 1.50147
\(70\) 0 0
\(71\) 10.9443 1.29885 0.649423 0.760427i \(-0.275010\pi\)
0.649423 + 0.760427i \(0.275010\pi\)
\(72\) 0 0
\(73\) 7.14590 0.836364 0.418182 0.908363i \(-0.362667\pi\)
0.418182 + 0.908363i \(0.362667\pi\)
\(74\) 0 0
\(75\) 7.47214 0.862808
\(76\) 0 0
\(77\) 0.763932 0.0870581
\(78\) 0 0
\(79\) 2.94427 0.331256 0.165628 0.986188i \(-0.447035\pi\)
0.165628 + 0.986188i \(0.447035\pi\)
\(80\) 0 0
\(81\) −7.70820 −0.856467
\(82\) 0 0
\(83\) 15.4164 1.69217 0.846085 0.533048i \(-0.178953\pi\)
0.846085 + 0.533048i \(0.178953\pi\)
\(84\) 0 0
\(85\) 0.618034 0.0670352
\(86\) 0 0
\(87\) 9.23607 0.990210
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) −1.23607 −0.129575
\(92\) 0 0
\(93\) −10.2361 −1.06143
\(94\) 0 0
\(95\) −5.23607 −0.537209
\(96\) 0 0
\(97\) 15.0902 1.53217 0.766087 0.642737i \(-0.222201\pi\)
0.766087 + 0.642737i \(0.222201\pi\)
\(98\) 0 0
\(99\) 0.291796 0.0293266
\(100\) 0 0
\(101\) −0.472136 −0.0469793 −0.0234896 0.999724i \(-0.507478\pi\)
−0.0234896 + 0.999724i \(0.507478\pi\)
\(102\) 0 0
\(103\) 15.4164 1.51902 0.759512 0.650493i \(-0.225438\pi\)
0.759512 + 0.650493i \(0.225438\pi\)
\(104\) 0 0
\(105\) 1.00000 0.0975900
\(106\) 0 0
\(107\) 10.7639 1.04059 0.520294 0.853987i \(-0.325822\pi\)
0.520294 + 0.853987i \(0.325822\pi\)
\(108\) 0 0
\(109\) −3.52786 −0.337908 −0.168954 0.985624i \(-0.554039\pi\)
−0.168954 + 0.985624i \(0.554039\pi\)
\(110\) 0 0
\(111\) −0.763932 −0.0725092
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) −4.76393 −0.444239
\(116\) 0 0
\(117\) −0.472136 −0.0436490
\(118\) 0 0
\(119\) −1.00000 −0.0916698
\(120\) 0 0
\(121\) −10.4164 −0.946946
\(122\) 0 0
\(123\) 0.145898 0.0131552
\(124\) 0 0
\(125\) −5.94427 −0.531672
\(126\) 0 0
\(127\) −7.85410 −0.696939 −0.348469 0.937320i \(-0.613298\pi\)
−0.348469 + 0.937320i \(0.613298\pi\)
\(128\) 0 0
\(129\) 19.5623 1.72236
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) 8.47214 0.734627
\(134\) 0 0
\(135\) 3.38197 0.291073
\(136\) 0 0
\(137\) 2.38197 0.203505 0.101753 0.994810i \(-0.467555\pi\)
0.101753 + 0.994810i \(0.467555\pi\)
\(138\) 0 0
\(139\) −14.0344 −1.19039 −0.595193 0.803583i \(-0.702924\pi\)
−0.595193 + 0.803583i \(0.702924\pi\)
\(140\) 0 0
\(141\) 13.7082 1.15444
\(142\) 0 0
\(143\) −0.944272 −0.0789640
\(144\) 0 0
\(145\) −3.52786 −0.292973
\(146\) 0 0
\(147\) −1.61803 −0.133453
\(148\) 0 0
\(149\) −13.6180 −1.11563 −0.557816 0.829964i \(-0.688361\pi\)
−0.557816 + 0.829964i \(0.688361\pi\)
\(150\) 0 0
\(151\) −15.1459 −1.23256 −0.616278 0.787529i \(-0.711360\pi\)
−0.616278 + 0.787529i \(0.711360\pi\)
\(152\) 0 0
\(153\) −0.381966 −0.0308801
\(154\) 0 0
\(155\) 3.90983 0.314045
\(156\) 0 0
\(157\) −2.18034 −0.174010 −0.0870050 0.996208i \(-0.527730\pi\)
−0.0870050 + 0.996208i \(0.527730\pi\)
\(158\) 0 0
\(159\) −17.4721 −1.38563
\(160\) 0 0
\(161\) 7.70820 0.607492
\(162\) 0 0
\(163\) 5.70820 0.447101 0.223551 0.974692i \(-0.428235\pi\)
0.223551 + 0.974692i \(0.428235\pi\)
\(164\) 0 0
\(165\) 0.763932 0.0594720
\(166\) 0 0
\(167\) 14.3820 1.11291 0.556455 0.830878i \(-0.312161\pi\)
0.556455 + 0.830878i \(0.312161\pi\)
\(168\) 0 0
\(169\) −11.4721 −0.882472
\(170\) 0 0
\(171\) 3.23607 0.247468
\(172\) 0 0
\(173\) −11.9098 −0.905488 −0.452744 0.891641i \(-0.649555\pi\)
−0.452744 + 0.891641i \(0.649555\pi\)
\(174\) 0 0
\(175\) 4.61803 0.349091
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −18.3820 −1.37393 −0.686966 0.726689i \(-0.741058\pi\)
−0.686966 + 0.726689i \(0.741058\pi\)
\(180\) 0 0
\(181\) −12.4721 −0.927047 −0.463523 0.886085i \(-0.653415\pi\)
−0.463523 + 0.886085i \(0.653415\pi\)
\(182\) 0 0
\(183\) 11.8541 0.876280
