Properties

Label 8001.2.a.p.1.6
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $0$
Dimension $14$
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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 5 x^{13} - 9 x^{12} + 76 x^{11} - 12 x^{10} - 414 x^{9} + 331 x^{8} + 959 x^{7} - 1067 x^{6} - 875 x^{5} + 1134 x^{4} + 301 x^{3} - 418 x^{2} - 42 x + 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5 \)
Twist minimal: no (minimal twist has level 2667)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-0.477041\) 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)
 
\(f(q)\) \(=\) \(q-0.477041 q^{2} -1.77243 q^{4} +1.98652 q^{5} -1.00000 q^{7} +1.79961 q^{8} +O(q^{10})\) \(q-0.477041 q^{2} -1.77243 q^{4} +1.98652 q^{5} -1.00000 q^{7} +1.79961 q^{8} -0.947652 q^{10} +0.129110 q^{11} +5.16963 q^{13} +0.477041 q^{14} +2.68638 q^{16} +2.68090 q^{17} +4.39995 q^{19} -3.52097 q^{20} -0.0615910 q^{22} +2.43926 q^{23} -1.05374 q^{25} -2.46613 q^{26} +1.77243 q^{28} -5.32140 q^{29} +2.06047 q^{31} -4.88072 q^{32} -1.27890 q^{34} -1.98652 q^{35} +2.29275 q^{37} -2.09896 q^{38} +3.57495 q^{40} +3.69549 q^{41} +6.38263 q^{43} -0.228839 q^{44} -1.16363 q^{46} +11.8866 q^{47} +1.00000 q^{49} +0.502678 q^{50} -9.16281 q^{52} +7.84774 q^{53} +0.256480 q^{55} -1.79961 q^{56} +2.53853 q^{58} -8.58540 q^{59} +3.08167 q^{61} -0.982930 q^{62} -3.04445 q^{64} +10.2696 q^{65} -5.38959 q^{67} -4.75171 q^{68} +0.947652 q^{70} -0.409103 q^{71} -9.61874 q^{73} -1.09373 q^{74} -7.79862 q^{76} -0.129110 q^{77} +4.71224 q^{79} +5.33654 q^{80} -1.76290 q^{82} +7.35722 q^{83} +5.32565 q^{85} -3.04478 q^{86} +0.232348 q^{88} -0.833032 q^{89} -5.16963 q^{91} -4.32342 q^{92} -5.67039 q^{94} +8.74059 q^{95} +2.28385 q^{97} -0.477041 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 5 q^{2} + 15 q^{4} + 4 q^{5} - 14 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 5 q^{2} + 15 q^{4} + 4 q^{5} - 14 q^{7} + 12 q^{8} + 4 q^{10} + 3 q^{11} - 13 q^{13} - 5 q^{14} + 13 q^{16} + 5 q^{17} + 21 q^{19} + 3 q^{20} - 3 q^{22} + 10 q^{23} + 4 q^{25} + 6 q^{26} - 15 q^{28} + 15 q^{29} + 33 q^{31} + 29 q^{32} + 28 q^{34} - 4 q^{35} - 29 q^{37} + 15 q^{38} + 3 q^{40} + q^{41} - 25 q^{43} + 26 q^{44} - 4 q^{46} + 9 q^{47} + 14 q^{49} + 28 q^{50} - 13 q^{52} + 35 q^{53} + 14 q^{55} - 12 q^{56} - 23 q^{58} - 10 q^{59} + q^{61} + 43 q^{62} - 2 q^{64} + 24 q^{65} - 38 q^{67} + 2 q^{68} - 4 q^{70} + 10 q^{71} + 8 q^{73} + 25 q^{74} + 26 q^{76} - 3 q^{77} + 26 q^{79} + 48 q^{80} + 6 q^{82} + 30 q^{83} - 32 q^{85} + 50 q^{86} - 29 q^{88} - 4 q^{89} + 13 q^{91} + 32 q^{92} - 7 q^{94} + 32 q^{95} + 15 q^{97} + 5 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.477041 −0.337319 −0.168660 0.985674i \(-0.553944\pi\)
−0.168660 + 0.985674i \(0.553944\pi\)
\(3\) 0 0
\(4\) −1.77243 −0.886216
\(5\) 1.98652 0.888398 0.444199 0.895928i \(-0.353488\pi\)
0.444199 + 0.895928i \(0.353488\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.79961 0.636257
\(9\) 0 0
\(10\) −0.947652 −0.299674
\(11\) 0.129110 0.0389283 0.0194641 0.999811i \(-0.493804\pi\)
0.0194641 + 0.999811i \(0.493804\pi\)
\(12\) 0 0
\(13\) 5.16963 1.43380 0.716898 0.697178i \(-0.245561\pi\)
0.716898 + 0.697178i \(0.245561\pi\)
\(14\) 0.477041 0.127495
\(15\) 0 0
\(16\) 2.68638 0.671594
\(17\) 2.68090 0.650213 0.325106 0.945677i \(-0.394600\pi\)
0.325106 + 0.945677i \(0.394600\pi\)
\(18\) 0 0
\(19\) 4.39995 1.00942 0.504709 0.863289i \(-0.331600\pi\)
0.504709 + 0.863289i \(0.331600\pi\)
\(20\) −3.52097 −0.787313
\(21\) 0 0
\(22\) −0.0615910 −0.0131312
\(23\) 2.43926 0.508621 0.254310 0.967123i \(-0.418151\pi\)
0.254310 + 0.967123i \(0.418151\pi\)
\(24\) 0 0
\(25\) −1.05374 −0.210748
\(26\) −2.46613 −0.483647
\(27\) 0 0
\(28\) 1.77243 0.334958
\(29\) −5.32140 −0.988158 −0.494079 0.869417i \(-0.664495\pi\)
−0.494079 + 0.869417i \(0.664495\pi\)
\(30\) 0 0
\(31\) 2.06047 0.370071 0.185036 0.982732i \(-0.440760\pi\)
0.185036 + 0.982732i \(0.440760\pi\)
\(32\) −4.88072 −0.862798
\(33\) 0 0
\(34\) −1.27890 −0.219329
\(35\) −1.98652 −0.335783
\(36\) 0 0
\(37\) 2.29275 0.376925 0.188463 0.982080i \(-0.439650\pi\)
0.188463 + 0.982080i \(0.439650\pi\)
\(38\) −2.09896 −0.340496
\(39\) 0 0
\(40\) 3.57495 0.565249
