Properties

Label 8044.2.a.a.1.15
Level $8044$
Weight $2$
Character 8044.1
Self dual yes
Analytic conductor $64.232$
Analytic rank $1$
Dimension $80$
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] = [8044,2,Mod(1,8044)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8044, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8044.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8044 = 2^{2} \cdot 2011 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8044.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: \(64.2316633859\)
Analytic rank: \(1\)
Dimension: \(80\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Character \(\chi\) \(=\) 8044.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.41417 q^{3} -0.367373 q^{5} +2.85158 q^{7} +2.82820 q^{9} +O(q^{10})\) \(q-2.41417 q^{3} -0.367373 q^{5} +2.85158 q^{7} +2.82820 q^{9} +3.63133 q^{11} +4.28112 q^{13} +0.886899 q^{15} -0.777112 q^{17} +3.65314 q^{19} -6.88420 q^{21} +0.929268 q^{23} -4.86504 q^{25} +0.414759 q^{27} -9.53312 q^{29} -5.17630 q^{31} -8.76663 q^{33} -1.04759 q^{35} -4.24185 q^{37} -10.3353 q^{39} +3.10567 q^{41} +4.09145 q^{43} -1.03900 q^{45} -11.6131 q^{47} +1.13153 q^{49} +1.87608 q^{51} -9.14880 q^{53} -1.33405 q^{55} -8.81929 q^{57} -9.67604 q^{59} -10.5770 q^{61} +8.06484 q^{63} -1.57277 q^{65} -5.44331 q^{67} -2.24341 q^{69} +2.68267 q^{71} +11.0473 q^{73} +11.7450 q^{75} +10.3550 q^{77} +6.12019 q^{79} -9.48589 q^{81} -15.8266 q^{83} +0.285490 q^{85} +23.0145 q^{87} +9.77794 q^{89} +12.2080 q^{91} +12.4964 q^{93} -1.34206 q^{95} -1.42721 q^{97} +10.2701 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9} - 34 q^{11} - q^{13} - 24 q^{15} - 35 q^{17} - 31 q^{19} - 3 q^{21} - 43 q^{23} + 58 q^{25} - 49 q^{27} - 5 q^{29} - 56 q^{31} - 23 q^{33} - 72 q^{35} - 11 q^{37} - 74 q^{39} - 81 q^{41} - 34 q^{43} - 14 q^{45} - 64 q^{47} + 40 q^{49} - 59 q^{51} + 3 q^{53} - 53 q^{55} - 34 q^{57} - 116 q^{59} - 13 q^{61} - 61 q^{63} - 55 q^{65} - 22 q^{67} - 10 q^{69} - 86 q^{71} - 32 q^{73} - 85 q^{75} + 4 q^{77} - 88 q^{79} + 12 q^{81} - 83 q^{83} - 2 q^{85} - 87 q^{87} - 72 q^{89} - 49 q^{91} - 102 q^{95} - 34 q^{97} - 103 q^{99}+O(q^{100}) \) 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
\(3\) −2.41417 −1.39382 −0.696910 0.717159i \(-0.745442\pi\)
−0.696910 + 0.717159i \(0.745442\pi\)
\(4\) 0 0
\(5\) −0.367373 −0.164294 −0.0821470 0.996620i \(-0.526178\pi\)
−0.0821470 + 0.996620i \(0.526178\pi\)
\(6\) 0 0
\(7\) 2.85158 1.07780 0.538899 0.842371i \(-0.318841\pi\)
0.538899 + 0.842371i \(0.318841\pi\)
\(8\) 0 0
\(9\) 2.82820 0.942733
\(10\) 0 0
\(11\) 3.63133 1.09489 0.547443 0.836843i \(-0.315601\pi\)
0.547443 + 0.836843i \(0.315601\pi\)
\(12\) 0 0
\(13\) 4.28112 1.18737 0.593684 0.804698i \(-0.297673\pi\)
0.593684 + 0.804698i \(0.297673\pi\)
\(14\) 0 0
\(15\) 0.886899 0.228996
\(16\) 0 0
\(17\) −0.777112 −0.188477 −0.0942387 0.995550i \(-0.530042\pi\)
−0.0942387 + 0.995550i \(0.530042\pi\)
\(18\) 0 0
\(19\) 3.65314 0.838088 0.419044 0.907966i \(-0.362365\pi\)
0.419044 + 0.907966i \(0.362365\pi\)
\(20\) 0 0
\(21\) −6.88420 −1.50225
\(22\) 0 0
\(23\) 0.929268 0.193766 0.0968829 0.995296i \(-0.469113\pi\)
0.0968829 + 0.995296i \(0.469113\pi\)
\(24\) 0 0
\(25\) −4.86504 −0.973007
\(26\) 0 0
\(27\) 0.414759 0.0798204
\(28\) 0 0
\(29\) −9.53312 −1.77026 −0.885128 0.465348i \(-0.845929\pi\)
−0.885128 + 0.465348i \(0.845929\pi\)
\(30\) 0 0
\(31\) −5.17630 −0.929691 −0.464845 0.885392i \(-0.653890\pi\)
−0.464845 + 0.885392i \(0.653890\pi\)
\(32\) 0 0
\(33\) −8.76663 −1.52607
\(34\) 0 0
\(35\) −1.04759 −0.177076
\(36\) 0 0
\(37\) −4.24185 −0.697355 −0.348678 0.937243i \(-0.613369\pi\)
−0.348678 + 0.937243i \(0.613369\pi\)
\(38\) 0 0
\(39\) −10.3353 −1.65498
\(40\) 0 0
\(41\) 3.10567 0.485024 0.242512 0.970148i \(-0.422029\pi\)
0.242512 + 0.970148i \(0.422029\pi\)
\(42\) 0 0
\(43\) 4.09145 0.623941 0.311970 0.950092i \(-0.399011\pi\)
0.311970 + 0.950092i \(0.399011\pi\)
\(44\) 0 0
\(45\) −1.03900 −0.154885
\(46\) 0 0
\(47\) −11.6131 −1.69394 −0.846970 0.531641i \(-0.821575\pi\)
−0.846970 + 0.531641i \(0.821575\pi\)
\(48\) 0 0
\(49\) 1.13153 0.161647
\(50\) 0 0
\(51\) 1.87608 0.262704
\(52\) 0 0
\(53\) −9.14880 −1.25668 −0.628342 0.777937i \(-0.716266\pi\)
