Properties

Label 8001.2.a.t.1.10
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.10
Root \(0.625678\) 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)
 
\(f(q)\) \(=\) \(q+0.625678 q^{2} -1.60853 q^{4} +0.920707 q^{5} -1.00000 q^{7} -2.25778 q^{8} +O(q^{10})\) \(q+0.625678 q^{2} -1.60853 q^{4} +0.920707 q^{5} -1.00000 q^{7} -2.25778 q^{8} +0.576066 q^{10} -0.186460 q^{11} +3.40951 q^{13} -0.625678 q^{14} +1.80441 q^{16} -6.65550 q^{17} -6.87673 q^{19} -1.48098 q^{20} -0.116664 q^{22} +4.83113 q^{23} -4.15230 q^{25} +2.13326 q^{26} +1.60853 q^{28} +5.29105 q^{29} -3.20074 q^{31} +5.64453 q^{32} -4.16420 q^{34} -0.920707 q^{35} -5.49716 q^{37} -4.30261 q^{38} -2.07875 q^{40} +0.995032 q^{41} -5.38517 q^{43} +0.299925 q^{44} +3.02273 q^{46} +12.1442 q^{47} +1.00000 q^{49} -2.59800 q^{50} -5.48429 q^{52} -2.21328 q^{53} -0.171675 q^{55} +2.25778 q^{56} +3.31049 q^{58} -0.266942 q^{59} +12.0006 q^{61} -2.00263 q^{62} -0.0771703 q^{64} +3.13916 q^{65} -4.28634 q^{67} +10.7056 q^{68} -0.576066 q^{70} +3.89809 q^{71} +3.02049 q^{73} -3.43945 q^{74} +11.0614 q^{76} +0.186460 q^{77} +12.4416 q^{79} +1.66134 q^{80} +0.622570 q^{82} +5.49252 q^{83} -6.12776 q^{85} -3.36938 q^{86} +0.420984 q^{88} +0.466080 q^{89} -3.40951 q^{91} -7.77100 q^{92} +7.59838 q^{94} -6.33145 q^{95} +10.2941 q^{97} +0.625678 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.625678 0.442421 0.221211 0.975226i \(-0.428999\pi\)
0.221211 + 0.975226i \(0.428999\pi\)
\(3\) 0 0
\(4\) −1.60853 −0.804264
\(5\) 0.920707 0.411753 0.205876 0.978578i \(-0.433996\pi\)
0.205876 + 0.978578i \(0.433996\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −2.25778 −0.798244
\(9\) 0 0
\(10\) 0.576066 0.182168
\(11\) −0.186460 −0.0562197 −0.0281098 0.999605i \(-0.508949\pi\)
−0.0281098 + 0.999605i \(0.508949\pi\)
\(12\) 0 0
\(13\) 3.40951 0.945628 0.472814 0.881162i \(-0.343238\pi\)
0.472814 + 0.881162i \(0.343238\pi\)
\(14\) −0.625678 −0.167219
\(15\) 0 0
\(16\) 1.80441 0.451104
\(17\) −6.65550 −1.61420 −0.807098 0.590418i \(-0.798963\pi\)
−0.807098 + 0.590418i \(0.798963\pi\)
\(18\) 0 0
\(19\) −6.87673 −1.57763 −0.788815 0.614631i \(-0.789305\pi\)
−0.788815 + 0.614631i \(0.789305\pi\)
\(20\) −1.48098 −0.331158
\(21\) 0 0
\(22\) −0.116664 −0.0248728
\(23\) 4.83113 1.00736 0.503680 0.863890i \(-0.331979\pi\)
0.503680 + 0.863890i \(0.331979\pi\)
\(24\) 0 0
\(25\) −4.15230 −0.830460
\(26\) 2.13326 0.418366
\(27\) 0 0
\(28\) 1.60853 0.303983
\(29\) 5.29105 0.982524 0.491262 0.871012i \(-0.336536\pi\)
0.491262 + 0.871012i \(0.336536\pi\)
\(30\) 0 0
\(31\) −3.20074 −0.574870 −0.287435 0.957800i \(-0.592803\pi\)
−0.287435 + 0.957800i \(0.592803\pi\)
\(32\) 5.64453 0.997822
\(33\) 0 0
\(34\) −4.16420 −0.714154
\(35\) −0.920707 −0.155628
\(36\) 0 0
\(37\) −5.49716 −0.903727 −0.451864 0.892087i \(-0.649241\pi\)
−0.451864 + 0.892087i \(0.649241\pi\)
\(38\) −4.30261 −0.697976
\(39\) 0 0
\(40\) −2.07875 −0.328679
\(41\) 0.995032 0.155398 0.0776990 0.996977i \(-0.475243\pi\)
0.0776990 + 0.996977i \(0.475243\pi\)
\(42\) 0 0
\(43\) −5.38517 −0.821231 −0.410616 0.911808i \(-0.634686\pi\)
−0.410616 + 0.911808i \(0.634686\pi\)
\(44\) 0.299925 0.0452154
\(45\) 0 0
\(46\) 3.02273 0.445677
\(47\) 12.1442 1.77142 0.885710 0.464239i \(-0.153672\pi\)
0.885710 + 0.464239i \(0.153672\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −2.59800 −0.367413
\(51\) 0 0
\(52\) −5.48429 −0.760534
\(53\) −2.21328 −0.304017 −0.152008 0.988379i \(-0.548574\pi\)
−0.152008 + 0.988379i \(0.548574\pi\)
\(54\) 0 0
\(55\) −0.171675 −0.0231486
\(56\) 2.25778 0.301708
\(57\) 0 0
\(58\) 3.31049 0.434689
\(59\) −0.266942 −0.0347529 −0.0173764 0.999849i \(-0.505531\pi\)
−0.0173764 + 0.999849i \(0.505531\pi\)
\(60\) 0 0
\(61\) 12.0006 1.53652 0.768261 0.640137i \(-0.221123\pi\)
0.768261 + 0.640137i \(0.221123\pi\)
\(62\) −2.00263 −0.254335
\(63\) 0 0
\(64\) −0.0771703 −0.00964629
\(65\) 3.13916 0.389365
\(66\) 0 0
\(67\) −4.28634 −0.523660 −0.261830 0.965114i \(-0.584326\pi\)
−0.261830 + 0.965114i \(0.584326\pi\)
\(68\) 10.7056 1.29824
\(69\) 0 0
\(70\) −0.576066 −0.0688530
\(71\) 3.89809 0.462618 0.231309 0.972880i \(-0.425699\pi\)
0.231309 + 0.972880i \(0.425699\pi\)
\(72\) 0 0
\(73\) 3.02049 0.353521 0.176761 0.984254i \(-0.443438\pi\)
0.176761 + 0.984254i \(0.443438\pi\)
\(74\) −3.43945 −0.399828
\(75\) 0 0
\(76\) 11.0614 1.26883
\(77\) 0.186460 0.0212490
\(78\) 0 0
\(79\) 12.4416 1.39979 0.699894 0.714247i \(-0.253231\pi\)