\(41\) 3.69549 0.577139 0.288570 0.957459i \(-0.406820\pi\)
0.288570 + 0.957459i \(0.406820\pi\)
\(42\) 0 0
\(43\) 6.38263 0.973342 0.486671 0.873585i \(-0.338211\pi\)
0.486671 + 0.873585i \(0.338211\pi\)
\(44\) −0.228839 −0.0344988
\(45\) 0 0
\(46\) −1.16363 −0.171568
\(47\) 11.8866 1.73384 0.866918 0.498451i \(-0.166098\pi\)
0.866918 + 0.498451i \(0.166098\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0.502678 0.0710894
\(51\) 0 0
\(52\) −9.16281 −1.27065
\(53\) 7.84774 1.07797 0.538985 0.842316i \(-0.318808\pi\)
0.538985 + 0.842316i \(0.318808\pi\)
\(54\) 0 0
\(55\) 0.256480 0.0345838
\(56\) −1.79961 −0.240482
\(57\) 0 0
\(58\) 2.53853 0.333325
\(59\) −8.58540 −1.11772 −0.558862 0.829261i \(-0.688762\pi\)
−0.558862 + 0.829261i \(0.688762\pi\)
\(60\) 0 0
\(61\) 3.08167 0.394567 0.197284 0.980346i \(-0.436788\pi\)
0.197284 + 0.980346i \(0.436788\pi\)
\(62\) −0.982930 −0.124832
\(63\) 0 0
\(64\) −3.04445 −0.380556
\(65\) 10.2696 1.27378
\(66\) 0 0
\(67\) −5.38959 −0.658444 −0.329222 0.944253i \(-0.606786\pi\)
−0.329222 + 0.944253i \(0.606786\pi\)
\(68\) −4.75171 −0.576229
\(69\) 0 0
\(70\) 0.947652 0.113266
\(71\) −0.409103 −0.0485516 −0.0242758 0.999705i \(-0.507728\pi\)
−0.0242758 + 0.999705i \(0.507728\pi\)
\(72\) 0 0
\(73\) −9.61874 −1.12579 −0.562894 0.826529i \(-0.690312\pi\)
−0.562894 + 0.826529i \(0.690312\pi\)
\(74\) −1.09373 −0.127144
\(75\) 0 0
\(76\) −7.79862 −0.894563
\(77\) −0.129110 −0.0147135
\(78\) 0 0
\(79\) 4.71224 0.530168 0.265084 0.964225i \(-0.414600\pi\)
0.265084 + 0.964225i \(0.414600\pi\)
\(80\) 5.33654 0.596643
\(81\) 0 0
\(82\) −1.76290 −0.194680
\(83\) 7.35722 0.807560 0.403780 0.914856i \(-0.367696\pi\)
0.403780 + 0.914856i \(0.367696\pi\)
\(84\) 0 0
\(85\) 5.32565 0.577648
\(86\) −3.04478 −0.328327
\(87\) 0 0
\(88\) 0.232348 0.0247684
\(89\) −0.833032 −0.0883013 −0.0441506 0.999025i \(-0.514058\pi\)
−0.0441506 + 0.999025i \(0.514058\pi\)
\(90\) 0 0
\(91\) −5.16963 −0.541924
\(92\) −4.32342 −0.450748
\(93\) 0 0
\(94\) −5.67039 −0.584856
\(95\) 8.74059 0.896766
\(96\) 0 0
\(97\) 2.28385 0.231890 0.115945 0.993256i \(-0.463010\pi\)
0.115945 + 0.993256i \(0.463010\pi\)
\(98\) −0.477041 −0.0481885
\(99\) 0 0
\(100\) 1.86768 0.186768
\(101\) 4.38532 0.436356 0.218178 0.975909i \(-0.429989\pi\)
0.218178 + 0.975909i \(0.429989\pi\)
\(102\) 0 0
\(103\) −6.21117 −0.612005 −0.306003 0.952031i \(-0.598992\pi\)
−0.306003 + 0.952031i \(0.598992\pi\)
\(104\) 9.30329 0.912263
\(105\) 0 0
\(106\) −3.74369 −0.363620
\(107\) 7.71277 0.745621 0.372811 0.927907i \(-0.378394\pi\)
0.372811 + 0.927907i \(0.378394\pi\)
\(108\) 0 0
\(109\) −19.1046 −1.82989 −0.914943 0.403582i \(-0.867765\pi\)
−0.914943 + 0.403582i \(0.867765\pi\)
\(110\) −0.122352 −0.0116658
\(111\) 0 0
\(112\) −2.68638 −0.253839
\(113\) 9.12949 0.858830 0.429415 0.903107i \(-0.358720\pi\)
0.429415 + 0.903107i \(0.358720\pi\)
\(114\) 0 0
\(115\) 4.84564 0.451858
\(116\) 9.43181 0.875722
\(117\) 0 0
\(118\) 4.09559 0.377029
\(119\) −2.68090 −0.245757
\(120\) 0 0
\(121\) −10.9833 −0.998485
\(122\) −1.47008 −0.133095
\(123\) 0 0
\(124\) −3.65204 −0.327963
\(125\) −12.0259 −1.07563
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) 11.2138 0.991167
\(129\) 0 0
\(130\) −4.89901 −0.429671
\(131\) −6.03668 −0.527428 −0.263714 0.964601i \(-0.584947\pi\)
−0.263714 + 0.964601i \(0.584947\pi\)
\(132\) 0 0
\(133\) −4.39995 −0.381524
\(134\) 2.57106 0.222106
\(135\) 0 0
\(136\) 4.82456 0.413702
\(137\) 5.11663 0.437143 0.218572 0.975821i \(-0.429860\pi\)
0.218572 + 0.975821i \(0.429860\pi\)
\(138\) 0 0
\(139\) −10.1869 −0.864038 −0.432019 0.901865i \(-0.642199\pi\)
−0.432019 + 0.901865i \(0.642199\pi\)
\(140\) 3.52097 0.297576
\(141\) 0 0
\(142\) 0.195159 0.0163774
\(143\) 0.667453 0.0558152
\(144\) 0 0
\(145\) −10.5711 −0.877878
\(146\) 4.58853 0.379750
\(147\) 0 0
\(148\) −4.06374 −0.334037
\(149\) −12.4575 −1.02056 −0.510278 0.860009i \(-0.670458\pi\)
−0.510278 + 0.860009i \(0.670458\pi\)
\(150\) 0 0
\(151\) −13.2748 −1.08029 −0.540145 0.841572i \(-0.681630\pi\)
−0.540145 + 0.841572i \(0.681630\pi\)
\(152\) 7.91818 0.642249
\(153\) 0 0
\(154\) 0.0615910 0.00496315
\(155\) 4.09316 0.328771
\(156\) 0 0
\(157\) −6.59894 −0.526653 −0.263326 0.964707i \(-0.584820\pi\)