−0.628342 + 0.777937i \(0.716266\pi\)
\(54\) 0 0
\(55\) −1.33405 −0.179883
\(56\) 0 0
\(57\) −8.81929 −1.16814
\(58\) 0 0
\(59\) −9.67604 −1.25971 −0.629857 0.776711i \(-0.716886\pi\)
−0.629857 + 0.776711i \(0.716886\pi\)
\(60\) 0 0
\(61\) −10.5770 −1.35425 −0.677125 0.735868i \(-0.736774\pi\)
−0.677125 + 0.735868i \(0.736774\pi\)
\(62\) 0 0
\(63\) 8.06484 1.01607
\(64\) 0 0
\(65\) −1.57277 −0.195078
\(66\) 0 0
\(67\) −5.44331 −0.665006 −0.332503 0.943102i \(-0.607893\pi\)
−0.332503 + 0.943102i \(0.607893\pi\)
\(68\) 0 0
\(69\) −2.24341 −0.270075
\(70\) 0 0
\(71\) 2.68267 0.318375 0.159187 0.987248i \(-0.449113\pi\)
0.159187 + 0.987248i \(0.449113\pi\)
\(72\) 0 0
\(73\) 11.0473 1.29299 0.646496 0.762917i \(-0.276234\pi\)
0.646496 + 0.762917i \(0.276234\pi\)
\(74\) 0 0
\(75\) 11.7450 1.35620
\(76\) 0 0
\(77\) 10.3550 1.18007
\(78\) 0 0
\(79\) 6.12019 0.688575 0.344287 0.938864i \(-0.388121\pi\)
0.344287 + 0.938864i \(0.388121\pi\)
\(80\) 0 0
\(81\) −9.48589 −1.05399
\(82\) 0 0
\(83\) −15.8266 −1.73720 −0.868598 0.495518i \(-0.834978\pi\)
−0.868598 + 0.495518i \(0.834978\pi\)
\(84\) 0 0
\(85\) 0.285490 0.0309657
\(86\) 0 0
\(87\) 23.0145 2.46742
\(88\) 0 0
\(89\) 9.77794 1.03646 0.518230 0.855241i \(-0.326591\pi\)
0.518230 + 0.855241i \(0.326591\pi\)
\(90\) 0 0
\(91\) 12.2080 1.27974
\(92\) 0 0
\(93\) 12.4964 1.29582
\(94\) 0 0
\(95\) −1.34206 −0.137693
\(96\) 0 0
\(97\) −1.42721 −0.144911 −0.0724555 0.997372i \(-0.523084\pi\)
−0.0724555 + 0.997372i \(0.523084\pi\)
\(98\) 0 0
\(99\) 10.2701 1.03219
\(100\) 0 0
\(101\) 6.34347 0.631199 0.315599 0.948893i \(-0.397794\pi\)
0.315599 + 0.948893i \(0.397794\pi\)
\(102\) 0 0
\(103\) −3.00942 −0.296527 −0.148264 0.988948i \(-0.547368\pi\)
−0.148264 + 0.988948i \(0.547368\pi\)
\(104\) 0 0
\(105\) 2.52907 0.246812
\(106\) 0 0
\(107\) 3.51267 0.339582 0.169791 0.985480i \(-0.445691\pi\)
0.169791 + 0.985480i \(0.445691\pi\)
\(108\) 0 0
\(109\) 9.24429 0.885442 0.442721 0.896659i \(-0.354013\pi\)
0.442721 + 0.896659i \(0.354013\pi\)
\(110\) 0 0
\(111\) 10.2405 0.971987
\(112\) 0 0
\(113\) −0.547548 −0.0515090 −0.0257545 0.999668i \(-0.508199\pi\)
−0.0257545 + 0.999668i \(0.508199\pi\)
\(114\) 0 0
\(115\) −0.341388 −0.0318346
\(116\) 0 0
\(117\) 12.1079 1.11937
\(118\) 0 0
\(119\) −2.21600 −0.203140
\(120\) 0 0
\(121\) 2.18655 0.198777
\(122\) 0 0
\(123\) −7.49760 −0.676036
\(124\) 0 0
\(125\) 3.62415 0.324153
\(126\) 0 0
\(127\) −20.6989 −1.83673 −0.918367 0.395730i \(-0.870492\pi\)
−0.918367 + 0.395730i \(0.870492\pi\)
\(128\) 0 0
\(129\) −9.87745 −0.869661
\(130\) 0 0
\(131\) −13.0504 −1.14022 −0.570109 0.821569i \(-0.693099\pi\)
−0.570109 + 0.821569i \(0.693099\pi\)
\(132\) 0 0
\(133\) 10.4172 0.903289
\(134\) 0 0
\(135\) −0.152371 −0.0131140
\(136\) 0 0
\(137\) −16.9610 −1.44908 −0.724539 0.689234i \(-0.757947\pi\)
−0.724539 + 0.689234i \(0.757947\pi\)
\(138\) 0 0
\(139\) −17.6667 −1.49847 −0.749236 0.662303i \(-0.769579\pi\)
−0.749236 + 0.662303i \(0.769579\pi\)
\(140\) 0 0
\(141\) 28.0359 2.36105
\(142\) 0 0
\(143\) 15.5462 1.30003
\(144\) 0 0
\(145\) 3.50221 0.290842
\(146\) 0 0
\(147\) −2.73170 −0.225307
\(148\) 0 0
\(149\) −15.8664 −1.29983 −0.649914 0.760007i \(-0.725195\pi\)
−0.649914 + 0.760007i \(0.725195\pi\)
\(150\) 0 0
\(151\) −2.29820 −0.187025 −0.0935123 0.995618i \(-0.529809\pi\)
−0.0935123 + 0.995618i \(0.529809\pi\)
\(152\) 0 0
\(153\) −2.19783 −0.177684
\(154\) 0 0
\(155\) 1.90163 0.152743
\(156\) 0 0
\(157\) −4.69403 −0.374624 −0.187312 0.982300i \(-0.559978\pi\)
−0.187312 + 0.982300i \(0.559978\pi\)
\(158\) 0 0
\(159\) 22.0867 1.75159
\(160\) 0 0
\(161\) 2.64989 0.208840
\(162\) 0 0
\(163\) 24.2952 1.90295 0.951473 0.307731i \(-0.0995697\pi\)
0.951473 + 0.307731i \(0.0995697\pi\)
\(164\) 0 0
\(165\) 3.22062 0.250725
\(166\) 0 0
\(167\) 1.62550 0.125785 0.0628926 0.998020i \(-0.479967\pi\)
0.0628926 + 0.998020i \(0.479967\pi\)
\(168\) 0 0
\(169\) 5.32798 0.409844
\(170\) 0 0
\(171\) 10.3318 0.790093
\(172\) 0 0
\(173\) 0.0184491 0.00140266 0.000701330 1.00000i \(-0.499777\pi\)
0.000701330 1.00000i \(0.499777\pi\)
\(174\) 0 0
\(175\) −13.8731 −1.04870
\(176\) 0 0
\(177\) 23.3596 1.75581
\(178\) 0 0
\(179\) 4.99441 0.373300 0.186650 0.982426i \(-0.440237\pi\)