0.699894 + 0.714247i \(0.253231\pi\)
\(80\) 1.66134 0.185743
\(81\) 0 0
\(82\) 0.622570 0.0687513
\(83\) 5.49252 0.602882 0.301441 0.953485i \(-0.402532\pi\)
0.301441 + 0.953485i \(0.402532\pi\)
\(84\) 0 0
\(85\) −6.12776 −0.664649
\(86\) −3.36938 −0.363330
\(87\) 0 0
\(88\) 0.420984 0.0448770
\(89\) 0.466080 0.0494043 0.0247022 0.999695i \(-0.492136\pi\)
0.0247022 + 0.999695i \(0.492136\pi\)
\(90\) 0 0
\(91\) −3.40951 −0.357414
\(92\) −7.77100 −0.810183
\(93\) 0 0
\(94\) 7.59838 0.783713
\(95\) −6.33145 −0.649593
\(96\) 0 0
\(97\) 10.2941 1.04521 0.522606 0.852574i \(-0.324960\pi\)
0.522606 + 0.852574i \(0.324960\pi\)
\(98\) 0.625678 0.0632030
\(99\) 0 0
\(100\) 6.67909 0.667909
\(101\) −10.6442 −1.05914 −0.529568 0.848268i \(-0.677646\pi\)
−0.529568 + 0.848268i \(0.677646\pi\)
\(102\) 0 0
\(103\) −8.97435 −0.884269 −0.442134 0.896949i \(-0.645779\pi\)
−0.442134 + 0.896949i \(0.645779\pi\)
\(104\) −7.69791 −0.754842
\(105\) 0 0
\(106\) −1.38480 −0.134503
\(107\) −8.95211 −0.865433 −0.432717 0.901530i \(-0.642445\pi\)
−0.432717 + 0.901530i \(0.642445\pi\)
\(108\) 0 0
\(109\) −7.57339 −0.725399 −0.362700 0.931906i \(-0.618145\pi\)
−0.362700 + 0.931906i \(0.618145\pi\)
\(110\) −0.107413 −0.0102414
\(111\) 0 0
\(112\) −1.80441 −0.170501
\(113\) 6.77283 0.637134 0.318567 0.947900i \(-0.396798\pi\)
0.318567 + 0.947900i \(0.396798\pi\)
\(114\) 0 0
\(115\) 4.44805 0.414783
\(116\) −8.51080 −0.790208
\(117\) 0 0
\(118\) −0.167020 −0.0153754
\(119\) 6.65550 0.610109
\(120\) 0 0
\(121\) −10.9652 −0.996839
\(122\) 7.50852 0.679789
\(123\) 0 0
\(124\) 5.14848 0.462347
\(125\) −8.42659 −0.753697
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) −11.3373 −1.00209
\(129\) 0 0
\(130\) 1.96410 0.172263
\(131\) −4.90013 −0.428126 −0.214063 0.976820i \(-0.568670\pi\)
−0.214063 + 0.976820i \(0.568670\pi\)
\(132\) 0 0
\(133\) 6.87673 0.596288
\(134\) −2.68187 −0.231678
\(135\) 0 0
\(136\) 15.0266 1.28852
\(137\) 7.01111 0.599000 0.299500 0.954096i \(-0.403180\pi\)
0.299500 + 0.954096i \(0.403180\pi\)
\(138\) 0 0
\(139\) 12.1478 1.03036 0.515181 0.857081i \(-0.327725\pi\)
0.515181 + 0.857081i \(0.327725\pi\)
\(140\) 1.48098 0.125166
\(141\) 0 0
\(142\) 2.43895 0.204672
\(143\) −0.635736 −0.0531629
\(144\) 0 0
\(145\) 4.87151 0.404557
\(146\) 1.88985 0.156405
\(147\) 0 0
\(148\) 8.84233 0.726835
\(149\) 13.8366 1.13354 0.566771 0.823876i \(-0.308193\pi\)
0.566771 + 0.823876i \(0.308193\pi\)
\(150\) 0 0
\(151\) 3.21945 0.261995 0.130998 0.991383i \(-0.458182\pi\)
0.130998 + 0.991383i \(0.458182\pi\)
\(152\) 15.5261 1.25933
\(153\) 0 0
\(154\) 0.116664 0.00940102
\(155\) −2.94694 −0.236704
\(156\) 0 0
\(157\) −8.19781 −0.654257 −0.327128 0.944980i \(-0.606081\pi\)
−0.327128 + 0.944980i \(0.606081\pi\)
\(158\) 7.78442 0.619295
\(159\) 0 0
\(160\) 5.19696 0.410856
\(161\) −4.83113 −0.380746
\(162\) 0 0
\(163\) 1.27054 0.0995167 0.0497583 0.998761i \(-0.484155\pi\)
0.0497583 + 0.998761i \(0.484155\pi\)
\(164\) −1.60054 −0.124981
\(165\) 0 0
\(166\) 3.43655 0.266728
\(167\) 15.9675 1.23560 0.617801 0.786334i \(-0.288024\pi\)
0.617801 + 0.786334i \(0.288024\pi\)
\(168\) 0 0
\(169\) −1.37523 −0.105787
\(170\) −3.83401 −0.294055
\(171\) 0 0
\(172\) 8.66220 0.660487
\(173\) 18.0461 1.37202 0.686011 0.727591i \(-0.259360\pi\)
0.686011 + 0.727591i \(0.259360\pi\)
\(174\) 0 0
\(175\) 4.15230 0.313884
\(176\) −0.336450 −0.0253609
\(177\) 0 0
\(178\) 0.291616 0.0218575
\(179\) 10.3836 0.776108 0.388054 0.921637i \(-0.373147\pi\)
0.388054 + 0.921637i \(0.373147\pi\)
\(180\) 0 0
\(181\) −10.3899 −0.772277 −0.386138 0.922441i \(-0.626191\pi\)
−0.386138 + 0.922441i \(0.626191\pi\)
\(182\) −2.13326 −0.158127
\(183\) 0 0
\(184\) −10.9076 −0.804119
\(185\) −5.06127 −0.372112
\(186\) 0 0
\(187\) 1.24098 0.0907496
\(188\) −19.5343 −1.42469
\(189\) 0 0
\(190\) −3.96145 −0.287394
\(191\) 12.5717 0.909654 0.454827 0.890580i \(-0.349701\pi\)
0.454827 + 0.890580i \(0.349701\pi\)
\(192\) 0 0
\(193\) 20.4024 1.46860 0.734298 0.678827i \(-0.237511\pi\)
0.734298 + 0.678827i \(0.237511\pi\)
\(194\) 6.44082 0.462424
\(195\) 0 0
\(196\) −1.60853 −0.114895
\(197\) −18.2917 −1.30323 −0.651613 0.758551i \(-0.725908\pi\)
−0.651613 + 0.758551i \(0.725908\pi\)
\(198\) 0 0
\(199\) 6.47355 0.458898 0.229449 0.973321i \(-0.426308\pi\)
0.229449 + 0.973321i \(0.426308\pi\)
\(200\) 9.37496 0.662910
\(201\) 0 0
\(202\) −6.65983 −0.468584
\(203\) −5.29105 −0.371359
