Properties

Label 8044.2.a.b.1.5
Level $8044$
Weight $2$
Character 8044.1
Self dual yes
Analytic conductor $64.232$
Analytic rank $0$
Dimension $87$
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: \(0\)
Dimension: \(87\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
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)
 
\(f(q)\) \(=\) \(q-2.93853 q^{3} +0.993611 q^{5} +0.0902794 q^{7} +5.63497 q^{9} +O(q^{10})\) \(q-2.93853 q^{3} +0.993611 q^{5} +0.0902794 q^{7} +5.63497 q^{9} -1.94385 q^{11} -4.99845 q^{13} -2.91976 q^{15} +1.90558 q^{17} +1.66802 q^{19} -0.265289 q^{21} +4.16543 q^{23} -4.01274 q^{25} -7.74293 q^{27} +0.760921 q^{29} +6.97918 q^{31} +5.71207 q^{33} +0.0897026 q^{35} -4.35287 q^{37} +14.6881 q^{39} -6.51969 q^{41} -8.15623 q^{43} +5.59897 q^{45} -3.47089 q^{47} -6.99185 q^{49} -5.59961 q^{51} -4.23286 q^{53} -1.93143 q^{55} -4.90154 q^{57} +3.46613 q^{59} +14.4967 q^{61} +0.508721 q^{63} -4.96652 q^{65} +6.30826 q^{67} -12.2402 q^{69} +5.76891 q^{71} +8.41521 q^{73} +11.7916 q^{75} -0.175490 q^{77} +8.49939 q^{79} +5.84795 q^{81} -11.0182 q^{83} +1.89341 q^{85} -2.23599 q^{87} +5.89197 q^{89} -0.451257 q^{91} -20.5085 q^{93} +1.65737 q^{95} -14.1368 q^{97} -10.9535 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9} + 36 q^{11} - q^{13} + 16 q^{15} + 31 q^{17} + 35 q^{19} - 3 q^{21} + 39 q^{23} + 93 q^{25} + 55 q^{27} - 5 q^{29} + 46 q^{31} + 25 q^{33} + 68 q^{35} - 11 q^{37} + 54 q^{39} + 83 q^{41} + 28 q^{43} - 14 q^{45} + 48 q^{47} + 103 q^{49} + 77 q^{51} + 3 q^{53} + 35 q^{55} + 14 q^{57} + 122 q^{59} - 13 q^{61} + 39 q^{63} + 41 q^{65} + 32 q^{67} - 10 q^{69} + 100 q^{71} + 34 q^{73} + 97 q^{75} + 4 q^{77} + 52 q^{79} + 131 q^{81} + 67 q^{83} - 2 q^{85} + 89 q^{87} + 68 q^{89} + 75 q^{91} + 138 q^{95} + 36 q^{97} + 107 q^{99}+O(q^{100}) \) 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
\(3\) −2.93853 −1.69656 −0.848281 0.529546i \(-0.822362\pi\)
−0.848281 + 0.529546i \(0.822362\pi\)
\(4\) 0 0
\(5\) 0.993611 0.444356 0.222178 0.975006i \(-0.428683\pi\)
0.222178 + 0.975006i \(0.428683\pi\)
\(6\) 0 0
\(7\) 0.0902794 0.0341224 0.0170612 0.999854i \(-0.494569\pi\)
0.0170612 + 0.999854i \(0.494569\pi\)
\(8\) 0 0
\(9\) 5.63497 1.87832
\(10\) 0 0
\(11\) −1.94385 −0.586093 −0.293047 0.956098i \(-0.594669\pi\)
−0.293047 + 0.956098i \(0.594669\pi\)
\(12\) 0 0
\(13\) −4.99845 −1.38632 −0.693160 0.720784i \(-0.743782\pi\)
−0.693160 + 0.720784i \(0.743782\pi\)
\(14\) 0 0
\(15\) −2.91976 −0.753878
\(16\) 0 0
\(17\) 1.90558 0.462171 0.231086 0.972933i \(-0.425772\pi\)
0.231086 + 0.972933i \(0.425772\pi\)
\(18\) 0 0
\(19\) 1.66802 0.382671 0.191335 0.981525i \(-0.438718\pi\)
0.191335 + 0.981525i \(0.438718\pi\)
\(20\) 0 0
\(21\) −0.265289 −0.0578907
\(22\) 0 0
\(23\) 4.16543 0.868552 0.434276 0.900780i \(-0.357004\pi\)
0.434276 + 0.900780i \(0.357004\pi\)
\(24\) 0 0
\(25\) −4.01274 −0.802547
\(26\) 0 0
\(27\) −7.74293 −1.49013
\(28\) 0 0
\(29\) 0.760921 0.141299 0.0706497 0.997501i \(-0.477493\pi\)
0.0706497 + 0.997501i \(0.477493\pi\)
\(30\) 0 0
\(31\) 6.97918 1.25350 0.626748 0.779222i \(-0.284385\pi\)
0.626748 + 0.779222i \(0.284385\pi\)
\(32\) 0 0
\(33\) 5.71207 0.994343
\(34\) 0 0
\(35\) 0.0897026 0.0151625
\(36\) 0 0
\(37\) −4.35287 −0.715607 −0.357804 0.933797i \(-0.616474\pi\)
−0.357804 + 0.933797i \(0.616474\pi\)
\(38\) 0 0
\(39\) 14.6881 2.35198
\(40\) 0 0
\(41\) −6.51969 −1.01821 −0.509103 0.860706i \(-0.670023\pi\)
−0.509103 + 0.860706i \(0.670023\pi\)
\(42\) 0 0
\(43\) −8.15623 −1.24381 −0.621906 0.783092i \(-0.713642\pi\)
−0.621906 + 0.783092i \(0.713642\pi\)
\(44\) 0 0
\(45\) 5.59897 0.834645
\(46\) 0 0
\(47\) −3.47089 −0.506281 −0.253141 0.967429i \(-0.581464\pi\)
−0.253141 + 0.967429i \(0.581464\pi\)
\(48\) 0 0
\(49\) −6.99185 −0.998836
\(50\) 0 0
\(51\) −5.59961 −0.784102
\(52\) 0 0
\(53\) −4.23286 −0.581428 −0.290714 0.956810i \(-0.593893\pi\)
−0.290714 + 0.956810i \(0.593893\pi\)
\(54\) 0 0
\(55\) −1.93143 −0.260434
\(56\) 0 0
\(57\) −4.90154 −0.649224
\(58\) 0 0
\(59\) 3.46613 0.451252 0.225626 0.974214i \(-0.427557\pi\)
0.225626 + 0.974214i \(0.427557\pi\)
\(60\) 0 0
\(61\) 14.4967 1.85611 0.928056 0.372441i \(-0.121479\pi\)
