Properties

Label 8001.2.a.t.1.16
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $0$
Dimension $16$
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] = [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: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} - 20 x^{14} + 38 x^{13} + 155 x^{12} - 275 x^{11} - 593 x^{10} + 957 x^{9} + 1177 x^{8} - 1655 x^{7} - 1150 x^{6} + 1279 x^{5} + 474 x^{4} - 280 x^{3} - 83 x^{2} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 889)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Root \(2.70451\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q+2.70451 q^{2} +5.31438 q^{4} +1.33706 q^{5} -1.00000 q^{7} +8.96376 q^{8} +O(q^{10})\) \(q+2.70451 q^{2} +5.31438 q^{4} +1.33706 q^{5} -1.00000 q^{7} +8.96376 q^{8} +3.61609 q^{10} +5.62240 q^{11} +3.64074 q^{13} -2.70451 q^{14} +13.6138 q^{16} +6.28684 q^{17} +2.90229 q^{19} +7.10563 q^{20} +15.2058 q^{22} -5.83444 q^{23} -3.21228 q^{25} +9.84641 q^{26} -5.31438 q^{28} +4.25704 q^{29} -9.19551 q^{31} +18.8912 q^{32} +17.0028 q^{34} -1.33706 q^{35} -7.71198 q^{37} +7.84927 q^{38} +11.9851 q^{40} -2.94873 q^{41} -9.41716 q^{43} +29.8796 q^{44} -15.7793 q^{46} -1.87098 q^{47} +1.00000 q^{49} -8.68764 q^{50} +19.3482 q^{52} -11.9975 q^{53} +7.51748 q^{55} -8.96376 q^{56} +11.5132 q^{58} -5.75850 q^{59} +0.977997 q^{61} -24.8693 q^{62} +23.8639 q^{64} +4.86787 q^{65} -0.352164 q^{67} +33.4107 q^{68} -3.61609 q^{70} +13.3286 q^{71} -11.9574 q^{73} -20.8571 q^{74} +15.4238 q^{76} -5.62240 q^{77} +0.00843871 q^{79} +18.2025 q^{80} -7.97486 q^{82} -0.0999682 q^{83} +8.40587 q^{85} -25.4688 q^{86} +50.3979 q^{88} -8.26088 q^{89} -3.64074 q^{91} -31.0064 q^{92} -5.06010 q^{94} +3.88053 q^{95} +3.01769 q^{97} +2.70451 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8} - 2 q^{10} + 22 q^{11} - 4 q^{13} - 2 q^{14} + 12 q^{16} + 18 q^{17} - 15 q^{19} + 40 q^{20} - 11 q^{22} + 5 q^{23} + 15 q^{25} + 24 q^{26} - 12 q^{28} + 12 q^{29} - 32 q^{31} + 9 q^{32} - 14 q^{34} - 9 q^{35} - 2 q^{37} - 3 q^{38} - 14 q^{40} + 45 q^{41} - 3 q^{43} + 54 q^{44} + 49 q^{47} + 16 q^{49} + 6 q^{50} + 38 q^{52} - 16 q^{53} + 7 q^{55} - 6 q^{56} + 16 q^{58} + 35 q^{59} - 11 q^{61} - 17 q^{62} - 2 q^{64} - 14 q^{65} + 17 q^{67} + 71 q^{68} + 2 q^{70} + 81 q^{71} - 15 q^{73} - 13 q^{74} + 14 q^{76} - 22 q^{77} - 34 q^{79} + 33 q^{80} - 14 q^{82} + 39 q^{83} - 17 q^{85} - 36 q^{86} + 61 q^{88} + 32 q^{89} + 4 q^{91} - 37 q^{92} + 13 q^{94} + 33 q^{95} - 4 q^{97} + 2 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.70451 1.91238 0.956189 0.292751i \(-0.0945706\pi\)
0.956189 + 0.292751i \(0.0945706\pi\)
\(3\) 0 0
\(4\) 5.31438 2.65719
\(5\) 1.33706 0.597950 0.298975 0.954261i \(-0.403355\pi\)
0.298975 + 0.954261i \(0.403355\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 8.96376 3.16917
\(9\) 0 0
\(10\) 3.61609 1.14351
\(11\) 5.62240 1.69522 0.847609 0.530621i \(-0.178041\pi\)
0.847609 + 0.530621i \(0.178041\pi\)
\(12\) 0 0
\(13\) 3.64074 1.00976 0.504879 0.863190i \(-0.331537\pi\)
0.504879 + 0.863190i \(0.331537\pi\)
\(14\) −2.70451 −0.722811
\(15\) 0 0
\(16\) 13.6138 3.40346
\(17\) 6.28684 1.52478 0.762392 0.647116i \(-0.224025\pi\)
0.762392 + 0.647116i \(0.224025\pi\)
\(18\) 0 0
\(19\) 2.90229 0.665830 0.332915 0.942957i \(-0.391968\pi\)
0.332915 + 0.942957i \(0.391968\pi\)
\(20\) 7.10563 1.58887
\(21\) 0 0
\(22\) 15.2058 3.24190
\(23\) −5.83444 −1.21656 −0.608282 0.793721i \(-0.708141\pi\)
−0.608282 + 0.793721i \(0.708141\pi\)
\(24\) 0 0
\(25\) −3.21228 −0.642455
\(26\) 9.84641 1.93104
\(27\) 0 0
\(28\) −5.31438 −1.00432
\(29\) 4.25704 0.790513 0.395256 0.918571i \(-0.370656\pi\)
0.395256 + 0.918571i \(0.370656\pi\)
\(30\) 0 0
\(31\) −9.19551 −1.65156 −0.825781 0.563991i \(-0.809265\pi\)
−0.825781 + 0.563991i \(0.809265\pi\)
\(32\) 18.8912 3.33953
\(33\) 0 0
\(34\) 17.0028 2.91596
\(35\) −1.33706 −0.226004
\(36\) 0 0
\(37\) −7.71198 −1.26784 −0.633921 0.773398i \(-0.718555\pi\)
−0.633921 + 0.773398i \(0.718555\pi\)
\(38\) 7.84927 1.27332
\(39\) 0 0
\(40\) 11.9851 1.89501
\(41\) −2.94873 −0.460514 −0.230257 0.973130i \(-0.573957\pi\)
−0.230257 + 0.973130i \(0.573957\pi\)
\(42\) 0 0
\(43\) −9.41716 −1.43610 −0.718051 0.695990i \(-0.754966\pi\)
−0.718051 + 0.695990i \(0.754966\pi\)
\(44\) 29.8796 4.50451
\(45\) 0 0
\(46\) −15.7793 −2.32653
\(47\) −1.87098 −0.272911 −0.136456 0.990646i \(-0.543571\pi\)
−0.136456 + 0.990646i \(0.543571\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −8.68764 −1.22862
\(51\) 0 0
\(52\) 19.3482 2.68312