\(204\) 0 0
\(205\) 0.916133 0.0639855
\(206\) −5.61505 −0.391219
\(207\) 0 0
\(208\) 6.15217 0.426576
\(209\) 1.28223 0.0886938
\(210\) 0 0
\(211\) 22.9921 1.58284 0.791419 0.611274i \(-0.209343\pi\)
0.791419 + 0.611274i \(0.209343\pi\)
\(212\) 3.56011 0.244510
\(213\) 0 0
\(214\) −5.60114 −0.382886
\(215\) −4.95817 −0.338144
\(216\) 0 0
\(217\) 3.20074 0.217280
\(218\) −4.73850 −0.320932
\(219\) 0 0
\(220\) 0.276143 0.0186176
\(221\) −22.6920 −1.52643
\(222\) 0 0
\(223\) 19.0825 1.27786 0.638929 0.769266i \(-0.279378\pi\)
0.638929 + 0.769266i \(0.279378\pi\)
\(224\) −5.64453 −0.377141
\(225\) 0 0
\(226\) 4.23761 0.281881
\(227\) 2.61933 0.173851 0.0869255 0.996215i \(-0.472296\pi\)
0.0869255 + 0.996215i \(0.472296\pi\)
\(228\) 0 0
\(229\) −12.9238 −0.854028 −0.427014 0.904245i \(-0.640434\pi\)
−0.427014 + 0.904245i \(0.640434\pi\)
\(230\) 2.78305 0.183509
\(231\) 0 0
\(232\) −11.9460 −0.784294
\(233\) −13.8782 −0.909188 −0.454594 0.890699i \(-0.650216\pi\)
−0.454594 + 0.890699i \(0.650216\pi\)
\(234\) 0 0
\(235\) 11.1813 0.729387
\(236\) 0.429384 0.0279505
\(237\) 0 0
\(238\) 4.16420 0.269925
\(239\) 14.3030 0.925187 0.462594 0.886570i \(-0.346919\pi\)
0.462594 + 0.886570i \(0.346919\pi\)
\(240\) 0 0
\(241\) 20.9198 1.34757 0.673783 0.738929i \(-0.264668\pi\)
0.673783 + 0.738929i \(0.264668\pi\)
\(242\) −6.86070 −0.441023
\(243\) 0 0
\(244\) −19.3033 −1.23577
\(245\) 0.920707 0.0588218
\(246\) 0 0
\(247\) −23.4463 −1.49185
\(248\) 7.22656 0.458887
\(249\) 0 0
\(250\) −5.27233 −0.333451
\(251\) −10.5122 −0.663523 −0.331761 0.943363i \(-0.607643\pi\)
−0.331761 + 0.943363i \(0.607643\pi\)
\(252\) 0 0
\(253\) −0.900810 −0.0566334
\(254\) −0.625678 −0.0392585
\(255\) 0 0
\(256\) −6.93919 −0.433699
\(257\) 16.2500 1.01365 0.506824 0.862049i \(-0.330819\pi\)
0.506824 + 0.862049i \(0.330819\pi\)
\(258\) 0 0
\(259\) 5.49716 0.341577
\(260\) −5.04943 −0.313152
\(261\) 0 0
\(262\) −3.06590 −0.189412
\(263\) −10.7447 −0.662546 −0.331273 0.943535i \(-0.607478\pi\)
−0.331273 + 0.943535i \(0.607478\pi\)
\(264\) 0 0
\(265\) −2.03778 −0.125180
\(266\) 4.30261 0.263810
\(267\) 0 0
\(268\) 6.89470 0.421161
\(269\) 10.2239 0.623364 0.311682 0.950186i \(-0.399108\pi\)
0.311682 + 0.950186i \(0.399108\pi\)
\(270\) 0 0
\(271\) 22.3620 1.35840 0.679199 0.733954i \(-0.262327\pi\)
0.679199 + 0.733954i \(0.262327\pi\)
\(272\) −12.0093 −0.728170
\(273\) 0 0
\(274\) 4.38670 0.265010
\(275\) 0.774236 0.0466882
\(276\) 0 0
\(277\) 5.95567 0.357842 0.178921 0.983863i \(-0.442739\pi\)
0.178921 + 0.983863i \(0.442739\pi\)
\(278\) 7.60060 0.455854
\(279\) 0 0
\(280\) 2.07875 0.124229
\(281\) 26.0667 1.55501 0.777503 0.628879i \(-0.216486\pi\)
0.777503 + 0.628879i \(0.216486\pi\)
\(282\) 0 0
\(283\) 9.23332 0.548864 0.274432 0.961607i \(-0.411510\pi\)
0.274432 + 0.961607i \(0.411510\pi\)
\(284\) −6.27018 −0.372067
\(285\) 0 0
\(286\) −0.397766 −0.0235204
\(287\) −0.995032 −0.0587349
\(288\) 0 0
\(289\) 27.2957 1.60563
\(290\) 3.04800 0.178984
\(291\) 0 0
\(292\) −4.85853 −0.284324
\(293\) −5.84040 −0.341200 −0.170600 0.985340i \(-0.554571\pi\)
−0.170600 + 0.985340i \(0.554571\pi\)
\(294\) 0 0
\(295\) −0.245775 −0.0143096
\(296\) 12.4113 0.721395
\(297\) 0 0
\(298\) 8.65728 0.501503
\(299\) 16.4718 0.952588
\(300\) 0 0
\(301\) 5.38517 0.310396
\(302\) 2.01434 0.115912
\(303\) 0 0
\(304\) −12.4085 −0.711674
\(305\) 11.0490 0.632667
\(306\) 0 0
\(307\) −30.3714 −1.73339 −0.866694 0.498841i \(-0.833759\pi\)
−0.866694 + 0.498841i \(0.833759\pi\)
\(308\) −0.299925 −0.0170898
\(309\) 0 0
\(310\) −1.84384 −0.104723
\(311\) −1.06093 −0.0601600 −0.0300800 0.999547i \(-0.509576\pi\)
−0.0300800 + 0.999547i \(0.509576\pi\)
\(312\) 0 0
\(313\) −2.40971 −0.136205 −0.0681024 0.997678i \(-0.521694\pi\)
−0.0681024 + 0.997678i \(0.521694\pi\)
\(314\) −5.12919 −0.289457
\(315\) 0 0
\(316\) −20.0126 −1.12580
\(317\) −5.11130 −0.287079 −0.143540 0.989645i \(-0.545848\pi\)
−0.143540 + 0.989645i \(0.545848\pi\)
\(318\) 0 0
\(319\) −0.986568 −0.0552372
\(320\) −0.0710512 −0.00397188
\(321\) 0 0
\(322\) −3.02273 −0.168450
\(323\) 45.7680 2.54660
\(324\) 0 0
\(325\) −14.1573 −0.785306
\(326\) 0.794951 0.0440283
\(327\) 0 0
\(328\) −2.24656 −0.124045
\(329\) −12.1442 −0.669534
\(330\) 0 0
\(331\) 5.99694 0.329621 0.164811 0.986325i \(-0.447299\pi\)
0.164811 + 0.986325i \(0.447299\pi\)
\(332\) −8.83487 −0.484876
\(333\) 0 0
\(334\) 9.99051 0.546656
\(335\) −3.94646 −0.215618