0.928056 + 0.372441i \(0.121479\pi\)
\(62\) 0 0
\(63\) 0.508721 0.0640928
\(64\) 0 0
\(65\) −4.96652 −0.616020
\(66\) 0 0
\(67\) 6.30826 0.770676 0.385338 0.922775i \(-0.374085\pi\)
0.385338 + 0.922775i \(0.374085\pi\)
\(68\) 0 0
\(69\) −12.2402 −1.47355
\(70\) 0 0
\(71\) 5.76891 0.684643 0.342322 0.939583i \(-0.388787\pi\)
0.342322 + 0.939583i \(0.388787\pi\)
\(72\) 0 0
\(73\) 8.41521 0.984925 0.492463 0.870334i \(-0.336097\pi\)
0.492463 + 0.870334i \(0.336097\pi\)
\(74\) 0 0
\(75\) 11.7916 1.36157
\(76\) 0 0
\(77\) −0.175490 −0.0199989
\(78\) 0 0
\(79\) 8.49939 0.956256 0.478128 0.878290i \(-0.341315\pi\)
0.478128 + 0.878290i \(0.341315\pi\)
\(80\) 0 0
\(81\) 5.84795 0.649773
\(82\) 0 0
\(83\) −11.0182 −1.20941 −0.604703 0.796451i \(-0.706708\pi\)
−0.604703 + 0.796451i \(0.706708\pi\)
\(84\) 0 0
\(85\) 1.89341 0.205369
\(86\) 0 0
\(87\) −2.23599 −0.239723
\(88\) 0 0
\(89\) 5.89197 0.624548 0.312274 0.949992i \(-0.398909\pi\)
0.312274 + 0.949992i \(0.398909\pi\)
\(90\) 0 0
\(91\) −0.451257 −0.0473046
\(92\) 0 0
\(93\) −20.5085 −2.12664
\(94\) 0 0
\(95\) 1.65737 0.170042
\(96\) 0 0
\(97\) −14.1368 −1.43538 −0.717688 0.696365i \(-0.754800\pi\)
−0.717688 + 0.696365i \(0.754800\pi\)
\(98\) 0 0
\(99\) −10.9535 −1.10087
\(100\) 0 0
\(101\) 6.49073 0.645852 0.322926 0.946424i \(-0.395334\pi\)
0.322926 + 0.946424i \(0.395334\pi\)
\(102\) 0 0
\(103\) 6.96176 0.685963 0.342981 0.939342i \(-0.388563\pi\)
0.342981 + 0.939342i \(0.388563\pi\)
\(104\) 0 0
\(105\) −0.263594 −0.0257241
\(106\) 0 0
\(107\) 16.2160 1.56766 0.783830 0.620976i \(-0.213263\pi\)
0.783830 + 0.620976i \(0.213263\pi\)
\(108\) 0 0
\(109\) −3.74889 −0.359079 −0.179539 0.983751i \(-0.557461\pi\)
−0.179539 + 0.983751i \(0.557461\pi\)
\(110\) 0 0
\(111\) 12.7910 1.21407
\(112\) 0 0
\(113\) −19.8492 −1.86726 −0.933629 0.358242i \(-0.883376\pi\)
−0.933629 + 0.358242i \(0.883376\pi\)
\(114\) 0 0
\(115\) 4.13882 0.385947
\(116\) 0 0
\(117\) −28.1661 −2.60396
\(118\) 0 0
\(119\) 0.172035 0.0157704
\(120\) 0 0
\(121\) −7.22144 −0.656495
\(122\) 0 0
\(123\) 19.1583 1.72745
\(124\) 0 0
\(125\) −8.95516 −0.800974
\(126\) 0 0
\(127\) 4.24961 0.377092 0.188546 0.982064i \(-0.439623\pi\)
0.188546 + 0.982064i \(0.439623\pi\)
\(128\) 0 0
\(129\) 23.9673 2.11021
\(130\) 0 0
\(131\) −18.5332 −1.61925 −0.809625 0.586948i \(-0.800329\pi\)
−0.809625 + 0.586948i \(0.800329\pi\)
\(132\) 0 0
\(133\) 0.150588 0.0130576
\(134\) 0 0
\(135\) −7.69347 −0.662148
\(136\) 0 0
\(137\) −4.82769 −0.412457 −0.206229 0.978504i \(-0.566119\pi\)
−0.206229 + 0.978504i \(0.566119\pi\)
\(138\) 0 0
\(139\) −19.8345 −1.68234 −0.841168 0.540774i \(-0.818132\pi\)
−0.841168 + 0.540774i \(0.818132\pi\)
\(140\) 0 0
\(141\) 10.1993 0.858937
\(142\) 0 0
\(143\) 9.71624 0.812513
\(144\) 0 0
\(145\) 0.756059 0.0627873
\(146\) 0 0
\(147\) 20.5458 1.69459
\(148\) 0 0
\(149\) 23.8187 1.95131 0.975653 0.219319i \(-0.0703837\pi\)
0.975653 + 0.219319i \(0.0703837\pi\)
\(150\) 0 0
\(151\) 16.6928 1.35844 0.679220 0.733935i \(-0.262318\pi\)
0.679220 + 0.733935i \(0.262318\pi\)
\(152\) 0 0
\(153\) 10.7379 0.868107
\(154\) 0 0
\(155\) 6.93459 0.556999
\(156\) 0 0
\(157\) −0.607785 −0.0485066 −0.0242533 0.999706i \(-0.507721\pi\)
−0.0242533 + 0.999706i \(0.507721\pi\)
\(158\) 0 0
\(159\) 12.4384 0.986429
\(160\) 0 0
\(161\) 0.376052 0.0296371
\(162\) 0 0
\(163\) −9.13567 −0.715561 −0.357780 0.933806i \(-0.616466\pi\)
−0.357780 + 0.933806i \(0.616466\pi\)
\(164\) 0 0
\(165\) 5.67557 0.441843
\(166\) 0 0
\(167\) −10.0373 −0.776712 −0.388356 0.921509i \(-0.626957\pi\)
−0.388356 + 0.921509i \(0.626957\pi\)
\(168\) 0 0
\(169\) 11.9845 0.921884
\(170\) 0 0
\(171\) 9.39925 0.718779
\(172\) 0 0
\(173\) 14.8825 1.13149 0.565746 0.824579i \(-0.308588\pi\)
0.565746 + 0.824579i \(0.308588\pi\)
\(174\) 0 0
\(175\) −0.362267 −0.0273848
\(176\) 0 0
\(177\) −10.1853 −0.765577
\(178\) 0 0
\(179\) −17.0439 −1.27392 −0.636962 0.770895i \(-0.719809\pi\)
−0.636962 + 0.770895i \(0.719809\pi\)
\(180\) 0 0
\(181\) −20.6336 −1.53368 −0.766840 0.641838i \(-0.778172\pi\)
−0.766840 + 0.641838i \(0.778172\pi\)
\(182\) 0 0
\(183\) −42.5990 −3.14901
\(184\) 0 0
\(185\) −4.32506 −0.317985
\(186\) 0 0