\(53\) −11.9975 −1.64799 −0.823993 0.566601i \(-0.808258\pi\)
−0.823993 + 0.566601i \(0.808258\pi\)
\(54\) 0 0
\(55\) 7.51748 1.01366
\(56\) −8.96376 −1.19783
\(57\) 0 0
\(58\) 11.5132 1.51176
\(59\) −5.75850 −0.749693 −0.374847 0.927087i \(-0.622305\pi\)
−0.374847 + 0.927087i \(0.622305\pi\)
\(60\) 0 0
\(61\) 0.977997 0.125220 0.0626098 0.998038i \(-0.480058\pi\)
0.0626098 + 0.998038i \(0.480058\pi\)
\(62\) −24.8693 −3.15841
\(63\) 0 0
\(64\) 23.8639 2.98298
\(65\) 4.86787 0.603785
\(66\) 0 0
\(67\) −0.352164 −0.0430237 −0.0215119 0.999769i \(-0.506848\pi\)
−0.0215119 + 0.999769i \(0.506848\pi\)
\(68\) 33.4107 4.05164
\(69\) 0 0
\(70\) −3.61609 −0.432205
\(71\) 13.3286 1.58181 0.790905 0.611940i \(-0.209610\pi\)
0.790905 + 0.611940i \(0.209610\pi\)
\(72\) 0 0
\(73\) −11.9574 −1.39951 −0.699753 0.714385i \(-0.746707\pi\)
−0.699753 + 0.714385i \(0.746707\pi\)
\(74\) −20.8571 −2.42459
\(75\) 0 0
\(76\) 15.4238 1.76924
\(77\) −5.62240 −0.640732
\(78\) 0 0
\(79\) 0.00843871 0.000949429 0 0.000474714 1.00000i \(-0.499849\pi\)
0.000474714 1.00000i \(0.499849\pi\)
\(80\) 18.2025 2.03510
\(81\) 0 0
\(82\) −7.97486 −0.880676
\(83\) −0.0999682 −0.0109729 −0.00548647 0.999985i \(-0.501746\pi\)
−0.00548647 + 0.999985i \(0.501746\pi\)
\(84\) 0 0
\(85\) 8.40587 0.911745
\(86\) −25.4688 −2.74637
\(87\) 0 0
\(88\) 50.3979 5.37243
\(89\) −8.26088 −0.875651 −0.437826 0.899060i \(-0.644251\pi\)
−0.437826 + 0.899060i \(0.644251\pi\)
\(90\) 0 0
\(91\) −3.64074 −0.381653
\(92\) −31.0064 −3.23264
\(93\) 0 0
\(94\) −5.06010 −0.521909
\(95\) 3.88053 0.398133
\(96\) 0 0
\(97\) 3.01769 0.306400 0.153200 0.988195i \(-0.451042\pi\)
0.153200 + 0.988195i \(0.451042\pi\)
\(98\) 2.70451 0.273197
\(99\) 0 0
\(100\) −17.0712 −1.70712
\(101\) 16.7322 1.66491 0.832457 0.554089i \(-0.186933\pi\)
0.832457 + 0.554089i \(0.186933\pi\)
\(102\) 0 0
\(103\) 5.76048 0.567597 0.283799 0.958884i \(-0.408405\pi\)
0.283799 + 0.958884i \(0.408405\pi\)
\(104\) 32.6347 3.20009
\(105\) 0 0
\(106\) −32.4474 −3.15157
\(107\) 4.24934 0.410799 0.205399 0.978678i \(-0.434151\pi\)
0.205399 + 0.978678i \(0.434151\pi\)
\(108\) 0 0
\(109\) −15.0625 −1.44273 −0.721365 0.692556i \(-0.756485\pi\)
−0.721365 + 0.692556i \(0.756485\pi\)
\(110\) 20.3311 1.93849
\(111\) 0 0
\(112\) −13.6138 −1.28639
\(113\) −11.1372 −1.04770 −0.523850 0.851810i \(-0.675505\pi\)
−0.523850 + 0.851810i \(0.675505\pi\)
\(114\) 0 0
\(115\) −7.80098 −0.727445
\(116\) 22.6235 2.10054
\(117\) 0 0
\(118\) −15.5739 −1.43370
\(119\) −6.28684 −0.576314
\(120\) 0 0
\(121\) 20.6114 1.87376
\(122\) 2.64500 0.239467
\(123\) 0 0
\(124\) −48.8684 −4.38851
\(125\) −10.9803 −0.982107
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) 26.7576 2.36506
\(129\) 0 0
\(130\) 13.1652 1.15467
\(131\) 10.4598 0.913879 0.456940 0.889498i \(-0.348946\pi\)
0.456940 + 0.889498i \(0.348946\pi\)
\(132\) 0 0
\(133\) −2.90229 −0.251660
\(134\) −0.952432 −0.0822776
\(135\) 0 0
\(136\) 56.3538 4.83230
\(137\) −8.31758 −0.710619 −0.355309 0.934749i \(-0.615624\pi\)
−0.355309 + 0.934749i \(0.615624\pi\)
\(138\) 0 0
\(139\) 10.6143 0.900292 0.450146 0.892955i \(-0.351372\pi\)
0.450146 + 0.892955i \(0.351372\pi\)
\(140\) −7.10563 −0.600535
\(141\) 0 0
\(142\) 36.0472 3.02502
\(143\) 20.4697 1.71176
\(144\) 0 0
\(145\) 5.69191 0.472687
\(146\) −32.3389 −2.67638
\(147\) 0 0
\(148\) −40.9844 −3.36889
\(149\) 0.259521 0.0212608 0.0106304 0.999943i \(-0.496616\pi\)
0.0106304 + 0.999943i \(0.496616\pi\)
\(150\) 0 0
\(151\) −13.6993 −1.11484 −0.557419 0.830232i \(-0.688208\pi\)
−0.557419 + 0.830232i \(0.688208\pi\)
\(152\) 26.0154 2.11013
\(153\) 0 0
\(154\) −15.2058 −1.22532
\(155\) −12.2949 −0.987552
\(156\) 0 0
\(157\) 17.4947 1.39623 0.698113 0.715988i \(-0.254023\pi\)
0.698113 + 0.715988i \(0.254023\pi\)
\(158\) 0.0228226 0.00181567
\(159\) 0 0
\(160\) 25.2587 1.99687
\(161\) 5.83444 0.459818
\(162\) 0 0
\(163\) 13.6923 1.07246 0.536232 0.844070i \(-0.319847\pi\)
0.536232 + 0.844070i \(0.319847\pi\)
\(164\) −15.6706 −1.22367
\(165\) 0 0
\(166\) −0.270365 −0.0209844
\(167\) −2.64032 −0.204314 −0.102157 0.994768i \(-0.532574\pi\)
−0.102157 + 0.994768i \(0.532574\pi\)
\(168\) 0 0
\(169\) 0.254955 0.0196119
\(170\) 22.7338 1.74360
\(171\) 0 0
\(172\) −50.0463 −3.81600
\(173\) 7.97546 0.606363 0.303181 0.952933i \(-0.401951\pi\)
0.303181 + 0.952933i \(0.401951\pi\)
\(174\) 0 0
\(175\) 3.21228 0.242825
\(176\) 76.5425 5.76961
\(177\) 0 0
\(178\) −22.3416 −1.67458
\(179\) 1.58809 0.118700 0.0593499 0.998237i \(-0.481097\pi\)
0.0593499 + 0.998237i \(0.481097\pi\)
