Properties

Label 8001.2.a.x.1.2
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $1$
Dimension $22$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8001,2,Mod(1,8001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8001, 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("8001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.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: \(63.8883066572\)
Analytic rank: \(1\)
Dimension: \(22\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Character \(\chi\) \(=\) 8001.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.32723 q^{2} +3.41602 q^{4} +1.82757 q^{5} +1.00000 q^{7} -3.29542 q^{8} +O(q^{10})\) \(q-2.32723 q^{2} +3.41602 q^{4} +1.82757 q^{5} +1.00000 q^{7} -3.29542 q^{8} -4.25318 q^{10} +3.25127 q^{11} +0.459637 q^{13} -2.32723 q^{14} +0.837162 q^{16} +0.923651 q^{17} +1.76737 q^{19} +6.24302 q^{20} -7.56646 q^{22} -8.20514 q^{23} -1.65999 q^{25} -1.06968 q^{26} +3.41602 q^{28} +7.06378 q^{29} -8.26211 q^{31} +4.64256 q^{32} -2.14955 q^{34} +1.82757 q^{35} +2.64461 q^{37} -4.11309 q^{38} -6.02260 q^{40} -5.05480 q^{41} -10.3966 q^{43} +11.1064 q^{44} +19.0953 q^{46} +6.87722 q^{47} +1.00000 q^{49} +3.86318 q^{50} +1.57013 q^{52} +1.18575 q^{53} +5.94191 q^{55} -3.29542 q^{56} -16.4391 q^{58} -4.54879 q^{59} -5.89946 q^{61} +19.2279 q^{62} -12.4786 q^{64} +0.840019 q^{65} -6.38576 q^{67} +3.15521 q^{68} -4.25318 q^{70} -4.59096 q^{71} -5.10778 q^{73} -6.15463 q^{74} +6.03738 q^{76} +3.25127 q^{77} -16.1766 q^{79} +1.52997 q^{80} +11.7637 q^{82} -10.0510 q^{83} +1.68804 q^{85} +24.1953 q^{86} -10.7143 q^{88} -16.1446 q^{89} +0.459637 q^{91} -28.0289 q^{92} -16.0049 q^{94} +3.23000 q^{95} +17.8106 q^{97} -2.32723 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q + 10 q^{4} + 22 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 22 q + 10 q^{4} + 22 q^{7} - 4 q^{10} - 10 q^{13} - 6 q^{16} - 18 q^{19} - 34 q^{22} - 26 q^{25} + 10 q^{28} - 42 q^{31} - 16 q^{34} - 36 q^{37} - 46 q^{40} - 54 q^{43} - 12 q^{46} + 22 q^{49} - 22 q^{52} - 28 q^{55} - 26 q^{58} - 22 q^{61} - 76 q^{64} - 48 q^{67} - 4 q^{70} - 48 q^{73} - 20 q^{76} - 94 q^{79} - 8 q^{82} - 40 q^{85} - 26 q^{88} - 10 q^{91} - 26 q^{94} - 56 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.32723 −1.64560 −0.822802 0.568328i \(-0.807590\pi\)
−0.822802 + 0.568328i \(0.807590\pi\)
\(3\) 0 0
\(4\) 3.41602 1.70801
\(5\) 1.82757 0.817314 0.408657 0.912688i \(-0.365997\pi\)
0.408657 + 0.912688i \(0.365997\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −3.29542 −1.16511
\(9\) 0 0
\(10\) −4.25318 −1.34497
\(11\) 3.25127 0.980293 0.490147 0.871640i \(-0.336943\pi\)
0.490147 + 0.871640i \(0.336943\pi\)
\(12\) 0 0
\(13\) 0.459637 0.127480 0.0637402 0.997967i \(-0.479697\pi\)
0.0637402 + 0.997967i \(0.479697\pi\)
\(14\) −2.32723 −0.621980
\(15\) 0 0
\(16\) 0.837162 0.209291
\(17\) 0.923651 0.224018 0.112009 0.993707i \(-0.464271\pi\)
0.112009 + 0.993707i \(0.464271\pi\)
\(18\) 0 0
\(19\) 1.76737 0.405463 0.202731 0.979234i \(-0.435018\pi\)
0.202731 + 0.979234i \(0.435018\pi\)
\(20\) 6.24302 1.39598
\(21\) 0 0
\(22\) −7.56646 −1.61317
\(23\) −8.20514 −1.71089 −0.855445 0.517894i \(-0.826716\pi\)
−0.855445 + 0.517894i \(0.826716\pi\)
\(24\) 0 0
\(25\) −1.65999 −0.331998
\(26\) −1.06968 −0.209782
\(27\) 0 0
\(28\) 3.41602 0.645567
\(29\) 7.06378 1.31171 0.655855 0.754887i \(-0.272308\pi\)
0.655855 + 0.754887i \(0.272308\pi\)
\(30\) 0 0
\(31\) −8.26211 −1.48392 −0.741959 0.670445i \(-0.766103\pi\)
−0.741959 + 0.670445i \(0.766103\pi\)
\(32\) 4.64256 0.820696
\(33\) 0 0
\(34\) −2.14955 −0.368645
\(35\) 1.82757 0.308916
\(36\) 0 0
\(37\) 2.64461 0.434771 0.217386 0.976086i \(-0.430247\pi\)
0.217386 + 0.976086i \(0.430247\pi\)
\(38\) −4.11309 −0.667231
\(39\) 0 0
\(40\) −6.02260 −0.952257
\(41\) −5.05480 −0.789428 −0.394714 0.918804i \(-0.629156\pi\)
−0.394714 + 0.918804i \(0.629156\pi\)
\(42\) 0 0
\(43\) −10.3966 −1.58546 −0.792732 0.609570i \(-0.791342\pi\)
−0.792732 + 0.609570i \(0.791342\pi\)
\(44\) 11.1064 1.67435
\(45\) 0 0
\(46\) 19.0953 2.81545
\(47\) 6.87722 1.00315 0.501573 0.865115i \(-0.332755\pi\)
0.501573 + 0.865115i \(0.332755\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 3.86318 0.546337
\(51\) 0 0
\(52\) 1.57013 0.217738
\(53\) 1.18575 0.162875 0.0814375 0.996678i \(-0.474049\pi\)
0.0814375 + 0.996678i \(0.474049\pi\)
\(54\) 0 0
\(55\) 5.94191 0.801208
\(56\) −3.29542 −0.440368
\(57\) 0 0
\(58\) −16.4391 −2.15856
\(59\) −4.54879 −0.592202 −0.296101 0.955157i \(-0.595686\pi\)