\(187\) −3.70417 −0.270875
\(188\) 0 0
\(189\) −0.699027 −0.0508467
\(190\) 0 0
\(191\) 17.6628 1.27803 0.639017 0.769193i \(-0.279341\pi\)
0.639017 + 0.769193i \(0.279341\pi\)
\(192\) 0 0
\(193\) 1.46575 0.105507 0.0527534 0.998608i \(-0.483200\pi\)
0.0527534 + 0.998608i \(0.483200\pi\)
\(194\) 0 0
\(195\) 14.5943 1.04512
\(196\) 0 0
\(197\) −10.1739 −0.724857 −0.362429 0.932012i \(-0.618052\pi\)
−0.362429 + 0.932012i \(0.618052\pi\)
\(198\) 0 0
\(199\) 16.9364 1.20059 0.600296 0.799778i \(-0.295049\pi\)
0.600296 + 0.799778i \(0.295049\pi\)
\(200\) 0 0
\(201\) −18.5370 −1.30750
\(202\) 0 0
\(203\) 0.0686954 0.00482147
\(204\) 0 0
\(205\) −6.47804 −0.452446
\(206\) 0 0
\(207\) 23.4721 1.63142
\(208\) 0 0
\(209\) −3.24239 −0.224281
\(210\) 0 0
\(211\) 17.3590 1.19504 0.597522 0.801853i \(-0.296152\pi\)
0.597522 + 0.801853i \(0.296152\pi\)
\(212\) 0 0
\(213\) −16.9521 −1.16154
\(214\) 0 0
\(215\) −8.10412 −0.552696
\(216\) 0 0
\(217\) 0.630075 0.0427723
\(218\) 0 0
\(219\) −24.7283 −1.67099
\(220\) 0 0
\(221\) −9.52495 −0.640717
\(222\) 0 0
\(223\) 24.3974 1.63377 0.816885 0.576800i \(-0.195699\pi\)
0.816885 + 0.576800i \(0.195699\pi\)
\(224\) 0 0
\(225\) −22.6116 −1.50744
\(226\) 0 0
\(227\) −15.6528 −1.03891 −0.519457 0.854497i \(-0.673866\pi\)
−0.519457 + 0.854497i \(0.673866\pi\)
\(228\) 0 0
\(229\) −3.34199 −0.220845 −0.110422 0.993885i \(-0.535220\pi\)
−0.110422 + 0.993885i \(0.535220\pi\)
\(230\) 0 0
\(231\) 0.515682 0.0339294
\(232\) 0 0
\(233\) −13.2902 −0.870668 −0.435334 0.900269i \(-0.643370\pi\)
−0.435334 + 0.900269i \(0.643370\pi\)
\(234\) 0 0
\(235\) −3.44871 −0.224969
\(236\) 0 0
\(237\) −24.9757 −1.62235
\(238\) 0 0
\(239\) 11.3709 0.735520 0.367760 0.929921i \(-0.380125\pi\)
0.367760 + 0.929921i \(0.380125\pi\)
\(240\) 0 0
\(241\) −4.69200 −0.302238 −0.151119 0.988516i \(-0.548288\pi\)
−0.151119 + 0.988516i \(0.548288\pi\)
\(242\) 0 0
\(243\) 6.04441 0.387749
\(244\) 0 0
\(245\) −6.94718 −0.443839
\(246\) 0 0
\(247\) −8.33753 −0.530504
\(248\) 0 0
\(249\) 32.3774 2.05183
\(250\) 0 0
\(251\) −29.8428 −1.88366 −0.941830 0.336089i \(-0.890896\pi\)
−0.941830 + 0.336089i \(0.890896\pi\)
\(252\) 0 0
\(253\) −8.09698 −0.509053
\(254\) 0 0
\(255\) −5.56383 −0.348421
\(256\) 0 0
\(257\) −0.498670 −0.0311062 −0.0155531 0.999879i \(-0.504951\pi\)
−0.0155531 + 0.999879i \(0.504951\pi\)
\(258\) 0 0
\(259\) −0.392974 −0.0244182
\(260\) 0 0
\(261\) 4.28776 0.265406
\(262\) 0 0
\(263\) 12.3452 0.761240 0.380620 0.924732i \(-0.375711\pi\)
0.380620 + 0.924732i \(0.375711\pi\)
\(264\) 0 0
\(265\) −4.20582 −0.258361
\(266\) 0 0
\(267\) −17.3137 −1.05958
\(268\) 0 0
\(269\) −17.1416 −1.04514 −0.522572 0.852595i \(-0.675027\pi\)
−0.522572 + 0.852595i \(0.675027\pi\)
\(270\) 0 0
\(271\) 20.2255 1.22861 0.614305 0.789069i \(-0.289436\pi\)
0.614305 + 0.789069i \(0.289436\pi\)
\(272\) 0 0
\(273\) 1.32603 0.0802551
\(274\) 0 0
\(275\) 7.80016 0.470367
\(276\) 0 0
\(277\) 19.6987 1.18358 0.591791 0.806092i \(-0.298421\pi\)
0.591791 + 0.806092i \(0.298421\pi\)
\(278\) 0 0
\(279\) 39.3274 2.35447
\(280\) 0 0
\(281\) −3.58456 −0.213837 −0.106918 0.994268i \(-0.534098\pi\)
−0.106918 + 0.994268i \(0.534098\pi\)
\(282\) 0 0
\(283\) 2.12723 0.126451 0.0632253 0.997999i \(-0.479861\pi\)
0.0632253 + 0.997999i \(0.479861\pi\)
\(284\) 0 0
\(285\) −4.87022 −0.288487
\(286\) 0 0
\(287\) −0.588594 −0.0347436
\(288\) 0 0
\(289\) −13.3688 −0.786398
\(290\) 0 0
\(291\) 41.5415 2.43520
\(292\) 0 0
\(293\) −6.56755 −0.383681 −0.191840 0.981426i \(-0.561446\pi\)
−0.191840 + 0.981426i \(0.561446\pi\)
\(294\) 0 0
\(295\) 3.44399 0.200517
\(296\) 0 0
\(297\) 15.0511 0.873354
\(298\) 0 0
\(299\) −20.8207 −1.20409
\(300\) 0 0
\(301\) −0.736339 −0.0424419
\(302\) 0 0
\(303\) −19.0732 −1.09573
\(304\) 0 0
\(305\) 14.4041 0.824775
\(306\) 0 0
\(307\) 6.76112 0.385877 0.192939 0.981211i \(-0.438198\pi\)
0.192939 + 0.981211i \(0.438198\pi\)
\(308\) 0 0
\(309\) −20.4574 −1.16378
\(310\) 0 0
\(311\) 16.5368 0.937713 0.468857 0.883274i \(-0.344666\pi\)
0.468857 + 0.883274i \(0.344666\pi\)
\(312\) 0 0
\(313\) 30.1580 1.70463 0.852314 0.523030i \(-0.175199\pi\)
0.852314 + 0.523030i \(0.175199\pi\)
\(314\) 0 0