\(180\) 0 0
\(181\) 1.41012 0.104814 0.0524068 0.998626i \(-0.483311\pi\)
0.0524068 + 0.998626i \(0.483311\pi\)
\(182\) −9.84641 −0.729864
\(183\) 0 0
\(184\) −52.2985 −3.85550
\(185\) −10.3114 −0.758106
\(186\) 0 0
\(187\) 35.3472 2.58484
\(188\) −9.94312 −0.725176
\(189\) 0 0
\(190\) 10.4949 0.761382
\(191\) 11.2562 0.814473 0.407236 0.913323i \(-0.366492\pi\)
0.407236 + 0.913323i \(0.366492\pi\)
\(192\) 0 0
\(193\) −4.69721 −0.338112 −0.169056 0.985606i \(-0.554072\pi\)
−0.169056 + 0.985606i \(0.554072\pi\)
\(194\) 8.16138 0.585953
\(195\) 0 0
\(196\) 5.31438 0.379598
\(197\) 8.77065 0.624883 0.312441 0.949937i \(-0.398853\pi\)
0.312441 + 0.949937i \(0.398853\pi\)
\(198\) 0 0
\(199\) 2.91493 0.206634 0.103317 0.994648i \(-0.467054\pi\)
0.103317 + 0.994648i \(0.467054\pi\)
\(200\) −28.7941 −2.03605
\(201\) 0 0
\(202\) 45.2524 3.18395
\(203\) −4.25704 −0.298786
\(204\) 0 0
\(205\) −3.94262 −0.275364
\(206\) 15.5793 1.08546
\(207\) 0 0
\(208\) 49.5644 3.43667
\(209\) 16.3178 1.12873
\(210\) 0 0
\(211\) 1.30272 0.0896832 0.0448416 0.998994i \(-0.485722\pi\)
0.0448416 + 0.998994i \(0.485722\pi\)
\(212\) −63.7593 −4.37901
\(213\) 0 0
\(214\) 11.4924 0.785602
\(215\) −12.5913 −0.858718
\(216\) 0 0
\(217\) 9.19551 0.624232
\(218\) −40.7368 −2.75904
\(219\) 0 0
\(220\) 39.9507 2.69347
\(221\) 22.8887 1.53966
\(222\) 0 0
\(223\) 2.75374 0.184404 0.0922019 0.995740i \(-0.470609\pi\)
0.0922019 + 0.995740i \(0.470609\pi\)
\(224\) −18.8912 −1.26222
\(225\) 0 0
\(226\) −30.1207 −2.00360
\(227\) 11.1830 0.742239 0.371119 0.928585i \(-0.378974\pi\)
0.371119 + 0.928585i \(0.378974\pi\)
\(228\) 0 0
\(229\) −18.8890 −1.24822 −0.624110 0.781336i \(-0.714538\pi\)
−0.624110 + 0.781336i \(0.714538\pi\)
\(230\) −21.0978 −1.39115
\(231\) 0 0
\(232\) 38.1591 2.50527
\(233\) −21.0749 −1.38066 −0.690332 0.723493i \(-0.742536\pi\)
−0.690332 + 0.723493i \(0.742536\pi\)
\(234\) 0 0
\(235\) −2.50161 −0.163187
\(236\) −30.6028 −1.99208
\(237\) 0 0
\(238\) −17.0028 −1.10213
\(239\) 23.5137 1.52097 0.760486 0.649354i \(-0.224961\pi\)
0.760486 + 0.649354i \(0.224961\pi\)
\(240\) 0 0
\(241\) 6.30607 0.406210 0.203105 0.979157i \(-0.434897\pi\)
0.203105 + 0.979157i \(0.434897\pi\)
\(242\) 55.7438 3.58334
\(243\) 0 0
\(244\) 5.19744 0.332732
\(245\) 1.33706 0.0854215
\(246\) 0 0
\(247\) 10.5665 0.672328
\(248\) −82.4264 −5.23408
\(249\) 0 0
\(250\) −29.6963 −1.87816
\(251\) −23.2776 −1.46927 −0.734635 0.678463i \(-0.762647\pi\)
−0.734635 + 0.678463i \(0.762647\pi\)
\(252\) 0 0
\(253\) −32.8036 −2.06234
\(254\) −2.70451 −0.169696
\(255\) 0 0
\(256\) 24.6385 1.53991
\(257\) 11.5727 0.721888 0.360944 0.932588i \(-0.382455\pi\)
0.360944 + 0.932588i \(0.382455\pi\)
\(258\) 0 0
\(259\) 7.71198 0.479199
\(260\) 25.8697 1.60437
\(261\) 0 0
\(262\) 28.2887 1.74768
\(263\) −21.4848 −1.32481 −0.662404 0.749147i \(-0.730464\pi\)
−0.662404 + 0.749147i \(0.730464\pi\)
\(264\) 0 0
\(265\) −16.0414 −0.985413
\(266\) −7.84927 −0.481269
\(267\) 0 0
\(268\) −1.87153 −0.114322
\(269\) 12.8595 0.784058 0.392029 0.919953i \(-0.371773\pi\)
0.392029 + 0.919953i \(0.371773\pi\)
\(270\) 0 0
\(271\) 26.0688 1.58357 0.791784 0.610802i \(-0.209153\pi\)
0.791784 + 0.610802i \(0.209153\pi\)
\(272\) 85.5881 5.18954
\(273\) 0 0
\(274\) −22.4950 −1.35897
\(275\) −18.0607 −1.08910
\(276\) 0 0
\(277\) −9.50628 −0.571177 −0.285589 0.958352i \(-0.592189\pi\)
−0.285589 + 0.958352i \(0.592189\pi\)
\(278\) 28.7064 1.72170
\(279\) 0 0
\(280\) −11.9851 −0.716245
\(281\) −20.6327 −1.23085 −0.615423 0.788197i \(-0.711015\pi\)
−0.615423 + 0.788197i \(0.711015\pi\)
\(282\) 0 0
\(283\) 31.5610 1.87611 0.938053 0.346491i \(-0.112627\pi\)
0.938053 + 0.346491i \(0.112627\pi\)
\(284\) 70.8330 4.20316
\(285\) 0 0
\(286\) 55.3605 3.27353
\(287\) 2.94873 0.174058
\(288\) 0 0
\(289\) 22.5244 1.32496
\(290\) 15.3938 0.903957
\(291\) 0 0
\(292\) −63.5460 −3.71875
\(293\) 15.4752 0.904068 0.452034 0.892001i \(-0.350699\pi\)
0.452034 + 0.892001i \(0.350699\pi\)
\(294\) 0 0
\(295\) −7.69945 −0.448279
\(296\) −69.1284 −4.01800
\(297\) 0 0
\(298\) 0.701878 0.0406587
\(299\) −21.2417 −1.22844
\(300\) 0 0
\(301\) 9.41716 0.542796
\(302\) −37.0500 −2.13199
\(303\) 0 0
\(304\) 39.5113 2.26613
\(305\) 1.30764 0.0748751
\(306\) 0 0
\(307\) 6.92375 0.395159 0.197580 0.980287i \(-0.436692\pi\)
0.197580 + 0.980287i \(0.436692\pi\)
\(308\) −29.8796 −1.70255
\(309\) 0 0
\(310\) −33.2517 −1.88857
\(311\) 30.2612 1.71596 0.857979 0.513685i \(-0.171720\pi\)
0.857979 + 0.513685i \(0.171720\pi\)
\(312\) 0 0