−0.263326 + 0.964707i \(0.584820\pi\)
\(158\) −2.24793 −0.178836
\(159\) 0 0
\(160\) −9.69565 −0.766509
\(161\) −2.43926 −0.192241
\(162\) 0 0
\(163\) 13.5374 1.06033 0.530166 0.847894i \(-0.322130\pi\)
0.530166 + 0.847894i \(0.322130\pi\)
\(164\) −6.55001 −0.511470
\(165\) 0 0
\(166\) −3.50970 −0.272405
\(167\) −16.6771 −1.29051 −0.645256 0.763966i \(-0.723249\pi\)
−0.645256 + 0.763966i \(0.723249\pi\)
\(168\) 0 0
\(169\) 13.7250 1.05577
\(170\) −2.54056 −0.194852
\(171\) 0 0
\(172\) −11.3128 −0.862591
\(173\) 0.0899016 0.00683509 0.00341754 0.999994i \(-0.498912\pi\)
0.00341754 + 0.999994i \(0.498912\pi\)
\(174\) 0 0
\(175\) 1.05374 0.0796553
\(176\) 0.346839 0.0261440
\(177\) 0 0
\(178\) 0.397391 0.0297857
\(179\) −2.08901 −0.156140 −0.0780698 0.996948i \(-0.524876\pi\)
−0.0780698 + 0.996948i \(0.524876\pi\)
\(180\) 0 0
\(181\) 9.62613 0.715505 0.357752 0.933817i \(-0.383543\pi\)
0.357752 + 0.933817i \(0.383543\pi\)
\(182\) 2.46613 0.182801
\(183\) 0 0
\(184\) 4.38971 0.323613
\(185\) 4.55459 0.334860
\(186\) 0 0
\(187\) 0.346132 0.0253117
\(188\) −21.0681 −1.53655
\(189\) 0 0
\(190\) −4.16962 −0.302496
\(191\) 8.89822 0.643853 0.321926 0.946765i \(-0.395670\pi\)
0.321926 + 0.946765i \(0.395670\pi\)
\(192\) 0 0
\(193\) −6.73789 −0.485004 −0.242502 0.970151i \(-0.577968\pi\)
−0.242502 + 0.970151i \(0.577968\pi\)
\(194\) −1.08949 −0.0782209
\(195\) 0 0
\(196\) −1.77243 −0.126602
\(197\) 18.1025 1.28975 0.644874 0.764289i \(-0.276910\pi\)
0.644874 + 0.764289i \(0.276910\pi\)
\(198\) 0 0
\(199\) 18.6429 1.32156 0.660778 0.750581i \(-0.270226\pi\)
0.660778 + 0.750581i \(0.270226\pi\)
\(200\) −1.89632 −0.134090
\(201\) 0 0
\(202\) −2.09198 −0.147191
\(203\) 5.32140 0.373489
\(204\) 0 0
\(205\) 7.34117 0.512729
\(206\) 2.96299 0.206441
\(207\) 0 0
\(208\) 13.8876 0.962929
\(209\) 0.568080 0.0392949
\(210\) 0 0
\(211\) −6.51918 −0.448799 −0.224399 0.974497i \(-0.572042\pi\)
−0.224399 + 0.974497i \(0.572042\pi\)
\(212\) −13.9096 −0.955313
\(213\) 0 0
\(214\) −3.67931 −0.251512
\(215\) 12.6792 0.864716
\(216\) 0 0
\(217\) −2.06047 −0.139874
\(218\) 9.11367 0.617256
\(219\) 0 0
\(220\) −0.454594 −0.0306487
\(221\) 13.8592 0.932273
\(222\) 0 0
\(223\) −2.81814 −0.188717 −0.0943584 0.995538i \(-0.530080\pi\)
−0.0943584 + 0.995538i \(0.530080\pi\)
\(224\) 4.88072 0.326107
\(225\) 0 0
\(226\) −4.35514 −0.289700
\(227\) −2.68150 −0.177977 −0.0889887 0.996033i \(-0.528363\pi\)
−0.0889887 + 0.996033i \(0.528363\pi\)
\(228\) 0 0
\(229\) −18.6506 −1.23246 −0.616232 0.787565i \(-0.711341\pi\)
−0.616232 + 0.787565i \(0.711341\pi\)
\(230\) −2.31157 −0.152420
\(231\) 0 0
\(232\) −9.57642 −0.628722
\(233\) 20.0572 1.31399 0.656994 0.753895i \(-0.271827\pi\)
0.656994 + 0.753895i \(0.271827\pi\)
\(234\) 0 0
\(235\) 23.6129 1.54034
\(236\) 15.2170 0.990544
\(237\) 0 0
\(238\) 1.27890 0.0828987
\(239\) 11.2371 0.726865 0.363432 0.931621i \(-0.381605\pi\)
0.363432 + 0.931621i \(0.381605\pi\)
\(240\) 0 0
\(241\) 22.6209 1.45714 0.728572 0.684970i \(-0.240185\pi\)
0.728572 + 0.684970i \(0.240185\pi\)
\(242\) 5.23950 0.336808
\(243\) 0 0
\(244\) −5.46205 −0.349672
\(245\) 1.98652 0.126914
\(246\) 0 0
\(247\) 22.7461 1.44730
\(248\) 3.70803 0.235460
\(249\) 0 0
\(250\) 5.73684 0.362830
\(251\) −16.1825 −1.02143 −0.510717 0.859749i \(-0.670620\pi\)
−0.510717 + 0.859749i \(0.670620\pi\)
\(252\) 0 0
\(253\) 0.314934 0.0197997
\(254\) 0.477041 0.0299322
\(255\) 0 0
\(256\) 0.739459 0.0462162
\(257\) −18.2153 −1.13624 −0.568119 0.822947i \(-0.692329\pi\)
−0.568119 + 0.822947i \(0.692329\pi\)
\(258\) 0 0
\(259\) −2.29275 −0.142464
\(260\) −18.2021 −1.12885
\(261\) 0 0
\(262\) 2.87975 0.177911
\(263\) 27.6166 1.70291 0.851457 0.524425i \(-0.175720\pi\)
0.851457 + 0.524425i \(0.175720\pi\)
\(264\) 0 0
\(265\) 15.5897 0.957666
\(266\) 2.09896 0.128695
\(267\) 0 0
\(268\) 9.55269 0.583523
\(269\) 1.60396 0.0977951 0.0488975 0.998804i \(-0.484429\pi\)
0.0488975 + 0.998804i \(0.484429\pi\)
\(270\) 0 0
\(271\) 6.86194 0.416833 0.208417 0.978040i \(-0.433169\pi\)
0.208417 + 0.978040i \(0.433169\pi\)
\(272\) 7.20190 0.436679
\(273\) 0 0
\(274\) −2.44084 −0.147457
\(275\) −0.136049 −0.00820406
\(276\) 0 0
\(277\) 11.6146 0.697854 0.348927 0.937150i \(-0.386546\pi\)