\(336\) 0 0
\(337\) 28.3255 1.54299 0.771494 0.636236i \(-0.219510\pi\)
0.771494 + 0.636236i \(0.219510\pi\)
\(338\) −0.860453 −0.0468025
\(339\) 0 0
\(340\) 9.85668 0.534553
\(341\) 0.596809 0.0323190
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 12.1585 0.655543
\(345\) 0 0
\(346\) 11.2911 0.607012
\(347\) −12.5610 −0.674311 −0.337156 0.941449i \(-0.609465\pi\)
−0.337156 + 0.941449i \(0.609465\pi\)
\(348\) 0 0
\(349\) −11.4710 −0.614030 −0.307015 0.951705i \(-0.599330\pi\)
−0.307015 + 0.951705i \(0.599330\pi\)
\(350\) 2.59800 0.138869
\(351\) 0 0
\(352\) −1.05248 −0.0560972
\(353\) 9.93044 0.528544 0.264272 0.964448i \(-0.414868\pi\)
0.264272 + 0.964448i \(0.414868\pi\)
\(354\) 0 0
\(355\) 3.58900 0.190484
\(356\) −0.749702 −0.0397341
\(357\) 0 0
\(358\) 6.49680 0.343366
\(359\) −7.77712 −0.410461 −0.205230 0.978714i \(-0.565794\pi\)
−0.205230 + 0.978714i \(0.565794\pi\)
\(360\) 0 0
\(361\) 28.2894 1.48891
\(362\) −6.50074 −0.341671
\(363\) 0 0
\(364\) 5.48429 0.287455
\(365\) 2.78098 0.145563
\(366\) 0 0
\(367\) −33.9549 −1.77243 −0.886217 0.463270i \(-0.846676\pi\)
−0.886217 + 0.463270i \(0.846676\pi\)
\(368\) 8.71736 0.454424
\(369\) 0 0
\(370\) −3.16672 −0.164630
\(371\) 2.21328 0.114908
\(372\) 0 0
\(373\) 19.6464 1.01725 0.508625 0.860988i \(-0.330154\pi\)
0.508625 + 0.860988i \(0.330154\pi\)
\(374\) 0.776455 0.0401495
\(375\) 0 0
\(376\) −27.4190 −1.41403
\(377\) 18.0399 0.929103
\(378\) 0 0
\(379\) 11.3610 0.583574 0.291787 0.956483i \(-0.405750\pi\)
0.291787 + 0.956483i \(0.405750\pi\)
\(380\) 10.1843 0.522444
\(381\) 0 0
\(382\) 7.86582 0.402450
\(383\) 35.3153 1.80453 0.902265 0.431182i \(-0.141903\pi\)
0.902265 + 0.431182i \(0.141903\pi\)
\(384\) 0 0
\(385\) 0.171675 0.00874935
\(386\) 12.7653 0.649738
\(387\) 0 0
\(388\) −16.5584 −0.840626
\(389\) 11.9216 0.604450 0.302225 0.953237i \(-0.402271\pi\)
0.302225 + 0.953237i \(0.402271\pi\)
\(390\) 0 0
\(391\) −32.1536 −1.62608
\(392\) −2.25778 −0.114035
\(393\) 0 0
\(394\) −11.4447 −0.576575
\(395\) 11.4550 0.576366
\(396\) 0 0
\(397\) −1.00894 −0.0506374 −0.0253187 0.999679i \(-0.508060\pi\)
−0.0253187 + 0.999679i \(0.508060\pi\)
\(398\) 4.05036 0.203026
\(399\) 0 0
\(400\) −7.49247 −0.374623
\(401\) 23.1876 1.15793 0.578967 0.815351i \(-0.303456\pi\)
0.578967 + 0.815351i \(0.303456\pi\)
\(402\) 0 0
\(403\) −10.9130 −0.543613
\(404\) 17.1214 0.851824
\(405\) 0 0
\(406\) −3.31049 −0.164297
\(407\) 1.02500 0.0508072
\(408\) 0 0
\(409\) 3.46733 0.171448 0.0857242 0.996319i \(-0.472680\pi\)
0.0857242 + 0.996319i \(0.472680\pi\)
\(410\) 0.573204 0.0283085
\(411\) 0 0
\(412\) 14.4355 0.711185
\(413\) 0.266942 0.0131354
\(414\) 0 0
\(415\) 5.05700 0.248238
\(416\) 19.2451 0.943569
\(417\) 0 0
\(418\) 0.802264 0.0392400
\(419\) 30.6708 1.49836 0.749182 0.662364i \(-0.230447\pi\)
0.749182 + 0.662364i \(0.230447\pi\)
\(420\) 0 0
\(421\) 18.9561 0.923866 0.461933 0.886915i \(-0.347156\pi\)
0.461933 + 0.886915i \(0.347156\pi\)
\(422\) 14.3856 0.700281
\(423\) 0 0
\(424\) 4.99708 0.242680
\(425\) 27.6356 1.34052
\(426\) 0 0
\(427\) −12.0006 −0.580750
\(428\) 14.3997 0.696037
\(429\) 0 0
\(430\) −3.10222 −0.149602
\(431\) −11.2659 −0.542661 −0.271330 0.962486i \(-0.587464\pi\)
−0.271330 + 0.962486i \(0.587464\pi\)
\(432\) 0 0
\(433\) 30.4858 1.46505 0.732527 0.680738i \(-0.238341\pi\)
0.732527 + 0.680738i \(0.238341\pi\)
\(434\) 2.00263 0.0961295
\(435\) 0 0
\(436\) 12.1820 0.583412
\(437\) −33.2223 −1.58924
\(438\) 0 0
\(439\) −21.6681 −1.03416 −0.517080 0.855937i \(-0.672981\pi\)
−0.517080 + 0.855937i \(0.672981\pi\)
\(440\) 0.387603 0.0184782
\(441\) 0 0
\(442\) −14.1979 −0.675324
\(443\) 20.7860 0.987572 0.493786 0.869584i \(-0.335613\pi\)
0.493786 + 0.869584i \(0.335613\pi\)
\(444\) 0 0
\(445\) 0.429123 0.0203424
\(446\) 11.9395 0.565351
\(447\) 0 0
\(448\) 0.0771703 0.00364595
\(449\) −37.3334 −1.76187 −0.880937 0.473234i \(-0.843087\pi\)
−0.880937 + 0.473234i \(0.843087\pi\)
\(450\) 0 0
\(451\) −0.185533 −0.00873642
\(452\) −10.8943 −0.512424
\(453\) 0 0
\(454\) 1.63886 0.0769153
\(455\) −3.13916 −0.147166
\(456\) 0 0
\(457\) −25.1132 −1.17475 −0.587374 0.809316i \(-0.699838\pi\)
−0.587374 + 0.809316i \(0.699838\pi\)
\(458\) −8.08613 −0.377840
\(459\) 0 0
\(460\) −7.15481 −0.333595
\(461\) −7.08390 −0.329930 −0.164965 0.986299i \(-0.552751\pi\)
−0.164965 + 0.986299i \(0.552751\pi\)
\(462\) 0 0
\(463\) −0.197820 −0.00919346 −0.00459673 0.999989i \(-0.501463\pi\)
