Properties

Label 7623.2.a.cx.1.5
Level $7623$
Weight $2$
Character 7623.1
Self dual yes
Analytic conductor $60.870$
Analytic rank $0$
Dimension $10$
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] = [7623,2,Mod(1,7623)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7623, 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("7623.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7623.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: \(60.8699614608\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 19x^{8} - x^{7} + 124x^{6} + 6x^{5} - 316x^{4} + 17x^{3} + 253x^{2} - 70x - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.473713\) of defining polynomial
Character \(\chi\) \(=\) 7623.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.473713 q^{2} -1.77560 q^{4} -3.75881 q^{5} -1.00000 q^{7} +1.78855 q^{8} +O(q^{10})\) \(q-0.473713 q^{2} -1.77560 q^{4} -3.75881 q^{5} -1.00000 q^{7} +1.78855 q^{8} +1.78060 q^{10} +2.70774 q^{13} +0.473713 q^{14} +2.70393 q^{16} -2.48283 q^{17} -1.74466 q^{19} +6.67413 q^{20} -3.79843 q^{23} +9.12868 q^{25} -1.28269 q^{26} +1.77560 q^{28} +5.80060 q^{29} +2.24763 q^{31} -4.85799 q^{32} +1.17615 q^{34} +3.75881 q^{35} -7.65311 q^{37} +0.826469 q^{38} -6.72282 q^{40} +4.18500 q^{41} +7.75162 q^{43} +1.79937 q^{46} -12.4071 q^{47} +1.00000 q^{49} -4.32437 q^{50} -4.80786 q^{52} -8.83956 q^{53} -1.78855 q^{56} -2.74782 q^{58} -4.54166 q^{59} -11.4825 q^{61} -1.06473 q^{62} -3.10658 q^{64} -10.1779 q^{65} -12.2041 q^{67} +4.40849 q^{68} -1.78060 q^{70} -7.75988 q^{71} +15.6796 q^{73} +3.62538 q^{74} +3.09781 q^{76} -7.05380 q^{79} -10.1636 q^{80} -1.98249 q^{82} -11.9004 q^{83} +9.33248 q^{85} -3.67204 q^{86} +1.69179 q^{89} -2.70774 q^{91} +6.74448 q^{92} +5.87739 q^{94} +6.55786 q^{95} +6.99638 q^{97} -0.473713 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 18 q^{4} - 5 q^{5} - 10 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 18 q^{4} - 5 q^{5} - 10 q^{7} - 3 q^{8} + 6 q^{10} - 6 q^{13} + 38 q^{16} + 8 q^{17} - 7 q^{20} + 31 q^{25} - q^{26} - 18 q^{28} - 14 q^{29} + 26 q^{31} - 41 q^{32} + 21 q^{34} + 5 q^{35} + 24 q^{37} - 8 q^{38} + 5 q^{40} + 19 q^{41} + 6 q^{43} + q^{46} - 15 q^{47} + 10 q^{49} - q^{50} + 25 q^{52} + q^{53} + 3 q^{56} + 11 q^{58} - 23 q^{59} + 11 q^{62} + 53 q^{64} - 29 q^{65} + 38 q^{67} + 87 q^{68} - 6 q^{70} - 26 q^{71} + q^{73} - 39 q^{74} + 2 q^{76} - 5 q^{79} - 6 q^{80} + 5 q^{82} + 6 q^{83} + q^{85} + 41 q^{86} + 9 q^{89} + 6 q^{91} + 48 q^{92} - 42 q^{95} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.473713 −0.334966 −0.167483 0.985875i \(-0.553564\pi\)
−0.167483 + 0.985875i \(0.553564\pi\)
\(3\) 0 0
\(4\) −1.77560 −0.887798
\(5\) −3.75881 −1.68099 −0.840496 0.541818i \(-0.817736\pi\)
−0.840496 + 0.541818i \(0.817736\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.78855 0.632348
\(9\) 0 0
\(10\) 1.78060 0.563075
\(11\) 0 0
\(12\) 0 0
\(13\) 2.70774 0.750993 0.375496 0.926824i \(-0.377472\pi\)
0.375496 + 0.926824i \(0.377472\pi\)
\(14\) 0.473713 0.126605
\(15\) 0 0
\(16\) 2.70393 0.675983
\(17\) −2.48283 −0.602174 −0.301087 0.953597i \(-0.597349\pi\)
−0.301087 + 0.953597i \(0.597349\pi\)
\(18\) 0 0
\(19\) −1.74466 −0.400253 −0.200126 0.979770i \(-0.564135\pi\)
−0.200126 + 0.979770i \(0.564135\pi\)
\(20\) 6.67413 1.49238
\(21\) 0 0
\(22\) 0 0
\(23\) −3.79843 −0.792028 −0.396014 0.918244i \(-0.629607\pi\)
−0.396014 + 0.918244i \(0.629607\pi\)
\(24\) 0 0
\(25\) 9.12868 1.82574
\(26\) −1.28269 −0.251557
\(27\) 0 0
\(28\) 1.77560 0.335556
\(29\) 5.80060 1.07714 0.538572 0.842580i \(-0.318964\pi\)
0.538572 + 0.842580i \(0.318964\pi\)
\(30\) 0 0
\(31\) 2.24763 0.403686 0.201843 0.979418i \(-0.435307\pi\)
0.201843 + 0.979418i \(0.435307\pi\)
\(32\) −4.85799 −0.858779
\(33\) 0 0
\(34\) 1.17615 0.201707
\(35\) 3.75881 0.635355
\(36\) 0 0
\(37\) −7.65311 −1.25816 −0.629082 0.777339i \(-0.716569\pi\)
−0.629082 + 0.777339i \(0.716569\pi\)
\(38\) 0.826469 0.134071
\(39\) 0 0
\(40\) −6.72282 −1.06297
\(41\) 4.18500 0.653587 0.326793 0.945096i \(-0.394032\pi\)
0.326793 + 0.945096i \(0.394032\pi\)
\(42\) 0 0
\(43\) 7.75162 1.18211 0.591056 0.806631i \(-0.298711\pi\)
0.591056 + 0.806631i \(0.298711\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 1.79937 0.265302
\(47\) −12.4071 −1.80976 −0.904878 0.425670i \(-0.860038\pi\)
−0.904878 + 0.425670i \(0.860038\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −4.32437 −0.611559
\(51\) 0 0
\(52\) −4.80786 −0.666730
\(53\) −8.83956 −1.21421 −0.607103 0.794623i \(-0.707669\pi\)