−0.296101 + 0.955157i \(0.595686\pi\)
\(60\) 0 0
\(61\) −5.89946 −0.755349 −0.377674 0.925939i \(-0.623276\pi\)
−0.377674 + 0.925939i \(0.623276\pi\)
\(62\) 19.2279 2.44194
\(63\) 0 0
\(64\) −12.4786 −1.55983
\(65\) 0.840019 0.104192
\(66\) 0 0
\(67\) −6.38576 −0.780144 −0.390072 0.920784i \(-0.627550\pi\)
−0.390072 + 0.920784i \(0.627550\pi\)
\(68\) 3.15521 0.382626
\(69\) 0 0
\(70\) −4.25318 −0.508353
\(71\) −4.59096 −0.544847 −0.272424 0.962177i \(-0.587825\pi\)
−0.272424 + 0.962177i \(0.587825\pi\)
\(72\) 0 0
\(73\) −5.10778 −0.597821 −0.298910 0.954281i \(-0.596623\pi\)
−0.298910 + 0.954281i \(0.596623\pi\)
\(74\) −6.15463 −0.715461
\(75\) 0 0
\(76\) 6.03738 0.692535
\(77\) 3.25127 0.370516
\(78\) 0 0
\(79\) −16.1766 −1.82001 −0.910007 0.414594i \(-0.863924\pi\)
−0.910007 + 0.414594i \(0.863924\pi\)
\(80\) 1.52997 0.171056
\(81\) 0 0
\(82\) 11.7637 1.29908
\(83\) −10.0510 −1.10324 −0.551620 0.834095i \(-0.685990\pi\)
−0.551620 + 0.834095i \(0.685990\pi\)
\(84\) 0 0
\(85\) 1.68804 0.183093
\(86\) 24.1953 2.60905
\(87\) 0 0
\(88\) −10.7143 −1.14215
\(89\) −16.1446 −1.71133 −0.855663 0.517533i \(-0.826850\pi\)
−0.855663 + 0.517533i \(0.826850\pi\)
\(90\) 0 0
\(91\) 0.459637 0.0481831
\(92\) −28.0289 −2.92222
\(93\) 0 0
\(94\) −16.0049 −1.65078
\(95\) 3.23000 0.331391
\(96\) 0 0
\(97\) 17.8106 1.80839 0.904196 0.427118i \(-0.140471\pi\)
0.904196 + 0.427118i \(0.140471\pi\)
\(98\) −2.32723 −0.235086
\(99\) 0 0
\(100\) −5.67056 −0.567056
\(101\) 9.57625 0.952872 0.476436 0.879209i \(-0.341928\pi\)
0.476436 + 0.879209i \(0.341928\pi\)
\(102\) 0 0
\(103\) −16.3872 −1.61468 −0.807342 0.590084i \(-0.799094\pi\)
−0.807342 + 0.590084i \(0.799094\pi\)
\(104\) −1.51470 −0.148528
\(105\) 0 0
\(106\) −2.75951 −0.268028
\(107\) 6.81709 0.659033 0.329516 0.944150i \(-0.393114\pi\)
0.329516 + 0.944150i \(0.393114\pi\)
\(108\) 0 0
\(109\) −2.06318 −0.197617 −0.0988083 0.995106i \(-0.531503\pi\)
−0.0988083 + 0.995106i \(0.531503\pi\)
\(110\) −13.8282 −1.31847
\(111\) 0 0
\(112\) 0.837162 0.0791044
\(113\) 1.12170 0.105520 0.0527601 0.998607i \(-0.483198\pi\)
0.0527601 + 0.998607i \(0.483198\pi\)
\(114\) 0 0
\(115\) −14.9955 −1.39833
\(116\) 24.1300 2.24042
\(117\) 0 0
\(118\) 10.5861 0.974529
\(119\) 0.923651 0.0846709
\(120\) 0 0
\(121\) −0.429271 −0.0390247
\(122\) 13.7294 1.24300
\(123\) 0 0
\(124\) −28.2235 −2.53455
\(125\) −12.1716 −1.08866
\(126\) 0 0
\(127\) 1.00000 0.0887357
\(128\) 19.7556 1.74617
\(129\) 0 0
\(130\) −1.95492 −0.171458
\(131\) 7.11191 0.621371 0.310685 0.950513i \(-0.399441\pi\)
0.310685 + 0.950513i \(0.399441\pi\)
\(132\) 0 0
\(133\) 1.76737 0.153251
\(134\) 14.8612 1.28381
\(135\) 0 0
\(136\) −3.04381 −0.261005
\(137\) 11.6913 0.998858 0.499429 0.866355i \(-0.333543\pi\)
0.499429 + 0.866355i \(0.333543\pi\)
\(138\) 0 0
\(139\) −3.50691 −0.297452 −0.148726 0.988878i \(-0.547517\pi\)
−0.148726 + 0.988878i \(0.547517\pi\)
\(140\) 6.24302 0.527631
\(141\) 0 0
\(142\) 10.6843 0.896603
\(143\) 1.49440 0.124968
\(144\) 0 0
\(145\) 12.9095 1.07208
\(146\) 11.8870 0.983776
\(147\) 0 0
\(148\) 9.03404 0.742594
\(149\) −6.88506 −0.564046 −0.282023 0.959408i \(-0.591005\pi\)
−0.282023 + 0.959408i \(0.591005\pi\)
\(150\) 0 0
\(151\) −8.92974 −0.726692 −0.363346 0.931654i \(-0.618366\pi\)
−0.363346 + 0.931654i \(0.618366\pi\)
\(152\) −5.82422 −0.472407
\(153\) 0 0
\(154\) −7.56646 −0.609723
\(155\) −15.0996 −1.21283
\(156\) 0 0
\(157\) −6.65410 −0.531055 −0.265527 0.964103i \(-0.585546\pi\)
−0.265527 + 0.964103i \(0.585546\pi\)
\(158\) 37.6468 2.99502
\(159\) 0 0
\(160\) 8.48460 0.670766
\(161\) −8.20514 −0.646655
\(162\) 0 0
\(163\) −22.0373 −1.72609 −0.863046 0.505126i \(-0.831446\pi\)
−0.863046 + 0.505126i \(0.831446\pi\)
\(164\) −17.2673 −1.34835
\(165\) 0 0
\(166\) 23.3910 1.81550
\(167\) −7.66316 −0.592993 −0.296497 0.955034i \(-0.595818\pi\)
−0.296497 + 0.955034i \(0.595818\pi\)
\(168\) 0 0
\(169\) −12.7887 −0.983749
\(170\) −3.92846 −0.301299
\(171\) 0 0
\(172\) −35.5150 −2.70799
\(173\) 4.68199 0.355965 0.177982 0.984034i \(-0.443043\pi\)
0.177982 + 0.984034i \(0.443043\pi\)
\(174\) 0 0
\(175\) −1.65999 −0.125483
\(176\) 2.72184 0.205166
\(177\) 0 0
\(178\) 37.5723 2.81616
\(179\) −12.8826 −0.962894 −0.481447 0.876475i \(-0.659889\pi\)
−0.481447 + 0.876475i \(0.659889\pi\)
\(180\) 0 0
\(181\) −11.8820 −0.883182 −0.441591 0.897216i \(-0.645586\pi\)
−0.441591 + 0.897216i \(0.645586\pi\)
\(182\) −1.06968 −0.0792903
\(183\) 0 0
\(184\) 27.0393 1.99337
\(185\) 4.83321 0.355345