0.186650 + 0.982426i \(0.440237\pi\)
\(180\) 0 0
\(181\) 22.6284 1.68195 0.840977 0.541071i \(-0.181981\pi\)
0.840977 + 0.541071i \(0.181981\pi\)
\(182\) 0 0
\(183\) 25.5347 1.88758
\(184\) 0 0
\(185\) 1.55834 0.114571
\(186\) 0 0
\(187\) −2.82195 −0.206361
\(188\) 0 0
\(189\) 1.18272 0.0860302
\(190\) 0 0
\(191\) 9.09224 0.657891 0.328946 0.944349i \(-0.393307\pi\)
0.328946 + 0.944349i \(0.393307\pi\)
\(192\) 0 0
\(193\) −4.77854 −0.343967 −0.171984 0.985100i \(-0.555018\pi\)
−0.171984 + 0.985100i \(0.555018\pi\)
\(194\) 0 0
\(195\) 3.79692 0.271903
\(196\) 0 0
\(197\) 9.32913 0.664673 0.332337 0.943161i \(-0.392163\pi\)
0.332337 + 0.943161i \(0.392163\pi\)
\(198\) 0 0
\(199\) 22.2474 1.57708 0.788540 0.614984i \(-0.210837\pi\)
0.788540 + 0.614984i \(0.210837\pi\)
\(200\) 0 0
\(201\) 13.1411 0.926898
\(202\) 0 0
\(203\) −27.1845 −1.90798
\(204\) 0 0
\(205\) −1.14094 −0.0796866
\(206\) 0 0
\(207\) 2.62815 0.182669
\(208\) 0 0
\(209\) 13.2658 0.917611
\(210\) 0 0
\(211\) −24.2416 −1.66886 −0.834430 0.551114i \(-0.814203\pi\)
−0.834430 + 0.551114i \(0.814203\pi\)
\(212\) 0 0
\(213\) −6.47642 −0.443757
\(214\) 0 0
\(215\) −1.50309 −0.102510
\(216\) 0 0
\(217\) −14.7607 −1.00202
\(218\) 0 0
\(219\) −26.6701 −1.80220
\(220\) 0 0
\(221\) −3.32691 −0.223792
\(222\) 0 0
\(223\) −24.7858 −1.65978 −0.829890 0.557927i \(-0.811597\pi\)
−0.829890 + 0.557927i \(0.811597\pi\)
\(224\) 0 0
\(225\) −13.7593 −0.917286
\(226\) 0 0
\(227\) −5.82621 −0.386699 −0.193350 0.981130i \(-0.561935\pi\)
−0.193350 + 0.981130i \(0.561935\pi\)
\(228\) 0 0
\(229\) 24.3234 1.60733 0.803666 0.595080i \(-0.202880\pi\)
0.803666 + 0.595080i \(0.202880\pi\)
\(230\) 0 0
\(231\) −24.9988 −1.64480
\(232\) 0 0
\(233\) −13.4156 −0.878883 −0.439442 0.898271i \(-0.644824\pi\)
−0.439442 + 0.898271i \(0.644824\pi\)
\(234\) 0 0
\(235\) 4.26632 0.278304
\(236\) 0 0
\(237\) −14.7752 −0.959749
\(238\) 0 0
\(239\) −11.2470 −0.727507 −0.363754 0.931495i \(-0.618505\pi\)
−0.363754 + 0.931495i \(0.618505\pi\)
\(240\) 0 0
\(241\) 2.55764 0.164752 0.0823760 0.996601i \(-0.473749\pi\)
0.0823760 + 0.996601i \(0.473749\pi\)
\(242\) 0 0
\(243\) 21.6562 1.38925
\(244\) 0 0
\(245\) −0.415693 −0.0265576
\(246\) 0 0
\(247\) 15.6395 0.995119
\(248\) 0 0
\(249\) 38.2080 2.42134
\(250\) 0 0
\(251\) −24.4374 −1.54248 −0.771238 0.636547i \(-0.780362\pi\)
−0.771238 + 0.636547i \(0.780362\pi\)
\(252\) 0 0
\(253\) 3.37448 0.212152
\(254\) 0 0
\(255\) −0.689220 −0.0431606
\(256\) 0 0
\(257\) −5.49352 −0.342677 −0.171338 0.985212i \(-0.554809\pi\)
−0.171338 + 0.985212i \(0.554809\pi\)
\(258\) 0 0
\(259\) −12.0960 −0.751607
\(260\) 0 0
\(261\) −26.9615 −1.66888
\(262\) 0 0
\(263\) 26.0407 1.60574 0.802868 0.596157i \(-0.203306\pi\)
0.802868 + 0.596157i \(0.203306\pi\)
\(264\) 0 0
\(265\) 3.36102 0.206466
\(266\) 0 0
\(267\) −23.6056 −1.44464
\(268\) 0 0
\(269\) 24.0104 1.46394 0.731969 0.681338i \(-0.238601\pi\)
0.731969 + 0.681338i \(0.238601\pi\)
\(270\) 0 0
\(271\) −29.4310 −1.78780 −0.893902 0.448262i \(-0.852043\pi\)
−0.893902 + 0.448262i \(0.852043\pi\)
\(272\) 0 0
\(273\) −29.4721 −1.78373
\(274\) 0 0
\(275\) −17.6666 −1.06533
\(276\) 0 0
\(277\) −2.76518 −0.166143 −0.0830716 0.996544i \(-0.526473\pi\)
−0.0830716 + 0.996544i \(0.526473\pi\)
\(278\) 0 0
\(279\) −14.6396 −0.876450
\(280\) 0 0
\(281\) −19.9931 −1.19269 −0.596343 0.802730i \(-0.703380\pi\)
−0.596343 + 0.802730i \(0.703380\pi\)
\(282\) 0 0
\(283\) 24.1579 1.43604 0.718018 0.696024i \(-0.245049\pi\)
0.718018 + 0.696024i \(0.245049\pi\)
\(284\) 0 0
\(285\) 3.23997 0.191919
\(286\) 0 0
\(287\) 8.85607 0.522758
\(288\) 0 0
\(289\) −16.3961 −0.964476
\(290\) 0 0
\(291\) 3.44552 0.201980
\(292\) 0 0
\(293\) −3.03723 −0.177437 −0.0887185 0.996057i \(-0.528277\pi\)
−0.0887185 + 0.996057i \(0.528277\pi\)
\(294\) 0 0
\(295\) 3.55471 0.206963
\(296\) 0 0
\(297\) 1.50613 0.0873943
\(298\) 0 0
\(299\) 3.97831 0.230072
\(300\) 0 0
\(301\) 11.6671 0.672482
\(302\) 0 0
\(303\) −15.3142 −0.879777
\(304\) 0 0
\(305\) 3.88571 0.222495
\(306\) 0 0
\(307\) 4.60560 0.262856 0.131428 0.991326i \(-0.458044\pi\)
0.131428 + 0.991326i \(0.458044\pi\)
\(308\) 0 0
\(309\) 7.26524 0.413305
\(310\) 0 0
\(311\) 5.97411 0.338761 0.169380 0.985551i \(-0.445823\pi\)