−0.607103 + 0.794623i \(0.707669\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.78855 −0.239005
\(57\) 0 0
\(58\) −2.74782 −0.360806
\(59\) −4.54166 −0.591274 −0.295637 0.955300i \(-0.595532\pi\)
−0.295637 + 0.955300i \(0.595532\pi\)
\(60\) 0 0
\(61\) −11.4825 −1.47019 −0.735093 0.677967i \(-0.762861\pi\)
−0.735093 + 0.677967i \(0.762861\pi\)
\(62\) −1.06473 −0.135221
\(63\) 0 0
\(64\) −3.10658 −0.388322
\(65\) −10.1779 −1.26241
\(66\) 0 0
\(67\) −12.2041 −1.49096 −0.745481 0.666527i \(-0.767780\pi\)
−0.745481 + 0.666527i \(0.767780\pi\)
\(68\) 4.40849 0.534609
\(69\) 0 0
\(70\) −1.78060 −0.212822
\(71\) −7.75988 −0.920928 −0.460464 0.887678i \(-0.652317\pi\)
−0.460464 + 0.887678i \(0.652317\pi\)
\(72\) 0 0
\(73\) 15.6796 1.83516 0.917580 0.397552i \(-0.130140\pi\)
0.917580 + 0.397552i \(0.130140\pi\)
\(74\) 3.62538 0.421441
\(75\) 0 0
\(76\) 3.09781 0.355344
\(77\) 0 0
\(78\) 0 0
\(79\) −7.05380 −0.793615 −0.396807 0.917902i \(-0.629882\pi\)
−0.396807 + 0.917902i \(0.629882\pi\)
\(80\) −10.1636 −1.13632
\(81\) 0 0
\(82\) −1.98249 −0.218929
\(83\) −11.9004 −1.30624 −0.653122 0.757253i \(-0.726541\pi\)
−0.653122 + 0.757253i \(0.726541\pi\)
\(84\) 0 0
\(85\) 9.33248 1.01225
\(86\) −3.67204 −0.395967
\(87\) 0 0
\(88\) 0 0
\(89\) 1.69179 0.179329 0.0896647 0.995972i \(-0.471420\pi\)
0.0896647 + 0.995972i \(0.471420\pi\)
\(90\) 0 0
\(91\) −2.70774 −0.283848
\(92\) 6.74448 0.703161
\(93\) 0 0
\(94\) 5.87739 0.606206
\(95\) 6.55786 0.672822
\(96\) 0 0
\(97\) 6.99638 0.710375 0.355188 0.934795i \(-0.384417\pi\)
0.355188 + 0.934795i \(0.384417\pi\)
\(98\) −0.473713 −0.0478522
\(99\) 0 0
\(100\) −16.2088 −1.62088
\(101\) −2.71683 −0.270335 −0.135167 0.990823i \(-0.543157\pi\)
−0.135167 + 0.990823i \(0.543157\pi\)
\(102\) 0 0
\(103\) −0.255788 −0.0252036 −0.0126018 0.999921i \(-0.504011\pi\)
−0.0126018 + 0.999921i \(0.504011\pi\)
\(104\) 4.84293 0.474888
\(105\) 0 0
\(106\) 4.18741 0.406717
\(107\) −11.8608 −1.14662 −0.573311 0.819338i \(-0.694341\pi\)
−0.573311 + 0.819338i \(0.694341\pi\)
\(108\) 0 0
\(109\) −15.8720 −1.52026 −0.760129 0.649772i \(-0.774864\pi\)
−0.760129 + 0.649772i \(0.774864\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.70393 −0.255498
\(113\) 5.11650 0.481320 0.240660 0.970609i \(-0.422636\pi\)
0.240660 + 0.970609i \(0.422636\pi\)
\(114\) 0 0
\(115\) 14.2776 1.33139
\(116\) −10.2995 −0.956286
\(117\) 0 0
\(118\) 2.15144 0.198056
\(119\) 2.48283 0.227600
\(120\) 0 0
\(121\) 0 0
\(122\) 5.43942 0.492462
\(123\) 0 0
\(124\) −3.99088 −0.358392
\(125\) −15.5189 −1.38805
\(126\) 0 0
\(127\) 0.203338 0.0180433 0.00902166 0.999959i \(-0.497128\pi\)
0.00902166 + 0.999959i \(0.497128\pi\)
\(128\) 11.1876 0.988853
\(129\) 0 0
\(130\) 4.82140 0.422865
\(131\) 10.2788 0.898067 0.449034 0.893515i \(-0.351768\pi\)
0.449034 + 0.893515i \(0.351768\pi\)
\(132\) 0 0
\(133\) 1.74466 0.151281
\(134\) 5.78122 0.499421
\(135\) 0 0
\(136\) −4.44065 −0.380783
\(137\) 17.1096 1.46177 0.730887 0.682499i \(-0.239107\pi\)
0.730887 + 0.682499i \(0.239107\pi\)
\(138\) 0 0
\(139\) 19.5009 1.65405 0.827023 0.562168i \(-0.190032\pi\)
0.827023 + 0.562168i \(0.190032\pi\)
\(140\) −6.67413 −0.564067
\(141\) 0 0
\(142\) 3.67596 0.308479
\(143\) 0 0
\(144\) 0 0
\(145\) −21.8034 −1.81067
\(146\) −7.42763 −0.614715
\(147\) 0 0
\(148\) 13.5888 1.11699
\(149\) 9.89037 0.810251 0.405125 0.914261i \(-0.367228\pi\)
0.405125 + 0.914261i \(0.367228\pi\)
\(150\) 0 0
\(151\) −15.5014 −1.26148 −0.630741 0.775993i \(-0.717249\pi\)
−0.630741 + 0.775993i \(0.717249\pi\)
\(152\) −3.12041 −0.253099
\(153\) 0 0
\(154\) 0 0
\(155\) −8.44843 −0.678594
\(156\) 0 0
\(157\) 16.9536 1.35305 0.676523 0.736422i \(-0.263486\pi\)
0.676523 + 0.736422i \(0.263486\pi\)
\(158\) 3.34148 0.265834
\(159\) 0 0
\(160\) 18.2603 1.44360
\(161\) 3.79843 0.299359
\(162\) 0 0
\(163\) 19.9739 1.56447 0.782237 0.622981i \(-0.214079\pi\)
0.782237 + 0.622981i \(0.214079\pi\)
\(164\) −7.43086 −0.580253
\(165\) 0 0
\(166\) 5.63739 0.437547
\(167\) −17.4516 −1.35044 −0.675222 0.737615i \(-0.735952\pi\)
−0.675222 + 0.737615i \(0.735952\pi\)
\(168\) 0 0
\(169\) −5.66813 −0.436010
\(170\) −4.42092 −0.339069
\(171\) 0 0
\(172\) −13.7637 −1.04948
\(173\) 9.35944 0.711585 0.355793 0.934565i \(-0.384211\pi\)
0.355793 + 0.934565i \(0.384211\pi\)
\(174\) 0 0
\(175\) −9.12868 −0.690063
\(176\) 0 0
\(177\) 0 0
\(178\) −0.801423 −0.0600692
\(179\) −5.49347 −0.410602 −0.205301 0.978699i \(-0.565817\pi\)
−0.205301 + 0.978699i \(0.565817\pi\)
\(180\) 0 0