\(184\) 0 0
\(185\) 0.291796 0.0214533
\(186\) 0 0
\(187\) −0.763932 −0.0558642
\(188\) 0 0
\(189\) −5.47214 −0.398039
\(190\) 0 0
\(191\) 20.7984 1.50492 0.752459 0.658639i \(-0.228868\pi\)
0.752459 + 0.658639i \(0.228868\pi\)
\(192\) 0 0
\(193\) −14.1803 −1.02072 −0.510362 0.859960i \(-0.670488\pi\)
−0.510362 + 0.859960i \(0.670488\pi\)
\(194\) 0 0
\(195\) −1.23607 −0.0885167
\(196\) 0 0
\(197\) 9.70820 0.691681 0.345840 0.938293i \(-0.387594\pi\)
0.345840 + 0.938293i \(0.387594\pi\)
\(198\) 0 0
\(199\) −12.0902 −0.857049 −0.428525 0.903530i \(-0.640967\pi\)
−0.428525 + 0.903530i \(0.640967\pi\)
\(200\) 0 0
\(201\) 21.1803 1.49395
\(202\) 0 0
\(203\) 5.70820 0.400637
\(204\) 0 0
\(205\) −0.0557281 −0.00389222
\(206\) 0 0
\(207\) 2.94427 0.204641
\(208\) 0 0
\(209\) 6.47214 0.447687
\(210\) 0 0
\(211\) −2.29180 −0.157774 −0.0788869 0.996884i \(-0.525137\pi\)
−0.0788869 + 0.996884i \(0.525137\pi\)
\(212\) 0 0
\(213\) −17.7082 −1.21335
\(214\) 0 0
\(215\) −7.47214 −0.509595
\(216\) 0 0
\(217\) −6.32624 −0.429453
\(218\) 0 0
\(219\) −11.5623 −0.781308
\(220\) 0 0
\(221\) 1.23607 0.0831469
\(222\) 0 0
\(223\) 2.94427 0.197163 0.0985815 0.995129i \(-0.468569\pi\)
0.0985815 + 0.995129i \(0.468569\pi\)
\(224\) 0 0
\(225\) 1.76393 0.117595
\(226\) 0 0
\(227\) 25.7426 1.70860 0.854300 0.519781i \(-0.173986\pi\)
0.854300 + 0.519781i \(0.173986\pi\)
\(228\) 0 0
\(229\) −19.2361 −1.27116 −0.635578 0.772037i \(-0.719238\pi\)
−0.635578 + 0.772037i \(0.719238\pi\)
\(230\) 0 0
\(231\) −1.23607 −0.0813273
\(232\) 0 0
\(233\) 6.47214 0.424004 0.212002 0.977269i \(-0.432002\pi\)
0.212002 + 0.977269i \(0.432002\pi\)
\(234\) 0 0
\(235\) −5.23607 −0.341563
\(236\) 0 0
\(237\) −4.76393 −0.309451
\(238\) 0 0
\(239\) 11.9098 0.770383 0.385191 0.922837i \(-0.374135\pi\)
0.385191 + 0.922837i \(0.374135\pi\)
\(240\) 0 0
\(241\) 4.03444 0.259881 0.129941 0.991522i \(-0.458521\pi\)
0.129941 + 0.991522i \(0.458521\pi\)
\(242\) 0 0
\(243\) −3.94427 −0.253025
\(244\) 0 0
\(245\) 0.618034 0.0394847
\(246\) 0 0
\(247\) −10.4721 −0.666326
\(248\) 0 0
\(249\) −24.9443 −1.58078
\(250\) 0 0
\(251\) −17.4164 −1.09931 −0.549657 0.835390i \(-0.685242\pi\)
−0.549657 + 0.835390i \(0.685242\pi\)
\(252\) 0 0
\(253\) 5.88854 0.370210
\(254\) 0 0
\(255\) −1.00000 −0.0626224
\(256\) 0 0
\(257\) 25.7082 1.60363 0.801817 0.597570i \(-0.203867\pi\)
0.801817 + 0.597570i \(0.203867\pi\)
\(258\) 0 0
\(259\) −0.472136 −0.0293371
\(260\) 0 0
\(261\) 2.18034 0.134960
\(262\) 0 0
\(263\) −16.9443 −1.04483 −0.522414 0.852692i \(-0.674969\pi\)
−0.522414 + 0.852692i \(0.674969\pi\)
\(264\) 0 0
\(265\) 6.67376 0.409966
\(266\) 0 0
\(267\) −3.23607 −0.198044
\(268\) 0 0
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) 3.70820 0.225257 0.112629 0.993637i \(-0.464073\pi\)
0.112629 + 0.993637i \(0.464073\pi\)
\(272\) 0 0
\(273\) 2.00000 0.121046
\(274\) 0 0
\(275\) 3.52786 0.212738
\(276\) 0 0
\(277\) 2.76393 0.166069 0.0830343 0.996547i \(-0.473539\pi\)
0.0830343 + 0.996547i \(0.473539\pi\)
\(278\) 0 0
\(279\) −2.41641 −0.144667
\(280\) 0 0
\(281\) −8.90983 −0.531516 −0.265758 0.964040i \(-0.585622\pi\)
−0.265758 + 0.964040i \(0.585622\pi\)
\(282\) 0 0
\(283\) −4.43769 −0.263794 −0.131897 0.991263i \(-0.542107\pi\)
−0.131897 + 0.991263i \(0.542107\pi\)
\(284\) 0 0
\(285\) 8.47214 0.501846
\(286\) 0 0
\(287\) 0.0901699 0.00532256
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) −24.4164 −1.43132
\(292\) 0 0
\(293\) −6.29180 −0.367571 −0.183785 0.982966i \(-0.558835\pi\)
−0.183785 + 0.982966i \(0.558835\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −4.18034 −0.242568
\(298\) 0 0