\(186\) 0 0
\(187\) 3.00303 0.219604
\(188\) 23.4927 1.71338
\(189\) 0 0
\(190\) −7.51696 −0.545337
\(191\) −7.32022 −0.529672 −0.264836 0.964293i \(-0.585318\pi\)
−0.264836 + 0.964293i \(0.585318\pi\)
\(192\) 0 0
\(193\) −5.18335 −0.373106 −0.186553 0.982445i \(-0.559732\pi\)
−0.186553 + 0.982445i \(0.559732\pi\)
\(194\) −41.4494 −2.97590
\(195\) 0 0
\(196\) 3.41602 0.244002
\(197\) −4.87691 −0.347465 −0.173733 0.984793i \(-0.555583\pi\)
−0.173733 + 0.984793i \(0.555583\pi\)
\(198\) 0 0
\(199\) −10.1824 −0.721808 −0.360904 0.932603i \(-0.617532\pi\)
−0.360904 + 0.932603i \(0.617532\pi\)
\(200\) 5.47035 0.386812
\(201\) 0 0
\(202\) −22.2862 −1.56805
\(203\) 7.06378 0.495780
\(204\) 0 0
\(205\) −9.23800 −0.645210
\(206\) 38.1370 2.65713
\(207\) 0 0
\(208\) 0.384791 0.0266805
\(209\) 5.74620 0.397473
\(210\) 0 0
\(211\) 17.1853 1.18308 0.591542 0.806274i \(-0.298519\pi\)
0.591542 + 0.806274i \(0.298519\pi\)
\(212\) 4.05054 0.278192
\(213\) 0 0
\(214\) −15.8650 −1.08451
\(215\) −19.0005 −1.29582
\(216\) 0 0
\(217\) −8.26211 −0.560868
\(218\) 4.80150 0.325199
\(219\) 0 0
\(220\) 20.2977 1.36847
\(221\) 0.424544 0.0285580
\(222\) 0 0
\(223\) 21.7429 1.45602 0.728008 0.685569i \(-0.240447\pi\)
0.728008 + 0.685569i \(0.240447\pi\)
\(224\) 4.64256 0.310194
\(225\) 0 0
\(226\) −2.61045 −0.173645
\(227\) −20.9089 −1.38777 −0.693885 0.720086i \(-0.744102\pi\)
−0.693885 + 0.720086i \(0.744102\pi\)
\(228\) 0 0
\(229\) −0.612842 −0.0404978 −0.0202489 0.999795i \(-0.506446\pi\)
−0.0202489 + 0.999795i \(0.506446\pi\)
\(230\) 34.8980 2.30110
\(231\) 0 0
\(232\) −23.2781 −1.52828
\(233\) −0.167173 −0.0109519 −0.00547593 0.999985i \(-0.501743\pi\)
−0.00547593 + 0.999985i \(0.501743\pi\)
\(234\) 0 0
\(235\) 12.5686 0.819885
\(236\) −15.5388 −1.01149
\(237\) 0 0
\(238\) −2.14955 −0.139335
\(239\) 14.2809 0.923752 0.461876 0.886945i \(-0.347177\pi\)
0.461876 + 0.886945i \(0.347177\pi\)
\(240\) 0 0
\(241\) −27.6063 −1.77828 −0.889138 0.457640i \(-0.848695\pi\)
−0.889138 + 0.457640i \(0.848695\pi\)
\(242\) 0.999015 0.0642191
\(243\) 0 0
\(244\) −20.1527 −1.29014
\(245\) 1.82757 0.116759
\(246\) 0 0
\(247\) 0.812350 0.0516886
\(248\) 27.2271 1.72892
\(249\) 0 0
\(250\) 28.3262 1.79150
\(251\) −9.82264 −0.620000 −0.310000 0.950737i \(-0.600329\pi\)
−0.310000 + 0.950737i \(0.600329\pi\)
\(252\) 0 0
\(253\) −26.6771 −1.67717
\(254\) −2.32723 −0.146024
\(255\) 0 0
\(256\) −21.0187 −1.31367
\(257\) 23.6795 1.47709 0.738544 0.674205i \(-0.235514\pi\)
0.738544 + 0.674205i \(0.235514\pi\)
\(258\) 0 0
\(259\) 2.64461 0.164328
\(260\) 2.86952 0.177960
\(261\) 0 0
\(262\) −16.5511 −1.02253
\(263\) 30.3269 1.87003 0.935017 0.354603i \(-0.115384\pi\)
0.935017 + 0.354603i \(0.115384\pi\)
\(264\) 0 0
\(265\) 2.16704 0.133120
\(266\) −4.11309 −0.252190
\(267\) 0 0
\(268\) −21.8139 −1.33249
\(269\) −20.2768 −1.23630 −0.618148 0.786062i \(-0.712117\pi\)
−0.618148 + 0.786062i \(0.712117\pi\)
\(270\) 0 0
\(271\) 8.68123 0.527347 0.263674 0.964612i \(-0.415066\pi\)
0.263674 + 0.964612i \(0.415066\pi\)
\(272\) 0.773246 0.0468849
\(273\) 0 0
\(274\) −27.2085 −1.64372
\(275\) −5.39706 −0.325455
\(276\) 0 0
\(277\) 28.4172 1.70742 0.853711 0.520747i \(-0.174347\pi\)
0.853711 + 0.520747i \(0.174347\pi\)
\(278\) 8.16141 0.489489
\(279\) 0 0
\(280\) −6.02260 −0.359919
\(281\) 6.58726 0.392963 0.196481 0.980508i \(-0.437048\pi\)
0.196481 + 0.980508i \(0.437048\pi\)
\(282\) 0 0
\(283\) 11.9138 0.708202 0.354101 0.935207i \(-0.384787\pi\)
0.354101 + 0.935207i \(0.384787\pi\)
\(284\) −15.6828 −0.930605
\(285\) 0 0
\(286\) −3.47783 −0.205648
\(287\) −5.05480 −0.298376
\(288\) 0 0
\(289\) −16.1469 −0.949816
\(290\) −30.0435 −1.76422
\(291\) 0 0
\(292\) −17.4483 −1.02108
\(293\) 16.9125 0.988037 0.494018 0.869451i \(-0.335528\pi\)
0.494018 + 0.869451i \(0.335528\pi\)
\(294\) 0 0
\(295\) −8.31323 −0.484015
\(296\) −8.71509 −0.506554
\(297\) 0 0
\(298\) 16.0231 0.928196
\(299\) −3.77139 −0.218105
\(300\) 0 0
\(301\) −10.3966 −0.599249
\(302\) 20.7816 1.19585
\(303\) 0 0
\(304\) 1.47958 0.0848595
\(305\) −10.7817 −0.617357
\(306\) 0 0
\(307\) 25.0202 1.42798 0.713989 0.700157i \(-0.246887\pi\)
0.713989 + 0.700157i \(0.246887\pi\)
\(308\) 11.1064 0.632846
\(309\) 0 0
\(310\) 35.1403 1.99583
\(311\) 11.7577 0.666718 0.333359 0.942800i \(-0.391818\pi\)
0.333359 + 0.942800i \(0.391818\pi\)
\(312\) 0 0
\(313\) −11.0636 −0.625351 −0.312675 0.949860i \(-0.601225\pi\)
−0.312675 + 0.949860i \(0.601225\pi\)
\(314\) 15.4856 0.873906
\(315\) 0 0
\(316\) −55.2597 −3.10860