−0.00459673 + 0.999989i \(0.501463\pi\)
\(464\) 9.54726 0.443220
\(465\) 0 0
\(466\) −8.68326 −0.402244
\(467\) 23.3627 1.08110 0.540549 0.841313i \(-0.318217\pi\)
0.540549 + 0.841313i \(0.318217\pi\)
\(468\) 0 0
\(469\) 4.28634 0.197925
\(470\) 6.99588 0.322696
\(471\) 0 0
\(472\) 0.602695 0.0277413
\(473\) 1.00412 0.0461694
\(474\) 0 0
\(475\) 28.5542 1.31016
\(476\) −10.7056 −0.490688
\(477\) 0 0
\(478\) 8.94910 0.409322
\(479\) 11.5549 0.527957 0.263978 0.964529i \(-0.414965\pi\)
0.263978 + 0.964529i \(0.414965\pi\)
\(480\) 0 0
\(481\) −18.7426 −0.854590
\(482\) 13.0891 0.596191
\(483\) 0 0
\(484\) 17.6379 0.801722
\(485\) 9.47789 0.430369
\(486\) 0 0
\(487\) 12.8422 0.581936 0.290968 0.956733i \(-0.406023\pi\)
0.290968 + 0.956733i \(0.406023\pi\)
\(488\) −27.0947 −1.22652
\(489\) 0 0
\(490\) 0.576066 0.0260240
\(491\) 12.0736 0.544874 0.272437 0.962174i \(-0.412170\pi\)
0.272437 + 0.962174i \(0.412170\pi\)
\(492\) 0 0
\(493\) −35.2146 −1.58599
\(494\) −14.6698 −0.660026
\(495\) 0 0
\(496\) −5.77546 −0.259326
\(497\) −3.89809 −0.174853
\(498\) 0 0
\(499\) 32.4440 1.45239 0.726197 0.687487i \(-0.241286\pi\)
0.726197 + 0.687487i \(0.241286\pi\)
\(500\) 13.5544 0.606171
\(501\) 0 0
\(502\) −6.57723 −0.293556
\(503\) 34.3236 1.53041 0.765207 0.643785i \(-0.222637\pi\)
0.765207 + 0.643785i \(0.222637\pi\)
\(504\) 0 0
\(505\) −9.80017 −0.436102
\(506\) −0.563617 −0.0250558
\(507\) 0 0
\(508\) 1.60853 0.0713669
\(509\) 4.59653 0.203738 0.101869 0.994798i \(-0.467518\pi\)
0.101869 + 0.994798i \(0.467518\pi\)
\(510\) 0 0
\(511\) −3.02049 −0.133618
\(512\) 18.3330 0.810212
\(513\) 0 0
\(514\) 10.1673 0.448460
\(515\) −8.26274 −0.364100
\(516\) 0 0
\(517\) −2.26441 −0.0995886
\(518\) 3.43945 0.151121
\(519\) 0 0
\(520\) −7.08752 −0.310808
\(521\) −4.84146 −0.212108 −0.106054 0.994360i \(-0.533822\pi\)
−0.106054 + 0.994360i \(0.533822\pi\)
\(522\) 0 0
\(523\) 20.2848 0.886992 0.443496 0.896276i \(-0.353738\pi\)
0.443496 + 0.896276i \(0.353738\pi\)
\(524\) 7.88199 0.344326
\(525\) 0 0
\(526\) −6.72271 −0.293124
\(527\) 21.3025 0.927953
\(528\) 0 0
\(529\) 0.339797 0.0147738
\(530\) −1.27499 −0.0553821
\(531\) 0 0
\(532\) −11.0614 −0.479573
\(533\) 3.39257 0.146949
\(534\) 0 0
\(535\) −8.24227 −0.356344
\(536\) 9.67759 0.418008
\(537\) 0 0
\(538\) 6.39688 0.275789
\(539\) −0.186460 −0.00803138
\(540\) 0 0
\(541\) 4.52831 0.194687 0.0973435 0.995251i \(-0.468965\pi\)
0.0973435 + 0.995251i \(0.468965\pi\)
\(542\) 13.9914 0.600984
\(543\) 0 0
\(544\) −37.5672 −1.61068
\(545\) −6.97287 −0.298685
\(546\) 0 0
\(547\) 16.4649 0.703988 0.351994 0.936002i \(-0.385504\pi\)
0.351994 + 0.936002i \(0.385504\pi\)
\(548\) −11.2776 −0.481754
\(549\) 0 0
\(550\) 0.484422 0.0206558
\(551\) −36.3851 −1.55006
\(552\) 0 0
\(553\) −12.4416 −0.529070
\(554\) 3.72633 0.158317
\(555\) 0 0
\(556\) −19.5401 −0.828683
\(557\) −7.78271 −0.329764 −0.164882 0.986313i \(-0.552724\pi\)
−0.164882 + 0.986313i \(0.552724\pi\)
\(558\) 0 0
\(559\) −18.3608 −0.776580
\(560\) −1.66134 −0.0702043
\(561\) 0 0
\(562\) 16.3093 0.687967
\(563\) 21.6814 0.913761 0.456881 0.889528i \(-0.348967\pi\)
0.456881 + 0.889528i \(0.348967\pi\)
\(564\) 0 0
\(565\) 6.23579 0.262342
\(566\) 5.77708 0.242829
\(567\) 0 0
\(568\) −8.80101 −0.369282
\(569\) −27.8221 −1.16636 −0.583181 0.812342i \(-0.698192\pi\)
−0.583181 + 0.812342i \(0.698192\pi\)
\(570\) 0 0
\(571\) −40.0581 −1.67638 −0.838190 0.545378i \(-0.816386\pi\)
−0.838190 + 0.545378i \(0.816386\pi\)
\(572\) 1.02260 0.0427570
\(573\) 0 0
\(574\) −0.622570 −0.0259856
\(575\) −20.0603 −0.836572
\(576\) 0 0
\(577\) 3.86000 0.160694 0.0803469 0.996767i \(-0.474397\pi\)
0.0803469 + 0.996767i \(0.474397\pi\)
\(578\) 17.0783 0.710363
\(579\) 0 0
\(580\) −7.83596 −0.325370
\(581\) −5.49252 −0.227868
\(582\) 0 0
\(583\) 0.412686 0.0170917
\(584\) −6.81958 −0.282196
\(585\) 0 0
\(586\) −3.65421 −0.150954
\(587\) 19.8612 0.819758 0.409879 0.912140i \(-0.365571\pi\)
0.409879 + 0.912140i \(0.365571\pi\)
\(588\) 0 0
\(589\) 22.0106 0.906932
\(590\) −0.153776 −0.00633087
\(591\) 0 0
\(592\) −9.91915 −0.407675
\(593\) −42.1555 −1.73112 −0.865559 0.500808i \(-0.833036\pi\)
−0.865559 + 0.500808i \(0.833036\pi\)
\(594\) 0 0
\(595\) 6.12776 0.251214
\(596\) −22.2566 −0.911666
\(597\) 0 0
\(598\) 10.3060 0.421445
\(599\) −28.6211 −1.16942 −0.584712 0.811241i \(-0.698793\pi\)
−0.584712 + 0.811241i \(0.698793\pi\)
\(600\) 0 0