\(299\) −9.52786 −0.551011
\(300\) 0 0
\(301\) 12.0902 0.696866
\(302\) 0 0
\(303\) 0.763932 0.0438867
\(304\) 0 0
\(305\) −4.52786 −0.259265
\(306\) 0 0
\(307\) 1.05573 0.0602536 0.0301268 0.999546i \(-0.490409\pi\)
0.0301268 + 0.999546i \(0.490409\pi\)
\(308\) 0 0
\(309\) −24.9443 −1.41903
\(310\) 0 0
\(311\) 26.6180 1.50937 0.754685 0.656087i \(-0.227790\pi\)
0.754685 + 0.656087i \(0.227790\pi\)
\(312\) 0 0
\(313\) −1.27051 −0.0718135 −0.0359067 0.999355i \(-0.511432\pi\)
−0.0359067 + 0.999355i \(0.511432\pi\)
\(314\) 0 0
\(315\) 0.236068 0.0133009
\(316\) 0 0
\(317\) −28.1803 −1.58277 −0.791383 0.611321i \(-0.790638\pi\)
−0.791383 + 0.611321i \(0.790638\pi\)
\(318\) 0 0
\(319\) 4.36068 0.244151
\(320\) 0 0
\(321\) −17.4164 −0.972089
\(322\) 0 0
\(323\) −8.47214 −0.471402
\(324\) 0 0
\(325\) −5.70820 −0.316634
\(326\) 0 0
\(327\) 5.70820 0.315664
\(328\) 0 0
\(329\) 8.47214 0.467084
\(330\) 0 0
\(331\) 18.0344 0.991263 0.495631 0.868533i \(-0.334937\pi\)
0.495631 + 0.868533i \(0.334937\pi\)
\(332\) 0 0
\(333\) −0.180340 −0.00988256
\(334\) 0 0
\(335\) −8.09017 −0.442013
\(336\) 0 0
\(337\) 9.52786 0.519016 0.259508 0.965741i \(-0.416440\pi\)
0.259508 + 0.965741i \(0.416440\pi\)
\(338\) 0 0
\(339\) −9.70820 −0.527277
\(340\) 0 0
\(341\) −4.83282 −0.261712
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 7.70820 0.414996
\(346\) 0 0
\(347\) 30.9443 1.66118 0.830588 0.556888i \(-0.188005\pi\)
0.830588 + 0.556888i \(0.188005\pi\)
\(348\) 0 0
\(349\) −15.5279 −0.831188 −0.415594 0.909550i \(-0.636426\pi\)
−0.415594 + 0.909550i \(0.636426\pi\)
\(350\) 0 0
\(351\) 6.76393 0.361032
\(352\) 0 0
\(353\) −12.7639 −0.679356 −0.339678 0.940542i \(-0.610318\pi\)
−0.339678 + 0.940542i \(0.610318\pi\)
\(354\) 0 0
\(355\) 6.76393 0.358992
\(356\) 0 0
\(357\) 1.61803 0.0856354
\(358\) 0 0
\(359\) 19.5623 1.03246 0.516230 0.856450i \(-0.327335\pi\)
0.516230 + 0.856450i \(0.327335\pi\)
\(360\) 0 0
\(361\) 52.7771 2.77774
\(362\) 0 0
\(363\) 16.8541 0.884611
\(364\) 0 0
\(365\) 4.41641 0.231165
\(366\) 0 0
\(367\) −19.5066 −1.01824 −0.509118 0.860697i \(-0.670028\pi\)
−0.509118 + 0.860697i \(0.670028\pi\)
\(368\) 0 0
\(369\) 0.0344419 0.00179297
\(370\) 0 0
\(371\) −10.7984 −0.560624
\(372\) 0 0
\(373\) −16.8541 −0.872672 −0.436336 0.899784i \(-0.643724\pi\)
−0.436336 + 0.899784i \(0.643724\pi\)
\(374\) 0 0
\(375\) 9.61803 0.496673
\(376\) 0 0
\(377\) −7.05573 −0.363388
\(378\) 0 0
\(379\) 10.6525 0.547181 0.273590 0.961846i \(-0.411789\pi\)
0.273590 + 0.961846i \(0.411789\pi\)
\(380\) 0 0
\(381\) 12.7082 0.651061
\(382\) 0 0
\(383\) −30.6525 −1.56627 −0.783134 0.621853i \(-0.786380\pi\)
−0.783134 + 0.621853i \(0.786380\pi\)
\(384\) 0 0
\(385\) 0.472136 0.0240623
\(386\) 0 0
\(387\) 4.61803 0.234748
\(388\) 0 0
\(389\) −21.5623 −1.09325 −0.546626 0.837377i \(-0.684088\pi\)
−0.546626 + 0.837377i \(0.684088\pi\)
\(390\) 0 0
\(391\) −7.70820 −0.389821
\(392\) 0 0
\(393\) 12.9443 0.652952
\(394\) 0 0
\(395\) 1.81966 0.0915570
\(396\) 0 0
\(397\) −6.79837 −0.341201 −0.170600 0.985340i \(-0.554571\pi\)
−0.170600 + 0.985340i \(0.554571\pi\)
\(398\) 0 0
\(399\) −13.7082 −0.686269
\(400\) 0 0
\(401\) −10.4721 −0.522954 −0.261477 0.965210i \(-0.584209\pi\)
−0.261477 + 0.965210i \(0.584209\pi\)
\(402\) 0 0
\(403\) 7.81966 0.389525
\(404\) 0 0
\(405\) −4.76393 −0.236722
\(406\) 0 0
\(407\) −0.360680 −0.0178782
\(408\) 0 0
\(409\) −8.18034 −0.404492 −0.202246 0.979335i \(-0.564824\pi\)
−0.202246 + 0.979335i \(0.564824\pi\)
\(410\) 0 0
\(411\) −3.85410 −0.190109
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 9.52786 0.467704
\(416\) 0 0
\(417\) 22.7082 1.11203
\(418\) 0 0