\(317\) −23.5641 −1.32349 −0.661746 0.749728i \(-0.730184\pi\)
−0.661746 + 0.749728i \(0.730184\pi\)
\(318\) 0 0
\(319\) 22.9662 1.28586
\(320\) −22.8056 −1.27487
\(321\) 0 0
\(322\) 19.0953 1.06414
\(323\) 1.63243 0.0908311
\(324\) 0 0
\(325\) −0.762993 −0.0423232
\(326\) 51.2859 2.84046
\(327\) 0 0
\(328\) 16.6577 0.919766
\(329\) 6.87722 0.379153
\(330\) 0 0
\(331\) 26.4687 1.45485 0.727427 0.686185i \(-0.240716\pi\)
0.727427 + 0.686185i \(0.240716\pi\)
\(332\) −34.3344 −1.88435
\(333\) 0 0
\(334\) 17.8340 0.975832
\(335\) −11.6704 −0.637623
\(336\) 0 0
\(337\) −6.94742 −0.378450 −0.189225 0.981934i \(-0.560598\pi\)
−0.189225 + 0.981934i \(0.560598\pi\)
\(338\) 29.7624 1.61886
\(339\) 0 0
\(340\) 5.76637 0.312725
\(341\) −26.8623 −1.45468
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 34.2611 1.84723
\(345\) 0 0
\(346\) −10.8961 −0.585777
\(347\) −8.93641 −0.479732 −0.239866 0.970806i \(-0.577103\pi\)
−0.239866 + 0.970806i \(0.577103\pi\)
\(348\) 0 0
\(349\) 23.0502 1.23385 0.616925 0.787022i \(-0.288378\pi\)
0.616925 + 0.787022i \(0.288378\pi\)
\(350\) 3.86318 0.206496
\(351\) 0 0
\(352\) 15.0942 0.804523
\(353\) 6.27976 0.334238 0.167119 0.985937i \(-0.446554\pi\)
0.167119 + 0.985937i \(0.446554\pi\)
\(354\) 0 0
\(355\) −8.39031 −0.445311
\(356\) −55.1504 −2.92296
\(357\) 0 0
\(358\) 29.9809 1.58454
\(359\) 1.98015 0.104508 0.0522541 0.998634i \(-0.483359\pi\)
0.0522541 + 0.998634i \(0.483359\pi\)
\(360\) 0 0
\(361\) −15.8764 −0.835600
\(362\) 27.6522 1.45337
\(363\) 0 0
\(364\) 1.57013 0.0822972
\(365\) −9.33483 −0.488607
\(366\) 0 0
\(367\) −6.94879 −0.362724 −0.181362 0.983416i \(-0.558051\pi\)
−0.181362 + 0.983416i \(0.558051\pi\)
\(368\) −6.86903 −0.358073
\(369\) 0 0
\(370\) −11.2480 −0.584756
\(371\) 1.18575 0.0615609
\(372\) 0 0
\(373\) 23.2338 1.20300 0.601500 0.798873i \(-0.294570\pi\)
0.601500 + 0.798873i \(0.294570\pi\)
\(374\) −6.98877 −0.361381
\(375\) 0 0
\(376\) −22.6633 −1.16877
\(377\) 3.24678 0.167217
\(378\) 0 0
\(379\) 8.06351 0.414195 0.207097 0.978320i \(-0.433598\pi\)
0.207097 + 0.978320i \(0.433598\pi\)
\(380\) 11.0337 0.566019
\(381\) 0 0
\(382\) 17.0359 0.871631
\(383\) −11.1000 −0.567182 −0.283591 0.958945i \(-0.591526\pi\)
−0.283591 + 0.958945i \(0.591526\pi\)
\(384\) 0 0
\(385\) 5.94191 0.302828
\(386\) 12.0629 0.613984
\(387\) 0 0
\(388\) 60.8414 3.08875
\(389\) 30.6647 1.55476 0.777381 0.629030i \(-0.216548\pi\)
0.777381 + 0.629030i \(0.216548\pi\)
\(390\) 0 0
\(391\) −7.57868 −0.383270
\(392\) −3.29542 −0.166444
\(393\) 0 0
\(394\) 11.3497 0.571790
\(395\) −29.5639 −1.48752
\(396\) 0 0
\(397\) 9.05032 0.454222 0.227111 0.973869i \(-0.427072\pi\)
0.227111 + 0.973869i \(0.427072\pi\)
\(398\) 23.6967 1.18781
\(399\) 0 0
\(400\) −1.38968 −0.0694840
\(401\) 32.6261 1.62927 0.814635 0.579974i \(-0.196937\pi\)
0.814635 + 0.579974i \(0.196937\pi\)
\(402\) 0 0
\(403\) −3.79757 −0.189171
\(404\) 32.7127 1.62752
\(405\) 0 0
\(406\) −16.4391 −0.815857
\(407\) 8.59833 0.426203
\(408\) 0 0
\(409\) 37.4178 1.85019 0.925095 0.379735i \(-0.123985\pi\)
0.925095 + 0.379735i \(0.123985\pi\)
\(410\) 21.4990 1.06176
\(411\) 0 0
\(412\) −55.9792 −2.75790
\(413\) −4.54879 −0.223831
\(414\) 0 0
\(415\) −18.3689 −0.901694
\(416\) 2.13389 0.104623
\(417\) 0 0
\(418\) −13.3727 −0.654082
\(419\) −36.3659 −1.77659 −0.888295 0.459274i \(-0.848110\pi\)
−0.888295 + 0.459274i \(0.848110\pi\)
\(420\) 0 0
\(421\) 37.7154 1.83814 0.919069 0.394096i \(-0.128942\pi\)
0.919069 + 0.394096i \(0.128942\pi\)
\(422\) −39.9942 −1.94689
\(423\) 0 0
\(424\) −3.90753 −0.189766
\(425\) −1.53325 −0.0743736
\(426\) 0 0
\(427\) −5.89946 −0.285495
\(428\) 23.2873 1.12564
\(429\) 0 0
\(430\) 44.2186 2.13241
\(431\) 2.24289 0.108036 0.0540181 0.998540i \(-0.482797\pi\)
0.0540181 + 0.998540i \(0.482797\pi\)
\(432\) 0 0
\(433\) 6.95449 0.334211 0.167106 0.985939i \(-0.446558\pi\)
0.167106 + 0.985939i \(0.446558\pi\)
\(434\) 19.2279 0.922967
\(435\) 0 0
\(436\) −7.04786 −0.337531
\(437\) −14.5015 −0.693702
\(438\) 0 0
\(439\) −12.6892 −0.605621 −0.302810 0.953051i \(-0.597925\pi\)
−0.302810 + 0.953051i \(0.597925\pi\)
\(440\) −19.5811 −0.933491
\(441\) 0 0
\(442\) −0.988015 −0.0469951
\(443\) −15.1946 −0.721918 −0.360959 0.932582i \(-0.617551\pi\)
−0.360959 + 0.932582i \(0.617551\pi\)
\(444\) 0 0
\(445\) −29.5054 −1.39869
\(446\) −50.6009 −2.39602
\(447\) 0 0
\(448\) −12.4786 −0.589561
\(449\) −9.95655 −0.469879 −0.234939 0.972010i \(-0.575489\pi\)
−0.234939 + 0.972010i \(0.575489\pi\)
\(450\) 0 0