0.169380 + 0.985551i \(0.445823\pi\)
\(312\) 0 0
\(313\) −12.6450 −0.714736 −0.357368 0.933964i \(-0.616326\pi\)
−0.357368 + 0.933964i \(0.616326\pi\)
\(314\) 0 0
\(315\) −2.96280 −0.166935
\(316\) 0 0
\(317\) 16.3654 0.919174 0.459587 0.888133i \(-0.347997\pi\)
0.459587 + 0.888133i \(0.347997\pi\)
\(318\) 0 0
\(319\) −34.6179 −1.93823
\(320\) 0 0
\(321\) −8.48016 −0.473317
\(322\) 0 0
\(323\) −2.83890 −0.157961
\(324\) 0 0
\(325\) −20.8278 −1.15532
\(326\) 0 0
\(327\) −22.3173 −1.23415
\(328\) 0 0
\(329\) −33.1156 −1.82572
\(330\) 0 0
\(331\) 14.7402 0.810194 0.405097 0.914274i \(-0.367238\pi\)
0.405097 + 0.914274i \(0.367238\pi\)
\(332\) 0 0
\(333\) −11.9968 −0.657419
\(334\) 0 0
\(335\) 1.99972 0.109257
\(336\) 0 0
\(337\) 23.2279 1.26530 0.632652 0.774436i \(-0.281966\pi\)
0.632652 + 0.774436i \(0.281966\pi\)
\(338\) 0 0
\(339\) 1.32187 0.0717942
\(340\) 0 0
\(341\) −18.7968 −1.01791
\(342\) 0 0
\(343\) −16.7344 −0.903575
\(344\) 0 0
\(345\) 0.824167 0.0443717
\(346\) 0 0
\(347\) −22.4020 −1.20260 −0.601301 0.799022i \(-0.705351\pi\)
−0.601301 + 0.799022i \(0.705351\pi\)
\(348\) 0 0
\(349\) 0.940983 0.0503697 0.0251848 0.999683i \(-0.491983\pi\)
0.0251848 + 0.999683i \(0.491983\pi\)
\(350\) 0 0
\(351\) 1.77563 0.0947763
\(352\) 0 0
\(353\) 7.07233 0.376422 0.188211 0.982129i \(-0.439731\pi\)
0.188211 + 0.982129i \(0.439731\pi\)
\(354\) 0 0
\(355\) −0.985541 −0.0523071
\(356\) 0 0
\(357\) 5.34979 0.283141
\(358\) 0 0
\(359\) 15.7926 0.833502 0.416751 0.909021i \(-0.363169\pi\)
0.416751 + 0.909021i \(0.363169\pi\)
\(360\) 0 0
\(361\) −5.65457 −0.297609
\(362\) 0 0
\(363\) −5.27870 −0.277060
\(364\) 0 0
\(365\) −4.05849 −0.212431
\(366\) 0 0
\(367\) −29.5186 −1.54086 −0.770430 0.637525i \(-0.779958\pi\)
−0.770430 + 0.637525i \(0.779958\pi\)
\(368\) 0 0
\(369\) 8.78345 0.457248
\(370\) 0 0
\(371\) −26.0886 −1.35445
\(372\) 0 0
\(373\) −2.15659 −0.111664 −0.0558321 0.998440i \(-0.517781\pi\)
−0.0558321 + 0.998440i \(0.517781\pi\)
\(374\) 0 0
\(375\) −8.74929 −0.451811
\(376\) 0 0
\(377\) −40.8124 −2.10195
\(378\) 0 0
\(379\) −4.56186 −0.234327 −0.117164 0.993113i \(-0.537380\pi\)
−0.117164 + 0.993113i \(0.537380\pi\)
\(380\) 0 0
\(381\) 49.9707 2.56008
\(382\) 0 0
\(383\) −23.7538 −1.21376 −0.606880 0.794793i \(-0.707579\pi\)
−0.606880 + 0.794793i \(0.707579\pi\)
\(384\) 0 0
\(385\) −3.80416 −0.193878
\(386\) 0 0
\(387\) 11.5714 0.588209
\(388\) 0 0
\(389\) 28.7427 1.45731 0.728657 0.684879i \(-0.240145\pi\)
0.728657 + 0.684879i \(0.240145\pi\)
\(390\) 0 0
\(391\) −0.722146 −0.0365205
\(392\) 0 0
\(393\) 31.5058 1.58926
\(394\) 0 0
\(395\) −2.24839 −0.113129
\(396\) 0 0
\(397\) 37.4648 1.88030 0.940152 0.340756i \(-0.110683\pi\)
0.940152 + 0.340756i \(0.110683\pi\)
\(398\) 0 0
\(399\) −25.1489 −1.25902
\(400\) 0 0
\(401\) 32.8520 1.64055 0.820274 0.571970i \(-0.193821\pi\)
0.820274 + 0.571970i \(0.193821\pi\)
\(402\) 0 0
\(403\) −22.1604 −1.10389
\(404\) 0 0
\(405\) 3.48486 0.173164
\(406\) 0 0
\(407\) −15.4035 −0.763525
\(408\) 0 0
\(409\) 3.16645 0.156571 0.0782854 0.996931i \(-0.475055\pi\)
0.0782854 + 0.996931i \(0.475055\pi\)
\(410\) 0 0
\(411\) 40.9467 2.01975
\(412\) 0 0
\(413\) −27.5920 −1.35772
\(414\) 0 0
\(415\) 5.81426 0.285411
\(416\) 0 0
\(417\) 42.6504 2.08860
\(418\) 0 0
\(419\) −23.1456 −1.13074 −0.565368 0.824838i \(-0.691266\pi\)
−0.565368 + 0.824838i \(0.691266\pi\)
\(420\) 0 0
\(421\) 20.2102 0.984987 0.492493 0.870316i \(-0.336086\pi\)
0.492493 + 0.870316i \(0.336086\pi\)
\(422\) 0 0
\(423\) −32.8440 −1.59693
\(424\) 0 0
\(425\) 3.78068 0.183390
\(426\) 0 0
\(427\) −30.1613 −1.45961
\(428\) 0 0
\(429\) −37.5310 −1.81201
\(430\) 0 0
\(431\) −28.6640 −1.38070 −0.690348 0.723478i \(-0.742543\pi\)
−0.690348 + 0.723478i \(0.742543\pi\)
\(432\) 0 0
\(433\) −17.7452 −0.852778 −0.426389 0.904540i \(-0.640215\pi\)
−0.426389 + 0.904540i \(0.640215\pi\)
\(434\) 0 0
\(435\) −8.45491 −0.405382
\(436\) 0 0
\(437\) 3.39475 0.162393
\(438\) 0 0
\(439\) −17.5014 −0.835295 −0.417648 0.908609i \(-0.637145\pi\)
−0.417648 + 0.908609i \(0.637145\pi\)
\(440\) 0 0
\(441\) 3.20019 0.152390
\(442\) 0 0
\(443\) −12.7660 −0.606533 −0.303266 0.952906i \(-0.598077\pi\)
−0.303266 + 0.952906i \(0.598077\pi\)