\(315\) 0.505471 0.0284801
\(316\) 0 0
\(317\) −17.8341 −1.00166 −0.500830 0.865546i \(-0.666972\pi\)
−0.500830 + 0.865546i \(0.666972\pi\)
\(318\) 0 0
\(319\) −1.47912 −0.0828146
\(320\) 0 0
\(321\) −47.6512 −2.65963
\(322\) 0 0
\(323\) 3.17855 0.176859
\(324\) 0 0
\(325\) 20.0575 1.11259
\(326\) 0 0
\(327\) 11.0162 0.609200
\(328\) 0 0
\(329\) −0.313350 −0.0172755
\(330\) 0 0
\(331\) −6.38602 −0.351008 −0.175504 0.984479i \(-0.556155\pi\)
−0.175504 + 0.984479i \(0.556155\pi\)
\(332\) 0 0
\(333\) −24.5283 −1.34414
\(334\) 0 0
\(335\) 6.26796 0.342455
\(336\) 0 0
\(337\) 5.60743 0.305456 0.152728 0.988268i \(-0.451194\pi\)
0.152728 + 0.988268i \(0.451194\pi\)
\(338\) 0 0
\(339\) 58.3275 3.16792
\(340\) 0 0
\(341\) −13.5665 −0.734666
\(342\) 0 0
\(343\) −1.26318 −0.0682050
\(344\) 0 0
\(345\) −12.1620 −0.654783
\(346\) 0 0
\(347\) 4.48735 0.240894 0.120447 0.992720i \(-0.461567\pi\)
0.120447 + 0.992720i \(0.461567\pi\)
\(348\) 0 0
\(349\) 6.57911 0.352172 0.176086 0.984375i \(-0.443656\pi\)
0.176086 + 0.984375i \(0.443656\pi\)
\(350\) 0 0
\(351\) 38.7027 2.06580
\(352\) 0 0
\(353\) 28.6755 1.52624 0.763120 0.646257i \(-0.223666\pi\)
0.763120 + 0.646257i \(0.223666\pi\)
\(354\) 0 0
\(355\) 5.73205 0.304226
\(356\) 0 0
\(357\) −0.505529 −0.0267554
\(358\) 0 0
\(359\) 16.2823 0.859349 0.429674 0.902984i \(-0.358628\pi\)
0.429674 + 0.902984i \(0.358628\pi\)
\(360\) 0 0
\(361\) −16.2177 −0.853563
\(362\) 0 0
\(363\) 21.2204 1.11378
\(364\) 0 0
\(365\) 8.36144 0.437658
\(366\) 0 0
\(367\) −21.0374 −1.09814 −0.549071 0.835776i \(-0.685018\pi\)
−0.549071 + 0.835776i \(0.685018\pi\)
\(368\) 0 0
\(369\) −36.7383 −1.91252
\(370\) 0 0
\(371\) −0.382140 −0.0198397
\(372\) 0 0
\(373\) −7.21288 −0.373469 −0.186734 0.982410i \(-0.559790\pi\)
−0.186734 + 0.982410i \(0.559790\pi\)
\(374\) 0 0
\(375\) 26.3150 1.35890
\(376\) 0 0
\(377\) −3.80342 −0.195886
\(378\) 0 0
\(379\) 13.4066 0.688653 0.344326 0.938850i \(-0.388107\pi\)
0.344326 + 0.938850i \(0.388107\pi\)
\(380\) 0 0
\(381\) −12.4876 −0.639760
\(382\) 0 0
\(383\) 19.1806 0.980083 0.490041 0.871699i \(-0.336982\pi\)
0.490041 + 0.871699i \(0.336982\pi\)
\(384\) 0 0
\(385\) −0.174368 −0.00888664
\(386\) 0 0
\(387\) −45.9601 −2.33628
\(388\) 0 0
\(389\) 33.2519 1.68594 0.842970 0.537961i \(-0.180805\pi\)
0.842970 + 0.537961i \(0.180805\pi\)
\(390\) 0 0
\(391\) 7.93757 0.401420
\(392\) 0 0
\(393\) 54.4603 2.74716
\(394\) 0 0
\(395\) 8.44509 0.424919
\(396\) 0 0
\(397\) 29.5765 1.48440 0.742202 0.670176i \(-0.233781\pi\)
0.742202 + 0.670176i \(0.233781\pi\)
\(398\) 0 0
\(399\) −0.442508 −0.0221531
\(400\) 0 0
\(401\) 36.9351 1.84445 0.922225 0.386653i \(-0.126369\pi\)
0.922225 + 0.386653i \(0.126369\pi\)
\(402\) 0 0
\(403\) −34.8851 −1.73775
\(404\) 0 0
\(405\) 5.81059 0.288731
\(406\) 0 0
\(407\) 8.46133 0.419413
\(408\) 0 0
\(409\) −27.1196 −1.34098 −0.670489 0.741920i \(-0.733915\pi\)
−0.670489 + 0.741920i \(0.733915\pi\)
\(410\) 0 0
\(411\) 14.1863 0.699759
\(412\) 0 0
\(413\) 0.312920 0.0153978
\(414\) 0 0
\(415\) −10.9478 −0.537408
\(416\) 0 0
\(417\) 58.2842 2.85419
\(418\) 0 0
\(419\) 7.77960 0.380059 0.190029 0.981778i \(-0.439142\pi\)
0.190029 + 0.981778i \(0.439142\pi\)
\(420\) 0 0
\(421\) 19.3117 0.941197 0.470598 0.882347i \(-0.344038\pi\)
0.470598 + 0.882347i \(0.344038\pi\)
\(422\) 0 0
\(423\) −19.5583 −0.950959
\(424\) 0 0
\(425\) −7.64659 −0.370914
\(426\) 0 0
\(427\) 1.30875 0.0633350
\(428\) 0 0
\(429\) −28.5515 −1.37848
\(430\) 0 0
\(431\) −0.444289 −0.0214006 −0.0107003 0.999943i \(-0.503406\pi\)
−0.0107003 + 0.999943i \(0.503406\pi\)
\(432\) 0 0
\(433\) 40.2096 1.93235 0.966176 0.257883i \(-0.0830250\pi\)
0.966176 + 0.257883i \(0.0830250\pi\)
\(434\) 0 0
\(435\) −2.22170 −0.106523
\(436\) 0 0
\(437\) 6.94803 0.332369
\(438\) 0 0
\(439\) 6.12968 0.292554 0.146277 0.989244i \(-0.453271\pi\)
0.146277 + 0.989244i \(0.453271\pi\)
\(440\) 0 0
\(441\) −39.3988 −1.87614
\(442\) 0 0
\(443\) 24.1176 1.14586 0.572931 0.819604i \(-0.305806\pi\)
0.572931 + 0.819604i \(0.305806\pi\)
\(444\) 0 0
\(445\) 5.85433 0.277522
\(446\) 0 0
\(447\) −69.9921 −3.31051
\(448\) 0 0
\(449\) 15.5748 0.735018 0.367509 0.930020i \(-0.380211\pi\)