\(181\) −11.1844 −0.831332 −0.415666 0.909517i \(-0.636451\pi\)
−0.415666 + 0.909517i \(0.636451\pi\)
\(182\) 1.28269 0.0950795
\(183\) 0 0
\(184\) −6.79369 −0.500837
\(185\) 28.7666 2.11496
\(186\) 0 0
\(187\) 0 0
\(188\) 22.0299 1.60670
\(189\) 0 0
\(190\) −3.10654 −0.225372
\(191\) 5.75822 0.416650 0.208325 0.978060i \(-0.433199\pi\)
0.208325 + 0.978060i \(0.433199\pi\)
\(192\) 0 0
\(193\) −14.8846 −1.07142 −0.535708 0.844404i \(-0.679955\pi\)
−0.535708 + 0.844404i \(0.679955\pi\)
\(194\) −3.31428 −0.237951
\(195\) 0 0
\(196\) −1.77560 −0.126828
\(197\) −12.5476 −0.893978 −0.446989 0.894540i \(-0.647504\pi\)
−0.446989 + 0.894540i \(0.647504\pi\)
\(198\) 0 0
\(199\) 0.834418 0.0591503 0.0295752 0.999563i \(-0.490585\pi\)
0.0295752 + 0.999563i \(0.490585\pi\)
\(200\) 16.3271 1.15450
\(201\) 0 0
\(202\) 1.28700 0.0905528
\(203\) −5.80060 −0.407122
\(204\) 0 0
\(205\) −15.7306 −1.09867
\(206\) 0.121170 0.00844233
\(207\) 0 0
\(208\) 7.32155 0.507658
\(209\) 0 0
\(210\) 0 0
\(211\) −26.8703 −1.84983 −0.924914 0.380176i \(-0.875863\pi\)
−0.924914 + 0.380176i \(0.875863\pi\)
\(212\) 15.6955 1.07797
\(213\) 0 0
\(214\) 5.61859 0.384079
\(215\) −29.1369 −1.98712
\(216\) 0 0
\(217\) −2.24763 −0.152579
\(218\) 7.51876 0.509234
\(219\) 0 0
\(220\) 0 0
\(221\) −6.72285 −0.452228
\(222\) 0 0
\(223\) 4.01918 0.269144 0.134572 0.990904i \(-0.457034\pi\)
0.134572 + 0.990904i \(0.457034\pi\)
\(224\) 4.85799 0.324588
\(225\) 0 0
\(226\) −2.42375 −0.161226
\(227\) 11.7020 0.776690 0.388345 0.921514i \(-0.373047\pi\)
0.388345 + 0.921514i \(0.373047\pi\)
\(228\) 0 0
\(229\) −6.54504 −0.432508 −0.216254 0.976337i \(-0.569384\pi\)
−0.216254 + 0.976337i \(0.569384\pi\)
\(230\) −6.76349 −0.445971
\(231\) 0 0
\(232\) 10.3746 0.681129
\(233\) −0.172567 −0.0113052 −0.00565262 0.999984i \(-0.501799\pi\)
−0.00565262 + 0.999984i \(0.501799\pi\)
\(234\) 0 0
\(235\) 46.6358 3.04219
\(236\) 8.06415 0.524932
\(237\) 0 0
\(238\) −1.17615 −0.0762383
\(239\) 10.4504 0.675982 0.337991 0.941149i \(-0.390253\pi\)
0.337991 + 0.941149i \(0.390253\pi\)
\(240\) 0 0
\(241\) −3.41821 −0.220186 −0.110093 0.993921i \(-0.535115\pi\)
−0.110093 + 0.993921i \(0.535115\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 20.3883 1.30523
\(245\) −3.75881 −0.240142
\(246\) 0 0
\(247\) −4.72409 −0.300587
\(248\) 4.02000 0.255270
\(249\) 0 0
\(250\) 7.35151 0.464951
\(251\) −12.6751 −0.800047 −0.400024 0.916505i \(-0.630998\pi\)
−0.400024 + 0.916505i \(0.630998\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −0.0963238 −0.00604389
\(255\) 0 0
\(256\) 0.913441 0.0570901
\(257\) 18.1852 1.13436 0.567182 0.823593i \(-0.308034\pi\)
0.567182 + 0.823593i \(0.308034\pi\)
\(258\) 0 0
\(259\) 7.65311 0.475541
\(260\) 18.0718 1.12077
\(261\) 0 0
\(262\) −4.86922 −0.300822
\(263\) −15.7837 −0.973264 −0.486632 0.873607i \(-0.661775\pi\)
−0.486632 + 0.873607i \(0.661775\pi\)
\(264\) 0 0
\(265\) 33.2262 2.04107
\(266\) −0.826469 −0.0506741
\(267\) 0 0
\(268\) 21.6695 1.32367
\(269\) −22.9896 −1.40170 −0.700851 0.713308i \(-0.747196\pi\)
−0.700851 + 0.713308i \(0.747196\pi\)
\(270\) 0 0
\(271\) 9.34021 0.567377 0.283689 0.958916i \(-0.408442\pi\)
0.283689 + 0.958916i \(0.408442\pi\)
\(272\) −6.71339 −0.407059
\(273\) 0 0
\(274\) −8.10505 −0.489644
\(275\) 0 0
\(276\) 0 0
\(277\) 19.4367 1.16784 0.583918 0.811813i \(-0.301519\pi\)
0.583918 + 0.811813i \(0.301519\pi\)
\(278\) −9.23784 −0.554049
\(279\) 0 0
\(280\) 6.72282 0.401765
\(281\) 4.44358 0.265082 0.132541 0.991178i \(-0.457686\pi\)
0.132541 + 0.991178i \(0.457686\pi\)
\(282\) 0 0
\(283\) 13.8486 0.823216 0.411608 0.911361i \(-0.364967\pi\)
0.411608 + 0.911361i \(0.364967\pi\)
\(284\) 13.7784 0.817598
\(285\) 0 0
\(286\) 0 0
\(287\) −4.18500 −0.247033
\(288\) 0 0
\(289\) −10.8356 −0.637387
\(290\) 10.3285 0.606512
\(291\) 0 0
\(292\) −27.8406 −1.62925
\(293\) 19.2359 1.12377 0.561887 0.827214i \(-0.310076\pi\)
0.561887 + 0.827214i \(0.310076\pi\)
\(294\) 0 0
\(295\) 17.0713 0.993927
\(296\) −13.6880 −0.795596
\(297\) 0 0
\(298\) −4.68520 −0.271406
\(299\) −10.2852 −0.594807
\(300\) 0 0
\(301\) −7.75162 −0.446796
\(302\) 7.34319 0.422553
\(303\) 0 0
\(304\) −4.71745 −0.270564
\(305\) 43.1606 2.47137
\(306\) 0 0
\(307\) 2.24474 0.128114 0.0640571 0.997946i \(-0.479596\pi\)
0.0640571 + 0.997946i \(0.479596\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 4.00213 0.227306
\(311\) −12.6291 −0.716129 −0.358065 0.933697i \(-0.616563\pi\)
−0.358065 + 0.933697i \(0.616563\pi\)
\(312\) 0 0
\(313\) 1.06646 0.0602797 0.0301398 0.999546i \(-0.490405\pi\)