\(444\) 0 0
\(445\) −3.59215 −0.170284
\(446\) 0 0
\(447\) 38.3042 1.81173
\(448\) 0 0
\(449\) 12.2187 0.576635 0.288317 0.957535i \(-0.406904\pi\)
0.288317 + 0.957535i \(0.406904\pi\)
\(450\) 0 0
\(451\) 11.2777 0.531046
\(452\) 0 0
\(453\) 5.54823 0.260679
\(454\) 0 0
\(455\) −4.48487 −0.210254
\(456\) 0 0
\(457\) 16.6417 0.778464 0.389232 0.921140i \(-0.372740\pi\)
0.389232 + 0.921140i \(0.372740\pi\)
\(458\) 0 0
\(459\) −0.322314 −0.0150444
\(460\) 0 0
\(461\) −33.4364 −1.55729 −0.778643 0.627467i \(-0.784092\pi\)
−0.778643 + 0.627467i \(0.784092\pi\)
\(462\) 0 0
\(463\) −27.3663 −1.27182 −0.635910 0.771763i \(-0.719375\pi\)
−0.635910 + 0.771763i \(0.719375\pi\)
\(464\) 0 0
\(465\) −4.59085 −0.212896
\(466\) 0 0
\(467\) 8.52445 0.394464 0.197232 0.980357i \(-0.436805\pi\)
0.197232 + 0.980357i \(0.436805\pi\)
\(468\) 0 0
\(469\) −15.5221 −0.716742
\(470\) 0 0
\(471\) 11.3322 0.522158
\(472\) 0 0
\(473\) 14.8574 0.683144
\(474\) 0 0
\(475\) −17.7727 −0.815466
\(476\) 0 0
\(477\) −25.8746 −1.18472
\(478\) 0 0
\(479\) 2.34014 0.106924 0.0534618 0.998570i \(-0.482974\pi\)
0.0534618 + 0.998570i \(0.482974\pi\)
\(480\) 0 0
\(481\) −18.1598 −0.828018
\(482\) 0 0
\(483\) −6.39727 −0.291086
\(484\) 0 0
\(485\) 0.524317 0.0238080
\(486\) 0 0
\(487\) 16.1539 0.732005 0.366003 0.930614i \(-0.380726\pi\)
0.366003 + 0.930614i \(0.380726\pi\)
\(488\) 0 0
\(489\) −58.6526 −2.65236
\(490\) 0 0
\(491\) −13.5042 −0.609436 −0.304718 0.952443i \(-0.598562\pi\)
−0.304718 + 0.952443i \(0.598562\pi\)
\(492\) 0 0
\(493\) 7.40830 0.333653
\(494\) 0 0
\(495\) −3.77296 −0.169582
\(496\) 0 0
\(497\) 7.64986 0.343143
\(498\) 0 0
\(499\) 2.99308 0.133989 0.0669944 0.997753i \(-0.478659\pi\)
0.0669944 + 0.997753i \(0.478659\pi\)
\(500\) 0 0
\(501\) −3.92424 −0.175322
\(502\) 0 0
\(503\) 10.8606 0.484250 0.242125 0.970245i \(-0.422156\pi\)
0.242125 + 0.970245i \(0.422156\pi\)
\(504\) 0 0
\(505\) −2.33042 −0.103702
\(506\) 0 0
\(507\) −12.8626 −0.571249
\(508\) 0 0
\(509\) −4.27449 −0.189463 −0.0947316 0.995503i \(-0.530199\pi\)
−0.0947316 + 0.995503i \(0.530199\pi\)
\(510\) 0 0
\(511\) 31.5024 1.39358
\(512\) 0 0
\(513\) 1.51517 0.0668965
\(514\) 0 0
\(515\) 1.10558 0.0487176
\(516\) 0 0
\(517\) −42.1709 −1.85467
\(518\) 0 0
\(519\) −0.0445392 −0.00195505
\(520\) 0 0
\(521\) −8.66847 −0.379773 −0.189886 0.981806i \(-0.560812\pi\)
−0.189886 + 0.981806i \(0.560812\pi\)
\(522\) 0 0
\(523\) 34.6824 1.51656 0.758279 0.651930i \(-0.226041\pi\)
0.758279 + 0.651930i \(0.226041\pi\)
\(524\) 0 0
\(525\) 33.4919 1.46171
\(526\) 0 0
\(527\) 4.02257 0.175226
\(528\) 0 0
\(529\) −22.1365 −0.962455
\(530\) 0 0
\(531\) −27.3658 −1.18757
\(532\) 0 0
\(533\) 13.2957 0.575902
\(534\) 0 0
\(535\) −1.29046 −0.0557914
\(536\) 0 0
\(537\) −12.0573 −0.520313
\(538\) 0 0
\(539\) 4.10895 0.176985
\(540\) 0 0
\(541\) 15.2221 0.654448 0.327224 0.944947i \(-0.393887\pi\)
0.327224 + 0.944947i \(0.393887\pi\)
\(542\) 0 0
\(543\) −54.6287 −2.34434
\(544\) 0 0
\(545\) −3.39610 −0.145473
\(546\) 0 0
\(547\) 11.6742 0.499152 0.249576 0.968355i \(-0.419709\pi\)
0.249576 + 0.968355i \(0.419709\pi\)
\(548\) 0 0
\(549\) −29.9139 −1.27670
\(550\) 0 0
\(551\) −34.8258 −1.48363
\(552\) 0 0
\(553\) 17.4522 0.742144
\(554\) 0 0
\(555\) −3.76209 −0.159692
\(556\) 0 0
\(557\) −28.6636 −1.21452 −0.607258 0.794505i \(-0.707731\pi\)
−0.607258 + 0.794505i \(0.707731\pi\)
\(558\) 0 0
\(559\) 17.5160 0.740848
\(560\) 0 0
\(561\) 6.81266 0.287631
\(562\) 0 0
\(563\) −31.5163 −1.32826 −0.664128 0.747619i \(-0.731197\pi\)
−0.664128 + 0.747619i \(0.731197\pi\)
\(564\) 0 0
\(565\) 0.201154 0.00846262
\(566\) 0 0
\(567\) −27.0498 −1.13599
\(568\) 0 0
\(569\) −9.96701 −0.417839 −0.208920 0.977933i \(-0.566995\pi\)
−0.208920 + 0.977933i \(0.566995\pi\)
\(570\) 0 0
\(571\) −3.32818 −0.139280 −0.0696401 0.997572i \(-0.522185\pi\)
−0.0696401 + 0.997572i \(0.522185\pi\)
\(572\) 0 0
\(573\) −21.9502 −0.916982
\(574\) 0 0
\(575\) −4.52093 −0.188536
\(576\) 0 0
\(577\) 18.9601 0.789318 0.394659 0.918828i \(-0.370863\pi\)
0.394659 + 0.918828i \(0.370863\pi\)
\(578\) 0 0
\(579\) 11.5362 0.479428
\(580\) 0 0
\(581\) −45.1309 −1.87234
\(582\) 0 0
\(583\) −33.2223 −1.37593
\(584\) 0 0
\(585\) −4.44809 −0.183906