0.367509 + 0.930020i \(0.380211\pi\)
\(450\) 0 0
\(451\) 12.6733 0.596763
\(452\) 0 0
\(453\) −49.0523 −2.30468
\(454\) 0 0
\(455\) −0.448374 −0.0210201
\(456\) 0 0
\(457\) −17.0953 −0.799687 −0.399843 0.916583i \(-0.630935\pi\)
−0.399843 + 0.916583i \(0.630935\pi\)
\(458\) 0 0
\(459\) −14.7548 −0.688694
\(460\) 0 0
\(461\) −6.72235 −0.313091 −0.156545 0.987671i \(-0.550036\pi\)
−0.156545 + 0.987671i \(0.550036\pi\)
\(462\) 0 0
\(463\) −0.254221 −0.0118147 −0.00590733 0.999983i \(-0.501880\pi\)
−0.00590733 + 0.999983i \(0.501880\pi\)
\(464\) 0 0
\(465\) −20.3775 −0.944984
\(466\) 0 0
\(467\) −2.98714 −0.138228 −0.0691142 0.997609i \(-0.522017\pi\)
−0.0691142 + 0.997609i \(0.522017\pi\)
\(468\) 0 0
\(469\) 0.569506 0.0262973
\(470\) 0 0
\(471\) 1.78600 0.0822944
\(472\) 0 0
\(473\) 15.8545 0.728990
\(474\) 0 0
\(475\) −6.69334 −0.307111
\(476\) 0 0
\(477\) −23.8520 −1.09211
\(478\) 0 0
\(479\) 8.77202 0.400804 0.200402 0.979714i \(-0.435775\pi\)
0.200402 + 0.979714i \(0.435775\pi\)
\(480\) 0 0
\(481\) 21.7576 0.992061
\(482\) 0 0
\(483\) −1.10504 −0.0502811
\(484\) 0 0
\(485\) −14.0465 −0.637819
\(486\) 0 0
\(487\) −7.52387 −0.340939 −0.170469 0.985363i \(-0.554528\pi\)
−0.170469 + 0.985363i \(0.554528\pi\)
\(488\) 0 0
\(489\) 26.8455 1.21399
\(490\) 0 0
\(491\) 12.5017 0.564194 0.282097 0.959386i \(-0.408970\pi\)
0.282097 + 0.959386i \(0.408970\pi\)
\(492\) 0 0
\(493\) 1.45000 0.0653045
\(494\) 0 0
\(495\) −10.8836 −0.489180
\(496\) 0 0
\(497\) 0.520813 0.0233617
\(498\) 0 0
\(499\) 38.3059 1.71481 0.857405 0.514642i \(-0.172075\pi\)
0.857405 + 0.514642i \(0.172075\pi\)
\(500\) 0 0
\(501\) 29.4950 1.31774
\(502\) 0 0
\(503\) 24.1748 1.07790 0.538950 0.842338i \(-0.318821\pi\)
0.538950 + 0.842338i \(0.318821\pi\)
\(504\) 0 0
\(505\) 6.44926 0.286988
\(506\) 0 0
\(507\) −35.2168 −1.56403
\(508\) 0 0
\(509\) 25.7740 1.14241 0.571205 0.820807i \(-0.306476\pi\)
0.571205 + 0.820807i \(0.306476\pi\)
\(510\) 0 0
\(511\) 0.759719 0.0336080
\(512\) 0 0
\(513\) −12.9154 −0.570228
\(514\) 0 0
\(515\) 6.91729 0.304812
\(516\) 0 0
\(517\) 6.74689 0.296728
\(518\) 0 0
\(519\) −43.7326 −1.91965
\(520\) 0 0
\(521\) 44.0797 1.93117 0.965583 0.260096i \(-0.0837541\pi\)
0.965583 + 0.260096i \(0.0837541\pi\)
\(522\) 0 0
\(523\) 26.2946 1.14978 0.574891 0.818230i \(-0.305045\pi\)
0.574891 + 0.818230i \(0.305045\pi\)
\(524\) 0 0
\(525\) 1.06453 0.0464601
\(526\) 0 0
\(527\) 13.2994 0.579330
\(528\) 0 0
\(529\) −5.64919 −0.245617
\(530\) 0 0
\(531\) 19.5315 0.847597
\(532\) 0 0
\(533\) 32.5884 1.41156
\(534\) 0 0
\(535\) 16.1124 0.696600
\(536\) 0 0
\(537\) 50.0842 2.16129
\(538\) 0 0
\(539\) 13.5911 0.585411
\(540\) 0 0
\(541\) −26.6723 −1.14673 −0.573367 0.819299i \(-0.694363\pi\)
−0.573367 + 0.819299i \(0.694363\pi\)
\(542\) 0 0
\(543\) 60.6324 2.60198
\(544\) 0 0
\(545\) −3.72494 −0.159559
\(546\) 0 0
\(547\) 26.8020 1.14597 0.572986 0.819565i \(-0.305785\pi\)
0.572986 + 0.819565i \(0.305785\pi\)
\(548\) 0 0
\(549\) 81.6884 3.48638
\(550\) 0 0
\(551\) 1.26923 0.0540711
\(552\) 0 0
\(553\) 0.767320 0.0326298
\(554\) 0 0
\(555\) 12.7093 0.539481
\(556\) 0 0
\(557\) −41.8469 −1.77311 −0.886555 0.462624i \(-0.846908\pi\)
−0.886555 + 0.462624i \(0.846908\pi\)
\(558\) 0 0
\(559\) 40.7685 1.72432
\(560\) 0 0
\(561\) 10.8848 0.459557
\(562\) 0 0
\(563\) 28.8189 1.21457 0.607286 0.794483i \(-0.292258\pi\)
0.607286 + 0.794483i \(0.292258\pi\)
\(564\) 0 0
\(565\) −19.7224 −0.829728
\(566\) 0 0
\(567\) 0.527950 0.0221718
\(568\) 0 0
\(569\) 15.3798 0.644757 0.322378 0.946611i \(-0.395518\pi\)
0.322378 + 0.946611i \(0.395518\pi\)
\(570\) 0 0
\(571\) −19.1127 −0.799842 −0.399921 0.916550i \(-0.630962\pi\)
−0.399921 + 0.916550i \(0.630962\pi\)
\(572\) 0 0
\(573\) −51.9026 −2.16826
\(574\) 0 0
\(575\) −16.7148 −0.697054
\(576\) 0 0
\(577\) 10.0005 0.416325 0.208163 0.978094i \(-0.433252\pi\)
0.208163 + 0.978094i \(0.433252\pi\)
\(578\) 0 0
\(579\) −4.30714 −0.178999
\(580\) 0 0
\(581\) −0.994718 −0.0412678
\(582\) 0 0
\(583\) 8.22806 0.340771
\(584\) 0 0
\(585\) −27.9862 −1.15708
\(586\) 0 0
\(587\) 2.85018 0.117640 0.0588198 0.998269i \(-0.481266\pi\)
0.0588198 + 0.998269i \(0.481266\pi\)
\(588\) 0 0
\(589\) 11.6414 0.479676