0.348927 + 0.937150i \(0.386546\pi\)
\(278\) 4.85955 0.291456
\(279\) 0 0
\(280\) −3.57495 −0.213644
\(281\) 22.9259 1.36764 0.683821 0.729650i \(-0.260317\pi\)
0.683821 + 0.729650i \(0.260317\pi\)
\(282\) 0 0
\(283\) −2.19161 −0.130277 −0.0651387 0.997876i \(-0.520749\pi\)
−0.0651387 + 0.997876i \(0.520749\pi\)
\(284\) 0.725107 0.0430272
\(285\) 0 0
\(286\) −0.318403 −0.0188275
\(287\) −3.69549 −0.218138
\(288\) 0 0
\(289\) −9.81279 −0.577223
\(290\) 5.04283 0.296125
\(291\) 0 0
\(292\) 17.0486 0.997691
\(293\) −11.8079 −0.689826 −0.344913 0.938635i \(-0.612092\pi\)
−0.344913 + 0.938635i \(0.612092\pi\)
\(294\) 0 0
\(295\) −17.0551 −0.992984
\(296\) 4.12604 0.239821
\(297\) 0 0
\(298\) 5.94274 0.344253
\(299\) 12.6101 0.729259
\(300\) 0 0
\(301\) −6.38263 −0.367889
\(302\) 6.33264 0.364402
\(303\) 0 0
\(304\) 11.8199 0.677920
\(305\) 6.12180 0.350533
\(306\) 0 0
\(307\) 26.2456 1.49791 0.748957 0.662619i \(-0.230555\pi\)
0.748957 + 0.662619i \(0.230555\pi\)
\(308\) 0.228839 0.0130393
\(309\) 0 0
\(310\) −1.95261 −0.110901
\(311\) 7.30728 0.414358 0.207179 0.978303i \(-0.433572\pi\)
0.207179 + 0.978303i \(0.433572\pi\)
\(312\) 0 0
\(313\) −11.1419 −0.629778 −0.314889 0.949128i \(-0.601967\pi\)
−0.314889 + 0.949128i \(0.601967\pi\)
\(314\) 3.14797 0.177650
\(315\) 0 0
\(316\) −8.35212 −0.469843
\(317\) 10.9722 0.616257 0.308129 0.951345i \(-0.400297\pi\)
0.308129 + 0.951345i \(0.400297\pi\)
\(318\) 0 0
\(319\) −0.687048 −0.0384673
\(320\) −6.04785 −0.338085
\(321\) 0 0
\(322\) 1.16363 0.0648464
\(323\) 11.7958 0.656337
\(324\) 0 0
\(325\) −5.44745 −0.302170
\(326\) −6.45791 −0.357670
\(327\) 0 0
\(328\) 6.65043 0.367209
\(329\) −11.8866 −0.655328
\(330\) 0 0
\(331\) −5.64958 −0.310529 −0.155264 0.987873i \(-0.549623\pi\)
−0.155264 + 0.987873i \(0.549623\pi\)
\(332\) −13.0402 −0.715672
\(333\) 0 0
\(334\) 7.95566 0.435314
\(335\) −10.7065 −0.584960
\(336\) 0 0
\(337\) −0.559535 −0.0304798 −0.0152399 0.999884i \(-0.504851\pi\)
−0.0152399 + 0.999884i \(0.504851\pi\)
\(338\) −6.54741 −0.356132
\(339\) 0 0
\(340\) −9.43936 −0.511921
\(341\) 0.266028 0.0144062
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 11.4862 0.619295
\(345\) 0 0
\(346\) −0.0428868 −0.00230561
\(347\) −2.34374 −0.125819 −0.0629094 0.998019i \(-0.520038\pi\)
−0.0629094 + 0.998019i \(0.520038\pi\)
\(348\) 0 0
\(349\) −0.675167 −0.0361409 −0.0180704 0.999837i \(-0.505752\pi\)
−0.0180704 + 0.999837i \(0.505752\pi\)
\(350\) −0.502678 −0.0268693
\(351\) 0 0
\(352\) −0.630152 −0.0335872
\(353\) 21.4329 1.14076 0.570379 0.821381i \(-0.306796\pi\)
0.570379 + 0.821381i \(0.306796\pi\)
\(354\) 0 0
\(355\) −0.812691 −0.0431332
\(356\) 1.47649 0.0782540
\(357\) 0 0
\(358\) 0.996542 0.0526689
\(359\) 3.26825 0.172492 0.0862459 0.996274i \(-0.472513\pi\)
0.0862459 + 0.996274i \(0.472513\pi\)
\(360\) 0 0
\(361\) 0.359596 0.0189261
\(362\) −4.59206 −0.241353
\(363\) 0 0
\(364\) 9.16281 0.480262
\(365\) −19.1078 −1.00015
\(366\) 0 0
\(367\) −7.41540 −0.387081 −0.193540 0.981092i \(-0.561997\pi\)
−0.193540 + 0.981092i \(0.561997\pi\)
\(368\) 6.55277 0.341587
\(369\) 0 0
\(370\) −2.17273 −0.112955
\(371\) −7.84774 −0.407434
\(372\) 0 0
\(373\) 3.75369 0.194358 0.0971792 0.995267i \(-0.469018\pi\)
0.0971792 + 0.995267i \(0.469018\pi\)
\(374\) −0.165119 −0.00853811
\(375\) 0 0
\(376\) 21.3912 1.10316
\(377\) −27.5096 −1.41682
\(378\) 0 0
\(379\) −5.35309 −0.274970 −0.137485 0.990504i \(-0.543902\pi\)
−0.137485 + 0.990504i \(0.543902\pi\)
\(380\) −15.4921 −0.794728
\(381\) 0 0
\(382\) −4.24482 −0.217184
\(383\) 9.90859 0.506305 0.253153 0.967426i \(-0.418533\pi\)
0.253153 + 0.967426i \(0.418533\pi\)
\(384\) 0 0
\(385\) −0.256480 −0.0130714
\(386\) 3.21425 0.163601
\(387\) 0 0
\(388\) −4.04797 −0.205505
\(389\) −26.0393 −1.32025 −0.660123 0.751157i \(-0.729496\pi\)
−0.660123 + 0.751157i \(0.729496\pi\)
\(390\) 0 0
\(391\) 6.53940 0.330712
\(392\) 1.79961 0.0908938
\(393\) 0 0
\(394\) −8.63562 −0.435056
\(395\) 9.36095 0.471000
\(396\) 0 0
\(397\) 9.39700 0.471622 0.235811 0.971799i \(-0.424225\pi\)
0.235811 + 0.971799i \(0.424225\pi\)
\(398\) −8.89341 −0.445786
\(399\) 0 0
\(400\) −2.83075 −0.141537
\(401\) 14.1417 0.706204 0.353102 0.935585i \(-0.385127\pi\)
0.353102 + 0.935585i \(0.385127\pi\)