\(419\) 37.4508 1.82959 0.914797 0.403914i \(-0.132351\pi\)
0.914797 + 0.403914i \(0.132351\pi\)
\(420\) 0 0
\(421\) −29.2148 −1.42384 −0.711921 0.702260i \(-0.752174\pi\)
−0.711921 + 0.702260i \(0.752174\pi\)
\(422\) 0 0
\(423\) 3.23607 0.157343
\(424\) 0 0
\(425\) −4.61803 −0.224008
\(426\) 0 0
\(427\) 7.32624 0.354542
\(428\) 0 0
\(429\) 1.52786 0.0737660
\(430\) 0 0
\(431\) 5.05573 0.243526 0.121763 0.992559i \(-0.461145\pi\)
0.121763 + 0.992559i \(0.461145\pi\)
\(432\) 0 0
\(433\) 16.9443 0.814290 0.407145 0.913364i \(-0.366524\pi\)
0.407145 + 0.913364i \(0.366524\pi\)
\(434\) 0 0
\(435\) 5.70820 0.273687
\(436\) 0 0
\(437\) 65.3050 3.12396
\(438\) 0 0
\(439\) −3.32624 −0.158753 −0.0793763 0.996845i \(-0.525293\pi\)
−0.0793763 + 0.996845i \(0.525293\pi\)
\(440\) 0 0
\(441\) −0.381966 −0.0181889
\(442\) 0 0
\(443\) 25.8885 1.23000 0.615001 0.788526i \(-0.289156\pi\)
0.615001 + 0.788526i \(0.289156\pi\)
\(444\) 0 0
\(445\) 1.23607 0.0585952
\(446\) 0 0
\(447\) 22.0344 1.04219
\(448\) 0 0
\(449\) 17.7082 0.835702 0.417851 0.908516i \(-0.362783\pi\)
0.417851 + 0.908516i \(0.362783\pi\)
\(450\) 0 0
\(451\) 0.0688837 0.00324361
\(452\) 0 0
\(453\) 24.5066 1.15142
\(454\) 0 0
\(455\) −0.763932 −0.0358137
\(456\) 0 0
\(457\) 0.618034 0.0289104 0.0144552 0.999896i \(-0.495399\pi\)
0.0144552 + 0.999896i \(0.495399\pi\)
\(458\) 0 0
\(459\) 5.47214 0.255417
\(460\) 0 0
\(461\) −12.9443 −0.602875 −0.301437 0.953486i \(-0.597466\pi\)
−0.301437 + 0.953486i \(0.597466\pi\)
\(462\) 0 0
\(463\) −13.0344 −0.605762 −0.302881 0.953028i \(-0.597948\pi\)
−0.302881 + 0.953028i \(0.597948\pi\)
\(464\) 0 0
\(465\) −6.32624 −0.293372
\(466\) 0 0
\(467\) 10.4721 0.484593 0.242296 0.970202i \(-0.422099\pi\)
0.242296 + 0.970202i \(0.422099\pi\)
\(468\) 0 0
\(469\) 13.0902 0.604448
\(470\) 0 0
\(471\) 3.52786 0.162555
\(472\) 0 0
\(473\) 9.23607 0.424675
\(474\) 0 0
\(475\) 39.1246 1.79516
\(476\) 0 0
\(477\) −4.12461 −0.188853
\(478\) 0 0
\(479\) −24.2705 −1.10895 −0.554474 0.832201i \(-0.687080\pi\)
−0.554474 + 0.832201i \(0.687080\pi\)
\(480\) 0 0
\(481\) 0.583592 0.0266095
\(482\) 0 0
\(483\) −12.4721 −0.567502
\(484\) 0 0
\(485\) 9.32624 0.423483
\(486\) 0 0
\(487\) −22.3607 −1.01326 −0.506630 0.862164i \(-0.669109\pi\)
−0.506630 + 0.862164i \(0.669109\pi\)
\(488\) 0 0
\(489\) −9.23607 −0.417669
\(490\) 0 0
\(491\) −1.09017 −0.0491987 −0.0245993 0.999697i \(-0.507831\pi\)
−0.0245993 + 0.999697i \(0.507831\pi\)
\(492\) 0 0
\(493\) −5.70820 −0.257085
\(494\) 0 0
\(495\) 0.180340 0.00810568
\(496\) 0 0
\(497\) −10.9443 −0.490918
\(498\) 0 0
\(499\) 17.0557 0.763519 0.381760 0.924262i \(-0.375318\pi\)
0.381760 + 0.924262i \(0.375318\pi\)
\(500\) 0 0
\(501\) −23.2705 −1.03965
\(502\) 0 0
\(503\) −10.6180 −0.473435 −0.236717 0.971579i \(-0.576072\pi\)
−0.236717 + 0.971579i \(0.576072\pi\)
\(504\) 0 0
\(505\) −0.291796 −0.0129848
\(506\) 0 0
\(507\) 18.5623 0.824381
\(508\) 0 0
\(509\) −0.763932 −0.0338607 −0.0169303 0.999857i \(-0.505389\pi\)
−0.0169303 + 0.999857i \(0.505389\pi\)
\(510\) 0 0
\(511\) −7.14590 −0.316116
\(512\) 0 0
\(513\) −46.3607 −2.04687
\(514\) 0 0
\(515\) 9.52786 0.419848
\(516\) 0 0
\(517\) 6.47214 0.284644
\(518\) 0 0
\(519\) 19.2705 0.845881
\(520\) 0 0
\(521\) −26.7426 −1.17162 −0.585808 0.810450i \(-0.699223\pi\)
−0.585808 + 0.810450i \(0.699223\pi\)
\(522\) 0 0
\(523\) 34.1803 1.49460 0.747301 0.664486i \(-0.231349\pi\)
0.747301 + 0.664486i \(0.231349\pi\)
\(524\) 0 0
\(525\) −7.47214 −0.326111
\(526\) 0 0
\(527\) 6.32624 0.275575
\(528\) 0 0
\(529\) 36.4164 1.58332
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −0.111456 −0.00482770
\(534\) 0 0