\(313\) 9.08717 0.513637 0.256819 0.966460i \(-0.417326\pi\)
0.256819 + 0.966460i \(0.417326\pi\)
\(314\) 47.3145 2.67011
\(315\) 0 0
\(316\) 0.0448465 0.00252281
\(317\) −10.5839 −0.594449 −0.297224 0.954808i \(-0.596061\pi\)
−0.297224 + 0.954808i \(0.596061\pi\)
\(318\) 0 0
\(319\) 23.9348 1.34009
\(320\) 31.9074 1.78368
\(321\) 0 0
\(322\) 15.7793 0.879346
\(323\) 18.2462 1.01525
\(324\) 0 0
\(325\) −11.6951 −0.648725
\(326\) 37.0310 2.05096
\(327\) 0 0
\(328\) −26.4317 −1.45945
\(329\) 1.87098 0.103151
\(330\) 0 0
\(331\) −15.1975 −0.835329 −0.417665 0.908601i \(-0.637151\pi\)
−0.417665 + 0.908601i \(0.637151\pi\)
\(332\) −0.531269 −0.0291572
\(333\) 0 0
\(334\) −7.14078 −0.390726
\(335\) −0.470864 −0.0257260
\(336\) 0 0
\(337\) 2.39847 0.130653 0.0653264 0.997864i \(-0.479191\pi\)
0.0653264 + 0.997864i \(0.479191\pi\)
\(338\) 0.689528 0.0375054
\(339\) 0 0
\(340\) 44.6720 2.42268
\(341\) −51.7008 −2.79976
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −84.4132 −4.55125
\(345\) 0 0
\(346\) 21.5697 1.15959
\(347\) −20.2731 −1.08832 −0.544158 0.838982i \(-0.683151\pi\)
−0.544158 + 0.838982i \(0.683151\pi\)
\(348\) 0 0
\(349\) −4.06968 −0.217845 −0.108922 0.994050i \(-0.534740\pi\)
−0.108922 + 0.994050i \(0.534740\pi\)
\(350\) 8.68764 0.464374
\(351\) 0 0
\(352\) 106.214 5.66123
\(353\) −23.2003 −1.23483 −0.617415 0.786638i \(-0.711820\pi\)
−0.617415 + 0.786638i \(0.711820\pi\)
\(354\) 0 0
\(355\) 17.8211 0.945843
\(356\) −43.9014 −2.32677
\(357\) 0 0
\(358\) 4.29502 0.226999
\(359\) 6.23773 0.329215 0.164607 0.986359i \(-0.447364\pi\)
0.164607 + 0.986359i \(0.447364\pi\)
\(360\) 0 0
\(361\) −10.5767 −0.556670
\(362\) 3.81369 0.200443
\(363\) 0 0
\(364\) −19.3482 −1.01412
\(365\) −15.9877 −0.836835
\(366\) 0 0
\(367\) −11.3363 −0.591749 −0.295874 0.955227i \(-0.595611\pi\)
−0.295874 + 0.955227i \(0.595611\pi\)
\(368\) −79.4291 −4.14053
\(369\) 0 0
\(370\) −27.8872 −1.44979
\(371\) 11.9975 0.622880
\(372\) 0 0
\(373\) −0.735946 −0.0381059 −0.0190529 0.999818i \(-0.506065\pi\)
−0.0190529 + 0.999818i \(0.506065\pi\)
\(374\) 95.5968 4.94319
\(375\) 0 0
\(376\) −16.7711 −0.864902
\(377\) 15.4988 0.798227
\(378\) 0 0
\(379\) −10.5206 −0.540408 −0.270204 0.962803i \(-0.587091\pi\)
−0.270204 + 0.962803i \(0.587091\pi\)
\(380\) 20.6226 1.05792
\(381\) 0 0
\(382\) 30.4426 1.55758
\(383\) −27.6080 −1.41070 −0.705350 0.708859i \(-0.749210\pi\)
−0.705350 + 0.708859i \(0.749210\pi\)
\(384\) 0 0
\(385\) −7.51748 −0.383126
\(386\) −12.7036 −0.646599
\(387\) 0 0
\(388\) 16.0372 0.814163
\(389\) −22.1840 −1.12477 −0.562386 0.826875i \(-0.690117\pi\)
−0.562386 + 0.826875i \(0.690117\pi\)
\(390\) 0 0
\(391\) −36.6802 −1.85500
\(392\) 8.96376 0.452738
\(393\) 0 0
\(394\) 23.7203 1.19501
\(395\) 0.0112830 0.000567711 0
\(396\) 0 0
\(397\) −1.54547 −0.0775650 −0.0387825 0.999248i \(-0.512348\pi\)
−0.0387825 + 0.999248i \(0.512348\pi\)
\(398\) 7.88347 0.395163
\(399\) 0 0
\(400\) −43.7314 −2.18657
\(401\) 15.0038 0.749256 0.374628 0.927175i \(-0.377771\pi\)
0.374628 + 0.927175i \(0.377771\pi\)
\(402\) 0 0
\(403\) −33.4784 −1.66768
\(404\) 88.9211 4.42399
\(405\) 0 0
\(406\) −11.5132 −0.571391
\(407\) −43.3598 −2.14927
\(408\) 0 0
\(409\) 2.77463 0.137197 0.0685984 0.997644i \(-0.478147\pi\)
0.0685984 + 0.997644i \(0.478147\pi\)
\(410\) −10.6628 −0.526600
\(411\) 0 0
\(412\) 30.6134 1.50821
\(413\) 5.75850 0.283357
\(414\) 0 0
\(415\) −0.133663 −0.00656127
\(416\) 68.7780 3.37212
\(417\) 0 0
\(418\) 44.1317 2.15855
\(419\) 0.450977 0.0220317 0.0110158 0.999939i \(-0.496493\pi\)
0.0110158 + 0.999939i \(0.496493\pi\)
\(420\) 0 0
\(421\) 30.4479 1.48394 0.741970 0.670433i \(-0.233892\pi\)
0.741970 + 0.670433i \(0.233892\pi\)
\(422\) 3.52323 0.171508
\(423\) 0 0
\(424\) −107.543 −5.22274
\(425\) −20.1951 −0.979605
\(426\) 0 0
\(427\) −0.977997 −0.0473286
\(428\) 22.5826 1.09157
\(429\) 0 0
\(430\) −34.0533 −1.64219
\(431\) 25.9282 1.24892 0.624459 0.781058i \(-0.285320\pi\)
0.624459 + 0.781058i \(0.285320\pi\)
\(432\) 0 0
\(433\) −32.5540 −1.56444 −0.782222 0.623000i \(-0.785914\pi\)
−0.782222 + 0.623000i \(0.785914\pi\)
\(434\) 24.8693 1.19377
\(435\) 0 0
\(436\) −80.0480 −3.83360
\(437\) −16.9332 −0.810026
\(438\) 0 0
\(439\) −13.2502 −0.632399 −0.316199 0.948693i \(-0.602407\pi\)
−0.316199 + 0.948693i \(0.602407\pi\)
\(440\) 67.3849 3.21245
\(441\) 0 0
\(442\) 61.9028 2.94442
\(443\) 2.91409 0.138453 0.0692264 0.997601i \(-0.477947\pi\)
0.0692264 + 0.997601i \(0.477947\pi\)
\(444\) 0 0
\(445\) −11.0453 −0.523596
\(446\) 7.44751 0.352650
\(447\) 0 0