\(402\) 0 0
\(403\) 10.6519 0.530607
\(404\) −7.77269 −0.386706
\(405\) 0 0
\(406\) −2.53853 −0.125985
\(407\) 0.296018 0.0146730
\(408\) 0 0
\(409\) −7.11432 −0.351781 −0.175890 0.984410i \(-0.556280\pi\)
−0.175890 + 0.984410i \(0.556280\pi\)
\(410\) −3.50204 −0.172953
\(411\) 0 0
\(412\) 11.0089 0.542369
\(413\) 8.58540 0.422460
\(414\) 0 0
\(415\) 14.6153 0.717435
\(416\) −25.2315 −1.23708
\(417\) 0 0
\(418\) −0.270998 −0.0132549
\(419\) 22.3886 1.09376 0.546879 0.837212i \(-0.315816\pi\)
0.546879 + 0.837212i \(0.315816\pi\)
\(420\) 0 0
\(421\) −34.5919 −1.68591 −0.842954 0.537986i \(-0.819186\pi\)
−0.842954 + 0.537986i \(0.819186\pi\)
\(422\) 3.10992 0.151388
\(423\) 0 0
\(424\) 14.1228 0.685865
\(425\) −2.82497 −0.137031
\(426\) 0 0
\(427\) −3.08167 −0.149132
\(428\) −13.6704 −0.660781
\(429\) 0 0
\(430\) −6.04851 −0.291685
\(431\) −14.9139 −0.718376 −0.359188 0.933265i \(-0.616946\pi\)
−0.359188 + 0.933265i \(0.616946\pi\)
\(432\) 0 0
\(433\) −13.1174 −0.630381 −0.315190 0.949028i \(-0.602068\pi\)
−0.315190 + 0.949028i \(0.602068\pi\)
\(434\) 0.982930 0.0471821
\(435\) 0 0
\(436\) 33.8616 1.62167
\(437\) 10.7326 0.513411
\(438\) 0 0
\(439\) −15.3284 −0.731584 −0.365792 0.930697i \(-0.619202\pi\)
−0.365792 + 0.930697i \(0.619202\pi\)
\(440\) 0.461564 0.0220042
\(441\) 0 0
\(442\) −6.61143 −0.314474
\(443\) 30.4768 1.44800 0.723999 0.689801i \(-0.242302\pi\)
0.723999 + 0.689801i \(0.242302\pi\)
\(444\) 0 0
\(445\) −1.65483 −0.0784467
\(446\) 1.34437 0.0636578
\(447\) 0 0
\(448\) 3.04445 0.143837
\(449\) −13.9932 −0.660382 −0.330191 0.943914i \(-0.607113\pi\)
−0.330191 + 0.943914i \(0.607113\pi\)
\(450\) 0 0
\(451\) 0.477127 0.0224670
\(452\) −16.1814 −0.761109
\(453\) 0 0
\(454\) 1.27919 0.0600352
\(455\) −10.2696 −0.481445
\(456\) 0 0
\(457\) 34.6734 1.62196 0.810978 0.585077i \(-0.198936\pi\)
0.810978 + 0.585077i \(0.198936\pi\)
\(458\) 8.89708 0.415733
\(459\) 0 0
\(460\) −8.58856 −0.400444
\(461\) 36.2406 1.68789 0.843945 0.536429i \(-0.180227\pi\)
0.843945 + 0.536429i \(0.180227\pi\)
\(462\) 0 0
\(463\) 22.1365 1.02877 0.514385 0.857559i \(-0.328020\pi\)
0.514385 + 0.857559i \(0.328020\pi\)
\(464\) −14.2953 −0.663642
\(465\) 0 0
\(466\) −9.56810 −0.443234
\(467\) 2.49566 0.115486 0.0577428 0.998331i \(-0.481610\pi\)
0.0577428 + 0.998331i \(0.481610\pi\)
\(468\) 0 0
\(469\) 5.38959 0.248868
\(470\) −11.2643 −0.519585
\(471\) 0 0
\(472\) −15.4503 −0.711159
\(473\) 0.824064 0.0378905
\(474\) 0 0
\(475\) −4.63641 −0.212733
\(476\) 4.75171 0.217794
\(477\) 0 0
\(478\) −5.36054 −0.245185
\(479\) 15.8007 0.721952 0.360976 0.932575i \(-0.382444\pi\)
0.360976 + 0.932575i \(0.382444\pi\)
\(480\) 0 0
\(481\) 11.8526 0.540434
\(482\) −10.7911 −0.491522
\(483\) 0 0
\(484\) 19.4672 0.884873
\(485\) 4.53692 0.206011
\(486\) 0 0
\(487\) −30.2573 −1.37109 −0.685545 0.728030i \(-0.740436\pi\)
−0.685545 + 0.728030i \(0.740436\pi\)
\(488\) 5.54579 0.251046
\(489\) 0 0
\(490\) −0.947652 −0.0428105
\(491\) −5.79434 −0.261495 −0.130747 0.991416i \(-0.541738\pi\)
−0.130747 + 0.991416i \(0.541738\pi\)
\(492\) 0 0
\(493\) −14.2661 −0.642513
\(494\) −10.8508 −0.488202
\(495\) 0 0
\(496\) 5.53520 0.248538
\(497\) 0.409103 0.0183508
\(498\) 0 0
\(499\) 33.3151 1.49139 0.745695 0.666288i \(-0.232118\pi\)
0.745695 + 0.666288i \(0.232118\pi\)
\(500\) 21.3150 0.953237
\(501\) 0 0
\(502\) 7.71974 0.344549
\(503\) −26.8954 −1.19920 −0.599602 0.800298i \(-0.704675\pi\)
−0.599602 + 0.800298i \(0.704675\pi\)
\(504\) 0 0
\(505\) 8.71153 0.387658
\(506\) −0.150237 −0.00667883
\(507\) 0 0
\(508\) 1.77243 0.0786389
\(509\) −10.0142 −0.443874 −0.221937 0.975061i \(-0.571238\pi\)
−0.221937 + 0.975061i \(0.571238\pi\)
\(510\) 0 0
\(511\) 9.61874 0.425508
\(512\) −22.7803 −1.00676
\(513\) 0 0
\(514\) 8.68944 0.383275
\(515\) −12.3386 −0.543704
\(516\) 0 0
\(517\) 1.53468 0.0674952
\(518\) 1.09373 0.0480560
\(519\) 0 0
\(520\) 18.4812 0.810453
\(521\) 33.9433 1.48708 0.743541 0.668690i \(-0.233145\pi\)
0.743541 + 0.668690i \(0.233145\pi\)
\(522\) 0 0
\(523\) 26.8700 1.17494 0.587471 0.809246i \(-0.300124\pi\)
0.587471 + 0.809246i \(0.300124\pi\)
\(524\) 10.6996 0.467415
\(525\) 0 0
\(526\) −13.1743 −0.574425
\(527\) 5.52391 0.240625
\(528\) 0 0
\(529\) −17.0500 −0.741305