0.0301398 + 0.999546i \(0.490405\pi\)
\(314\) −8.03115 −0.453224
\(315\) 0 0
\(316\) 12.5247 0.704569
\(317\) 25.1710 1.41374 0.706872 0.707342i \(-0.250106\pi\)
0.706872 + 0.707342i \(0.250106\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 11.6770 0.652766
\(321\) 0 0
\(322\) −1.79937 −0.100275
\(323\) 4.33169 0.241022
\(324\) 0 0
\(325\) 24.7181 1.37111
\(326\) −9.46188 −0.524045
\(327\) 0 0
\(328\) 7.48507 0.413294
\(329\) 12.4071 0.684024
\(330\) 0 0
\(331\) −8.53640 −0.469203 −0.234601 0.972092i \(-0.575378\pi\)
−0.234601 + 0.972092i \(0.575378\pi\)
\(332\) 21.1304 1.15968
\(333\) 0 0
\(334\) 8.26704 0.452352
\(335\) 45.8728 2.50630
\(336\) 0 0
\(337\) −17.2565 −0.940020 −0.470010 0.882661i \(-0.655750\pi\)
−0.470010 + 0.882661i \(0.655750\pi\)
\(338\) 2.68507 0.146048
\(339\) 0 0
\(340\) −16.5707 −0.898673
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 13.8642 0.747505
\(345\) 0 0
\(346\) −4.43369 −0.238357
\(347\) −4.15698 −0.223159 −0.111579 0.993756i \(-0.535591\pi\)
−0.111579 + 0.993756i \(0.535591\pi\)
\(348\) 0 0
\(349\) 7.70785 0.412592 0.206296 0.978490i \(-0.433859\pi\)
0.206296 + 0.978490i \(0.433859\pi\)
\(350\) 4.32437 0.231147
\(351\) 0 0
\(352\) 0 0
\(353\) 4.84242 0.257736 0.128868 0.991662i \(-0.458866\pi\)
0.128868 + 0.991662i \(0.458866\pi\)
\(354\) 0 0
\(355\) 29.1679 1.54807
\(356\) −3.00393 −0.159208
\(357\) 0 0
\(358\) 2.60233 0.137537
\(359\) −4.41834 −0.233191 −0.116596 0.993179i \(-0.537198\pi\)
−0.116596 + 0.993179i \(0.537198\pi\)
\(360\) 0 0
\(361\) −15.9562 −0.839798
\(362\) 5.29821 0.278468
\(363\) 0 0
\(364\) 4.80786 0.252000
\(365\) −58.9367 −3.08489
\(366\) 0 0
\(367\) −29.5315 −1.54153 −0.770766 0.637118i \(-0.780126\pi\)
−0.770766 + 0.637118i \(0.780126\pi\)
\(368\) −10.2707 −0.535398
\(369\) 0 0
\(370\) −13.6271 −0.708440
\(371\) 8.83956 0.458927
\(372\) 0 0
\(373\) 7.64965 0.396084 0.198042 0.980194i \(-0.436542\pi\)
0.198042 + 0.980194i \(0.436542\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −22.1906 −1.14440
\(377\) 15.7065 0.808927
\(378\) 0 0
\(379\) −11.2094 −0.575788 −0.287894 0.957662i \(-0.592955\pi\)
−0.287894 + 0.957662i \(0.592955\pi\)
\(380\) −11.6441 −0.597330
\(381\) 0 0
\(382\) −2.72774 −0.139563
\(383\) 24.7447 1.26440 0.632199 0.774806i \(-0.282153\pi\)
0.632199 + 0.774806i \(0.282153\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 7.05102 0.358887
\(387\) 0 0
\(388\) −12.4228 −0.630670
\(389\) 29.0347 1.47212 0.736060 0.676917i \(-0.236684\pi\)
0.736060 + 0.676917i \(0.236684\pi\)
\(390\) 0 0
\(391\) 9.43085 0.476939
\(392\) 1.78855 0.0903354
\(393\) 0 0
\(394\) 5.94395 0.299452
\(395\) 26.5139 1.33406
\(396\) 0 0
\(397\) −8.27461 −0.415291 −0.207645 0.978204i \(-0.566580\pi\)
−0.207645 + 0.978204i \(0.566580\pi\)
\(398\) −0.395275 −0.0198133
\(399\) 0 0
\(400\) 24.6833 1.23417
\(401\) 26.7390 1.33528 0.667642 0.744483i \(-0.267304\pi\)
0.667642 + 0.744483i \(0.267304\pi\)
\(402\) 0 0
\(403\) 6.08601 0.303166
\(404\) 4.82399 0.240003
\(405\) 0 0
\(406\) 2.74782 0.136372
\(407\) 0 0
\(408\) 0 0
\(409\) 34.6829 1.71496 0.857479 0.514519i \(-0.172030\pi\)
0.857479 + 0.514519i \(0.172030\pi\)
\(410\) 7.45180 0.368018
\(411\) 0 0
\(412\) 0.454177 0.0223757
\(413\) 4.54166 0.223480
\(414\) 0 0
\(415\) 44.7315 2.19578
\(416\) −13.1542 −0.644936
\(417\) 0 0
\(418\) 0 0
\(419\) 4.34546 0.212290 0.106145 0.994351i \(-0.466149\pi\)
0.106145 + 0.994351i \(0.466149\pi\)
\(420\) 0 0
\(421\) −4.73841 −0.230936 −0.115468 0.993311i \(-0.536837\pi\)
−0.115468 + 0.993311i \(0.536837\pi\)
\(422\) 12.7288 0.619629
\(423\) 0 0
\(424\) −15.8100 −0.767800
\(425\) −22.6649 −1.09941
\(426\) 0 0
\(427\) 11.4825 0.555678
\(428\) 21.0599 1.01797
\(429\) 0 0
\(430\) 13.8025 0.665617
\(431\) 29.8622 1.43841 0.719206 0.694797i \(-0.244506\pi\)
0.719206 + 0.694797i \(0.244506\pi\)
\(432\) 0 0
\(433\) −21.7239 −1.04399 −0.521993 0.852950i \(-0.674811\pi\)
−0.521993 + 0.852950i \(0.674811\pi\)
\(434\) 1.06473 0.0511088
\(435\) 0 0
\(436\) 28.1822 1.34968
\(437\) 6.62698 0.317012
\(438\) 0 0
\(439\) −40.3447 −1.92555 −0.962773 0.270313i \(-0.912873\pi\)
−0.962773 + 0.270313i \(0.912873\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 3.18470 0.151481
\(443\) 13.8789 0.659408 0.329704 0.944084i \(-0.393051\pi\)
0.329704 + 0.944084i \(0.393051\pi\)
\(444\) 0 0
\(445\) −6.35912 −0.301451
\(446\) −1.90394 −0.0901540
\(447\) 0 0
\(448\) 3.10658 0.146772
\(449\) 14.3023 0.674966 0.337483 0.941332i \(-0.390425\pi\)
0.337483 + 0.941332i \(0.390425\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −9.08484 −0.427315