\(448\) −23.8639 −1.12746
\(449\) −31.6089 −1.49172 −0.745859 0.666104i \(-0.767961\pi\)
−0.745859 + 0.666104i \(0.767961\pi\)
\(450\) 0 0
\(451\) −16.5789 −0.780671
\(452\) −59.1873 −2.78394
\(453\) 0 0
\(454\) 30.2444 1.41944
\(455\) −4.86787 −0.228209
\(456\) 0 0
\(457\) −2.01306 −0.0941671 −0.0470835 0.998891i \(-0.514993\pi\)
−0.0470835 + 0.998891i \(0.514993\pi\)
\(458\) −51.0855 −2.38707
\(459\) 0 0
\(460\) −41.4574 −1.93296
\(461\) −18.0592 −0.841100 −0.420550 0.907269i \(-0.638163\pi\)
−0.420550 + 0.907269i \(0.638163\pi\)
\(462\) 0 0
\(463\) 19.4181 0.902438 0.451219 0.892413i \(-0.350989\pi\)
0.451219 + 0.892413i \(0.350989\pi\)
\(464\) 57.9547 2.69048
\(465\) 0 0
\(466\) −56.9973 −2.64035
\(467\) 25.5878 1.18406 0.592032 0.805914i \(-0.298326\pi\)
0.592032 + 0.805914i \(0.298326\pi\)
\(468\) 0 0
\(469\) 0.352164 0.0162614
\(470\) −6.76564 −0.312076
\(471\) 0 0
\(472\) −51.6178 −2.37590
\(473\) −52.9470 −2.43451
\(474\) 0 0
\(475\) −9.32295 −0.427766
\(476\) −33.4107 −1.53137
\(477\) 0 0
\(478\) 63.5929 2.90867
\(479\) 25.6165 1.17045 0.585223 0.810872i \(-0.301007\pi\)
0.585223 + 0.810872i \(0.301007\pi\)
\(480\) 0 0
\(481\) −28.0773 −1.28021
\(482\) 17.0548 0.776826
\(483\) 0 0
\(484\) 109.537 4.97894
\(485\) 4.03483 0.183212
\(486\) 0 0
\(487\) −20.7048 −0.938225 −0.469112 0.883139i \(-0.655426\pi\)
−0.469112 + 0.883139i \(0.655426\pi\)
\(488\) 8.76653 0.396842
\(489\) 0 0
\(490\) 3.61609 0.163358
\(491\) 24.8187 1.12005 0.560026 0.828475i \(-0.310791\pi\)
0.560026 + 0.828475i \(0.310791\pi\)
\(492\) 0 0
\(493\) 26.7634 1.20536
\(494\) 28.5771 1.28574
\(495\) 0 0
\(496\) −125.186 −5.62102
\(497\) −13.3286 −0.597868
\(498\) 0 0
\(499\) 37.6772 1.68666 0.843332 0.537393i \(-0.180591\pi\)
0.843332 + 0.537393i \(0.180591\pi\)
\(500\) −58.3534 −2.60964
\(501\) 0 0
\(502\) −62.9545 −2.80980
\(503\) −20.7543 −0.925390 −0.462695 0.886517i \(-0.653118\pi\)
−0.462695 + 0.886517i \(0.653118\pi\)
\(504\) 0 0
\(505\) 22.3719 0.995536
\(506\) −88.7176 −3.94398
\(507\) 0 0
\(508\) −5.31438 −0.235787
\(509\) −12.3887 −0.549119 −0.274559 0.961570i \(-0.588532\pi\)
−0.274559 + 0.961570i \(0.588532\pi\)
\(510\) 0 0
\(511\) 11.9574 0.528964
\(512\) 13.1199 0.579821
\(513\) 0 0
\(514\) 31.2986 1.38052
\(515\) 7.70210 0.339395
\(516\) 0 0
\(517\) −10.5194 −0.462644
\(518\) 20.8571 0.916409
\(519\) 0 0
\(520\) 43.6345 1.91350
\(521\) 4.91067 0.215141 0.107570 0.994197i \(-0.465693\pi\)
0.107570 + 0.994197i \(0.465693\pi\)
\(522\) 0 0
\(523\) 27.1334 1.18646 0.593231 0.805033i \(-0.297852\pi\)
0.593231 + 0.805033i \(0.297852\pi\)
\(524\) 55.5875 2.42835
\(525\) 0 0
\(526\) −58.1058 −2.53353
\(527\) −57.8107 −2.51827
\(528\) 0 0
\(529\) 11.0407 0.480030
\(530\) −43.3840 −1.88448
\(531\) 0 0
\(532\) −15.4238 −0.668709
\(533\) −10.7355 −0.465007
\(534\) 0 0
\(535\) 5.68161 0.245637
\(536\) −3.15672 −0.136349
\(537\) 0 0
\(538\) 34.7787 1.49942
\(539\) 5.62240 0.242174
\(540\) 0 0
\(541\) 18.8380 0.809907 0.404954 0.914337i \(-0.367288\pi\)
0.404954 + 0.914337i \(0.367288\pi\)
\(542\) 70.5034 3.02838
\(543\) 0 0
\(544\) 118.766 5.09206
\(545\) −20.1395 −0.862680
\(546\) 0 0
\(547\) −35.8740 −1.53386 −0.766930 0.641731i \(-0.778217\pi\)
−0.766930 + 0.641731i \(0.778217\pi\)
\(548\) −44.2027 −1.88825
\(549\) 0 0
\(550\) −48.8454 −2.08277
\(551\) 12.3552 0.526347
\(552\) 0 0
\(553\) −0.00843871 −0.000358850 0
\(554\) −25.7098 −1.09231
\(555\) 0 0
\(556\) 56.4083 2.39224
\(557\) 27.0609 1.14661 0.573304 0.819343i \(-0.305661\pi\)
0.573304 + 0.819343i \(0.305661\pi\)
\(558\) 0 0
\(559\) −34.2854 −1.45012
\(560\) −18.2025 −0.769195
\(561\) 0 0
\(562\) −55.8014 −2.35384
\(563\) −14.9787 −0.631278 −0.315639 0.948879i \(-0.602219\pi\)
−0.315639 + 0.948879i \(0.602219\pi\)
\(564\) 0 0
\(565\) −14.8911 −0.626473
\(566\) 85.3570 3.58782
\(567\) 0 0
\(568\) 119.474 5.01302
\(569\) −2.07662 −0.0870566 −0.0435283 0.999052i \(-0.513860\pi\)
−0.0435283 + 0.999052i \(0.513860\pi\)
\(570\) 0 0
\(571\) −16.6703 −0.697632 −0.348816 0.937191i \(-0.613416\pi\)
−0.348816 + 0.937191i \(0.613416\pi\)
\(572\) 108.784 4.54847
\(573\) 0 0
\(574\) 7.97486 0.332864
\(575\) 18.7418 0.781589
\(576\) 0 0
\(577\) −13.3733 −0.556737 −0.278368 0.960474i \(-0.589794\pi\)
−0.278368 + 0.960474i \(0.589794\pi\)
\(578\) 60.9175 2.53383
\(579\) 0 0
\(580\) 30.2490 1.25602
\(581\) 0.0999682 0.00414738
\(582\) 0 0
\(583\) −67.4548 −2.79369
\(584\) −107.183 −4.43527
\(585\) 0 0
\(586\) 41.8527 1.72892
\(587\) 0.270658 0.0111713 0.00558563 0.999984i \(-0.498222\pi\)