\(586\) 0 0
\(587\) 44.3196 1.82927 0.914633 0.404286i \(-0.132480\pi\)
0.914633 + 0.404286i \(0.132480\pi\)
\(588\) 0 0
\(589\) −18.9097 −0.779163
\(590\) 0 0
\(591\) −22.5221 −0.926434
\(592\) 0 0
\(593\) −16.7045 −0.685971 −0.342986 0.939341i \(-0.611438\pi\)
−0.342986 + 0.939341i \(0.611438\pi\)
\(594\) 0 0
\(595\) 0.814098 0.0333748
\(596\) 0 0
\(597\) −53.7090 −2.19816
\(598\) 0 0
\(599\) 6.51457 0.266178 0.133089 0.991104i \(-0.457510\pi\)
0.133089 + 0.991104i \(0.457510\pi\)
\(600\) 0 0
\(601\) −39.7093 −1.61978 −0.809888 0.586585i \(-0.800472\pi\)
−0.809888 + 0.586585i \(0.800472\pi\)
\(602\) 0 0
\(603\) −15.3948 −0.626923
\(604\) 0 0
\(605\) −0.803279 −0.0326579
\(606\) 0 0
\(607\) 40.7313 1.65323 0.826616 0.562767i \(-0.190263\pi\)
0.826616 + 0.562767i \(0.190263\pi\)
\(608\) 0 0
\(609\) 65.6278 2.65937
\(610\) 0 0
\(611\) −49.7169 −2.01133
\(612\) 0 0
\(613\) 13.7368 0.554824 0.277412 0.960751i \(-0.410523\pi\)
0.277412 + 0.960751i \(0.410523\pi\)
\(614\) 0 0
\(615\) 2.75441 0.111069
\(616\) 0 0
\(617\) 7.06795 0.284545 0.142273 0.989828i \(-0.454559\pi\)
0.142273 + 0.989828i \(0.454559\pi\)
\(618\) 0 0
\(619\) −6.29328 −0.252948 −0.126474 0.991970i \(-0.540366\pi\)
−0.126474 + 0.991970i \(0.540366\pi\)
\(620\) 0 0
\(621\) 0.385423 0.0154665
\(622\) 0 0
\(623\) 27.8826 1.11709
\(624\) 0 0
\(625\) 22.9938 0.919751
\(626\) 0 0
\(627\) −32.0257 −1.27898
\(628\) 0 0
\(629\) 3.29639 0.131436
\(630\) 0 0
\(631\) 43.0056 1.71202 0.856012 0.516956i \(-0.172935\pi\)
0.856012 + 0.516956i \(0.172935\pi\)
\(632\) 0 0
\(633\) 58.5232 2.32609
\(634\) 0 0
\(635\) 7.60423 0.301765
\(636\) 0 0
\(637\) 4.84421 0.191934
\(638\) 0 0
\(639\) 7.58713 0.300142
\(640\) 0 0
\(641\) −10.1266 −0.399977 −0.199988 0.979798i \(-0.564090\pi\)
−0.199988 + 0.979798i \(0.564090\pi\)
\(642\) 0 0
\(643\) 16.2279 0.639964 0.319982 0.947424i \(-0.396323\pi\)
0.319982 + 0.947424i \(0.396323\pi\)
\(644\) 0 0
\(645\) 3.62870 0.142880
\(646\) 0 0
\(647\) −23.6203 −0.928611 −0.464305 0.885675i \(-0.653696\pi\)
−0.464305 + 0.885675i \(0.653696\pi\)
\(648\) 0 0
\(649\) −35.1369 −1.37924
\(650\) 0 0
\(651\) 35.6347 1.39663
\(652\) 0 0
\(653\) −13.2350 −0.517925 −0.258962 0.965887i \(-0.583381\pi\)
−0.258962 + 0.965887i \(0.583381\pi\)
\(654\) 0 0
\(655\) 4.79436 0.187331
\(656\) 0 0
\(657\) 31.2440 1.21895
\(658\) 0 0
\(659\) 44.9188 1.74979 0.874894 0.484315i \(-0.160931\pi\)
0.874894 + 0.484315i \(0.160931\pi\)
\(660\) 0 0
\(661\) −31.7701 −1.23572 −0.617858 0.786290i \(-0.711999\pi\)
−0.617858 + 0.786290i \(0.711999\pi\)
\(662\) 0 0
\(663\) 8.03171 0.311926
\(664\) 0 0
\(665\) −3.82701 −0.148405
\(666\) 0 0
\(667\) −8.85882 −0.343015
\(668\) 0 0
\(669\) 59.8370 2.31343
\(670\) 0 0
\(671\) −38.4087 −1.48275
\(672\) 0 0
\(673\) −8.43016 −0.324959 −0.162479 0.986712i \(-0.551949\pi\)
−0.162479 + 0.986712i \(0.551949\pi\)
\(674\) 0 0
\(675\) −2.01782 −0.0776659
\(676\) 0 0
\(677\) 7.57365 0.291079 0.145540 0.989352i \(-0.453508\pi\)
0.145540 + 0.989352i \(0.453508\pi\)
\(678\) 0 0
\(679\) −4.06980 −0.156185
\(680\) 0 0
\(681\) 14.0654 0.538989
\(682\) 0 0
\(683\) 32.3196 1.23668 0.618338 0.785912i \(-0.287806\pi\)
0.618338 + 0.785912i \(0.287806\pi\)
\(684\) 0 0
\(685\) 6.23102 0.238075
\(686\) 0 0
\(687\) −58.7206 −2.24033
\(688\) 0 0
\(689\) −39.1671 −1.49215
\(690\) 0 0
\(691\) 25.3266 0.963468 0.481734 0.876318i \(-0.340007\pi\)
0.481734 + 0.876318i \(0.340007\pi\)
\(692\) 0 0
\(693\) 29.2861 1.11249
\(694\) 0 0
\(695\) 6.49028 0.246190
\(696\) 0 0
\(697\) −2.41345 −0.0914161
\(698\) 0 0
\(699\) 32.3874 1.22500
\(700\) 0 0
\(701\) −3.65906 −0.138201 −0.0691003 0.997610i \(-0.522013\pi\)
−0.0691003 + 0.997610i \(0.522013\pi\)
\(702\) 0 0
\(703\) −15.4961 −0.584445
\(704\) 0 0
\(705\) −10.2996 −0.387906
\(706\) 0 0
\(707\) 18.0889 0.680304
\(708\) 0 0
\(709\) −16.9484 −0.636510 −0.318255 0.948005i \(-0.603097\pi\)
−0.318255 + 0.948005i \(0.603097\pi\)
\(710\) 0 0
\(711\) 17.3091 0.649142
\(712\) 0 0
\(713\) −4.81017 −0.180142
\(714\) 0 0
\(715\) −5.71123 −0.213588
\(716\) 0 0
\(717\) 27.1521 1.01401
\(718\) 0 0
\(719\) 2.41889 0.0902094 0.0451047 0.998982i \(-0.485638\pi\)
0.0451047 + 0.998982i \(0.485638\pi\)
\(720\) 0 0
\(721\) −8.58161 −0.319596