\(535\) 6.65248 0.287612
\(536\) 0 0
\(537\) 29.7426 1.28349
\(538\) 0 0
\(539\) −0.763932 −0.0329049
\(540\) 0 0
\(541\) −5.23607 −0.225116 −0.112558 0.993645i \(-0.535904\pi\)
−0.112558 + 0.993645i \(0.535904\pi\)
\(542\) 0 0
\(543\) 20.1803 0.866021
\(544\) 0 0
\(545\) −2.18034 −0.0933955
\(546\) 0 0
\(547\) −40.6525 −1.73817 −0.869087 0.494659i \(-0.835293\pi\)
−0.869087 + 0.494659i \(0.835293\pi\)
\(548\) 0 0
\(549\) 2.79837 0.119432
\(550\) 0 0
\(551\) 48.3607 2.06023
\(552\) 0 0
\(553\) −2.94427 −0.125203
\(554\) 0 0
\(555\) −0.472136 −0.0200411
\(556\) 0 0
\(557\) 34.3607 1.45591 0.727954 0.685626i \(-0.240471\pi\)
0.727954 + 0.685626i \(0.240471\pi\)
\(558\) 0 0
\(559\) −14.9443 −0.632075
\(560\) 0 0
\(561\) 1.23607 0.0521868
\(562\) 0 0
\(563\) −29.1246 −1.22746 −0.613728 0.789518i \(-0.710331\pi\)
−0.613728 + 0.789518i \(0.710331\pi\)
\(564\) 0 0
\(565\) 3.70820 0.156005
\(566\) 0 0
\(567\) 7.70820 0.323714
\(568\) 0 0
\(569\) −43.5066 −1.82389 −0.911945 0.410312i \(-0.865420\pi\)
−0.911945 + 0.410312i \(0.865420\pi\)
\(570\) 0 0
\(571\) 8.47214 0.354548 0.177274 0.984162i \(-0.443272\pi\)
0.177274 + 0.984162i \(0.443272\pi\)
\(572\) 0 0
\(573\) −33.6525 −1.40585
\(574\) 0 0
\(575\) 35.5967 1.48449
\(576\) 0 0
\(577\) 17.1246 0.712907 0.356453 0.934313i \(-0.383986\pi\)
0.356453 + 0.934313i \(0.383986\pi\)
\(578\) 0 0
\(579\) 22.9443 0.953531
\(580\) 0 0
\(581\) −15.4164 −0.639580
\(582\) 0 0
\(583\) −8.24922 −0.341648
\(584\) 0 0
\(585\) −0.291796 −0.0120643
\(586\) 0 0
\(587\) −42.8328 −1.76790 −0.883950 0.467582i \(-0.845125\pi\)
−0.883950 + 0.467582i \(0.845125\pi\)
\(588\) 0 0
\(589\) −53.5967 −2.20842
\(590\) 0 0
\(591\) −15.7082 −0.646149
\(592\) 0 0
\(593\) −22.3607 −0.918243 −0.459122 0.888373i \(-0.651836\pi\)
−0.459122 + 0.888373i \(0.651836\pi\)
\(594\) 0 0
\(595\) −0.618034 −0.0253369
\(596\) 0 0
\(597\) 19.5623 0.800632
\(598\) 0 0
\(599\) 36.0902 1.47460 0.737302 0.675563i \(-0.236099\pi\)
0.737302 + 0.675563i \(0.236099\pi\)
\(600\) 0 0
\(601\) 48.2492 1.96813 0.984063 0.177818i \(-0.0569037\pi\)
0.984063 + 0.177818i \(0.0569037\pi\)
\(602\) 0 0
\(603\) 5.00000 0.203616
\(604\) 0 0
\(605\) −6.43769 −0.261729
\(606\) 0 0
\(607\) −39.8541 −1.61763 −0.808814 0.588064i \(-0.799890\pi\)
−0.808814 + 0.588064i \(0.799890\pi\)
\(608\) 0 0
\(609\) −9.23607 −0.374264
\(610\) 0 0
\(611\) −10.4721 −0.423657
\(612\) 0 0
\(613\) −17.5623 −0.709335 −0.354667 0.934993i \(-0.615406\pi\)
−0.354667 + 0.934993i \(0.615406\pi\)
\(614\) 0 0
\(615\) 0.0901699 0.00363600
\(616\) 0 0
\(617\) −35.0132 −1.40958 −0.704788 0.709418i \(-0.748958\pi\)
−0.704788 + 0.709418i \(0.748958\pi\)
\(618\) 0 0
\(619\) −32.0000 −1.28619 −0.643094 0.765787i \(-0.722350\pi\)
−0.643094 + 0.765787i \(0.722350\pi\)
\(620\) 0 0
\(621\) −42.1803 −1.69264
\(622\) 0 0
\(623\) −2.00000 −0.0801283
\(624\) 0 0
\(625\) 19.4164 0.776656
\(626\) 0 0
\(627\) −10.4721 −0.418217
\(628\) 0 0
\(629\) 0.472136 0.0188253
\(630\) 0 0
\(631\) −37.0902 −1.47654 −0.738268 0.674507i \(-0.764356\pi\)
−0.738268 + 0.674507i \(0.764356\pi\)
\(632\) 0 0
\(633\) 3.70820 0.147388
\(634\) 0 0
\(635\) −4.85410 −0.192629
\(636\) 0 0
\(637\) 1.23607 0.0489748
\(638\) 0 0
\(639\) −4.18034 −0.165372
\(640\) 0 0
\(641\) 21.0557 0.831651 0.415826 0.909444i \(-0.363493\pi\)
0.415826 + 0.909444i \(0.363493\pi\)
\(642\) 0 0
\(643\) 3.14590 0.124062 0.0620311 0.998074i \(-0.480242\pi\)
0.0620311 + 0.998074i \(0.480242\pi\)
\(644\) 0 0
\(645\) 12.0902 0.476050
\(646\) 0 0
\(647\) −20.7639 −0.816314 −0.408157 0.912912i \(-0.633828\pi\)
−0.408157 + 0.912912i \(0.633828\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 10.2361 0.401183