0.00558563 + 0.999984i \(0.498222\pi\)
\(588\) 0 0
\(589\) −26.6880 −1.09966
\(590\) −20.8232 −0.857279
\(591\) 0 0
\(592\) −104.990 −4.31505
\(593\) −4.07885 −0.167498 −0.0837492 0.996487i \(-0.526689\pi\)
−0.0837492 + 0.996487i \(0.526689\pi\)
\(594\) 0 0
\(595\) −8.40587 −0.344607
\(596\) 1.37919 0.0564939
\(597\) 0 0
\(598\) −57.4483 −2.34923
\(599\) 6.85182 0.279958 0.139979 0.990154i \(-0.455297\pi\)
0.139979 + 0.990154i \(0.455297\pi\)
\(600\) 0 0
\(601\) 0.956812 0.0390292 0.0195146 0.999810i \(-0.493788\pi\)
0.0195146 + 0.999810i \(0.493788\pi\)
\(602\) 25.4688 1.03803
\(603\) 0 0
\(604\) −72.8035 −2.96233
\(605\) 27.5586 1.12042
\(606\) 0 0
\(607\) 17.9541 0.728734 0.364367 0.931255i \(-0.381285\pi\)
0.364367 + 0.931255i \(0.381285\pi\)
\(608\) 54.8278 2.22356
\(609\) 0 0
\(610\) 3.53652 0.143190
\(611\) −6.81176 −0.275574
\(612\) 0 0
\(613\) 45.8740 1.85283 0.926417 0.376500i \(-0.122873\pi\)
0.926417 + 0.376500i \(0.122873\pi\)
\(614\) 18.7253 0.755693
\(615\) 0 0
\(616\) −50.3979 −2.03059
\(617\) −42.3860 −1.70640 −0.853198 0.521588i \(-0.825340\pi\)
−0.853198 + 0.521588i \(0.825340\pi\)
\(618\) 0 0
\(619\) 1.43782 0.0577910 0.0288955 0.999582i \(-0.490801\pi\)
0.0288955 + 0.999582i \(0.490801\pi\)
\(620\) −65.3398 −2.62411
\(621\) 0 0
\(622\) 81.8418 3.28156
\(623\) 8.26088 0.330965
\(624\) 0 0
\(625\) 1.38011 0.0552043
\(626\) 24.5763 0.982268
\(627\) 0 0
\(628\) 92.9732 3.71003
\(629\) −48.4840 −1.93318
\(630\) 0 0
\(631\) 19.4516 0.774356 0.387178 0.922005i \(-0.373450\pi\)
0.387178 + 0.922005i \(0.373450\pi\)
\(632\) 0.0756426 0.00300890
\(633\) 0 0
\(634\) −28.6242 −1.13681
\(635\) −1.33706 −0.0530595
\(636\) 0 0
\(637\) 3.64074 0.144251
\(638\) 64.7319 2.56276
\(639\) 0 0
\(640\) 35.7765 1.41419
\(641\) 28.1948 1.11363 0.556813 0.830638i \(-0.312024\pi\)
0.556813 + 0.830638i \(0.312024\pi\)
\(642\) 0 0
\(643\) −3.11642 −0.122900 −0.0614499 0.998110i \(-0.519572\pi\)
−0.0614499 + 0.998110i \(0.519572\pi\)
\(644\) 31.0064 1.22182
\(645\) 0 0
\(646\) 49.3471 1.94154
\(647\) 48.4587 1.90511 0.952553 0.304372i \(-0.0984465\pi\)
0.952553 + 0.304372i \(0.0984465\pi\)
\(648\) 0 0
\(649\) −32.3766 −1.27089
\(650\) −31.6294 −1.24061
\(651\) 0 0
\(652\) 72.7661 2.84974
\(653\) 32.8131 1.28408 0.642038 0.766673i \(-0.278089\pi\)
0.642038 + 0.766673i \(0.278089\pi\)
\(654\) 0 0
\(655\) 13.9854 0.546455
\(656\) −40.1435 −1.56734
\(657\) 0 0
\(658\) 5.06010 0.197263
\(659\) 41.8057 1.62852 0.814259 0.580502i \(-0.197144\pi\)
0.814259 + 0.580502i \(0.197144\pi\)
\(660\) 0 0
\(661\) 38.9351 1.51440 0.757201 0.653182i \(-0.226566\pi\)
0.757201 + 0.653182i \(0.226566\pi\)
\(662\) −41.1018 −1.59747
\(663\) 0 0
\(664\) −0.896091 −0.0347751
\(665\) −3.88053 −0.150480
\(666\) 0 0
\(667\) −24.8375 −0.961710
\(668\) −14.0317 −0.542902
\(669\) 0 0
\(670\) −1.27346 −0.0491979
\(671\) 5.49869 0.212275
\(672\) 0 0
\(673\) 10.7975 0.416213 0.208106 0.978106i \(-0.433270\pi\)
0.208106 + 0.978106i \(0.433270\pi\)
\(674\) 6.48668 0.249858
\(675\) 0 0
\(676\) 1.35493 0.0521125
\(677\) −5.24314 −0.201510 −0.100755 0.994911i \(-0.532126\pi\)
−0.100755 + 0.994911i \(0.532126\pi\)
\(678\) 0 0
\(679\) −3.01769 −0.115808
\(680\) 75.3483 2.88947
\(681\) 0 0
\(682\) −139.825 −5.35419
\(683\) −11.9241 −0.456263 −0.228132 0.973630i \(-0.573262\pi\)
−0.228132 + 0.973630i \(0.573262\pi\)
\(684\) 0 0
\(685\) −11.1211 −0.424915
\(686\) −2.70451 −0.103259
\(687\) 0 0
\(688\) −128.204 −4.88772
\(689\) −43.6798 −1.66407
\(690\) 0 0
\(691\) −38.7809 −1.47529 −0.737647 0.675187i \(-0.764063\pi\)
−0.737647 + 0.675187i \(0.764063\pi\)
\(692\) 42.3846 1.61122
\(693\) 0 0
\(694\) −54.8288 −2.08127
\(695\) 14.1919 0.538330
\(696\) 0 0
\(697\) −18.5382 −0.702183
\(698\) −11.0065 −0.416601
\(699\) 0 0
\(700\) 17.0712 0.645233
\(701\) −17.9182 −0.676761 −0.338381 0.941009i \(-0.609879\pi\)
−0.338381 + 0.941009i \(0.609879\pi\)
\(702\) 0 0
\(703\) −22.3824 −0.844167
\(704\) 134.172 5.05681
\(705\) 0 0
\(706\) −62.7456 −2.36146
\(707\) −16.7322 −0.629279
\(708\) 0 0
\(709\) 29.4343 1.10543 0.552715 0.833371i \(-0.313592\pi\)
0.552715 + 0.833371i \(0.313592\pi\)
\(710\) 48.1972 1.80881
\(711\) 0 0
\(712\) −74.0485 −2.77509
\(713\) 53.6506 2.00923
\(714\) 0 0
\(715\) 27.3691 1.02355
\(716\) 8.43973 0.315407
\(717\) 0 0
\(718\) 16.8700 0.629583
\(719\) 47.0977 1.75645 0.878224 0.478249i \(-0.158729\pi\)
0.878224 + 0.478249i \(0.158729\pi\)
\(720\) 0 0
\(721\) −5.76048 −0.214532
\(722\) −28.6049 −1.06456
\(723\) 0 0
\(724\) 7.49393 0.278510
\(725\) −13.6748 −0.507869
\(726\) 0 0