\(530\) −7.43692 −0.323039
\(531\) 0 0
\(532\) 7.79862 0.338113
\(533\) 19.1043 0.827500
\(534\) 0 0
\(535\) 15.3216 0.662409
\(536\) −9.69914 −0.418939
\(537\) 0 0
\(538\) −0.765154 −0.0329881
\(539\) 0.129110 0.00556118
\(540\) 0 0
\(541\) 36.8389 1.58383 0.791914 0.610632i \(-0.209085\pi\)
0.791914 + 0.610632i \(0.209085\pi\)
\(542\) −3.27343 −0.140606
\(543\) 0 0
\(544\) −13.0847 −0.561003
\(545\) −37.9516 −1.62567
\(546\) 0 0
\(547\) −41.1644 −1.76006 −0.880031 0.474917i \(-0.842478\pi\)
−0.880031 + 0.474917i \(0.842478\pi\)
\(548\) −9.06888 −0.387403
\(549\) 0 0
\(550\) 0.0649010 0.00276739
\(551\) −23.4139 −0.997466
\(552\) 0 0
\(553\) −4.71224 −0.200385
\(554\) −5.54065 −0.235399
\(555\) 0 0
\(556\) 18.0555 0.765724
\(557\) 41.8639 1.77383 0.886915 0.461933i \(-0.152844\pi\)
0.886915 + 0.461933i \(0.152844\pi\)
\(558\) 0 0
\(559\) 32.9958 1.39557
\(560\) −5.33654 −0.225510
\(561\) 0 0
\(562\) −10.9366 −0.461332
\(563\) 11.9735 0.504624 0.252312 0.967646i \(-0.418809\pi\)
0.252312 + 0.967646i \(0.418809\pi\)
\(564\) 0 0
\(565\) 18.1359 0.762983
\(566\) 1.04549 0.0439451
\(567\) 0 0
\(568\) −0.736224 −0.0308913
\(569\) −15.6036 −0.654138 −0.327069 0.945001i \(-0.606061\pi\)
−0.327069 + 0.945001i \(0.606061\pi\)
\(570\) 0 0
\(571\) 5.18440 0.216960 0.108480 0.994099i \(-0.465402\pi\)
0.108480 + 0.994099i \(0.465402\pi\)
\(572\) −1.18301 −0.0494643
\(573\) 0 0
\(574\) 1.76290 0.0735821
\(575\) −2.57035 −0.107191
\(576\) 0 0
\(577\) 0.288300 0.0120021 0.00600105 0.999982i \(-0.498090\pi\)
0.00600105 + 0.999982i \(0.498090\pi\)
\(578\) 4.68111 0.194708
\(579\) 0 0
\(580\) 18.7365 0.777990
\(581\) −7.35722 −0.305229
\(582\) 0 0
\(583\) 1.01322 0.0419635
\(584\) −17.3099 −0.716290
\(585\) 0 0
\(586\) 5.63287 0.232692
\(587\) 1.34227 0.0554013 0.0277006 0.999616i \(-0.491181\pi\)
0.0277006 + 0.999616i \(0.491181\pi\)
\(588\) 0 0
\(589\) 9.06598 0.373557
\(590\) 8.13597 0.334952
\(591\) 0 0
\(592\) 6.15918 0.253141
\(593\) 25.2541 1.03706 0.518530 0.855060i \(-0.326480\pi\)
0.518530 + 0.855060i \(0.326480\pi\)
\(594\) 0 0
\(595\) −5.32565 −0.218330
\(596\) 22.0800 0.904434
\(597\) 0 0
\(598\) −6.01552 −0.245993
\(599\) 35.6941 1.45842 0.729211 0.684289i \(-0.239887\pi\)
0.729211 + 0.684289i \(0.239887\pi\)
\(600\) 0 0
\(601\) 1.21440 0.0495364 0.0247682 0.999693i \(-0.492115\pi\)
0.0247682 + 0.999693i \(0.492115\pi\)
\(602\) 3.04478 0.124096
\(603\) 0 0
\(604\) 23.5287 0.957369
\(605\) −21.8186 −0.887052
\(606\) 0 0
\(607\) −3.81330 −0.154777 −0.0773886 0.997001i \(-0.524658\pi\)
−0.0773886 + 0.997001i \(0.524658\pi\)
\(608\) −21.4750 −0.870925
\(609\) 0 0
\(610\) −2.92035 −0.118242
\(611\) 61.4492 2.48597
\(612\) 0 0
\(613\) −31.3128 −1.26471 −0.632357 0.774677i \(-0.717912\pi\)
−0.632357 + 0.774677i \(0.717912\pi\)
\(614\) −12.5202 −0.505275
\(615\) 0 0
\(616\) −0.232348 −0.00936156
\(617\) 18.9750 0.763904 0.381952 0.924182i \(-0.375252\pi\)
0.381952 + 0.924182i \(0.375252\pi\)
\(618\) 0 0
\(619\) −28.4920 −1.14519 −0.572596 0.819838i \(-0.694064\pi\)
−0.572596 + 0.819838i \(0.694064\pi\)
\(620\) −7.25485 −0.291362
\(621\) 0 0
\(622\) −3.48587 −0.139771
\(623\) 0.833032 0.0333747
\(624\) 0 0
\(625\) −18.6209 −0.744837
\(626\) 5.31515 0.212436
\(627\) 0 0
\(628\) 11.6962 0.466728
\(629\) 6.14662 0.245082
\(630\) 0 0
\(631\) −12.7097 −0.505966 −0.252983 0.967471i \(-0.581412\pi\)
−0.252983 + 0.967471i \(0.581412\pi\)
\(632\) 8.48017 0.337323
\(633\) 0 0
\(634\) −5.23417 −0.207875
\(635\) −1.98652 −0.0788326
\(636\) 0 0
\(637\) 5.16963 0.204828
\(638\) 0.327750 0.0129758
\(639\) 0 0
\(640\) 22.2764 0.880551
\(641\) −5.36641 −0.211961 −0.105980 0.994368i \(-0.533798\pi\)
−0.105980 + 0.994368i \(0.533798\pi\)
\(642\) 0 0
\(643\) 7.99046 0.315113 0.157557 0.987510i \(-0.449638\pi\)
0.157557 + 0.987510i \(0.449638\pi\)
\(644\) 4.32342 0.170367
\(645\) 0 0
\(646\) −5.62709 −0.221395
\(647\) 3.76942 0.148191 0.0740956 0.997251i \(-0.476393\pi\)
0.0740956 + 0.997251i \(0.476393\pi\)
\(648\) 0 0
\(649\) −1.10846 −0.0435110
\(650\) 2.59866 0.101928
\(651\) 0 0
\(652\) −23.9941 −0.939683
\(653\) 17.1656 0.671742 0.335871 0.941908i \(-0.390969\pi\)
0.335871 + 0.941908i \(0.390969\pi\)
\(654\) 0 0
\(655\) −11.9920 −0.468566
\(656\) 9.92749 0.387603