\(601\) −22.7380 −0.927501 −0.463751 0.885966i \(-0.653497\pi\)
−0.463751 + 0.885966i \(0.653497\pi\)
\(602\) 3.36938 0.137326
\(603\) 0 0
\(604\) −5.17858 −0.210713
\(605\) −10.0958 −0.410451
\(606\) 0 0
\(607\) 4.36454 0.177151 0.0885756 0.996069i \(-0.471769\pi\)
0.0885756 + 0.996069i \(0.471769\pi\)
\(608\) −38.8159 −1.57419
\(609\) 0 0
\(610\) 6.91314 0.279905
\(611\) 41.4059 1.67510
\(612\) 0 0
\(613\) −12.4785 −0.504001 −0.252001 0.967727i \(-0.581088\pi\)
−0.252001 + 0.967727i \(0.581088\pi\)
\(614\) −19.0027 −0.766887
\(615\) 0 0
\(616\) −0.420984 −0.0169619
\(617\) −33.3765 −1.34369 −0.671845 0.740692i \(-0.734498\pi\)
−0.671845 + 0.740692i \(0.734498\pi\)
\(618\) 0 0
\(619\) 21.2133 0.852637 0.426318 0.904573i \(-0.359810\pi\)
0.426318 + 0.904573i \(0.359810\pi\)
\(620\) 4.74024 0.190373
\(621\) 0 0
\(622\) −0.663802 −0.0266160
\(623\) −0.466080 −0.0186731
\(624\) 0 0
\(625\) 13.0031 0.520123
\(626\) −1.50770 −0.0602599
\(627\) 0 0
\(628\) 13.1864 0.526195
\(629\) 36.5863 1.45879
\(630\) 0 0
\(631\) 16.7036 0.664960 0.332480 0.943110i \(-0.392115\pi\)
0.332480 + 0.943110i \(0.392115\pi\)
\(632\) −28.0903 −1.11737
\(633\) 0 0
\(634\) −3.19803 −0.127010
\(635\) −0.920707 −0.0365371
\(636\) 0 0
\(637\) 3.40951 0.135090
\(638\) −0.617274 −0.0244381
\(639\) 0 0
\(640\) −10.4384 −0.412613
\(641\) 9.20283 0.363490 0.181745 0.983346i \(-0.441825\pi\)
0.181745 + 0.983346i \(0.441825\pi\)
\(642\) 0 0
\(643\) −43.1262 −1.70073 −0.850365 0.526193i \(-0.823619\pi\)
−0.850365 + 0.526193i \(0.823619\pi\)
\(644\) 7.77100 0.306220
\(645\) 0 0
\(646\) 28.6360 1.12667
\(647\) −13.7067 −0.538866 −0.269433 0.963019i \(-0.586836\pi\)
−0.269433 + 0.963019i \(0.586836\pi\)
\(648\) 0 0
\(649\) 0.0497739 0.00195380
\(650\) −8.85791 −0.347436
\(651\) 0 0
\(652\) −2.04370 −0.0800376
\(653\) 6.63237 0.259545 0.129772 0.991544i \(-0.458575\pi\)
0.129772 + 0.991544i \(0.458575\pi\)
\(654\) 0 0
\(655\) −4.51158 −0.176282
\(656\) 1.79545 0.0701006
\(657\) 0 0
\(658\) −7.59838 −0.296216
\(659\) −29.2407 −1.13906 −0.569528 0.821972i \(-0.692874\pi\)
−0.569528 + 0.821972i \(0.692874\pi\)
\(660\) 0 0
\(661\) −46.2116 −1.79742 −0.898711 0.438541i \(-0.855495\pi\)
−0.898711 + 0.438541i \(0.855495\pi\)
\(662\) 3.75215 0.145831
\(663\) 0 0
\(664\) −12.4009 −0.481247
\(665\) 6.33145 0.245523
\(666\) 0 0
\(667\) 25.5618 0.989755
\(668\) −25.6842 −0.993750
\(669\) 0 0
\(670\) −2.46921 −0.0953941
\(671\) −2.23763 −0.0863827
\(672\) 0 0
\(673\) 24.6842 0.951506 0.475753 0.879579i \(-0.342176\pi\)
0.475753 + 0.879579i \(0.342176\pi\)
\(674\) 17.7226 0.682651
\(675\) 0 0
\(676\) 2.21210 0.0850808
\(677\) −26.3763 −1.01372 −0.506862 0.862027i \(-0.669194\pi\)
−0.506862 + 0.862027i \(0.669194\pi\)
\(678\) 0 0
\(679\) −10.2941 −0.395053
\(680\) 13.8351 0.530552
\(681\) 0 0
\(682\) 0.373410 0.0142986
\(683\) −37.5965 −1.43859 −0.719296 0.694704i \(-0.755535\pi\)
−0.719296 + 0.694704i \(0.755535\pi\)
\(684\) 0 0
\(685\) 6.45518 0.246640
\(686\) −0.625678 −0.0238885
\(687\) 0 0
\(688\) −9.71709 −0.370461
\(689\) −7.54619 −0.287487
\(690\) 0 0
\(691\) −38.6125 −1.46889 −0.734444 0.678669i \(-0.762557\pi\)
−0.734444 + 0.678669i \(0.762557\pi\)
\(692\) −29.0277 −1.10347
\(693\) 0 0
\(694\) −7.85915 −0.298329
\(695\) 11.1846 0.424254
\(696\) 0 0
\(697\) −6.62244 −0.250843
\(698\) −7.17717 −0.271660
\(699\) 0 0
\(700\) −6.67909 −0.252446
\(701\) −4.49148 −0.169641 −0.0848204 0.996396i \(-0.527032\pi\)
−0.0848204 + 0.996396i \(0.527032\pi\)
\(702\) 0 0
\(703\) 37.8024 1.42575
\(704\) 0.0143891 0.000542311 0
\(705\) 0 0
\(706\) 6.21326 0.233839
\(707\) 10.6442 0.400315
\(708\) 0 0
\(709\) −53.1269 −1.99522 −0.997612 0.0690674i \(-0.977998\pi\)
−0.997612 + 0.0690674i \(0.977998\pi\)
\(710\) 2.24556 0.0842742
\(711\) 0 0
\(712\) −1.05230 −0.0394367
\(713\) −15.4632 −0.579101
\(714\) 0 0
\(715\) −0.585327 −0.0218900
\(716\) −16.7023 −0.624195
\(717\) 0 0
\(718\) −4.86597 −0.181596
\(719\) −33.4034 −1.24574 −0.622869 0.782327i \(-0.714033\pi\)
−0.622869 + 0.782327i \(0.714033\pi\)
\(720\) 0 0
\(721\) 8.97435 0.334222
\(722\) 17.7000 0.658727
\(723\) 0 0
\(724\) 16.7125 0.621114
\(725\) −21.9700 −0.815947
\(726\) 0 0
\(727\) 21.1975 0.786173 0.393086 0.919502i \(-0.371407\pi\)
0.393086 + 0.919502i \(0.371407\pi\)
\(728\) 7.69791 0.285304
\(729\) 0 0
\(730\) 1.74000 0.0644002
\(731\) 35.8410 1.32563
\(732\) 0 0
\(733\) −6.14227 −0.226870 −0.113435 0.993545i \(-0.536185\pi\)