\(451\) −16.4345 −0.773871
\(452\) 3.83174 0.180230
\(453\) 0 0
\(454\) 48.6598 2.28372
\(455\) 0.840019 0.0393807
\(456\) 0 0
\(457\) 21.7339 1.01667 0.508334 0.861160i \(-0.330261\pi\)
0.508334 + 0.861160i \(0.330261\pi\)
\(458\) 1.42623 0.0666433
\(459\) 0 0
\(460\) −51.2248 −2.38837
\(461\) 30.8514 1.43690 0.718448 0.695581i \(-0.244853\pi\)
0.718448 + 0.695581i \(0.244853\pi\)
\(462\) 0 0
\(463\) −14.6055 −0.678773 −0.339386 0.940647i \(-0.610219\pi\)
−0.339386 + 0.940647i \(0.610219\pi\)
\(464\) 5.91353 0.274529
\(465\) 0 0
\(466\) 0.389051 0.0180224
\(467\) 4.61228 0.213431 0.106715 0.994290i \(-0.465967\pi\)
0.106715 + 0.994290i \(0.465967\pi\)
\(468\) 0 0
\(469\) −6.38576 −0.294867
\(470\) −29.2501 −1.34921
\(471\) 0 0
\(472\) 14.9901 0.689978
\(473\) −33.8021 −1.55422
\(474\) 0 0
\(475\) −2.93382 −0.134613
\(476\) 3.15521 0.144619
\(477\) 0 0
\(478\) −33.2349 −1.52013
\(479\) 35.3743 1.61629 0.808147 0.588981i \(-0.200471\pi\)
0.808147 + 0.588981i \(0.200471\pi\)
\(480\) 0 0
\(481\) 1.21556 0.0554248
\(482\) 64.2462 2.92634
\(483\) 0 0
\(484\) −1.46640 −0.0666545
\(485\) 32.5501 1.47802
\(486\) 0 0
\(487\) 17.3071 0.784259 0.392129 0.919910i \(-0.371739\pi\)
0.392129 + 0.919910i \(0.371739\pi\)
\(488\) 19.4412 0.880061
\(489\) 0 0
\(490\) −4.25318 −0.192139
\(491\) −3.61049 −0.162939 −0.0814696 0.996676i \(-0.525961\pi\)
−0.0814696 + 0.996676i \(0.525961\pi\)
\(492\) 0 0
\(493\) 6.52447 0.293847
\(494\) −1.89053 −0.0850589
\(495\) 0 0
\(496\) −6.91672 −0.310570
\(497\) −4.59096 −0.205933
\(498\) 0 0
\(499\) 8.71652 0.390205 0.195103 0.980783i \(-0.437496\pi\)
0.195103 + 0.980783i \(0.437496\pi\)
\(500\) −41.5784 −1.85944
\(501\) 0 0
\(502\) 22.8596 1.02027
\(503\) −17.0106 −0.758463 −0.379232 0.925302i \(-0.623812\pi\)
−0.379232 + 0.925302i \(0.623812\pi\)
\(504\) 0 0
\(505\) 17.5013 0.778796
\(506\) 62.0838 2.75996
\(507\) 0 0
\(508\) 3.41602 0.151561
\(509\) −33.1772 −1.47055 −0.735276 0.677768i \(-0.762947\pi\)
−0.735276 + 0.677768i \(0.762947\pi\)
\(510\) 0 0
\(511\) −5.10778 −0.225955
\(512\) 9.40417 0.415609
\(513\) 0 0
\(514\) −55.1078 −2.43070
\(515\) −29.9488 −1.31970
\(516\) 0 0
\(517\) 22.3597 0.983377
\(518\) −6.15463 −0.270419
\(519\) 0 0
\(520\) −2.76821 −0.121394
\(521\) −3.54384 −0.155259 −0.0776293 0.996982i \(-0.524735\pi\)
−0.0776293 + 0.996982i \(0.524735\pi\)
\(522\) 0 0
\(523\) 1.14095 0.0498901 0.0249450 0.999689i \(-0.492059\pi\)
0.0249450 + 0.999689i \(0.492059\pi\)
\(524\) 24.2945 1.06131
\(525\) 0 0
\(526\) −70.5777 −3.07733
\(527\) −7.63130 −0.332425
\(528\) 0 0
\(529\) 44.3243 1.92714
\(530\) −5.04320 −0.219063
\(531\) 0 0
\(532\) 6.03738 0.261754
\(533\) −2.32338 −0.100637
\(534\) 0 0
\(535\) 12.4587 0.538637
\(536\) 21.0437 0.908950
\(537\) 0 0
\(538\) 47.1888 2.03445
\(539\) 3.25127 0.140042
\(540\) 0 0
\(541\) −27.1946 −1.16919 −0.584594 0.811326i \(-0.698746\pi\)
−0.584594 + 0.811326i \(0.698746\pi\)
\(542\) −20.2033 −0.867804
\(543\) 0 0
\(544\) 4.28810 0.183851
\(545\) −3.77060 −0.161515
\(546\) 0 0
\(547\) −15.6059 −0.667262 −0.333631 0.942704i \(-0.608274\pi\)
−0.333631 + 0.942704i \(0.608274\pi\)
\(548\) 39.9378 1.70606
\(549\) 0 0
\(550\) 12.5602 0.535570
\(551\) 12.4843 0.531850
\(552\) 0 0
\(553\) −16.1766 −0.687900
\(554\) −66.1334 −2.80974
\(555\) 0 0
\(556\) −11.9797 −0.508052
\(557\) −24.7064 −1.04685 −0.523423 0.852073i \(-0.675345\pi\)
−0.523423 + 0.852073i \(0.675345\pi\)
\(558\) 0 0
\(559\) −4.77866 −0.202116
\(560\) 1.52997 0.0646531
\(561\) 0 0
\(562\) −15.3301 −0.646661
\(563\) −34.7837 −1.46596 −0.732979 0.680251i \(-0.761871\pi\)
−0.732979 + 0.680251i \(0.761871\pi\)
\(564\) 0 0
\(565\) 2.04998 0.0862432
\(566\) −27.7262 −1.16542
\(567\) 0 0
\(568\) 15.1291 0.634804
\(569\) −12.2389 −0.513080 −0.256540 0.966534i \(-0.582582\pi\)
−0.256540 + 0.966534i \(0.582582\pi\)
\(570\) 0 0
\(571\) 3.51258 0.146997 0.0734985 0.997295i \(-0.476584\pi\)
0.0734985 + 0.997295i \(0.476584\pi\)
\(572\) 5.10491 0.213447
\(573\) 0 0
\(574\) 11.7637 0.491008
\(575\) 13.6204 0.568011
\(576\) 0 0
\(577\) 35.0474 1.45904 0.729521 0.683958i \(-0.239743\pi\)
0.729521 + 0.683958i \(0.239743\pi\)
\(578\) 37.5776 1.56302
\(579\) 0 0
\(580\) 44.0993 1.83112
\(581\) −10.0510 −0.416986
\(582\) 0 0
\(583\) 3.85518 0.159665
\(584\) 16.8323 0.696524
\(585\) 0 0
\(586\) −39.3593 −1.62592
\(587\) 27.3306 1.12805 0.564027 0.825756i \(-0.309251\pi\)
0.564027 + 0.825756i \(0.309251\pi\)
\(588\) 0 0
\(589\) −14.6022 −0.601674
\(590\) 19.3468 0.796497
\(591\) 0 0
\(592\) 2.21397 0.0909935