\(722\) 0 0
\(723\) −6.17456 −0.229634
\(724\) 0 0
\(725\) 46.3790 1.72247
\(726\) 0 0
\(727\) −3.06509 −0.113678 −0.0568389 0.998383i \(-0.518102\pi\)
−0.0568389 + 0.998383i \(0.518102\pi\)
\(728\) 0 0
\(729\) −23.8241 −0.882373
\(730\) 0 0
\(731\) −3.17952 −0.117599
\(732\) 0 0
\(733\) −7.06199 −0.260841 −0.130420 0.991459i \(-0.541633\pi\)
−0.130420 + 0.991459i \(0.541633\pi\)
\(734\) 0 0
\(735\) 1.00355 0.0370165
\(736\) 0 0
\(737\) −19.7664 −0.728106
\(738\) 0 0
\(739\) 27.5968 1.01517 0.507583 0.861603i \(-0.330539\pi\)
0.507583 + 0.861603i \(0.330539\pi\)
\(740\) 0 0
\(741\) −37.7564 −1.38702
\(742\) 0 0
\(743\) −32.9671 −1.20945 −0.604723 0.796436i \(-0.706716\pi\)
−0.604723 + 0.796436i \(0.706716\pi\)
\(744\) 0 0
\(745\) 5.82889 0.213554
\(746\) 0 0
\(747\) −44.7608 −1.63771
\(748\) 0 0
\(749\) 10.0167 0.366001
\(750\) 0 0
\(751\) −41.4940 −1.51414 −0.757069 0.653334i \(-0.773370\pi\)
−0.757069 + 0.653334i \(0.773370\pi\)
\(752\) 0 0
\(753\) 58.9960 2.14993
\(754\) 0 0
\(755\) 0.844295 0.0307271
\(756\) 0 0
\(757\) −6.79135 −0.246836 −0.123418 0.992355i \(-0.539386\pi\)
−0.123418 + 0.992355i \(0.539386\pi\)
\(758\) 0 0
\(759\) −8.14655 −0.295701
\(760\) 0 0
\(761\) −38.1721 −1.38374 −0.691870 0.722023i \(-0.743213\pi\)
−0.691870 + 0.722023i \(0.743213\pi\)
\(762\) 0 0
\(763\) 26.3609 0.954327
\(764\) 0 0
\(765\) 0.807422 0.0291924
\(766\) 0 0
\(767\) −41.4243 −1.49574
\(768\) 0 0
\(769\) 9.54758 0.344295 0.172147 0.985071i \(-0.444930\pi\)
0.172147 + 0.985071i \(0.444930\pi\)
\(770\) 0 0
\(771\) 13.2623 0.477629
\(772\) 0 0
\(773\) 13.0303 0.468667 0.234333 0.972156i \(-0.424709\pi\)
0.234333 + 0.972156i \(0.424709\pi\)
\(774\) 0 0
\(775\) 25.1829 0.904596
\(776\) 0 0
\(777\) 29.2017 1.04761
\(778\) 0 0
\(779\) 11.3454 0.406493
\(780\) 0 0
\(781\) 9.74167 0.348584
\(782\) 0 0
\(783\) −3.95395 −0.141303
\(784\) 0 0
\(785\) 1.72446 0.0615485
\(786\) 0 0
\(787\) 10.5859 0.377346 0.188673 0.982040i \(-0.439581\pi\)
0.188673 + 0.982040i \(0.439581\pi\)
\(788\) 0 0
\(789\) −62.8665 −2.23811
\(790\) 0 0
\(791\) −1.56138 −0.0555162
\(792\) 0 0
\(793\) −45.2815 −1.60799
\(794\) 0 0
\(795\) −8.11406 −0.287776
\(796\) 0 0
\(797\) 27.3773 0.969754 0.484877 0.874582i \(-0.338864\pi\)
0.484877 + 0.874582i \(0.338864\pi\)
\(798\) 0 0
\(799\) 9.02465 0.319269
\(800\) 0 0
\(801\) 27.6540 0.977104
\(802\) 0 0
\(803\) 40.1165 1.41568
\(804\) 0 0
\(805\) −0.973496 −0.0343112
\(806\) 0 0
\(807\) −57.9650 −2.04047
\(808\) 0 0
\(809\) 8.77858 0.308638 0.154319 0.988021i \(-0.450682\pi\)
0.154319 + 0.988021i \(0.450682\pi\)
\(810\) 0 0
\(811\) −46.3590 −1.62788 −0.813942 0.580946i \(-0.802683\pi\)
−0.813942 + 0.580946i \(0.802683\pi\)
\(812\) 0 0
\(813\) 71.0513 2.49188
\(814\) 0 0
\(815\) −8.92539 −0.312643
\(816\) 0 0
\(817\) 14.9466 0.522917
\(818\) 0 0
\(819\) 34.5265 1.20646
\(820\) 0 0
\(821\) 16.6208 0.580070 0.290035 0.957016i \(-0.406333\pi\)
0.290035 + 0.957016i \(0.406333\pi\)
\(822\) 0 0
\(823\) 30.7490 1.07184 0.535922 0.844268i \(-0.319964\pi\)
0.535922 + 0.844268i \(0.319964\pi\)
\(824\) 0 0
\(825\) 42.6500 1.48488
\(826\) 0 0
\(827\) −53.8130 −1.87126 −0.935630 0.352982i \(-0.885168\pi\)
−0.935630 + 0.352982i \(0.885168\pi\)
\(828\) 0 0
\(829\) −46.9817 −1.63174 −0.815870 0.578235i \(-0.803742\pi\)
−0.815870 + 0.578235i \(0.803742\pi\)
\(830\) 0 0
\(831\) 6.67559 0.231574
\(832\) 0 0
\(833\) −0.879325 −0.0304668
\(834\) 0 0
\(835\) −0.597166 −0.0206658
\(836\) 0 0
\(837\) −2.14692 −0.0742083
\(838\) 0 0
\(839\) −1.36976 −0.0472894 −0.0236447 0.999720i \(-0.507527\pi\)
−0.0236447 + 0.999720i \(0.507527\pi\)
\(840\) 0 0
\(841\) 61.8803 2.13380
\(842\) 0 0
\(843\) 48.2666 1.66239
\(844\) 0 0
\(845\) −1.95735 −0.0673350
\(846\) 0 0
\(847\) 6.23513 0.214242
\(848\) 0 0
\(849\) −58.3211 −2.00158
\(850\) 0 0
\(851\) −3.94181 −0.135124
\(852\) 0 0
\(853\) 41.8613 1.43330 0.716652 0.697431i \(-0.245674\pi\)
0.716652 + 0.697431i \(0.245674\pi\)
\(854\) 0 0
\(855\) −3.79562 −0.129808
\(856\) 0 0
\(857\) 28.5582 0.975529 0.487765 0.872975i \(-0.337812\pi\)
0.487765 + 0.872975i \(0.337812\pi\)
\(858\) 0 0
\(859\) −37.0371 −1.26369 −0.631845 0.775095i \(-0.717702\pi\)
−0.631845 + 0.775095i \(0.717702\pi\)
\(860\) 0 0
\(861\) −21.3800 −0.728630