\(453\) 0 0
\(454\) −5.54340 −0.260164
\(455\) 10.1779 0.477147
\(456\) 0 0
\(457\) −14.1278 −0.660869 −0.330435 0.943829i \(-0.607195\pi\)
−0.330435 + 0.943829i \(0.607195\pi\)
\(458\) 3.10047 0.144875
\(459\) 0 0
\(460\) −25.3513 −1.18201
\(461\) 0.966736 0.0450254 0.0225127 0.999747i \(-0.492833\pi\)
0.0225127 + 0.999747i \(0.492833\pi\)
\(462\) 0 0
\(463\) 31.8119 1.47842 0.739212 0.673473i \(-0.235198\pi\)
0.739212 + 0.673473i \(0.235198\pi\)
\(464\) 15.6844 0.728131
\(465\) 0 0
\(466\) 0.0817473 0.00378687
\(467\) 29.6133 1.37034 0.685169 0.728384i \(-0.259728\pi\)
0.685169 + 0.728384i \(0.259728\pi\)
\(468\) 0 0
\(469\) 12.2041 0.563531
\(470\) −22.0920 −1.01903
\(471\) 0 0
\(472\) −8.12298 −0.373891
\(473\) 0 0
\(474\) 0 0
\(475\) −15.9264 −0.730756
\(476\) −4.40849 −0.202063
\(477\) 0 0
\(478\) −4.95051 −0.226431
\(479\) 26.5853 1.21471 0.607356 0.794430i \(-0.292230\pi\)
0.607356 + 0.794430i \(0.292230\pi\)
\(480\) 0 0
\(481\) −20.7226 −0.944871
\(482\) 1.61925 0.0737548
\(483\) 0 0
\(484\) 0 0
\(485\) −26.2981 −1.19414
\(486\) 0 0
\(487\) 14.1343 0.640486 0.320243 0.947335i \(-0.396235\pi\)
0.320243 + 0.947335i \(0.396235\pi\)
\(488\) −20.5370 −0.929668
\(489\) 0 0
\(490\) 1.78060 0.0804392
\(491\) −15.4186 −0.695831 −0.347916 0.937526i \(-0.613110\pi\)
−0.347916 + 0.937526i \(0.613110\pi\)
\(492\) 0 0
\(493\) −14.4019 −0.648627
\(494\) 2.23786 0.100686
\(495\) 0 0
\(496\) 6.07744 0.272885
\(497\) 7.75988 0.348078
\(498\) 0 0
\(499\) 30.7024 1.37443 0.687214 0.726455i \(-0.258833\pi\)
0.687214 + 0.726455i \(0.258833\pi\)
\(500\) 27.5553 1.23231
\(501\) 0 0
\(502\) 6.00438 0.267988
\(503\) 18.1890 0.811005 0.405503 0.914094i \(-0.367096\pi\)
0.405503 + 0.914094i \(0.367096\pi\)
\(504\) 0 0
\(505\) 10.2121 0.454431
\(506\) 0 0
\(507\) 0 0
\(508\) −0.361046 −0.0160188
\(509\) 32.1744 1.42611 0.713054 0.701110i \(-0.247312\pi\)
0.713054 + 0.701110i \(0.247312\pi\)
\(510\) 0 0
\(511\) −15.6796 −0.693625
\(512\) −22.8079 −1.00798
\(513\) 0 0
\(514\) −8.61458 −0.379973
\(515\) 0.961461 0.0423670
\(516\) 0 0
\(517\) 0 0
\(518\) −3.62538 −0.159290
\(519\) 0 0
\(520\) −18.2037 −0.798284
\(521\) 27.8238 1.21898 0.609490 0.792793i \(-0.291374\pi\)
0.609490 + 0.792793i \(0.291374\pi\)
\(522\) 0 0
\(523\) −19.1282 −0.836419 −0.418209 0.908351i \(-0.637342\pi\)
−0.418209 + 0.908351i \(0.637342\pi\)
\(524\) −18.2511 −0.797302
\(525\) 0 0
\(526\) 7.47694 0.326010
\(527\) −5.58048 −0.243089
\(528\) 0 0
\(529\) −8.57190 −0.372691
\(530\) −15.7397 −0.683689
\(531\) 0 0
\(532\) −3.09781 −0.134307
\(533\) 11.3319 0.490839
\(534\) 0 0
\(535\) 44.5824 1.92746
\(536\) −21.8275 −0.942806
\(537\) 0 0
\(538\) 10.8905 0.469522
\(539\) 0 0
\(540\) 0 0
\(541\) 42.6820 1.83504 0.917521 0.397688i \(-0.130187\pi\)
0.917521 + 0.397688i \(0.130187\pi\)
\(542\) −4.42458 −0.190052
\(543\) 0 0
\(544\) 12.0615 0.517134
\(545\) 59.6597 2.55554
\(546\) 0 0
\(547\) −25.1162 −1.07389 −0.536946 0.843617i \(-0.680422\pi\)
−0.536946 + 0.843617i \(0.680422\pi\)
\(548\) −30.3798 −1.29776
\(549\) 0 0
\(550\) 0 0
\(551\) −10.1201 −0.431130
\(552\) 0 0
\(553\) 7.05380 0.299958
\(554\) −9.20740 −0.391185
\(555\) 0 0
\(556\) −34.6258 −1.46846
\(557\) 4.25720 0.180383 0.0901917 0.995924i \(-0.471252\pi\)
0.0901917 + 0.995924i \(0.471252\pi\)
\(558\) 0 0
\(559\) 20.9894 0.887757
\(560\) 10.1636 0.429490
\(561\) 0 0
\(562\) −2.10498 −0.0887933
\(563\) 7.43714 0.313438 0.156719 0.987643i \(-0.449908\pi\)
0.156719 + 0.987643i \(0.449908\pi\)
\(564\) 0 0
\(565\) −19.2320 −0.809096
\(566\) −6.56028 −0.275749
\(567\) 0 0
\(568\) −13.8789 −0.582347
\(569\) −43.6293 −1.82904 −0.914518 0.404546i \(-0.867430\pi\)
−0.914518 + 0.404546i \(0.867430\pi\)
\(570\) 0 0
\(571\) −34.1074 −1.42735 −0.713676 0.700476i \(-0.752971\pi\)
−0.713676 + 0.700476i \(0.752971\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 1.98249 0.0827474
\(575\) −34.6747 −1.44603
\(576\) 0 0
\(577\) 28.0804 1.16900 0.584501 0.811393i \(-0.301290\pi\)
0.584501 + 0.811393i \(0.301290\pi\)
\(578\) 5.13295 0.213503
\(579\) 0 0
\(580\) 38.7140 1.60751
\(581\) 11.9004 0.493713
\(582\) 0 0
\(583\) 0 0
\(584\) 28.0437 1.16046
\(585\) 0 0
\(586\) −9.11230 −0.376426
\(587\) 6.72735 0.277668 0.138834 0.990316i \(-0.455665\pi\)
0.138834 + 0.990316i \(0.455665\pi\)
\(588\) 0 0
\(589\) −3.92136 −0.161577
\(590\) −8.08687 −0.332931
\(591\) 0 0
\(592\) −20.6935 −0.850497
\(593\) −20.8442 −0.855969 −0.427984 0.903786i \(-0.640776\pi\)
−0.427984 + 0.903786i \(0.640776\pi\)