\(590\) 0 0
\(591\) 29.8962 1.22977
\(592\) 0 0
\(593\) 32.6777 1.34191 0.670956 0.741497i \(-0.265884\pi\)
0.670956 + 0.741497i \(0.265884\pi\)
\(594\) 0 0
\(595\) 0.170936 0.00700767
\(596\) 0 0
\(597\) −49.7683 −2.03688
\(598\) 0 0
\(599\) 42.9216 1.75373 0.876864 0.480739i \(-0.159632\pi\)
0.876864 + 0.480739i \(0.159632\pi\)
\(600\) 0 0
\(601\) −39.6295 −1.61652 −0.808260 0.588826i \(-0.799590\pi\)
−0.808260 + 0.588826i \(0.799590\pi\)
\(602\) 0 0
\(603\) 35.5468 1.44758
\(604\) 0 0
\(605\) −7.17531 −0.291718
\(606\) 0 0
\(607\) 34.6352 1.40580 0.702899 0.711289i \(-0.251888\pi\)
0.702899 + 0.711289i \(0.251888\pi\)
\(608\) 0 0
\(609\) −0.201864 −0.00817993
\(610\) 0 0
\(611\) 17.3491 0.701868
\(612\) 0 0
\(613\) −24.0502 −0.971378 −0.485689 0.874132i \(-0.661431\pi\)
−0.485689 + 0.874132i \(0.661431\pi\)
\(614\) 0 0
\(615\) 19.0359 0.767603
\(616\) 0 0
\(617\) 20.0804 0.808405 0.404202 0.914670i \(-0.367549\pi\)
0.404202 + 0.914670i \(0.367549\pi\)
\(618\) 0 0
\(619\) 31.0284 1.24714 0.623569 0.781768i \(-0.285682\pi\)
0.623569 + 0.781768i \(0.285682\pi\)
\(620\) 0 0
\(621\) −32.2527 −1.29425
\(622\) 0 0
\(623\) 0.531923 0.0213111
\(624\) 0 0
\(625\) 11.1657 0.446630
\(626\) 0 0
\(627\) 9.52786 0.380506
\(628\) 0 0
\(629\) −8.29475 −0.330733
\(630\) 0 0
\(631\) 11.5762 0.460839 0.230420 0.973091i \(-0.425990\pi\)
0.230420 + 0.973091i \(0.425990\pi\)
\(632\) 0 0
\(633\) −51.0100 −2.02747
\(634\) 0 0
\(635\) 4.22246 0.167563
\(636\) 0 0
\(637\) 34.9484 1.38471
\(638\) 0 0
\(639\) 32.5076 1.28598
\(640\) 0 0
\(641\) −7.85742 −0.310350 −0.155175 0.987887i \(-0.549594\pi\)
−0.155175 + 0.987887i \(0.549594\pi\)
\(642\) 0 0
\(643\) −34.3808 −1.35585 −0.677923 0.735133i \(-0.737120\pi\)
−0.677923 + 0.735133i \(0.737120\pi\)
\(644\) 0 0
\(645\) 23.8142 0.937684
\(646\) 0 0
\(647\) −37.8578 −1.48834 −0.744172 0.667988i \(-0.767156\pi\)
−0.744172 + 0.667988i \(0.767156\pi\)
\(648\) 0 0
\(649\) −6.73765 −0.264476
\(650\) 0 0
\(651\) −1.85150 −0.0725659
\(652\) 0 0
\(653\) 30.4658 1.19222 0.596110 0.802903i \(-0.296712\pi\)
0.596110 + 0.802903i \(0.296712\pi\)
\(654\) 0 0
\(655\) −18.4148 −0.719524
\(656\) 0 0
\(657\) 47.4194 1.85001
\(658\) 0 0
\(659\) −36.4945 −1.42162 −0.710812 0.703382i \(-0.751672\pi\)
−0.710812 + 0.703382i \(0.751672\pi\)
\(660\) 0 0
\(661\) 41.7959 1.62567 0.812836 0.582492i \(-0.197922\pi\)
0.812836 + 0.582492i \(0.197922\pi\)
\(662\) 0 0
\(663\) 27.9894 1.08702
\(664\) 0 0
\(665\) 0.149626 0.00580224
\(666\) 0 0
\(667\) 3.16956 0.122726
\(668\) 0 0
\(669\) −71.6925 −2.77179
\(670\) 0 0
\(671\) −28.1794 −1.08785
\(672\) 0 0
\(673\) 2.81464 0.108497 0.0542483 0.998527i \(-0.482724\pi\)
0.0542483 + 0.998527i \(0.482724\pi\)
\(674\) 0 0
\(675\) 31.0704 1.19590
\(676\) 0 0
\(677\) 11.1290 0.427722 0.213861 0.976864i \(-0.431396\pi\)
0.213861 + 0.976864i \(0.431396\pi\)
\(678\) 0 0
\(679\) −1.27626 −0.0489785
\(680\) 0 0
\(681\) 45.9963 1.76258
\(682\) 0 0
\(683\) 21.7571 0.832513 0.416257 0.909247i \(-0.363342\pi\)
0.416257 + 0.909247i \(0.363342\pi\)
\(684\) 0 0
\(685\) −4.79684 −0.183278
\(686\) 0 0
\(687\) 9.82054 0.374677
\(688\) 0 0
\(689\) 21.1578 0.806046
\(690\) 0 0
\(691\) 0.958497 0.0364629 0.0182315 0.999834i \(-0.494196\pi\)
0.0182315 + 0.999834i \(0.494196\pi\)
\(692\) 0 0
\(693\) −0.988878 −0.0375644
\(694\) 0 0
\(695\) −19.7077 −0.747557
\(696\) 0 0
\(697\) −12.4238 −0.470585
\(698\) 0 0
\(699\) 39.0536 1.47714
\(700\) 0 0
\(701\) 43.0162 1.62470 0.812350 0.583170i \(-0.198188\pi\)
0.812350 + 0.583170i \(0.198188\pi\)
\(702\) 0 0
\(703\) −7.26068 −0.273842
\(704\) 0 0
\(705\) 10.1342 0.381674
\(706\) 0 0
\(707\) 0.585979 0.0220380
\(708\) 0 0
\(709\) −35.2585 −1.32416 −0.662080 0.749434i \(-0.730326\pi\)
−0.662080 + 0.749434i \(0.730326\pi\)
\(710\) 0 0
\(711\) 47.8938 1.79616
\(712\) 0 0
\(713\) 29.0713 1.08873
\(714\) 0 0
\(715\) 9.65417 0.361045
\(716\) 0 0
\(717\) −33.4136 −1.24786
\(718\) 0 0
\(719\) −31.0227 −1.15695 −0.578476 0.815699i \(-0.696353\pi\)
−0.578476 + 0.815699i \(0.696353\pi\)
\(720\) 0 0
\(721\) 0.628504 0.0234067
\(722\) 0 0
\(723\) 13.7876 0.512765
\(724\) 0 0
\(725\) −3.05337 −0.113399
\(726\) 0 0