\(657\) 0 0
\(658\) 5.67039 0.221055
\(659\) −20.9072 −0.814430 −0.407215 0.913332i \(-0.633500\pi\)
−0.407215 + 0.913332i \(0.633500\pi\)
\(660\) 0 0
\(661\) −30.9980 −1.20568 −0.602841 0.797862i \(-0.705965\pi\)
−0.602841 + 0.797862i \(0.705965\pi\)
\(662\) 2.69508 0.104747
\(663\) 0 0
\(664\) 13.2401 0.513815
\(665\) −8.74059 −0.338946
\(666\) 0 0
\(667\) −12.9803 −0.502598
\(668\) 29.5590 1.14367
\(669\) 0 0
\(670\) 5.10746 0.197318
\(671\) 0.397876 0.0153598
\(672\) 0 0
\(673\) −14.7276 −0.567707 −0.283853 0.958868i \(-0.591613\pi\)
−0.283853 + 0.958868i \(0.591613\pi\)
\(674\) 0.266922 0.0102814
\(675\) 0 0
\(676\) −24.3267 −0.935642
\(677\) −35.4202 −1.36131 −0.680654 0.732605i \(-0.738304\pi\)
−0.680654 + 0.732605i \(0.738304\pi\)
\(678\) 0 0
\(679\) −2.28385 −0.0876462
\(680\) 9.58408 0.367533
\(681\) 0 0
\(682\) −0.126906 −0.00485950
\(683\) 25.0084 0.956920 0.478460 0.878109i \(-0.341195\pi\)
0.478460 + 0.878109i \(0.341195\pi\)
\(684\) 0 0
\(685\) 10.1643 0.388357
\(686\) 0.477041 0.0182135
\(687\) 0 0
\(688\) 17.1462 0.653691
\(689\) 40.5699 1.54559
\(690\) 0 0
\(691\) 33.3999 1.27059 0.635295 0.772269i \(-0.280878\pi\)
0.635295 + 0.772269i \(0.280878\pi\)
\(692\) −0.159344 −0.00605736
\(693\) 0 0
\(694\) 1.11806 0.0424411
\(695\) −20.2364 −0.767610
\(696\) 0 0
\(697\) 9.90724 0.375263
\(698\) 0.322083 0.0121910
\(699\) 0 0
\(700\) −1.86768 −0.0705918
\(701\) −3.96654 −0.149814 −0.0749070 0.997191i \(-0.523866\pi\)
−0.0749070 + 0.997191i \(0.523866\pi\)
\(702\) 0 0
\(703\) 10.0880 0.380475
\(704\) −0.393070 −0.0148144
\(705\) 0 0
\(706\) −10.2244 −0.384800
\(707\) −4.38532 −0.164927
\(708\) 0 0
\(709\) −28.7457 −1.07957 −0.539783 0.841804i \(-0.681494\pi\)
−0.539783 + 0.841804i \(0.681494\pi\)
\(710\) 0.387687 0.0145496
\(711\) 0 0
\(712\) −1.49913 −0.0561823
\(713\) 5.02602 0.188226
\(714\) 0 0
\(715\) 1.32591 0.0495861
\(716\) 3.70262 0.138373
\(717\) 0 0
\(718\) −1.55909 −0.0581848
\(719\) −25.8862 −0.965391 −0.482696 0.875788i \(-0.660342\pi\)
−0.482696 + 0.875788i \(0.660342\pi\)
\(720\) 0 0
\(721\) 6.21117 0.231316
\(722\) −0.171542 −0.00638413
\(723\) 0 0
\(724\) −17.0617 −0.634091
\(725\) 5.60737 0.208253
\(726\) 0 0
\(727\) 34.6071 1.28350 0.641752 0.766912i \(-0.278208\pi\)
0.641752 + 0.766912i \(0.278208\pi\)
\(728\) −9.30329 −0.344803
\(729\) 0 0
\(730\) 9.11521 0.337369
\(731\) 17.1112 0.632880
\(732\) 0 0
\(733\) −31.4119 −1.16023 −0.580113 0.814536i \(-0.696992\pi\)
−0.580113 + 0.814536i \(0.696992\pi\)
\(734\) 3.53745 0.130570
\(735\) 0 0
\(736\) −11.9054 −0.438837
\(737\) −0.695853 −0.0256321
\(738\) 0 0
\(739\) 43.6874 1.60707 0.803534 0.595259i \(-0.202951\pi\)
0.803534 + 0.595259i \(0.202951\pi\)
\(740\) −8.07269 −0.296758
\(741\) 0 0
\(742\) 3.74369 0.137435
\(743\) 35.4888 1.30196 0.650978 0.759097i \(-0.274359\pi\)
0.650978 + 0.759097i \(0.274359\pi\)
\(744\) 0 0
\(745\) −24.7470 −0.906661
\(746\) −1.79066 −0.0655608
\(747\) 0 0
\(748\) −0.613495 −0.0224316
\(749\) −7.71277 −0.281818
\(750\) 0 0
\(751\) 41.8825 1.52831 0.764157 0.645030i \(-0.223155\pi\)
0.764157 + 0.645030i \(0.223155\pi\)
\(752\) 31.9318 1.16443
\(753\) 0 0
\(754\) 13.1232 0.477920
\(755\) −26.3707 −0.959727
\(756\) 0 0
\(757\) 28.1857 1.02443 0.512213 0.858858i \(-0.328826\pi\)
0.512213 + 0.858858i \(0.328826\pi\)
\(758\) 2.55364 0.0927525
\(759\) 0 0
\(760\) 15.7296 0.570573
\(761\) −21.1198 −0.765591 −0.382796 0.923833i \(-0.625039\pi\)
−0.382796 + 0.923833i \(0.625039\pi\)
\(762\) 0 0
\(763\) 19.1046 0.691632
\(764\) −15.7715 −0.570593
\(765\) 0 0
\(766\) −4.72681 −0.170786
\(767\) −44.3833 −1.60259
\(768\) 0 0
\(769\) −41.2389 −1.48711 −0.743556 0.668673i \(-0.766862\pi\)
−0.743556 + 0.668673i \(0.766862\pi\)
\(770\) 0.122352 0.00440925
\(771\) 0 0
\(772\) 11.9425 0.429818
\(773\) −10.8885 −0.391631 −0.195815 0.980641i \(-0.562735\pi\)
−0.195815 + 0.980641i \(0.562735\pi\)
\(774\) 0 0
\(775\) −2.17120 −0.0779919
\(776\) 4.11003 0.147542
\(777\) 0 0
\(778\) 12.4218 0.445345
\(779\) 16.2600 0.582575
\(780\) 0 0
\(781\) −0.0528195 −0.00189003
\(782\) −3.11957 −0.111555
\(783\) 0 0
\(784\) 2.68638 0.0959420
\(785\) −13.1089 −0.467877
\(786\) 0 0
\(787\) 50.3691 1.79546 0.897732 0.440542i \(-0.145214\pi\)
0.897732 + 0.440542i \(0.145214\pi\)