\(594\) 0 0
\(595\) −9.33248 −0.382594
\(596\) −17.5613 −0.719339
\(597\) 0 0
\(598\) 4.87222 0.199240
\(599\) −0.180675 −0.00738218 −0.00369109 0.999993i \(-0.501175\pi\)
−0.00369109 + 0.999993i \(0.501175\pi\)
\(600\) 0 0
\(601\) 27.6043 1.12600 0.563001 0.826456i \(-0.309647\pi\)
0.563001 + 0.826456i \(0.309647\pi\)
\(602\) 3.67204 0.149661
\(603\) 0 0
\(604\) 27.5241 1.11994
\(605\) 0 0
\(606\) 0 0
\(607\) 23.3280 0.946854 0.473427 0.880833i \(-0.343017\pi\)
0.473427 + 0.880833i \(0.343017\pi\)
\(608\) 8.47554 0.343729
\(609\) 0 0
\(610\) −20.4457 −0.827824
\(611\) −33.5951 −1.35911
\(612\) 0 0
\(613\) −19.5003 −0.787608 −0.393804 0.919194i \(-0.628841\pi\)
−0.393804 + 0.919194i \(0.628841\pi\)
\(614\) −1.06336 −0.0429139
\(615\) 0 0
\(616\) 0 0
\(617\) 23.2566 0.936276 0.468138 0.883655i \(-0.344925\pi\)
0.468138 + 0.883655i \(0.344925\pi\)
\(618\) 0 0
\(619\) −28.5491 −1.14749 −0.573743 0.819035i \(-0.694509\pi\)
−0.573743 + 0.819035i \(0.694509\pi\)
\(620\) 15.0010 0.602454
\(621\) 0 0
\(622\) 5.98256 0.239879
\(623\) −1.69179 −0.0677801
\(624\) 0 0
\(625\) 12.6893 0.507574
\(626\) −0.505194 −0.0201916
\(627\) 0 0
\(628\) −30.1028 −1.20123
\(629\) 19.0013 0.757633
\(630\) 0 0
\(631\) 5.30112 0.211034 0.105517 0.994417i \(-0.466350\pi\)
0.105517 + 0.994417i \(0.466350\pi\)
\(632\) −12.6161 −0.501840
\(633\) 0 0
\(634\) −11.9238 −0.473556
\(635\) −0.764309 −0.0303307
\(636\) 0 0
\(637\) 2.70774 0.107285
\(638\) 0 0
\(639\) 0 0
\(640\) −42.0521 −1.66225
\(641\) 29.8881 1.18051 0.590255 0.807217i \(-0.299027\pi\)
0.590255 + 0.807217i \(0.299027\pi\)
\(642\) 0 0
\(643\) 38.4403 1.51594 0.757969 0.652291i \(-0.226192\pi\)
0.757969 + 0.652291i \(0.226192\pi\)
\(644\) −6.74448 −0.265770
\(645\) 0 0
\(646\) −2.05198 −0.0807340
\(647\) 32.3602 1.27221 0.636105 0.771603i \(-0.280545\pi\)
0.636105 + 0.771603i \(0.280545\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −11.7093 −0.459276
\(651\) 0 0
\(652\) −35.4655 −1.38894
\(653\) −23.9046 −0.935459 −0.467729 0.883872i \(-0.654928\pi\)
−0.467729 + 0.883872i \(0.654928\pi\)
\(654\) 0 0
\(655\) −38.6363 −1.50964
\(656\) 11.3160 0.441814
\(657\) 0 0
\(658\) −5.87739 −0.229124
\(659\) −34.5124 −1.34441 −0.672205 0.740365i \(-0.734653\pi\)
−0.672205 + 0.740365i \(0.734653\pi\)
\(660\) 0 0
\(661\) −2.56237 −0.0996647 −0.0498324 0.998758i \(-0.515869\pi\)
−0.0498324 + 0.998758i \(0.515869\pi\)
\(662\) 4.04380 0.157167
\(663\) 0 0
\(664\) −21.2845 −0.826000
\(665\) −6.55786 −0.254303
\(666\) 0 0
\(667\) −22.0332 −0.853128
\(668\) 30.9869 1.19892
\(669\) 0 0
\(670\) −21.7305 −0.839523
\(671\) 0 0
\(672\) 0 0
\(673\) 36.6670 1.41341 0.706704 0.707509i \(-0.250181\pi\)
0.706704 + 0.707509i \(0.250181\pi\)
\(674\) 8.17462 0.314874
\(675\) 0 0
\(676\) 10.0643 0.387089
\(677\) −31.9812 −1.22914 −0.614568 0.788864i \(-0.710670\pi\)
−0.614568 + 0.788864i \(0.710670\pi\)
\(678\) 0 0
\(679\) −6.99638 −0.268497
\(680\) 16.6916 0.640093
\(681\) 0 0
\(682\) 0 0
\(683\) −6.96871 −0.266650 −0.133325 0.991072i \(-0.542565\pi\)
−0.133325 + 0.991072i \(0.542565\pi\)
\(684\) 0 0
\(685\) −64.3119 −2.45723
\(686\) 0.473713 0.0180864
\(687\) 0 0
\(688\) 20.9599 0.799087
\(689\) −23.9352 −0.911860
\(690\) 0 0
\(691\) −5.07203 −0.192949 −0.0964746 0.995335i \(-0.530757\pi\)
−0.0964746 + 0.995335i \(0.530757\pi\)
\(692\) −16.6186 −0.631744
\(693\) 0 0
\(694\) 1.96922 0.0747505
\(695\) −73.3003 −2.78044
\(696\) 0 0
\(697\) −10.3906 −0.393573
\(698\) −3.65131 −0.138204
\(699\) 0 0
\(700\) 16.2088 0.612637
\(701\) −0.418223 −0.0157961 −0.00789803 0.999969i \(-0.502514\pi\)
−0.00789803 + 0.999969i \(0.502514\pi\)
\(702\) 0 0
\(703\) 13.3521 0.503583
\(704\) 0 0
\(705\) 0 0
\(706\) −2.29392 −0.0863328
\(707\) 2.71683 0.102177
\(708\) 0 0
\(709\) −3.60664 −0.135450 −0.0677251 0.997704i \(-0.521574\pi\)
−0.0677251 + 0.997704i \(0.521574\pi\)
\(710\) −13.8172 −0.518551
\(711\) 0 0
\(712\) 3.02585 0.113398
\(713\) −8.53748 −0.319731
\(714\) 0 0
\(715\) 0 0
\(716\) 9.75419 0.364531
\(717\) 0 0
\(718\) 2.09303 0.0781110
\(719\) 17.9697 0.670155 0.335078 0.942190i \(-0.391237\pi\)
0.335078 + 0.942190i \(0.391237\pi\)
\(720\) 0 0
\(721\) 0.255788 0.00952606
\(722\) 7.55864 0.281303
\(723\) 0 0
\(724\) 19.8590 0.738055
\(725\) 52.9518 1.96658
\(726\) 0 0
\(727\) 10.0774 0.373752 0.186876 0.982384i \(-0.440164\pi\)
0.186876 + 0.982384i \(0.440164\pi\)
\(728\) −4.84293 −0.179491
\(729\) 0 0
\(730\) 27.9191 1.03333
\(731\) −19.2459 −0.711836
\(732\) 0 0
\(733\) −12.2275 −0.451631 −0.225816 0.974170i \(-0.572505\pi\)