\(727\) −25.9640 −0.962953 −0.481477 0.876459i \(-0.659899\pi\)
−0.481477 + 0.876459i \(0.659899\pi\)
\(728\) 0 0
\(729\) −35.3055 −1.30761
\(730\) 0 0
\(731\) −15.5423 −0.574855
\(732\) 0 0
\(733\) 39.5111 1.45937 0.729687 0.683781i \(-0.239666\pi\)
0.729687 + 0.683781i \(0.239666\pi\)
\(734\) 0 0
\(735\) 20.4145 0.753000
\(736\) 0 0
\(737\) −12.2623 −0.451688
\(738\) 0 0
\(739\) −1.53098 −0.0563178 −0.0281589 0.999603i \(-0.508964\pi\)
−0.0281589 + 0.999603i \(0.508964\pi\)
\(740\) 0 0
\(741\) 24.5001 0.900033
\(742\) 0 0
\(743\) −9.06503 −0.332564 −0.166282 0.986078i \(-0.553176\pi\)
−0.166282 + 0.986078i \(0.553176\pi\)
\(744\) 0 0
\(745\) 23.6666 0.867076
\(746\) 0 0
\(747\) −62.0873 −2.27166
\(748\) 0 0
\(749\) 1.46397 0.0534923
\(750\) 0 0
\(751\) 4.20210 0.153337 0.0766684 0.997057i \(-0.475572\pi\)
0.0766684 + 0.997057i \(0.475572\pi\)
\(752\) 0 0
\(753\) 87.6940 3.19575
\(754\) 0 0
\(755\) 16.5861 0.603632
\(756\) 0 0
\(757\) −29.9627 −1.08901 −0.544507 0.838756i \(-0.683283\pi\)
−0.544507 + 0.838756i \(0.683283\pi\)
\(758\) 0 0
\(759\) 23.7932 0.863639
\(760\) 0 0
\(761\) 34.5367 1.25195 0.625977 0.779841i \(-0.284700\pi\)
0.625977 + 0.779841i \(0.284700\pi\)
\(762\) 0 0
\(763\) −0.338448 −0.0122526
\(764\) 0 0
\(765\) 10.6693 0.385749
\(766\) 0 0
\(767\) −17.3253 −0.625580
\(768\) 0 0
\(769\) 51.8849 1.87102 0.935510 0.353302i \(-0.114941\pi\)
0.935510 + 0.353302i \(0.114941\pi\)
\(770\) 0 0
\(771\) 1.46536 0.0527735
\(772\) 0 0
\(773\) 12.2464 0.440473 0.220237 0.975447i \(-0.429317\pi\)
0.220237 + 0.975447i \(0.429317\pi\)
\(774\) 0 0
\(775\) −28.0056 −1.00599
\(776\) 0 0
\(777\) 1.15477 0.0414270
\(778\) 0 0
\(779\) −10.8750 −0.389637
\(780\) 0 0
\(781\) −11.2139 −0.401265
\(782\) 0 0
\(783\) −5.89176 −0.210554
\(784\) 0 0
\(785\) −0.603902 −0.0215542
\(786\) 0 0
\(787\) −17.6784 −0.630166 −0.315083 0.949064i \(-0.602032\pi\)
−0.315083 + 0.949064i \(0.602032\pi\)
\(788\) 0 0
\(789\) −36.2769 −1.29149
\(790\) 0 0
\(791\) −1.79197 −0.0637153
\(792\) 0 0
\(793\) −72.4610 −2.57317
\(794\) 0 0
\(795\) 12.3589 0.438326
\(796\) 0 0
\(797\) −12.4972 −0.442672 −0.221336 0.975198i \(-0.571042\pi\)
−0.221336 + 0.975198i \(0.571042\pi\)
\(798\) 0 0
\(799\) −6.61406 −0.233989
\(800\) 0 0
\(801\) 33.2011 1.17310
\(802\) 0 0
\(803\) −16.3579 −0.577258
\(804\) 0 0
\(805\) 0.373650 0.0131694
\(806\) 0 0
\(807\) 50.3712 1.77315
\(808\) 0 0
\(809\) 32.5348 1.14386 0.571932 0.820301i \(-0.306194\pi\)
0.571932 + 0.820301i \(0.306194\pi\)
\(810\) 0 0
\(811\) −45.4673 −1.59657 −0.798287 0.602277i \(-0.794260\pi\)
−0.798287 + 0.602277i \(0.794260\pi\)
\(812\) 0 0
\(813\) −59.4332 −2.08441
\(814\) 0 0
\(815\) −9.07730 −0.317964
\(816\) 0 0
\(817\) −13.6048 −0.475971
\(818\) 0 0
\(819\) −2.54282 −0.0888532
\(820\) 0 0
\(821\) 13.0616 0.455852 0.227926 0.973678i \(-0.426806\pi\)
0.227926 + 0.973678i \(0.426806\pi\)
\(822\) 0 0
\(823\) −29.9980 −1.04566 −0.522832 0.852436i \(-0.675125\pi\)
−0.522832 + 0.852436i \(0.675125\pi\)
\(824\) 0 0
\(825\) −22.9210 −0.798008
\(826\) 0 0
\(827\) −34.9927 −1.21682 −0.608408 0.793625i \(-0.708191\pi\)
−0.608408 + 0.793625i \(0.708191\pi\)
\(828\) 0 0
\(829\) −55.1169 −1.91429 −0.957145 0.289609i \(-0.906475\pi\)
−0.957145 + 0.289609i \(0.906475\pi\)
\(830\) 0 0
\(831\) −57.8853 −2.00802
\(832\) 0 0
\(833\) −13.3235 −0.461633
\(834\) 0 0
\(835\) −9.97321 −0.345137
\(836\) 0 0
\(837\) −54.0393 −1.86787
\(838\) 0 0
\(839\) 7.53512 0.260141 0.130071 0.991505i \(-0.458480\pi\)
0.130071 + 0.991505i \(0.458480\pi\)
\(840\) 0 0
\(841\) −28.4210 −0.980034
\(842\) 0 0
\(843\) 10.5333 0.362787
\(844\) 0 0
\(845\) 11.9079 0.409645
\(846\) 0 0
\(847\) −0.651947 −0.0224012
\(848\) 0 0
\(849\) −6.25092 −0.214531
\(850\) 0 0
\(851\) −18.1316 −0.621542
\(852\) 0 0
\(853\) −11.3575 −0.388872 −0.194436 0.980915i \(-0.562288\pi\)
−0.194436 + 0.980915i \(0.562288\pi\)
\(854\) 0 0
\(855\) 9.33920 0.319394
\(856\) 0 0
\(857\) −29.3738 −1.00339 −0.501694 0.865045i \(-0.667290\pi\)
−0.501694 + 0.865045i \(0.667290\pi\)
\(858\) 0 0
\(859\) 54.9469 1.87477 0.937383 0.348301i \(-0.113241\pi\)
0.937383 + 0.348301i \(0.113241\pi\)
\(860\) 0 0
\(861\) 1.72960 0.0589447
\(862\) 0 0