−0.113435 + 0.993545i \(0.536185\pi\)
\(734\) −21.2449 −0.784162
\(735\) 0 0
\(736\) 27.2695 1.00517
\(737\) 0.799229 0.0294400
\(738\) 0 0
\(739\) 2.53793 0.0933592 0.0466796 0.998910i \(-0.485136\pi\)
0.0466796 + 0.998910i \(0.485136\pi\)
\(740\) 8.14119 0.299276
\(741\) 0 0
\(742\) 1.38480 0.0508375
\(743\) 37.7650 1.38546 0.692731 0.721196i \(-0.256407\pi\)
0.692731 + 0.721196i \(0.256407\pi\)
\(744\) 0 0
\(745\) 12.7395 0.466739
\(746\) 12.2923 0.450053
\(747\) 0 0
\(748\) −1.99615 −0.0729866
\(749\) 8.95211 0.327103
\(750\) 0 0
\(751\) −15.0073 −0.547625 −0.273812 0.961783i \(-0.588285\pi\)
−0.273812 + 0.961783i \(0.588285\pi\)
\(752\) 21.9132 0.799094
\(753\) 0 0
\(754\) 11.2872 0.411054
\(755\) 2.96417 0.107877
\(756\) 0 0
\(757\) −6.16228 −0.223972 −0.111986 0.993710i \(-0.535721\pi\)
−0.111986 + 0.993710i \(0.535721\pi\)
\(758\) 7.10831 0.258185
\(759\) 0 0
\(760\) 14.2950 0.518534
\(761\) −27.0694 −0.981265 −0.490632 0.871367i \(-0.663234\pi\)
−0.490632 + 0.871367i \(0.663234\pi\)
\(762\) 0 0
\(763\) 7.57339 0.274175
\(764\) −20.2219 −0.731602
\(765\) 0 0
\(766\) 22.0960 0.798362
\(767\) −0.910142 −0.0328633
\(768\) 0 0
\(769\) −16.2404 −0.585644 −0.292822 0.956167i \(-0.594594\pi\)
−0.292822 + 0.956167i \(0.594594\pi\)
\(770\) 0.107413 0.00387090
\(771\) 0 0
\(772\) −32.8178 −1.18114
\(773\) 41.9505 1.50886 0.754428 0.656383i \(-0.227915\pi\)
0.754428 + 0.656383i \(0.227915\pi\)
\(774\) 0 0
\(775\) 13.2904 0.477406
\(776\) −23.2419 −0.834335
\(777\) 0 0
\(778\) 7.45909 0.267421
\(779\) −6.84256 −0.245160
\(780\) 0 0
\(781\) −0.726836 −0.0260082
\(782\) −20.1178 −0.719410
\(783\) 0 0
\(784\) 1.80441 0.0644434
\(785\) −7.54778 −0.269392
\(786\) 0 0
\(787\) 34.0139 1.21247 0.606233 0.795287i \(-0.292680\pi\)
0.606233 + 0.795287i \(0.292680\pi\)
\(788\) 29.4226 1.04814
\(789\) 0 0
\(790\) 7.16717 0.254996
\(791\) −6.77283 −0.240814
\(792\) 0 0
\(793\) 40.9162 1.45298
\(794\) −0.631273 −0.0224030
\(795\) 0 0
\(796\) −10.4129 −0.369075
\(797\) 19.6244 0.695130 0.347565 0.937656i \(-0.387009\pi\)
0.347565 + 0.937656i \(0.387009\pi\)
\(798\) 0 0
\(799\) −80.8260 −2.85942
\(800\) −23.4378 −0.828651
\(801\) 0 0
\(802\) 14.5080 0.512294
\(803\) −0.563198 −0.0198748
\(804\) 0 0
\(805\) −4.44805 −0.156773
\(806\) −6.82800 −0.240506
\(807\) 0 0
\(808\) 24.0322 0.845448
\(809\) 52.5671 1.84816 0.924080 0.382200i \(-0.124833\pi\)
0.924080 + 0.382200i \(0.124833\pi\)
\(810\) 0 0
\(811\) 9.33906 0.327939 0.163969 0.986465i \(-0.447570\pi\)
0.163969 + 0.986465i \(0.447570\pi\)
\(812\) 8.51080 0.298671
\(813\) 0 0
\(814\) 0.641318 0.0224782
\(815\) 1.16980 0.0409763
\(816\) 0 0
\(817\) 37.0324 1.29560
\(818\) 2.16943 0.0758524
\(819\) 0 0
\(820\) −1.47363 −0.0514612
\(821\) −0.683311 −0.0238477 −0.0119239 0.999929i \(-0.503796\pi\)
−0.0119239 + 0.999929i \(0.503796\pi\)
\(822\) 0 0
\(823\) −40.8593 −1.42427 −0.712133 0.702045i \(-0.752271\pi\)
−0.712133 + 0.702045i \(0.752271\pi\)
\(824\) 20.2621 0.705862
\(825\) 0 0
\(826\) 0.167020 0.00581136
\(827\) −42.8159 −1.48885 −0.744427 0.667704i \(-0.767277\pi\)
−0.744427 + 0.667704i \(0.767277\pi\)
\(828\) 0 0
\(829\) 33.7742 1.17303 0.586514 0.809939i \(-0.300500\pi\)
0.586514 + 0.809939i \(0.300500\pi\)
\(830\) 3.16405 0.109826
\(831\) 0 0
\(832\) −0.263113 −0.00912180
\(833\) −6.65550 −0.230599
\(834\) 0 0
\(835\) 14.7014 0.508763
\(836\) −2.06250 −0.0713332
\(837\) 0 0
\(838\) 19.1900 0.662908
\(839\) 11.7315 0.405017 0.202508 0.979281i \(-0.435091\pi\)
0.202508 + 0.979281i \(0.435091\pi\)
\(840\) 0 0
\(841\) −1.00475 −0.0346465
\(842\) 11.8604 0.408738
\(843\) 0 0
\(844\) −36.9833 −1.27302
\(845\) −1.26619 −0.0435582
\(846\) 0 0
\(847\) 10.9652 0.376770
\(848\) −3.99367 −0.137143
\(849\) 0 0
\(850\) 17.2910 0.593076
\(851\) −26.5575 −0.910378
\(852\) 0 0
\(853\) −6.49984 −0.222550 −0.111275 0.993790i \(-0.535493\pi\)
−0.111275 + 0.993790i \(0.535493\pi\)
\(854\) −7.50852 −0.256936
\(855\) 0 0
\(856\) 20.2119 0.690827
\(857\) 7.34627 0.250944 0.125472 0.992097i \(-0.459956\pi\)
0.125472 + 0.992097i \(0.459956\pi\)
\(858\) 0 0
\(859\) 11.6811 0.398553 0.199276 0.979943i \(-0.436141\pi\)
0.199276 + 0.979943i \(0.436141\pi\)
\(860\) 7.97535 0.271957
\(861\) 0 0
\(862\) −7.04884 −0.240085
\(863\) 54.3317 1.84947 0.924736 0.380610i \(-0.124286\pi\)
0.924736 + 0.380610i \(0.124286\pi\)
\(864\) 0 0
\(865\) 16.6152 0.564934
\(866\) 19.0743 0.648170
\(867\) 0 0
\(868\) −5.14848 −0.174751