−0.225816 + 0.974170i \(0.572505\pi\)
\(734\) 13.9895 0.516360
\(735\) 0 0
\(736\) 18.4527 0.680177
\(737\) 0 0
\(738\) 0 0
\(739\) 3.49821 0.128684 0.0643419 0.997928i \(-0.479505\pi\)
0.0643419 + 0.997928i \(0.479505\pi\)
\(740\) −51.0779 −1.87766
\(741\) 0 0
\(742\) −4.18741 −0.153725
\(743\) −2.73642 −0.100389 −0.0501947 0.998739i \(-0.515984\pi\)
−0.0501947 + 0.998739i \(0.515984\pi\)
\(744\) 0 0
\(745\) −37.1761 −1.36203
\(746\) −3.62374 −0.132674
\(747\) 0 0
\(748\) 0 0
\(749\) 11.8608 0.433382
\(750\) 0 0
\(751\) −10.0480 −0.366655 −0.183327 0.983052i \(-0.558687\pi\)
−0.183327 + 0.983052i \(0.558687\pi\)
\(752\) −33.5479 −1.22337
\(753\) 0 0
\(754\) −7.44038 −0.270963
\(755\) 58.2667 2.12054
\(756\) 0 0
\(757\) 15.1490 0.550600 0.275300 0.961358i \(-0.411223\pi\)
0.275300 + 0.961358i \(0.411223\pi\)
\(758\) 5.31004 0.192869
\(759\) 0 0
\(760\) 11.7290 0.425457
\(761\) −35.4413 −1.28475 −0.642373 0.766392i \(-0.722050\pi\)
−0.642373 + 0.766392i \(0.722050\pi\)
\(762\) 0 0
\(763\) 15.8720 0.574604
\(764\) −10.2243 −0.369901
\(765\) 0 0
\(766\) −11.7219 −0.423530
\(767\) −12.2976 −0.444042
\(768\) 0 0
\(769\) 38.7592 1.39769 0.698845 0.715273i \(-0.253698\pi\)
0.698845 + 0.715273i \(0.253698\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 26.4290 0.951200
\(773\) −26.5419 −0.954646 −0.477323 0.878728i \(-0.658393\pi\)
−0.477323 + 0.878728i \(0.658393\pi\)
\(774\) 0 0
\(775\) 20.5179 0.737025
\(776\) 12.5134 0.449204
\(777\) 0 0
\(778\) −13.7541 −0.493109
\(779\) −7.30140 −0.261600
\(780\) 0 0
\(781\) 0 0
\(782\) −4.46752 −0.159758
\(783\) 0 0
\(784\) 2.70393 0.0965690
\(785\) −63.7255 −2.27446
\(786\) 0 0
\(787\) 19.6802 0.701524 0.350762 0.936465i \(-0.385923\pi\)
0.350762 + 0.936465i \(0.385923\pi\)
\(788\) 22.2794 0.793672
\(789\) 0 0
\(790\) −12.5600 −0.446864
\(791\) −5.11650 −0.181922
\(792\) 0 0
\(793\) −31.0917 −1.10410
\(794\) 3.91979 0.139108
\(795\) 0 0
\(796\) −1.48159 −0.0525135
\(797\) 25.2797 0.895451 0.447726 0.894171i \(-0.352234\pi\)
0.447726 + 0.894171i \(0.352234\pi\)
\(798\) 0 0
\(799\) 30.8046 1.08979
\(800\) −44.3470 −1.56790
\(801\) 0 0
\(802\) −12.6666 −0.447274
\(803\) 0 0
\(804\) 0 0
\(805\) −14.2776 −0.503219
\(806\) −2.88302 −0.101550
\(807\) 0 0
\(808\) −4.85918 −0.170945
\(809\) 35.7891 1.25828 0.629138 0.777293i \(-0.283408\pi\)
0.629138 + 0.777293i \(0.283408\pi\)
\(810\) 0 0
\(811\) 20.9415 0.735355 0.367677 0.929953i \(-0.380153\pi\)
0.367677 + 0.929953i \(0.380153\pi\)
\(812\) 10.2995 0.361442
\(813\) 0 0
\(814\) 0 0
\(815\) −75.0780 −2.62987
\(816\) 0 0
\(817\) −13.5240 −0.473143
\(818\) −16.4297 −0.574452
\(819\) 0 0
\(820\) 27.9312 0.975401
\(821\) 8.06663 0.281527 0.140764 0.990043i \(-0.455044\pi\)
0.140764 + 0.990043i \(0.455044\pi\)
\(822\) 0 0
\(823\) −20.1359 −0.701895 −0.350948 0.936395i \(-0.614140\pi\)
−0.350948 + 0.936395i \(0.614140\pi\)
\(824\) −0.457490 −0.0159374
\(825\) 0 0
\(826\) −2.15144 −0.0748583
\(827\) 21.4181 0.744780 0.372390 0.928076i \(-0.378538\pi\)
0.372390 + 0.928076i \(0.378538\pi\)
\(828\) 0 0
\(829\) −27.3960 −0.951504 −0.475752 0.879580i \(-0.657824\pi\)
−0.475752 + 0.879580i \(0.657824\pi\)
\(830\) −21.1899 −0.735512
\(831\) 0 0
\(832\) −8.41181 −0.291627
\(833\) −2.48283 −0.0860248
\(834\) 0 0
\(835\) 65.5972 2.27008
\(836\) 0 0
\(837\) 0 0
\(838\) −2.05850 −0.0711097
\(839\) 20.3757 0.703446 0.351723 0.936104i \(-0.385596\pi\)
0.351723 + 0.936104i \(0.385596\pi\)
\(840\) 0 0
\(841\) 4.64691 0.160238
\(842\) 2.24465 0.0773556
\(843\) 0 0
\(844\) 47.7108 1.64227
\(845\) 21.3055 0.732930
\(846\) 0 0
\(847\) 0 0
\(848\) −23.9016 −0.820783
\(849\) 0 0
\(850\) 10.7367 0.368264
\(851\) 29.0698 0.996501
\(852\) 0 0
\(853\) −8.28686 −0.283737 −0.141868 0.989886i \(-0.545311\pi\)
−0.141868 + 0.989886i \(0.545311\pi\)
\(854\) −5.43942 −0.186133
\(855\) 0 0
\(856\) −21.2135 −0.725064
\(857\) 31.0459 1.06051 0.530254 0.847839i \(-0.322097\pi\)
0.530254 + 0.847839i \(0.322097\pi\)
\(858\) 0 0
\(859\) −1.63124 −0.0556573 −0.0278287 0.999613i \(-0.508859\pi\)
−0.0278287 + 0.999613i \(0.508859\pi\)
\(860\) 51.7354 1.76416
\(861\) 0 0
\(862\) −14.1461 −0.481819
\(863\) −5.32862 −0.181388 −0.0906942 0.995879i \(-0.528909\pi\)
−0.0906942 + 0.995879i \(0.528909\pi\)
\(864\) 0 0
\(865\) −35.1804 −1.19617
\(866\) 10.2909 0.349699
\(867\) 0 0
\(868\) 3.99088 0.135459
\(869\) 0 0
\(870\) 0 0
\(871\) −33.0454 −1.11970
\(872\) −28.3878 −0.961332
\(873\) 0 0
\(874\) −3.13929 −0.106188
\(875\) 15.5189 0.524635
\(876\) 0 0