\(863\) −9.27307 −0.315659 −0.157830 0.987466i \(-0.550450\pi\)
−0.157830 + 0.987466i \(0.550450\pi\)
\(864\) 0 0
\(865\) 14.7874 0.502786
\(866\) 0 0
\(867\) 39.2845 1.33417
\(868\) 0 0
\(869\) −16.5216 −0.560455
\(870\) 0 0
\(871\) −31.5315 −1.06840
\(872\) 0 0
\(873\) −79.6605 −2.69610
\(874\) 0 0
\(875\) −0.808466 −0.0273311
\(876\) 0 0
\(877\) 18.6606 0.630123 0.315062 0.949071i \(-0.397975\pi\)
0.315062 + 0.949071i \(0.397975\pi\)
\(878\) 0 0
\(879\) 19.2990 0.650938
\(880\) 0 0
\(881\) 31.4878 1.06085 0.530425 0.847732i \(-0.322032\pi\)
0.530425 + 0.847732i \(0.322032\pi\)
\(882\) 0 0
\(883\) 21.9879 0.739951 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(884\) 0 0
\(885\) −10.1203 −0.340189
\(886\) 0 0
\(887\) −11.7514 −0.394572 −0.197286 0.980346i \(-0.563213\pi\)
−0.197286 + 0.980346i \(0.563213\pi\)
\(888\) 0 0
\(889\) 0.383652 0.0128673
\(890\) 0 0
\(891\) −11.3676 −0.380827
\(892\) 0 0
\(893\) −5.78952 −0.193739
\(894\) 0 0
\(895\) −16.9351 −0.566076
\(896\) 0 0
\(897\) 61.1823 2.04282
\(898\) 0 0
\(899\) 5.31060 0.177118
\(900\) 0 0
\(901\) −8.06606 −0.268720
\(902\) 0 0
\(903\) 2.16376 0.0720053
\(904\) 0 0
\(905\) −20.5017 −0.681501
\(906\) 0 0
\(907\) 45.4075 1.50773 0.753865 0.657029i \(-0.228187\pi\)
0.753865 + 0.657029i \(0.228187\pi\)
\(908\) 0 0
\(909\) 36.5750 1.21312
\(910\) 0 0
\(911\) −24.9445 −0.826447 −0.413224 0.910630i \(-0.635597\pi\)
−0.413224 + 0.910630i \(0.635597\pi\)
\(912\) 0 0
\(913\) 21.4178 0.708825
\(914\) 0 0
\(915\) −42.3268 −1.39928
\(916\) 0 0
\(917\) −1.67316 −0.0552527
\(918\) 0 0
\(919\) −47.9718 −1.58244 −0.791221 0.611530i \(-0.790554\pi\)
−0.791221 + 0.611530i \(0.790554\pi\)
\(920\) 0 0
\(921\) −19.8677 −0.654665
\(922\) 0 0
\(923\) −28.8356 −0.949135
\(924\) 0 0
\(925\) 17.4669 0.574309
\(926\) 0 0
\(927\) 39.2293 1.28846
\(928\) 0 0
\(929\) −39.9180 −1.30967 −0.654833 0.755774i \(-0.727261\pi\)
−0.654833 + 0.755774i \(0.727261\pi\)
\(930\) 0 0
\(931\) −11.6626 −0.382225
\(932\) 0 0
\(933\) −48.5938 −1.59089
\(934\) 0 0
\(935\) −3.68050 −0.120365
\(936\) 0 0
\(937\) −37.3385 −1.21980 −0.609899 0.792480i \(-0.708790\pi\)
−0.609899 + 0.792480i \(0.708790\pi\)
\(938\) 0 0
\(939\) −88.6201 −2.89201
\(940\) 0 0
\(941\) 14.2129 0.463329 0.231664 0.972796i \(-0.425583\pi\)
0.231664 + 0.972796i \(0.425583\pi\)
\(942\) 0 0
\(943\) −27.1573 −0.884364
\(944\) 0 0
\(945\) −0.694561 −0.0225941
\(946\) 0 0
\(947\) −20.0125 −0.650318 −0.325159 0.945659i \(-0.605418\pi\)
−0.325159 + 0.945659i \(0.605418\pi\)
\(948\) 0 0
\(949\) −42.0630 −1.36542
\(950\) 0 0
\(951\) 52.4059 1.69938
\(952\) 0 0
\(953\) −9.36568 −0.303384 −0.151692 0.988428i \(-0.548472\pi\)
−0.151692 + 0.988428i \(0.548472\pi\)
\(954\) 0 0
\(955\) 17.5499 0.567903
\(956\) 0 0
\(957\) 4.34643 0.140500
\(958\) 0 0
\(959\) −0.435840 −0.0140740
\(960\) 0 0
\(961\) 17.7089 0.571255
\(962\) 0 0
\(963\) 91.3766 2.94457
\(964\) 0 0
\(965\) 1.45638 0.0468826
\(966\) 0 0
\(967\) −13.5061 −0.434328 −0.217164 0.976135i \(-0.569681\pi\)
−0.217164 + 0.976135i \(0.569681\pi\)
\(968\) 0 0
\(969\) −9.34027 −0.300053
\(970\) 0 0
\(971\) −46.8866 −1.50466 −0.752331 0.658785i \(-0.771071\pi\)
−0.752331 + 0.658785i \(0.771071\pi\)
\(972\) 0 0
\(973\) −1.79064 −0.0574053
\(974\) 0 0
\(975\) −58.9395 −1.88757
\(976\) 0 0
\(977\) −12.1818 −0.389731 −0.194866 0.980830i \(-0.562427\pi\)
−0.194866 + 0.980830i \(0.562427\pi\)
\(978\) 0 0
\(979\) −11.4531 −0.366043
\(980\) 0 0
\(981\) −21.1249 −0.674466
\(982\) 0 0
\(983\) −26.3909 −0.841740 −0.420870 0.907121i \(-0.638275\pi\)
−0.420870 + 0.907121i \(0.638275\pi\)
\(984\) 0 0
\(985\) −10.1089 −0.322095
\(986\) 0 0
\(987\) 0.920788 0.0293090
\(988\) 0 0
\(989\) −33.9742 −1.08032
\(990\) 0 0
\(991\) 14.7104 0.467291 0.233645 0.972322i \(-0.424935\pi\)
0.233645 + 0.972322i \(0.424935\pi\)
\(992\) 0 0
\(993\) 18.7655 0.595506
\(994\) 0 0
\(995\) 16.8282 0.533491
\(996\) 0 0
\(997\) 24.2442 0.767821 0.383911 0.923370i \(-0.374577\pi\)
0.383911 + 0.923370i \(0.374577\pi\)
\(998\) 0 0
\(999\) 33.7040 1.06635
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.b.1.5 87
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8044.2.a.b.1.5 87 1.1 even 1 trivial