\(652\) 0 0
\(653\) −32.4721 −1.27073 −0.635366 0.772211i \(-0.719151\pi\)
−0.635366 + 0.772211i \(0.719151\pi\)
\(654\) 0 0
\(655\) −4.94427 −0.193189
\(656\) 0 0
\(657\) −2.72949 −0.106488
\(658\) 0 0
\(659\) 1.74265 0.0678838 0.0339419 0.999424i \(-0.489194\pi\)
0.0339419 + 0.999424i \(0.489194\pi\)
\(660\) 0 0
\(661\) 30.8328 1.19926 0.599629 0.800278i \(-0.295315\pi\)
0.599629 + 0.800278i \(0.295315\pi\)
\(662\) 0 0
\(663\) −2.00000 −0.0776736
\(664\) 0 0
\(665\) 5.23607 0.203046
\(666\) 0 0
\(667\) 44.0000 1.70369
\(668\) 0 0
\(669\) −4.76393 −0.184184
\(670\) 0 0
\(671\) 5.59675 0.216060
\(672\) 0 0
\(673\) −14.4721 −0.557860 −0.278930 0.960311i \(-0.589980\pi\)
−0.278930 + 0.960311i \(0.589980\pi\)
\(674\) 0 0
\(675\) −25.2705 −0.972662
\(676\) 0 0
\(677\) −33.7771 −1.29816 −0.649079 0.760721i \(-0.724846\pi\)
−0.649079 + 0.760721i \(0.724846\pi\)
\(678\) 0 0
\(679\) −15.0902 −0.579108
\(680\) 0 0
\(681\) −41.6525 −1.59613
\(682\) 0 0
\(683\) 28.1803 1.07829 0.539145 0.842213i \(-0.318747\pi\)
0.539145 + 0.842213i \(0.318747\pi\)
\(684\) 0 0
\(685\) 1.47214 0.0562474
\(686\) 0 0
\(687\) 31.1246 1.18748
\(688\) 0 0
\(689\) 13.3475 0.508500
\(690\) 0 0
\(691\) 8.49342 0.323105 0.161553 0.986864i \(-0.448350\pi\)
0.161553 + 0.986864i \(0.448350\pi\)
\(692\) 0 0
\(693\) −0.291796 −0.0110844
\(694\) 0 0
\(695\) −8.67376 −0.329015
\(696\) 0 0
\(697\) −0.0901699 −0.00341543
\(698\) 0 0
\(699\) −10.4721 −0.396093
\(700\) 0 0
\(701\) −45.4164 −1.71535 −0.857677 0.514189i \(-0.828093\pi\)
−0.857677 + 0.514189i \(0.828093\pi\)
\(702\) 0 0
\(703\) −4.00000 −0.150863
\(704\) 0 0
\(705\) 8.47214 0.319079
\(706\) 0 0
\(707\) 0.472136 0.0177565
\(708\) 0 0
\(709\) 40.6525 1.52674 0.763368 0.645964i \(-0.223544\pi\)
0.763368 + 0.645964i \(0.223544\pi\)
\(710\) 0 0
\(711\) −1.12461 −0.0421762
\(712\) 0 0
\(713\) −48.7639 −1.82622
\(714\) 0 0
\(715\) −0.583592 −0.0218251
\(716\) 0 0
\(717\) −19.2705 −0.719670
\(718\) 0 0
\(719\) −39.1459 −1.45990 −0.729948 0.683503i \(-0.760456\pi\)
−0.729948 + 0.683503i \(0.760456\pi\)
\(720\) 0 0
\(721\) −15.4164 −0.574137
\(722\) 0 0
\(723\) −6.52786 −0.242774
\(724\) 0 0
\(725\) 26.3607 0.979011
\(726\) 0 0
\(727\) 16.4721 0.610918 0.305459 0.952205i \(-0.401190\pi\)
0.305459 + 0.952205i \(0.401190\pi\)
\(728\) 0 0
\(729\) 29.5066 1.09284
\(730\) 0 0
\(731\) −12.0902 −0.447171
\(732\) 0 0
\(733\) 36.0000 1.32969 0.664845 0.746981i \(-0.268498\pi\)
0.664845 + 0.746981i \(0.268498\pi\)
\(734\) 0 0
\(735\) −1.00000 −0.0368856
\(736\) 0 0
\(737\) 10.0000 0.368355
\(738\) 0 0
\(739\) −1.43769 −0.0528864 −0.0264432 0.999650i \(-0.508418\pi\)
−0.0264432 + 0.999650i \(0.508418\pi\)
\(740\) 0 0
\(741\) 16.9443 0.622463
\(742\) 0 0
\(743\) 4.65248 0.170683 0.0853414 0.996352i \(-0.472802\pi\)
0.0853414 + 0.996352i \(0.472802\pi\)
\(744\) 0 0
\(745\) −8.41641 −0.308353
\(746\) 0 0
\(747\) −5.88854 −0.215451
\(748\) 0 0
\(749\) −10.7639 −0.393306
\(750\) 0 0
\(751\) 10.9443 0.399362 0.199681 0.979861i \(-0.436009\pi\)
0.199681 + 0.979861i \(0.436009\pi\)
\(752\) 0 0
\(753\) 28.1803 1.02695
\(754\) 0 0
\(755\) −9.36068 −0.340670
\(756\) 0 0
\(757\) −9.43769 −0.343019 −0.171509 0.985182i \(-0.554864\pi\)
−0.171509 + 0.985182i \(0.554864\pi\)
\(758\) 0 0
\(759\) −9.52786 −0.345840
\(760\) 0 0
\(761\) 30.1803 1.09404 0.547018 0.837121i \(-0.315763\pi\)
0.547018 + 0.837121i \(0.315763\pi\)
\(762\) 0 0
\(763\) 3.52786 0.127717
\(764\) 0 0
\(765\) −0.236068 −0.00853506
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −7.12461 −0.256920 −0.128460 0.991715i \(-0.541003\pi\)
−0.128460 + 0.991715i \(0.541003\pi\)