\(593\) −13.2955 −0.545980 −0.272990 0.962017i \(-0.588013\pi\)
−0.272990 + 0.962017i \(0.588013\pi\)
\(594\) 0 0
\(595\) 1.68804 0.0692028
\(596\) −23.5195 −0.963396
\(597\) 0 0
\(598\) 8.77690 0.358914
\(599\) 30.9629 1.26511 0.632556 0.774515i \(-0.282006\pi\)
0.632556 + 0.774515i \(0.282006\pi\)
\(600\) 0 0
\(601\) −7.39614 −0.301695 −0.150847 0.988557i \(-0.548200\pi\)
−0.150847 + 0.988557i \(0.548200\pi\)
\(602\) 24.1953 0.986127
\(603\) 0 0
\(604\) −30.5042 −1.24120
\(605\) −0.784523 −0.0318954
\(606\) 0 0
\(607\) 12.5650 0.509998 0.254999 0.966941i \(-0.417925\pi\)
0.254999 + 0.966941i \(0.417925\pi\)
\(608\) 8.20513 0.332762
\(609\) 0 0
\(610\) 25.0915 1.01592
\(611\) 3.16103 0.127881
\(612\) 0 0
\(613\) −17.4801 −0.706017 −0.353008 0.935620i \(-0.614841\pi\)
−0.353008 + 0.935620i \(0.614841\pi\)
\(614\) −58.2278 −2.34988
\(615\) 0 0
\(616\) −10.7143 −0.431690
\(617\) 43.9764 1.77042 0.885212 0.465188i \(-0.154013\pi\)
0.885212 + 0.465188i \(0.154013\pi\)
\(618\) 0 0
\(619\) 22.9321 0.921717 0.460858 0.887474i \(-0.347542\pi\)
0.460858 + 0.887474i \(0.347542\pi\)
\(620\) −51.5805 −2.07152
\(621\) 0 0
\(622\) −27.3629 −1.09715
\(623\) −16.1446 −0.646821
\(624\) 0 0
\(625\) −13.9445 −0.557780
\(626\) 25.7476 1.02908
\(627\) 0 0
\(628\) −22.7305 −0.907048
\(629\) 2.44270 0.0973967
\(630\) 0 0
\(631\) 1.27656 0.0508189 0.0254094 0.999677i \(-0.491911\pi\)
0.0254094 + 0.999677i \(0.491911\pi\)
\(632\) 53.3087 2.12051
\(633\) 0 0
\(634\) 54.8392 2.17794
\(635\) 1.82757 0.0725249
\(636\) 0 0
\(637\) 0.459637 0.0182115
\(638\) −53.4478 −2.11602
\(639\) 0 0
\(640\) 36.1048 1.42717
\(641\) −19.4343 −0.767610 −0.383805 0.923414i \(-0.625386\pi\)
−0.383805 + 0.923414i \(0.625386\pi\)
\(642\) 0 0
\(643\) 28.5251 1.12492 0.562459 0.826825i \(-0.309855\pi\)
0.562459 + 0.826825i \(0.309855\pi\)
\(644\) −28.0289 −1.10449
\(645\) 0 0
\(646\) −3.79906 −0.149472
\(647\) 49.8165 1.95849 0.979243 0.202688i \(-0.0649676\pi\)
0.979243 + 0.202688i \(0.0649676\pi\)
\(648\) 0 0
\(649\) −14.7893 −0.580532
\(650\) 1.77566 0.0696472
\(651\) 0 0
\(652\) −75.2798 −2.94818
\(653\) −21.2095 −0.829994 −0.414997 0.909823i \(-0.636217\pi\)
−0.414997 + 0.909823i \(0.636217\pi\)
\(654\) 0 0
\(655\) 12.9975 0.507855
\(656\) −4.23169 −0.165220
\(657\) 0 0
\(658\) −16.0049 −0.623936
\(659\) 17.4609 0.680179 0.340089 0.940393i \(-0.389543\pi\)
0.340089 + 0.940393i \(0.389543\pi\)
\(660\) 0 0
\(661\) 40.0081 1.55613 0.778067 0.628182i \(-0.216200\pi\)
0.778067 + 0.628182i \(0.216200\pi\)
\(662\) −61.5990 −2.39411
\(663\) 0 0
\(664\) 33.1222 1.28539
\(665\) 3.23000 0.125254
\(666\) 0 0
\(667\) −57.9593 −2.24419
\(668\) −26.1775 −1.01284
\(669\) 0 0
\(670\) 27.1598 1.04927
\(671\) −19.1807 −0.740463
\(672\) 0 0
\(673\) −32.7911 −1.26400 −0.632001 0.774967i \(-0.717766\pi\)
−0.632001 + 0.774967i \(0.717766\pi\)
\(674\) 16.1683 0.622779
\(675\) 0 0
\(676\) −43.6866 −1.68025
\(677\) −11.8714 −0.456254 −0.228127 0.973631i \(-0.573260\pi\)
−0.228127 + 0.973631i \(0.573260\pi\)
\(678\) 0 0
\(679\) 17.8106 0.683508
\(680\) −5.56278 −0.213323
\(681\) 0 0
\(682\) 62.5149 2.39382
\(683\) 36.6310 1.40165 0.700824 0.713334i \(-0.252816\pi\)
0.700824 + 0.713334i \(0.252816\pi\)
\(684\) 0 0
\(685\) 21.3667 0.816380
\(686\) −2.32723 −0.0888542
\(687\) 0 0
\(688\) −8.70363 −0.331823
\(689\) 0.545014 0.0207634
\(690\) 0 0
\(691\) −16.1927 −0.616000 −0.308000 0.951386i \(-0.599660\pi\)
−0.308000 + 0.951386i \(0.599660\pi\)
\(692\) 15.9938 0.607992
\(693\) 0 0
\(694\) 20.7971 0.789448
\(695\) −6.40913 −0.243112
\(696\) 0 0
\(697\) −4.66887 −0.176846
\(698\) −53.6432 −2.03043
\(699\) 0 0
\(700\) −5.67056 −0.214327
\(701\) −38.7730 −1.46444 −0.732218 0.681071i \(-0.761515\pi\)
−0.732218 + 0.681071i \(0.761515\pi\)
\(702\) 0 0
\(703\) 4.67401 0.176284
\(704\) −40.5714 −1.52909
\(705\) 0 0
\(706\) −14.6145 −0.550023
\(707\) 9.57625 0.360152
\(708\) 0 0
\(709\) −36.5278 −1.37183 −0.685915 0.727682i \(-0.740598\pi\)
−0.685915 + 0.727682i \(0.740598\pi\)
\(710\) 19.5262 0.732806
\(711\) 0 0
\(712\) 53.2032 1.99388
\(713\) 67.7917 2.53882
\(714\) 0 0
\(715\) 2.73113 0.102138
\(716\) −44.0074 −1.64463
\(717\) 0 0
\(718\) −4.60827 −0.171979
\(719\) −41.7503 −1.55702 −0.778512 0.627630i \(-0.784025\pi\)
−0.778512 + 0.627630i \(0.784025\pi\)
\(720\) 0 0
\(721\) −16.3872 −0.610293
\(722\) 36.9481 1.37507
\(723\) 0 0
\(724\) −40.5892 −1.50848
\(725\) −11.7258 −0.435485
\(726\) 0 0
\(727\) −17.1822 −0.637251 −0.318626 0.947881i \(-0.603221\pi\)