\(727\) −34.1781 −1.26760 −0.633798 0.773499i \(-0.718505\pi\)
−0.633798 + 0.773499i \(0.718505\pi\)
\(728\) −32.6347 −1.20952
\(729\) 0 0
\(730\) −43.2389 −1.60034
\(731\) −59.2042 −2.18975
\(732\) 0 0
\(733\) −40.1865 −1.48432 −0.742162 0.670221i \(-0.766199\pi\)
−0.742162 + 0.670221i \(0.766199\pi\)
\(734\) −30.6591 −1.13165
\(735\) 0 0
\(736\) −110.220 −4.06276
\(737\) −1.98001 −0.0729346
\(738\) 0 0
\(739\) 26.2533 0.965742 0.482871 0.875691i \(-0.339594\pi\)
0.482871 + 0.875691i \(0.339594\pi\)
\(740\) −54.7984 −2.01443
\(741\) 0 0
\(742\) 32.4474 1.19118
\(743\) −15.6848 −0.575421 −0.287711 0.957717i \(-0.592894\pi\)
−0.287711 + 0.957717i \(0.592894\pi\)
\(744\) 0 0
\(745\) 0.346995 0.0127129
\(746\) −1.99037 −0.0728728
\(747\) 0 0
\(748\) 187.848 6.86841
\(749\) −4.24934 −0.155267
\(750\) 0 0
\(751\) −18.5963 −0.678588 −0.339294 0.940680i \(-0.610188\pi\)
−0.339294 + 0.940680i \(0.610188\pi\)
\(752\) −25.4713 −0.928842
\(753\) 0 0
\(754\) 41.9166 1.52651
\(755\) −18.3168 −0.666617
\(756\) 0 0
\(757\) −15.8501 −0.576083 −0.288042 0.957618i \(-0.593004\pi\)
−0.288042 + 0.957618i \(0.593004\pi\)
\(758\) −28.4531 −1.03346
\(759\) 0 0
\(760\) 34.7841 1.26175
\(761\) 24.2645 0.879588 0.439794 0.898099i \(-0.355051\pi\)
0.439794 + 0.898099i \(0.355051\pi\)
\(762\) 0 0
\(763\) 15.0625 0.545300
\(764\) 59.8199 2.16421
\(765\) 0 0
\(766\) −74.6660 −2.69779
\(767\) −20.9652 −0.757009
\(768\) 0 0
\(769\) 22.6349 0.816236 0.408118 0.912929i \(-0.366185\pi\)
0.408118 + 0.912929i \(0.366185\pi\)
\(770\) −20.3311 −0.732682
\(771\) 0 0
\(772\) −24.9627 −0.898428
\(773\) −13.3581 −0.480459 −0.240230 0.970716i \(-0.577223\pi\)
−0.240230 + 0.970716i \(0.577223\pi\)
\(774\) 0 0
\(775\) 29.5385 1.06105
\(776\) 27.0499 0.971034
\(777\) 0 0
\(778\) −59.9968 −2.15099
\(779\) −8.55805 −0.306624
\(780\) 0 0
\(781\) 74.9385 2.68151
\(782\) −99.2020 −3.54746
\(783\) 0 0
\(784\) 13.6138 0.486209
\(785\) 23.3914 0.834873
\(786\) 0 0
\(787\) −39.0191 −1.39088 −0.695439 0.718585i \(-0.744790\pi\)
−0.695439 + 0.718585i \(0.744790\pi\)
\(788\) 46.6105 1.66043
\(789\) 0 0
\(790\) 0.0305151 0.00108568
\(791\) 11.1372 0.395993
\(792\) 0 0
\(793\) 3.56063 0.126442
\(794\) −4.17975 −0.148334
\(795\) 0 0
\(796\) 15.4911 0.549066
\(797\) 49.7364 1.76175 0.880877 0.473345i \(-0.156954\pi\)
0.880877 + 0.473345i \(0.156954\pi\)
\(798\) 0 0
\(799\) −11.7626 −0.416131
\(800\) −60.6839 −2.14550
\(801\) 0 0
\(802\) 40.5780 1.43286
\(803\) −67.2292 −2.37247
\(804\) 0 0
\(805\) 7.80098 0.274949
\(806\) −90.5427 −3.18923
\(807\) 0 0
\(808\) 149.983 5.27640
\(809\) −47.5739 −1.67261 −0.836304 0.548266i \(-0.815288\pi\)
−0.836304 + 0.548266i \(0.815288\pi\)
\(810\) 0 0
\(811\) −13.6412 −0.479008 −0.239504 0.970895i \(-0.576985\pi\)
−0.239504 + 0.970895i \(0.576985\pi\)
\(812\) −22.6235 −0.793930
\(813\) 0 0
\(814\) −117.267 −4.11021
\(815\) 18.3074 0.641281
\(816\) 0 0
\(817\) −27.3313 −0.956201
\(818\) 7.50403 0.262372
\(819\) 0 0
\(820\) −20.9525 −0.731695
\(821\) −2.47458 −0.0863635 −0.0431818 0.999067i \(-0.513749\pi\)
−0.0431818 + 0.999067i \(0.513749\pi\)
\(822\) 0 0
\(823\) 0.220843 0.00769810 0.00384905 0.999993i \(-0.498775\pi\)
0.00384905 + 0.999993i \(0.498775\pi\)
\(824\) 51.6356 1.79881
\(825\) 0 0
\(826\) 15.5739 0.541886
\(827\) 43.5850 1.51560 0.757799 0.652488i \(-0.226275\pi\)
0.757799 + 0.652488i \(0.226275\pi\)
\(828\) 0 0
\(829\) −15.2871 −0.530944 −0.265472 0.964119i \(-0.585528\pi\)
−0.265472 + 0.964119i \(0.585528\pi\)
\(830\) −0.361494 −0.0125476
\(831\) 0 0
\(832\) 86.8821 3.01209
\(833\) 6.28684 0.217826
\(834\) 0 0
\(835\) −3.53026 −0.122170
\(836\) 86.7191 2.99924
\(837\) 0 0
\(838\) 1.21967 0.0421328
\(839\) 33.9366 1.17162 0.585811 0.810448i \(-0.300776\pi\)
0.585811 + 0.810448i \(0.300776\pi\)
\(840\) 0 0
\(841\) −10.8776 −0.375089
\(842\) 82.3467 2.83785
\(843\) 0 0
\(844\) 6.92316 0.238305
\(845\) 0.340889 0.0117269
\(846\) 0 0
\(847\) −20.6114 −0.708216
\(848\) −163.332 −5.60885
\(849\) 0 0
\(850\) −54.6178 −1.87338
\(851\) 44.9951 1.54241
\(852\) 0 0
\(853\) 0.144164 0.00493609 0.00246804 0.999997i \(-0.499214\pi\)
0.00246804 + 0.999997i \(0.499214\pi\)
\(854\) −2.64500 −0.0905101
\(855\) 0 0
\(856\) 38.0900 1.30189
\(857\) −18.4457 −0.630093 −0.315047 0.949076i \(-0.602020\pi\)
−0.315047 + 0.949076i \(0.602020\pi\)
\(858\) 0 0
\(859\) −29.4383 −1.00442 −0.502210 0.864746i \(-0.667480\pi\)
−0.502210 + 0.864746i \(0.667480\pi\)
\(860\) −66.9148 −2.28178
\(861\) 0 0
\(862\) 70.1231 2.38840
\(863\) 32.8559 1.11843 0.559214 0.829023i \(-0.311103\pi\)