\(770\) 0 0
\(771\) −41.5967 −1.49807
\(772\) 0 0
\(773\) 5.41641 0.194815 0.0974073 0.995245i \(-0.468945\pi\)
0.0974073 + 0.995245i \(0.468945\pi\)
\(774\) 0 0
\(775\) −29.2148 −1.04943
\(776\) 0 0
\(777\) 0.763932 0.0274059
\(778\) 0 0
\(779\) 0.763932 0.0273707
\(780\) 0 0
\(781\) −8.36068 −0.299169
\(782\) 0 0
\(783\) −31.2361 −1.11629
\(784\) 0 0
\(785\) −1.34752 −0.0480952
\(786\) 0 0
\(787\) −12.0000 −0.427754 −0.213877 0.976861i \(-0.568609\pi\)
−0.213877 + 0.976861i \(0.568609\pi\)
\(788\) 0 0
\(789\) 27.4164 0.976050
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) −9.05573 −0.321578
\(794\) 0 0
\(795\) −10.7984 −0.382979
\(796\) 0 0
\(797\) 15.4164 0.546077 0.273039 0.962003i \(-0.411971\pi\)
0.273039 + 0.962003i \(0.411971\pi\)
\(798\) 0 0
\(799\) −8.47214 −0.299723
\(800\) 0 0
\(801\) −0.763932 −0.0269922
\(802\) 0 0
\(803\) −5.45898 −0.192643
\(804\) 0 0
\(805\) 4.76393 0.167907
\(806\) 0 0
\(807\) 16.1803 0.569575
\(808\) 0 0
\(809\) −53.4853 −1.88044 −0.940221 0.340564i \(-0.889382\pi\)
−0.940221 + 0.340564i \(0.889382\pi\)
\(810\) 0 0
\(811\) −2.96556 −0.104135 −0.0520674 0.998644i \(-0.516581\pi\)
−0.0520674 + 0.998644i \(0.516581\pi\)
\(812\) 0 0
\(813\) −6.00000 −0.210429
\(814\) 0 0
\(815\) 3.52786 0.123576
\(816\) 0 0
\(817\) 102.430 3.58356
\(818\) 0 0
\(819\) 0.472136 0.0164978
\(820\) 0 0
\(821\) 14.3607 0.501191 0.250596 0.968092i \(-0.419374\pi\)
0.250596 + 0.968092i \(0.419374\pi\)
\(822\) 0 0
\(823\) 47.4853 1.65523 0.827617 0.561294i \(-0.189696\pi\)
0.827617 + 0.561294i \(0.189696\pi\)
\(824\) 0 0
\(825\) −5.70820 −0.198734
\(826\) 0 0
\(827\) 27.5279 0.957238 0.478619 0.878023i \(-0.341138\pi\)
0.478619 + 0.878023i \(0.341138\pi\)
\(828\) 0 0
\(829\) −6.18034 −0.214652 −0.107326 0.994224i \(-0.534229\pi\)
−0.107326 + 0.994224i \(0.534229\pi\)
\(830\) 0 0
\(831\) −4.47214 −0.155137
\(832\) 0 0
\(833\) 1.00000 0.0346479
\(834\) 0 0
\(835\) 8.88854 0.307601
\(836\) 0 0
\(837\) 34.6180 1.19657
\(838\) 0 0
\(839\) −13.5279 −0.467034 −0.233517 0.972353i \(-0.575023\pi\)
−0.233517 + 0.972353i \(0.575023\pi\)
\(840\) 0 0
\(841\) 3.58359 0.123572
\(842\) 0 0
\(843\) 14.4164 0.496527
\(844\) 0 0
\(845\) −7.09017 −0.243909
\(846\) 0 0
\(847\) 10.4164 0.357912
\(848\) 0 0
\(849\) 7.18034 0.246429
\(850\) 0 0
\(851\) −3.63932 −0.124754
\(852\) 0 0
\(853\) −38.3607 −1.31344 −0.656722 0.754132i \(-0.728058\pi\)
−0.656722 + 0.754132i \(0.728058\pi\)
\(854\) 0 0
\(855\) 2.00000 0.0683986
\(856\) 0 0
\(857\) −15.0902 −0.515470 −0.257735 0.966216i \(-0.582976\pi\)
−0.257735 + 0.966216i \(0.582976\pi\)
\(858\) 0 0
\(859\) 14.9443 0.509892 0.254946 0.966955i \(-0.417942\pi\)
0.254946 + 0.966955i \(0.417942\pi\)
\(860\) 0 0
\(861\) −0.145898 −0.00497219
\(862\) 0 0
\(863\) −46.3262 −1.57696 −0.788482 0.615058i \(-0.789133\pi\)
−0.788482 + 0.615058i \(0.789133\pi\)
\(864\) 0 0
\(865\) −7.36068 −0.250271
\(866\) 0 0
\(867\) −1.61803 −0.0549513
\(868\) 0 0
\(869\) −2.24922 −0.0762997
\(870\) 0 0
\(871\) −16.1803 −0.548250
\(872\) 0 0
\(873\) −5.76393 −0.195080
\(874\) 0 0
\(875\) 5.94427 0.200953
\(876\) 0 0
\(877\) 18.2918 0.617670 0.308835 0.951116i \(-0.400061\pi\)
0.308835 + 0.951116i \(0.400061\pi\)
\(878\) 0 0
\(879\) 10.1803 0.343374
\(880\) 0 0
\(881\) −19.6738 −0.662826 −0.331413 0.943486i \(-0.607525\pi\)
−0.331413 + 0.943486i \(0.607525\pi\)
\(882\) 0 0
\(883\) 14.9787 0.504074 0.252037 0.967718i \(-0.418900\pi\)
0.252037 + 0.967718i \(0.418900\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −16.5066 −0.554237 −0.277118 0.960836i \(-0.589379\pi\)
−0.277118 + 0.960836i \(0.589379\pi\)
\(888\) 0 0
\(889\) 7.85410 0.263418