−0.318626 + 0.947881i \(0.603221\pi\)
\(728\) −1.51470 −0.0561384
\(729\) 0 0
\(730\) 21.7243 0.804054
\(731\) −9.60282 −0.355173
\(732\) 0 0
\(733\) −0.994796 −0.0367436 −0.0183718 0.999831i \(-0.505848\pi\)
−0.0183718 + 0.999831i \(0.505848\pi\)
\(734\) 16.1715 0.596899
\(735\) 0 0
\(736\) −38.0928 −1.40412
\(737\) −20.7618 −0.764770
\(738\) 0 0
\(739\) −11.0471 −0.406373 −0.203187 0.979140i \(-0.565130\pi\)
−0.203187 + 0.979140i \(0.565130\pi\)
\(740\) 16.5103 0.606932
\(741\) 0 0
\(742\) −2.75951 −0.101305
\(743\) 19.7759 0.725507 0.362753 0.931885i \(-0.381837\pi\)
0.362753 + 0.931885i \(0.381837\pi\)
\(744\) 0 0
\(745\) −12.5829 −0.461003
\(746\) −54.0705 −1.97966
\(747\) 0 0
\(748\) 10.2584 0.375085
\(749\) 6.81709 0.249091
\(750\) 0 0
\(751\) −48.4494 −1.76794 −0.883972 0.467540i \(-0.845140\pi\)
−0.883972 + 0.467540i \(0.845140\pi\)
\(752\) 5.75735 0.209949
\(753\) 0 0
\(754\) −7.55601 −0.275174
\(755\) −16.3197 −0.593935
\(756\) 0 0
\(757\) −42.1784 −1.53300 −0.766500 0.642244i \(-0.778004\pi\)
−0.766500 + 0.642244i \(0.778004\pi\)
\(758\) −18.7657 −0.681600
\(759\) 0 0
\(760\) −10.6442 −0.386105
\(761\) −26.8590 −0.973639 −0.486819 0.873503i \(-0.661843\pi\)
−0.486819 + 0.873503i \(0.661843\pi\)
\(762\) 0 0
\(763\) −2.06318 −0.0746921
\(764\) −25.0060 −0.904686
\(765\) 0 0
\(766\) 25.8322 0.933357
\(767\) −2.09079 −0.0754942
\(768\) 0 0
\(769\) −39.8390 −1.43663 −0.718316 0.695717i \(-0.755087\pi\)
−0.718316 + 0.695717i \(0.755087\pi\)
\(770\) −13.8282 −0.498335
\(771\) 0 0
\(772\) −17.7064 −0.637269
\(773\) 39.5219 1.42151 0.710753 0.703442i \(-0.248355\pi\)
0.710753 + 0.703442i \(0.248355\pi\)
\(774\) 0 0
\(775\) 13.7150 0.492658
\(776\) −58.6933 −2.10697
\(777\) 0 0
\(778\) −71.3639 −2.55852
\(779\) −8.93372 −0.320084
\(780\) 0 0
\(781\) −14.9264 −0.534110
\(782\) 17.6374 0.630711
\(783\) 0 0
\(784\) 0.837162 0.0298986
\(785\) −12.1608 −0.434039
\(786\) 0 0
\(787\) −13.8077 −0.492191 −0.246095 0.969246i \(-0.579148\pi\)
−0.246095 + 0.969246i \(0.579148\pi\)
\(788\) −16.6596 −0.593474
\(789\) 0 0
\(790\) 68.8022 2.44787
\(791\) 1.12170 0.0398829
\(792\) 0 0
\(793\) −2.71161 −0.0962922
\(794\) −21.0622 −0.747470
\(795\) 0 0
\(796\) −34.7832 −1.23286
\(797\) 0.963524 0.0341298 0.0170649 0.999854i \(-0.494568\pi\)
0.0170649 + 0.999854i \(0.494568\pi\)
\(798\) 0 0
\(799\) 6.35215 0.224723
\(800\) −7.70659 −0.272469
\(801\) 0 0
\(802\) −75.9286 −2.68113
\(803\) −16.6068 −0.586040
\(804\) 0 0
\(805\) −14.9955 −0.528520
\(806\) 8.83784 0.311300
\(807\) 0 0
\(808\) −31.5577 −1.11020
\(809\) 3.44050 0.120961 0.0604807 0.998169i \(-0.480737\pi\)
0.0604807 + 0.998169i \(0.480737\pi\)
\(810\) 0 0
\(811\) −51.4147 −1.80542 −0.902708 0.430255i \(-0.858424\pi\)
−0.902708 + 0.430255i \(0.858424\pi\)
\(812\) 24.1300 0.846798
\(813\) 0 0
\(814\) −20.0103 −0.701362
\(815\) −40.2746 −1.41076
\(816\) 0 0
\(817\) −18.3746 −0.642847
\(818\) −87.0800 −3.04468
\(819\) 0 0
\(820\) −31.5572 −1.10203
\(821\) −8.94184 −0.312072 −0.156036 0.987751i \(-0.549872\pi\)
−0.156036 + 0.987751i \(0.549872\pi\)
\(822\) 0 0
\(823\) −24.5739 −0.856593 −0.428296 0.903638i \(-0.640886\pi\)
−0.428296 + 0.903638i \(0.640886\pi\)
\(824\) 54.0028 1.88128
\(825\) 0 0
\(826\) 10.5861 0.368337
\(827\) −21.9314 −0.762628 −0.381314 0.924446i \(-0.624528\pi\)
−0.381314 + 0.924446i \(0.624528\pi\)
\(828\) 0 0
\(829\) −15.4559 −0.536805 −0.268402 0.963307i \(-0.586496\pi\)
−0.268402 + 0.963307i \(0.586496\pi\)
\(830\) 42.7487 1.48383
\(831\) 0 0
\(832\) −5.73565 −0.198848
\(833\) 0.923651 0.0320026
\(834\) 0 0
\(835\) −14.0050 −0.484662
\(836\) 19.6291 0.678888
\(837\) 0 0
\(838\) 84.6319 2.92356
\(839\) 5.25125 0.181293 0.0906466 0.995883i \(-0.471107\pi\)
0.0906466 + 0.995883i \(0.471107\pi\)
\(840\) 0 0
\(841\) 20.8970 0.720585
\(842\) −87.7727 −3.02485
\(843\) 0 0
\(844\) 58.7054 2.02072
\(845\) −23.3723 −0.804032
\(846\) 0 0
\(847\) −0.429271 −0.0147499
\(848\) 0.992663 0.0340882
\(849\) 0 0
\(850\) 3.56823 0.122389
\(851\) −21.6994 −0.743845
\(852\) 0 0
\(853\) 0.692697 0.0237175 0.0118588 0.999930i \(-0.496225\pi\)
0.0118588 + 0.999930i \(0.496225\pi\)
\(854\) 13.7294 0.469811
\(855\) 0 0
\(856\) −22.4651 −0.767843
\(857\) 29.2977 1.00079 0.500395 0.865797i \(-0.333188\pi\)
0.500395 + 0.865797i \(0.333188\pi\)
\(858\) 0 0
\(859\) 28.3475 0.967206 0.483603 0.875288i \(-0.339328\pi\)
0.483603 + 0.875288i \(0.339328\pi\)
\(860\) −64.9061 −2.21328
\(861\) 0 0
\(862\) −5.21974 −0.177785
\(863\) 30.8219 1.04919 0.524595 0.851352i \(-0.324217\pi\)