0.559214 + 0.829023i \(0.311103\pi\)
\(864\) 0 0
\(865\) 10.6636 0.362575
\(866\) −88.0426 −2.99181
\(867\) 0 0
\(868\) 48.8684 1.65870
\(869\) 0.0474458 0.00160949
\(870\) 0 0
\(871\) −1.28214 −0.0434435
\(872\) −135.017 −4.57225
\(873\) 0 0
\(874\) −45.7961 −1.54908
\(875\) 10.9803 0.371201
\(876\) 0 0
\(877\) 24.3527 0.822332 0.411166 0.911561i \(-0.365122\pi\)
0.411166 + 0.911561i \(0.365122\pi\)
\(878\) −35.8354 −1.20939
\(879\) 0 0
\(880\) 102.342 3.44994
\(881\) 23.4022 0.788439 0.394219 0.919016i \(-0.371015\pi\)
0.394219 + 0.919016i \(0.371015\pi\)
\(882\) 0 0
\(883\) −33.2982 −1.12057 −0.560287 0.828299i \(-0.689309\pi\)
−0.560287 + 0.828299i \(0.689309\pi\)
\(884\) 121.639 4.09117
\(885\) 0 0
\(886\) 7.88120 0.264774
\(887\) 5.13157 0.172301 0.0861507 0.996282i \(-0.472543\pi\)
0.0861507 + 0.996282i \(0.472543\pi\)
\(888\) 0 0
\(889\) 1.00000 0.0335389
\(890\) −29.8720 −1.00131
\(891\) 0 0
\(892\) 14.6344 0.489996
\(893\) −5.43014 −0.181713
\(894\) 0 0
\(895\) 2.12337 0.0709765
\(896\) −26.7576 −0.893909
\(897\) 0 0
\(898\) −85.4867 −2.85273
\(899\) −39.1457 −1.30558
\(900\) 0 0
\(901\) −75.4265 −2.51282
\(902\) −44.8379 −1.49294
\(903\) 0 0
\(904\) −99.8313 −3.32034
\(905\) 1.88542 0.0626734
\(906\) 0 0
\(907\) 36.5415 1.21334 0.606670 0.794954i \(-0.292505\pi\)
0.606670 + 0.794954i \(0.292505\pi\)
\(908\) 59.4304 1.97227
\(909\) 0 0
\(910\) −13.1652 −0.436423
\(911\) −15.3920 −0.509961 −0.254980 0.966946i \(-0.582069\pi\)
−0.254980 + 0.966946i \(0.582069\pi\)
\(912\) 0 0
\(913\) −0.562061 −0.0186015
\(914\) −5.44435 −0.180083
\(915\) 0 0
\(916\) −100.383 −3.31676
\(917\) −10.4598 −0.345414
\(918\) 0 0
\(919\) −17.7665 −0.586061 −0.293031 0.956103i \(-0.594664\pi\)
−0.293031 + 0.956103i \(0.594664\pi\)
\(920\) −69.9262 −2.30540
\(921\) 0 0
\(922\) −48.8412 −1.60850
\(923\) 48.5258 1.59724
\(924\) 0 0
\(925\) 24.7730 0.814532
\(926\) 52.5166 1.72580
\(927\) 0 0
\(928\) 80.4208 2.63994
\(929\) 39.8633 1.30787 0.653936 0.756550i \(-0.273117\pi\)
0.653936 + 0.756550i \(0.273117\pi\)
\(930\) 0 0
\(931\) 2.90229 0.0951186
\(932\) −112.000 −3.66868
\(933\) 0 0
\(934\) 69.2026 2.26438
\(935\) 47.2612 1.54561
\(936\) 0 0
\(937\) −15.0762 −0.492518 −0.246259 0.969204i \(-0.579201\pi\)
−0.246259 + 0.969204i \(0.579201\pi\)
\(938\) 0.952432 0.0310980
\(939\) 0 0
\(940\) −13.2945 −0.433619
\(941\) 1.47329 0.0480279 0.0240140 0.999712i \(-0.492355\pi\)
0.0240140 + 0.999712i \(0.492355\pi\)
\(942\) 0 0
\(943\) 17.2042 0.560245
\(944\) −78.3953 −2.55155
\(945\) 0 0
\(946\) −143.196 −4.65570
\(947\) 5.38384 0.174951 0.0874756 0.996167i \(-0.472120\pi\)
0.0874756 + 0.996167i \(0.472120\pi\)
\(948\) 0 0
\(949\) −43.5337 −1.41316
\(950\) −25.2140 −0.818051
\(951\) 0 0
\(952\) −56.3538 −1.82644
\(953\) −38.9729 −1.26246 −0.631228 0.775598i \(-0.717449\pi\)
−0.631228 + 0.775598i \(0.717449\pi\)
\(954\) 0 0
\(955\) 15.0502 0.487014
\(956\) 124.960 4.04151
\(957\) 0 0
\(958\) 69.2800 2.23833
\(959\) 8.31758 0.268589
\(960\) 0 0
\(961\) 53.5573 1.72766
\(962\) −75.9353 −2.44825
\(963\) 0 0
\(964\) 33.5128 1.07938
\(965\) −6.28044 −0.202174
\(966\) 0 0
\(967\) −39.5360 −1.27139 −0.635696 0.771940i \(-0.719287\pi\)
−0.635696 + 0.771940i \(0.719287\pi\)
\(968\) 184.756 5.93827
\(969\) 0 0
\(970\) 10.9122 0.350371
\(971\) 3.27748 0.105179 0.0525896 0.998616i \(-0.483252\pi\)
0.0525896 + 0.998616i \(0.483252\pi\)
\(972\) 0 0
\(973\) −10.6143 −0.340278
\(974\) −55.9964 −1.79424
\(975\) 0 0
\(976\) 13.3143 0.426180
\(977\) 23.4451 0.750075 0.375037 0.927010i \(-0.377630\pi\)
0.375037 + 0.927010i \(0.377630\pi\)
\(978\) 0 0
\(979\) −46.4460 −1.48442
\(980\) 7.10563 0.226981
\(981\) 0 0
\(982\) 67.1224 2.14196
\(983\) 45.8347 1.46190 0.730950 0.682431i \(-0.239077\pi\)
0.730950 + 0.682431i \(0.239077\pi\)
\(984\) 0 0
\(985\) 11.7269 0.373649
\(986\) 72.3818 2.30511
\(987\) 0 0
\(988\) 56.1541 1.78650
\(989\) 54.9438 1.74711
\(990\) 0 0
\(991\) −9.15029 −0.290669 −0.145334 0.989383i \(-0.546426\pi\)
−0.145334 + 0.989383i \(0.546426\pi\)
\(992\) −173.715 −5.51544
\(993\) 0 0
\(994\) −36.0472 −1.14335
\(995\) 3.89744 0.123557
\(996\) 0 0
\(997\) −11.9523 −0.378532 −0.189266 0.981926i \(-0.560611\pi\)
−0.189266 + 0.981926i \(0.560611\pi\)
\(998\) 101.898 3.22554
\(999\) 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.t.1.16 16
3.2 odd 2 889.2.a.c.1.1 16
21.20 even 2 6223.2.a.k.1.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
889.2.a.c.1.1 16 3.2 odd 2
6223.2.a.k.1.1 16 21.20 even 2
8001.2.a.t.1.16 16 1.1 even 1 trivial