\(877\) −16.1267 −0.544561 −0.272281 0.962218i \(-0.587778\pi\)
−0.272281 + 0.962218i \(0.587778\pi\)
\(878\) 19.1118 0.644992
\(879\) 0 0
\(880\) 0 0
\(881\) 35.1173 1.18313 0.591565 0.806257i \(-0.298510\pi\)
0.591565 + 0.806257i \(0.298510\pi\)
\(882\) 0 0
\(883\) 0.480232 0.0161611 0.00808054 0.999967i \(-0.497428\pi\)
0.00808054 + 0.999967i \(0.497428\pi\)
\(884\) 11.9371 0.401487
\(885\) 0 0
\(886\) −6.57463 −0.220879
\(887\) −36.7238 −1.23306 −0.616532 0.787330i \(-0.711463\pi\)
−0.616532 + 0.787330i \(0.711463\pi\)
\(888\) 0 0
\(889\) −0.203338 −0.00681973
\(890\) 3.01240 0.100976
\(891\) 0 0
\(892\) −7.13644 −0.238946
\(893\) 21.6461 0.724360
\(894\) 0 0
\(895\) 20.6489 0.690218
\(896\) −11.1876 −0.373751
\(897\) 0 0
\(898\) −6.77517 −0.226090
\(899\) 13.0376 0.434828
\(900\) 0 0
\(901\) 21.9471 0.731163
\(902\) 0 0
\(903\) 0 0
\(904\) 9.15112 0.304362
\(905\) 42.0402 1.39746
\(906\) 0 0
\(907\) 12.1917 0.404818 0.202409 0.979301i \(-0.435123\pi\)
0.202409 + 0.979301i \(0.435123\pi\)
\(908\) −20.7781 −0.689544
\(909\) 0 0
\(910\) −4.82140 −0.159828
\(911\) 8.11848 0.268977 0.134489 0.990915i \(-0.457061\pi\)
0.134489 + 0.990915i \(0.457061\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 6.69251 0.221369
\(915\) 0 0
\(916\) 11.6213 0.383980
\(917\) −10.2788 −0.339438
\(918\) 0 0
\(919\) −49.6539 −1.63793 −0.818965 0.573843i \(-0.805452\pi\)
−0.818965 + 0.573843i \(0.805452\pi\)
\(920\) 25.5362 0.841903
\(921\) 0 0
\(922\) −0.457955 −0.0150820
\(923\) −21.0118 −0.691610
\(924\) 0 0
\(925\) −69.8627 −2.29707
\(926\) −15.0697 −0.495221
\(927\) 0 0
\(928\) −28.1792 −0.925028
\(929\) 0.373776 0.0122632 0.00613160 0.999981i \(-0.498048\pi\)
0.00613160 + 0.999981i \(0.498048\pi\)
\(930\) 0 0
\(931\) −1.74466 −0.0571790
\(932\) 0.306409 0.0100368
\(933\) 0 0
\(934\) −14.0282 −0.459016
\(935\) 0 0
\(936\) 0 0
\(937\) −46.0667 −1.50493 −0.752467 0.658630i \(-0.771136\pi\)
−0.752467 + 0.658630i \(0.771136\pi\)
\(938\) −5.78122 −0.188763
\(939\) 0 0
\(940\) −82.8064 −2.70085
\(941\) 19.3374 0.630380 0.315190 0.949029i \(-0.397932\pi\)
0.315190 + 0.949029i \(0.397932\pi\)
\(942\) 0 0
\(943\) −15.8964 −0.517659
\(944\) −12.2803 −0.399691
\(945\) 0 0
\(946\) 0 0
\(947\) 8.06969 0.262230 0.131115 0.991367i \(-0.458144\pi\)
0.131115 + 0.991367i \(0.458144\pi\)
\(948\) 0 0
\(949\) 42.4563 1.37819
\(950\) 7.54457 0.244778
\(951\) 0 0
\(952\) 4.44065 0.143922
\(953\) −34.6358 −1.12196 −0.560982 0.827828i \(-0.689576\pi\)
−0.560982 + 0.827828i \(0.689576\pi\)
\(954\) 0 0
\(955\) −21.6441 −0.700385
\(956\) −18.5557 −0.600136
\(957\) 0 0
\(958\) −12.5938 −0.406887
\(959\) −17.1096 −0.552498
\(960\) 0 0
\(961\) −25.9482 −0.837037
\(962\) 9.81659 0.316499
\(963\) 0 0
\(964\) 6.06935 0.195481
\(965\) 55.9483 1.80104
\(966\) 0 0
\(967\) 8.15074 0.262110 0.131055 0.991375i \(-0.458164\pi\)
0.131055 + 0.991375i \(0.458164\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 12.4578 0.399994
\(971\) −33.4330 −1.07292 −0.536458 0.843927i \(-0.680238\pi\)
−0.536458 + 0.843927i \(0.680238\pi\)
\(972\) 0 0
\(973\) −19.5009 −0.625171
\(974\) −6.69560 −0.214541
\(975\) 0 0
\(976\) −31.0480 −0.993821
\(977\) 5.48798 0.175576 0.0877880 0.996139i \(-0.472020\pi\)
0.0877880 + 0.996139i \(0.472020\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 6.67413 0.213197
\(981\) 0 0
\(982\) 7.30398 0.233080
\(983\) −14.5404 −0.463767 −0.231884 0.972744i \(-0.574489\pi\)
−0.231884 + 0.972744i \(0.574489\pi\)
\(984\) 0 0
\(985\) 47.1640 1.50277
\(986\) 6.82235 0.217268
\(987\) 0 0
\(988\) 8.38808 0.266860
\(989\) −29.4440 −0.936266
\(990\) 0 0
\(991\) −9.68326 −0.307599 −0.153799 0.988102i \(-0.549151\pi\)
−0.153799 + 0.988102i \(0.549151\pi\)
\(992\) −10.9190 −0.346677
\(993\) 0 0
\(994\) −3.67596 −0.116594
\(995\) −3.13642 −0.0994312
\(996\) 0 0
\(997\) −25.2216 −0.798775 −0.399388 0.916782i \(-0.630777\pi\)
−0.399388 + 0.916782i \(0.630777\pi\)
\(998\) −14.5441 −0.460387
\(999\) 0 0
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 7623.2.a.cx.1.5 10
3.2 odd 2 2541.2.a.bq.1.6 10
11.5 even 5 693.2.m.j.190.3 20
11.9 even 5 693.2.m.j.631.3 20
11.10 odd 2 7623.2.a.cy.1.6 10
33.5 odd 10 231.2.j.g.190.3 yes 20
33.20 odd 10 231.2.j.g.169.3 20
33.32 even 2 2541.2.a.br.1.5 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.j.g.169.3 20 33.20 odd 10
231.2.j.g.190.3 yes 20 33.5 odd 10
693.2.m.j.190.3 20 11.5 even 5
693.2.m.j.631.3 20 11.9 even 5
2541.2.a.bq.1.6 10 3.2 odd 2
2541.2.a.br.1.5 10 33.32 even 2
7623.2.a.cx.1.5 10 1.1 even 1 trivial
7623.2.a.cy.1.6 10 11.10 odd 2