0.524595 + 0.851352i \(0.324217\pi\)
\(864\) 0 0
\(865\) 8.55666 0.290935
\(866\) −16.1847 −0.549979
\(867\) 0 0
\(868\) −28.2235 −0.957969
\(869\) −52.5945 −1.78415
\(870\) 0 0
\(871\) −2.93513 −0.0994531
\(872\) 6.79903 0.230244
\(873\) 0 0
\(874\) 33.7485 1.14156
\(875\) −12.1716 −0.411475
\(876\) 0 0
\(877\) −30.7468 −1.03825 −0.519123 0.854700i \(-0.673741\pi\)
−0.519123 + 0.854700i \(0.673741\pi\)
\(878\) 29.5307 0.996611
\(879\) 0 0
\(880\) 4.97435 0.167685
\(881\) −44.5343 −1.50040 −0.750200 0.661211i \(-0.770043\pi\)
−0.750200 + 0.661211i \(0.770043\pi\)
\(882\) 0 0
\(883\) 22.5101 0.757525 0.378763 0.925494i \(-0.376350\pi\)
0.378763 + 0.925494i \(0.376350\pi\)
\(884\) 1.45025 0.0487773
\(885\) 0 0
\(886\) 35.3614 1.18799
\(887\) −10.1771 −0.341713 −0.170857 0.985296i \(-0.554653\pi\)
−0.170857 + 0.985296i \(0.554653\pi\)
\(888\) 0 0
\(889\) 1.00000 0.0335389
\(890\) 68.6660 2.30169
\(891\) 0 0
\(892\) 74.2744 2.48689
\(893\) 12.1546 0.406738
\(894\) 0 0
\(895\) −23.5439 −0.786987
\(896\) 19.7556 0.659989
\(897\) 0 0
\(898\) 23.1712 0.773234
\(899\) −58.3617 −1.94647
\(900\) 0 0
\(901\) 1.09522 0.0364870
\(902\) 38.2470 1.27348
\(903\) 0 0
\(904\) −3.69645 −0.122942
\(905\) −21.7152 −0.721837
\(906\) 0 0
\(907\) −45.9867 −1.52696 −0.763482 0.645829i \(-0.776512\pi\)
−0.763482 + 0.645829i \(0.776512\pi\)
\(908\) −71.4251 −2.37033
\(909\) 0 0
\(910\) −1.95492 −0.0648050
\(911\) −19.6478 −0.650961 −0.325480 0.945549i \(-0.605526\pi\)
−0.325480 + 0.945549i \(0.605526\pi\)
\(912\) 0 0
\(913\) −32.6785 −1.08150
\(914\) −50.5799 −1.67303
\(915\) 0 0
\(916\) −2.09348 −0.0691706
\(917\) 7.11191 0.234856
\(918\) 0 0
\(919\) 43.5078 1.43519 0.717594 0.696461i \(-0.245243\pi\)
0.717594 + 0.696461i \(0.245243\pi\)
\(920\) 49.4163 1.62921
\(921\) 0 0
\(922\) −71.7986 −2.36456
\(923\) −2.11018 −0.0694574
\(924\) 0 0
\(925\) −4.39002 −0.144343
\(926\) 33.9903 1.11699
\(927\) 0 0
\(928\) 32.7940 1.07652
\(929\) −24.1917 −0.793706 −0.396853 0.917882i \(-0.629898\pi\)
−0.396853 + 0.917882i \(0.629898\pi\)
\(930\) 0 0
\(931\) 1.76737 0.0579233
\(932\) −0.571066 −0.0187059
\(933\) 0 0
\(934\) −10.7339 −0.351222
\(935\) 5.48826 0.179485
\(936\) 0 0
\(937\) 33.6803 1.10029 0.550144 0.835069i \(-0.314573\pi\)
0.550144 + 0.835069i \(0.314573\pi\)
\(938\) 14.8612 0.485234
\(939\) 0 0
\(940\) 42.9346 1.40037
\(941\) 36.5879 1.19273 0.596366 0.802712i \(-0.296611\pi\)
0.596366 + 0.802712i \(0.296611\pi\)
\(942\) 0 0
\(943\) 41.4753 1.35062
\(944\) −3.80807 −0.123942
\(945\) 0 0
\(946\) 78.6653 2.55763
\(947\) 0.972952 0.0316167 0.0158084 0.999875i \(-0.494968\pi\)
0.0158084 + 0.999875i \(0.494968\pi\)
\(948\) 0 0
\(949\) −2.34773 −0.0762105
\(950\) 6.82768 0.221519
\(951\) 0 0
\(952\) −3.04381 −0.0986506
\(953\) −7.66231 −0.248207 −0.124103 0.992269i \(-0.539605\pi\)
−0.124103 + 0.992269i \(0.539605\pi\)
\(954\) 0 0
\(955\) −13.3782 −0.432909
\(956\) 48.7837 1.57778
\(957\) 0 0
\(958\) −82.3243 −2.65978
\(959\) 11.6913 0.377533
\(960\) 0 0
\(961\) 37.2624 1.20201
\(962\) −2.82890 −0.0912073
\(963\) 0 0
\(964\) −94.3036 −3.03731
\(965\) −9.47293 −0.304945
\(966\) 0 0
\(967\) −9.37101 −0.301351 −0.150676 0.988583i \(-0.548145\pi\)
−0.150676 + 0.988583i \(0.548145\pi\)
\(968\) 1.41463 0.0454678
\(969\) 0 0
\(970\) −75.7517 −2.43224
\(971\) 4.76122 0.152795 0.0763974 0.997077i \(-0.475658\pi\)
0.0763974 + 0.997077i \(0.475658\pi\)
\(972\) 0 0
\(973\) −3.50691 −0.112426
\(974\) −40.2776 −1.29058
\(975\) 0 0
\(976\) −4.93880 −0.158087
\(977\) 52.6338 1.68391 0.841953 0.539551i \(-0.181406\pi\)
0.841953 + 0.539551i \(0.181406\pi\)
\(978\) 0 0
\(979\) −52.4904 −1.67760
\(980\) 6.24302 0.199426
\(981\) 0 0
\(982\) 8.40246 0.268133
\(983\) −54.7160 −1.74517 −0.872585 0.488462i \(-0.837558\pi\)
−0.872585 + 0.488462i \(0.837558\pi\)
\(984\) 0 0
\(985\) −8.91289 −0.283988
\(986\) −15.1840 −0.483556
\(987\) 0 0
\(988\) 2.77501 0.0882847
\(989\) 85.3054 2.71255
\(990\) 0 0
\(991\) 44.5606 1.41552 0.707758 0.706455i \(-0.249707\pi\)
0.707758 + 0.706455i \(0.249707\pi\)
\(992\) −38.3573 −1.21785
\(993\) 0 0
\(994\) 10.6843 0.338884
\(995\) −18.6090 −0.589944
\(996\) 0 0
\(997\) 27.0547 0.856830 0.428415 0.903582i \(-0.359072\pi\)
0.428415 + 0.903582i \(0.359072\pi\)
\(998\) −20.2854 −0.642123
\(999\) 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 8001.2.a.x.1.2 22
3.2 odd 2 inner 8001.2.a.x.1.21 yes 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8001.2.a.x.1.2 22 1.1 even 1 trivial
8001.2.a.x.1.21 yes 22 3.2 odd 2 inner