\(862\) 0 0
\(863\) −55.2897 −1.88208 −0.941042 0.338290i \(-0.890152\pi\)
−0.941042 + 0.338290i \(0.890152\pi\)
\(864\) 0 0
\(865\) −0.00677770 −0.000230449 0
\(866\) 0 0
\(867\) 39.5829 1.34431
\(868\) 0 0
\(869\) 22.2244 0.753912
\(870\) 0 0
\(871\) −23.3035 −0.789607
\(872\) 0 0
\(873\) −4.03643 −0.136612
\(874\) 0 0
\(875\) 10.3346 0.349372
\(876\) 0 0
\(877\) 22.4835 0.759215 0.379607 0.925148i \(-0.376059\pi\)
0.379607 + 0.925148i \(0.376059\pi\)
\(878\) 0 0
\(879\) 7.33238 0.247315
\(880\) 0 0
\(881\) −39.0250 −1.31479 −0.657393 0.753548i \(-0.728341\pi\)
−0.657393 + 0.753548i \(0.728341\pi\)
\(882\) 0 0
\(883\) −25.0904 −0.844358 −0.422179 0.906513i \(-0.638735\pi\)
−0.422179 + 0.906513i \(0.638735\pi\)
\(884\) 0 0
\(885\) −8.58167 −0.288470
\(886\) 0 0
\(887\) 0.896277 0.0300940 0.0150470 0.999887i \(-0.495210\pi\)
0.0150470 + 0.999887i \(0.495210\pi\)
\(888\) 0 0
\(889\) −59.0248 −1.97963
\(890\) 0 0
\(891\) −34.4464 −1.15400
\(892\) 0 0
\(893\) −42.4241 −1.41967
\(894\) 0 0
\(895\) −1.83481 −0.0613310
\(896\) 0 0
\(897\) −9.60430 −0.320678
\(898\) 0 0
\(899\) 49.3463 1.64579
\(900\) 0 0
\(901\) 7.10964 0.236857
\(902\) 0 0
\(903\) −28.1664 −0.937318
\(904\) 0 0
\(905\) −8.31305 −0.276335
\(906\) 0 0
\(907\) 53.5332 1.77754 0.888771 0.458352i \(-0.151560\pi\)
0.888771 + 0.458352i \(0.151560\pi\)
\(908\) 0 0
\(909\) 17.9406 0.595052
\(910\) 0 0
\(911\) −37.8448 −1.25385 −0.626926 0.779079i \(-0.715687\pi\)
−0.626926 + 0.779079i \(0.715687\pi\)
\(912\) 0 0
\(913\) −57.4716 −1.90203
\(914\) 0 0
\(915\) −9.38075 −0.310118
\(916\) 0 0
\(917\) −37.2143 −1.22892
\(918\) 0 0
\(919\) 29.6564 0.978275 0.489138 0.872207i \(-0.337312\pi\)
0.489138 + 0.872207i \(0.337312\pi\)
\(920\) 0 0
\(921\) −11.1187 −0.366373
\(922\) 0 0
\(923\) 11.4848 0.378028
\(924\) 0 0
\(925\) 20.6367 0.678532
\(926\) 0 0
\(927\) −8.51124 −0.279546
\(928\) 0 0
\(929\) 3.56393 0.116929 0.0584644 0.998289i \(-0.481380\pi\)
0.0584644 + 0.998289i \(0.481380\pi\)
\(930\) 0 0
\(931\) 4.13363 0.135474
\(932\) 0 0
\(933\) −14.4225 −0.472171
\(934\) 0 0
\(935\) 1.03671 0.0339040
\(936\) 0 0
\(937\) 11.9586 0.390672 0.195336 0.980736i \(-0.437420\pi\)
0.195336 + 0.980736i \(0.437420\pi\)
\(938\) 0 0
\(939\) 30.5271 0.996213
\(940\) 0 0
\(941\) 16.7537 0.546156 0.273078 0.961992i \(-0.411958\pi\)
0.273078 + 0.961992i \(0.411958\pi\)
\(942\) 0 0
\(943\) 2.88600 0.0939811
\(944\) 0 0
\(945\) −0.434499 −0.0141343
\(946\) 0 0
\(947\) −41.1667 −1.33774 −0.668870 0.743380i \(-0.733222\pi\)
−0.668870 + 0.743380i \(0.733222\pi\)
\(948\) 0 0
\(949\) 47.2950 1.53526
\(950\) 0 0
\(951\) −39.5089 −1.28116
\(952\) 0 0
\(953\) 24.4059 0.790585 0.395292 0.918555i \(-0.370643\pi\)
0.395292 + 0.918555i \(0.370643\pi\)
\(954\) 0 0
\(955\) −3.34024 −0.108088
\(956\) 0 0
\(957\) 83.5733 2.70154
\(958\) 0 0
\(959\) −48.3658 −1.56181
\(960\) 0 0
\(961\) −4.20592 −0.135675
\(962\) 0 0
\(963\) 9.93452 0.320135
\(964\) 0 0
\(965\) 1.75551 0.0565118
\(966\) 0 0
\(967\) 59.0370 1.89850 0.949251 0.314519i \(-0.101843\pi\)
0.949251 + 0.314519i \(0.101843\pi\)
\(968\) 0 0
\(969\) 6.85358 0.220169
\(970\) 0 0
\(971\) 51.0766 1.63913 0.819563 0.572990i \(-0.194216\pi\)
0.819563 + 0.572990i \(0.194216\pi\)
\(972\) 0 0
\(973\) −50.3782 −1.61505
\(974\) 0 0
\(975\) 50.2818 1.61031
\(976\) 0 0
\(977\) −4.27766 −0.136854 −0.0684272 0.997656i \(-0.521798\pi\)
−0.0684272 + 0.997656i \(0.521798\pi\)
\(978\) 0 0
\(979\) 35.5069 1.13481
\(980\) 0 0
\(981\) 26.1447 0.834735
\(982\) 0 0
\(983\) −49.4130 −1.57603 −0.788014 0.615657i \(-0.788891\pi\)
−0.788014 + 0.615657i \(0.788891\pi\)
\(984\) 0 0
\(985\) −3.42727 −0.109202
\(986\) 0 0
\(987\) 79.9466 2.54473
\(988\) 0 0
\(989\) 3.80206 0.120898
\(990\) 0 0
\(991\) −4.37601 −0.139009 −0.0695043 0.997582i \(-0.522142\pi\)
−0.0695043 + 0.997582i \(0.522142\pi\)
\(992\) 0 0
\(993\) −35.5853 −1.12926
\(994\) 0 0
\(995\) −8.17311 −0.259105
\(996\) 0 0
\(997\) 20.0002 0.633412 0.316706 0.948524i \(-0.397423\pi\)
0.316706 + 0.948524i \(0.397423\pi\)
\(998\) 0 0
\(999\) −1.75934 −0.0556632
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 8044.2.a.a.1.15 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8044.2.a.a.1.15 80 1.1 even 1 trivial