\(890\) 0 0
\(891\) 5.88854 0.197274
\(892\) 0 0
\(893\) 71.7771 2.40193
\(894\) 0 0
\(895\) −11.3607 −0.379746
\(896\) 0 0
\(897\) 15.4164 0.514739
\(898\) 0 0
\(899\) −36.1115 −1.20438
\(900\) 0 0
\(901\) 10.7984 0.359746
\(902\) 0 0
\(903\) −19.5623 −0.650993
\(904\) 0 0
\(905\) −7.70820 −0.256229
\(906\) 0 0
\(907\) −31.3050 −1.03946 −0.519732 0.854329i \(-0.673968\pi\)
−0.519732 + 0.854329i \(0.673968\pi\)
\(908\) 0 0
\(909\) 0.180340 0.00598150
\(910\) 0 0
\(911\) −14.4721 −0.479483 −0.239742 0.970837i \(-0.577063\pi\)
−0.239742 + 0.970837i \(0.577063\pi\)
\(912\) 0 0
\(913\) −11.7771 −0.389765
\(914\) 0 0
\(915\) 7.32624 0.242198
\(916\) 0 0
\(917\) 8.00000 0.264183
\(918\) 0 0
\(919\) −30.1591 −0.994855 −0.497428 0.867505i \(-0.665722\pi\)
−0.497428 + 0.867505i \(0.665722\pi\)
\(920\) 0 0
\(921\) −1.70820 −0.0562872
\(922\) 0 0
\(923\) 13.5279 0.445275
\(924\) 0 0
\(925\) −2.18034 −0.0716891
\(926\) 0 0
\(927\) −5.88854 −0.193405
\(928\) 0 0
\(929\) 21.2705 0.697863 0.348931 0.937148i \(-0.386545\pi\)
0.348931 + 0.937148i \(0.386545\pi\)
\(930\) 0 0
\(931\) −8.47214 −0.277663
\(932\) 0 0
\(933\) −43.0689 −1.41001
\(934\) 0 0
\(935\) −0.472136 −0.0154405
\(936\) 0 0
\(937\) −5.30495 −0.173305 −0.0866526 0.996239i \(-0.527617\pi\)
−0.0866526 + 0.996239i \(0.527617\pi\)
\(938\) 0 0
\(939\) 2.05573 0.0670862
\(940\) 0 0
\(941\) 45.9787 1.49886 0.749432 0.662082i \(-0.230327\pi\)
0.749432 + 0.662082i \(0.230327\pi\)
\(942\) 0 0
\(943\) 0.695048 0.0226339
\(944\) 0 0
\(945\) −3.38197 −0.110015
\(946\) 0 0
\(947\) 6.87539 0.223420 0.111710 0.993741i \(-0.464367\pi\)
0.111710 + 0.993741i \(0.464367\pi\)
\(948\) 0 0
\(949\) 8.83282 0.286725
\(950\) 0 0
\(951\) 45.5967 1.47858
\(952\) 0 0
\(953\) 32.3262 1.04715 0.523575 0.851980i \(-0.324598\pi\)
0.523575 + 0.851980i \(0.324598\pi\)
\(954\) 0 0
\(955\) 12.8541 0.415949
\(956\) 0 0
\(957\) −7.05573 −0.228079
\(958\) 0 0
\(959\) −2.38197 −0.0769177
\(960\) 0 0
\(961\) 9.02129 0.291009
\(962\) 0 0
\(963\) −4.11146 −0.132490
\(964\) 0 0
\(965\) −8.76393 −0.282121
\(966\) 0 0
\(967\) 37.7984 1.21551 0.607757 0.794123i \(-0.292070\pi\)
0.607757 + 0.794123i \(0.292070\pi\)
\(968\) 0 0
\(969\) 13.7082 0.440371
\(970\) 0 0
\(971\) 21.1246 0.677921 0.338961 0.940801i \(-0.389925\pi\)
0.338961 + 0.940801i \(0.389925\pi\)
\(972\) 0 0
\(973\) 14.0344 0.449924
\(974\) 0 0
\(975\) 9.23607 0.295791
\(976\) 0 0
\(977\) −2.21478 −0.0708571 −0.0354286 0.999372i \(-0.511280\pi\)
−0.0354286 + 0.999372i \(0.511280\pi\)
\(978\) 0 0
\(979\) −1.52786 −0.0488307
\(980\) 0 0
\(981\) 1.34752 0.0430231
\(982\) 0 0
\(983\) 6.90983 0.220389 0.110195 0.993910i \(-0.464853\pi\)
0.110195 + 0.993910i \(0.464853\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 0 0
\(987\) −13.7082 −0.436337
\(988\) 0 0
\(989\) 93.1935 2.96338
\(990\) 0 0
\(991\) −28.2492 −0.897366 −0.448683 0.893691i \(-0.648107\pi\)
−0.448683 + 0.893691i \(0.648107\pi\)
\(992\) 0 0
\(993\) −29.1803 −0.926010
\(994\) 0 0
\(995\) −7.47214 −0.236883
\(996\) 0 0
\(997\) 29.0344 0.919530 0.459765 0.888041i \(-0.347934\pi\)
0.459765 + 0.888041i \(0.347934\pi\)
\(998\) 0 0
\(999\) 2.58359 0.0817412
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 476.2.a.b.1.1 2
3.2 odd 2 4284.2.a.m.1.1 2
4.3 odd 2 1904.2.a.j.1.2 2
7.6 odd 2 3332.2.a.l.1.2 2
8.3 odd 2 7616.2.a.p.1.1 2
8.5 even 2 7616.2.a.u.1.2 2
17.16 even 2 8092.2.a.m.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
476.2.a.b.1.1 2 1.1 even 1 trivial
1904.2.a.j.1.2 2 4.3 odd 2
3332.2.a.l.1.2 2 7.6 odd 2
4284.2.a.m.1.1 2 3.2 odd 2
7616.2.a.p.1.1 2 8.3 odd 2
7616.2.a.u.1.2 2 8.5 even 2
8092.2.a.m.1.2 2 17.16 even 2