\(869\) −2.31985 −0.0786956
\(870\) 0 0
\(871\) −14.6143 −0.495188
\(872\) 17.0990 0.579046
\(873\) 0 0
\(874\) −20.7865 −0.703113
\(875\) 8.42659 0.284871
\(876\) 0 0
\(877\) −45.5254 −1.53728 −0.768642 0.639679i \(-0.779067\pi\)
−0.768642 + 0.639679i \(0.779067\pi\)
\(878\) −13.5572 −0.457534
\(879\) 0 0
\(880\) −0.309772 −0.0104424
\(881\) 26.9306 0.907316 0.453658 0.891176i \(-0.350119\pi\)
0.453658 + 0.891176i \(0.350119\pi\)
\(882\) 0 0
\(883\) 48.7174 1.63947 0.819735 0.572743i \(-0.194121\pi\)
0.819735 + 0.572743i \(0.194121\pi\)
\(884\) 36.5007 1.22765
\(885\) 0 0
\(886\) 13.0053 0.436922
\(887\) 43.8984 1.47396 0.736982 0.675913i \(-0.236250\pi\)
0.736982 + 0.675913i \(0.236250\pi\)
\(888\) 0 0
\(889\) 1.00000 0.0335389
\(890\) 0.268493 0.00899989
\(891\) 0 0
\(892\) −30.6947 −1.02773
\(893\) −83.5126 −2.79464
\(894\) 0 0
\(895\) 9.56027 0.319565
\(896\) 11.3373 0.378754
\(897\) 0 0
\(898\) −23.3587 −0.779490
\(899\) −16.9353 −0.564824
\(900\) 0 0
\(901\) 14.7305 0.490742
\(902\) −0.116084 −0.00386518
\(903\) 0 0
\(904\) −15.2915 −0.508588
\(905\) −9.56607 −0.317987
\(906\) 0 0
\(907\) −40.3143 −1.33861 −0.669307 0.742986i \(-0.733409\pi\)
−0.669307 + 0.742986i \(0.733409\pi\)
\(908\) −4.21326 −0.139822
\(909\) 0 0
\(910\) −1.96410 −0.0651094
\(911\) 52.0297 1.72382 0.861910 0.507062i \(-0.169268\pi\)
0.861910 + 0.507062i \(0.169268\pi\)
\(912\) 0 0
\(913\) −1.02413 −0.0338939
\(914\) −15.7128 −0.519733
\(915\) 0 0
\(916\) 20.7883 0.686863
\(917\) 4.90013 0.161817
\(918\) 0 0
\(919\) 37.4975 1.23693 0.618464 0.785813i \(-0.287755\pi\)
0.618464 + 0.785813i \(0.287755\pi\)
\(920\) −10.0427 −0.331098
\(921\) 0 0
\(922\) −4.43224 −0.145968
\(923\) 13.2906 0.437465
\(924\) 0 0
\(925\) 22.8258 0.750509
\(926\) −0.123771 −0.00406738
\(927\) 0 0
\(928\) 29.8655 0.980384
\(929\) 44.8965 1.47300 0.736502 0.676435i \(-0.236476\pi\)
0.736502 + 0.676435i \(0.236476\pi\)
\(930\) 0 0
\(931\) −6.87673 −0.225376
\(932\) 22.3234 0.731227
\(933\) 0 0
\(934\) 14.6175 0.478300
\(935\) 1.14258 0.0373664
\(936\) 0 0
\(937\) 20.2780 0.662453 0.331226 0.943551i \(-0.392538\pi\)
0.331226 + 0.943551i \(0.392538\pi\)
\(938\) 2.68187 0.0875661
\(939\) 0 0
\(940\) −17.9854 −0.586619
\(941\) −11.3416 −0.369725 −0.184863 0.982764i \(-0.559184\pi\)
−0.184863 + 0.982764i \(0.559184\pi\)
\(942\) 0 0
\(943\) 4.80713 0.156542
\(944\) −0.481674 −0.0156772
\(945\) 0 0
\(946\) 0.628254 0.0204263
\(947\) −46.8755 −1.52325 −0.761624 0.648019i \(-0.775598\pi\)
−0.761624 + 0.648019i \(0.775598\pi\)
\(948\) 0 0
\(949\) 10.2984 0.334300
\(950\) 17.8657 0.579641
\(951\) 0 0
\(952\) −15.0266 −0.487016
\(953\) −52.6734 −1.70626 −0.853129 0.521699i \(-0.825298\pi\)
−0.853129 + 0.521699i \(0.825298\pi\)
\(954\) 0 0
\(955\) 11.5748 0.374553
\(956\) −23.0068 −0.744095
\(957\) 0 0
\(958\) 7.22964 0.233579
\(959\) −7.01111 −0.226401
\(960\) 0 0
\(961\) −20.7553 −0.669524
\(962\) −11.7268 −0.378089
\(963\) 0 0
\(964\) −33.6501 −1.08380
\(965\) 18.7846 0.604699
\(966\) 0 0
\(967\) −21.2720 −0.684062 −0.342031 0.939689i \(-0.611115\pi\)
−0.342031 + 0.939689i \(0.611115\pi\)
\(968\) 24.7570 0.795721
\(969\) 0 0
\(970\) 5.93011 0.190404
\(971\) 10.1693 0.326347 0.163174 0.986597i \(-0.447827\pi\)
0.163174 + 0.986597i \(0.447827\pi\)
\(972\) 0 0
\(973\) −12.1478 −0.389440
\(974\) 8.03508 0.257461
\(975\) 0 0
\(976\) 21.6541 0.693130
\(977\) 31.3392 1.00263 0.501314 0.865265i \(-0.332850\pi\)
0.501314 + 0.865265i \(0.332850\pi\)
\(978\) 0 0
\(979\) −0.0869050 −0.00277750
\(980\) −1.48098 −0.0473082
\(981\) 0 0
\(982\) 7.55419 0.241064
\(983\) −8.93368 −0.284940 −0.142470 0.989799i \(-0.545504\pi\)
−0.142470 + 0.989799i \(0.545504\pi\)
\(984\) 0 0
\(985\) −16.8413 −0.536607
\(986\) −22.0330 −0.701674
\(987\) 0 0
\(988\) 37.7140 1.19984
\(989\) −26.0165 −0.827276
\(990\) 0 0
\(991\) −59.2418 −1.88188 −0.940939 0.338576i \(-0.890055\pi\)
−0.940939 + 0.338576i \(0.890055\pi\)
\(992\) −18.0667 −0.573618
\(993\) 0 0
\(994\) −2.43895 −0.0773587
\(995\) 5.96025 0.188953
\(996\) 0 0
\(997\) 27.7574 0.879086 0.439543 0.898222i \(-0.355140\pi\)
0.439543 + 0.898222i \(0.355140\pi\)
\(998\) 20.2995 0.642569
\(999\) 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.1.10 16
3.2 odd 2 889.2.a.c.1.7 16
21.20 even 2 6223.2.a.k.1.7 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
889.2.a.c.1.7 16 3.2 odd 2
6223.2.a.k.1.7 16 21.20 even 2
8001.2.a.t.1.10 16 1.1 even 1 trivial