\(788\) −32.0854 −1.14299
\(789\) 0 0
\(790\) −4.46556 −0.158877
\(791\) −9.12949 −0.324607
\(792\) 0 0
\(793\) 15.9311 0.565729
\(794\) −4.48276 −0.159087
\(795\) 0 0
\(796\) −33.0432 −1.17118
\(797\) −19.5899 −0.693909 −0.346954 0.937882i \(-0.612784\pi\)
−0.346954 + 0.937882i \(0.612784\pi\)
\(798\) 0 0
\(799\) 31.8667 1.12736
\(800\) 5.14302 0.181833
\(801\) 0 0
\(802\) −6.74618 −0.238216
\(803\) −1.24188 −0.0438250
\(804\) 0 0
\(805\) −4.84564 −0.170786
\(806\) −5.08138 −0.178984
\(807\) 0 0
\(808\) 7.89185 0.277634
\(809\) −3.78018 −0.132904 −0.0664520 0.997790i \(-0.521168\pi\)
−0.0664520 + 0.997790i \(0.521168\pi\)
\(810\) 0 0
\(811\) −5.20558 −0.182793 −0.0913963 0.995815i \(-0.529133\pi\)
−0.0913963 + 0.995815i \(0.529133\pi\)
\(812\) −9.43181 −0.330992
\(813\) 0 0
\(814\) −0.141213 −0.00494950
\(815\) 26.8923 0.941997
\(816\) 0 0
\(817\) 28.0833 0.982510
\(818\) 3.39383 0.118662
\(819\) 0 0
\(820\) −13.0117 −0.454389
\(821\) 9.88339 0.344933 0.172466 0.985015i \(-0.444826\pi\)
0.172466 + 0.985015i \(0.444826\pi\)
\(822\) 0 0
\(823\) 36.4906 1.27198 0.635991 0.771697i \(-0.280592\pi\)
0.635991 + 0.771697i \(0.280592\pi\)
\(824\) −11.1777 −0.389392
\(825\) 0 0
\(826\) −4.09559 −0.142504
\(827\) 11.6976 0.406765 0.203383 0.979099i \(-0.434806\pi\)
0.203383 + 0.979099i \(0.434806\pi\)
\(828\) 0 0
\(829\) 12.0192 0.417443 0.208722 0.977975i \(-0.433070\pi\)
0.208722 + 0.977975i \(0.433070\pi\)
\(830\) −6.97208 −0.242005
\(831\) 0 0
\(832\) −15.7387 −0.545640
\(833\) 2.68090 0.0928876
\(834\) 0 0
\(835\) −33.1294 −1.14649
\(836\) −1.00688 −0.0348238
\(837\) 0 0
\(838\) −10.6803 −0.368945
\(839\) −15.6959 −0.541883 −0.270941 0.962596i \(-0.587335\pi\)
−0.270941 + 0.962596i \(0.587335\pi\)
\(840\) 0 0
\(841\) −0.682742 −0.0235428
\(842\) 16.5018 0.568689
\(843\) 0 0
\(844\) 11.5548 0.397733
\(845\) 27.2651 0.937947
\(846\) 0 0
\(847\) 10.9833 0.377392
\(848\) 21.0820 0.723958
\(849\) 0 0
\(850\) 1.34763 0.0462233
\(851\) 5.59261 0.191712
\(852\) 0 0
\(853\) −56.3980 −1.93103 −0.965515 0.260346i \(-0.916163\pi\)
−0.965515 + 0.260346i \(0.916163\pi\)
\(854\) 1.47008 0.0503052
\(855\) 0 0
\(856\) 13.8799 0.474407
\(857\) 4.53943 0.155064 0.0775320 0.996990i \(-0.475296\pi\)
0.0775320 + 0.996990i \(0.475296\pi\)
\(858\) 0 0
\(859\) −20.6839 −0.705726 −0.352863 0.935675i \(-0.614792\pi\)
−0.352863 + 0.935675i \(0.614792\pi\)
\(860\) −22.4730 −0.766325
\(861\) 0 0
\(862\) 7.11453 0.242322
\(863\) 3.53249 0.120247 0.0601237 0.998191i \(-0.480850\pi\)
0.0601237 + 0.998191i \(0.480850\pi\)
\(864\) 0 0
\(865\) 0.178591 0.00607228
\(866\) 6.25753 0.212640
\(867\) 0 0
\(868\) 3.65204 0.123958
\(869\) 0.608399 0.0206385
\(870\) 0 0
\(871\) −27.8622 −0.944074
\(872\) −34.3807 −1.16428
\(873\) 0 0
\(874\) −5.11991 −0.173184
\(875\) 12.0259 0.406549
\(876\) 0 0
\(877\) 47.7631 1.61285 0.806423 0.591340i \(-0.201401\pi\)
0.806423 + 0.591340i \(0.201401\pi\)
\(878\) 7.31227 0.246777
\(879\) 0 0
\(880\) 0.689003 0.0232263
\(881\) −9.77071 −0.329184 −0.164592 0.986362i \(-0.552631\pi\)
−0.164592 + 0.986362i \(0.552631\pi\)
\(882\) 0 0
\(883\) −16.4589 −0.553886 −0.276943 0.960886i \(-0.589321\pi\)
−0.276943 + 0.960886i \(0.589321\pi\)
\(884\) −24.5645 −0.826195
\(885\) 0 0
\(886\) −14.5387 −0.488437
\(887\) −16.5289 −0.554985 −0.277493 0.960728i \(-0.589503\pi\)
−0.277493 + 0.960728i \(0.589503\pi\)
\(888\) 0 0
\(889\) 1.00000 0.0335389
\(890\) 0.789425 0.0264616
\(891\) 0 0
\(892\) 4.99497 0.167244
\(893\) 52.3004 1.75017
\(894\) 0 0
\(895\) −4.14985 −0.138714
\(896\) −11.2138 −0.374626
\(897\) 0 0
\(898\) 6.67535 0.222759
\(899\) −10.9646 −0.365689
\(900\) 0 0
\(901\) 21.0390 0.700910
\(902\) −0.227609 −0.00757856
\(903\) 0 0
\(904\) 16.4295 0.546436
\(905\) 19.1225 0.635653
\(906\) 0 0
\(907\) 5.12502 0.170173 0.0850867 0.996374i \(-0.472883\pi\)
0.0850867 + 0.996374i \(0.472883\pi\)
\(908\) 4.75277 0.157726
\(909\) 0 0
\(910\) 4.89901 0.162400
\(911\) 49.3661 1.63557 0.817787 0.575521i \(-0.195201\pi\)
0.817787 + 0.575521i \(0.195201\pi\)
\(912\) 0 0
\(913\) 0.949894 0.0314369
\(914\) −16.5407 −0.547117
\(915\) 0 0
\(916\) 33.0568 1.09223
\(917\) 6.03668 0.199349
\(918\) 0 0
\(919\) 20.9866 0.692283 0.346141 0.938182i \(-0.387492\pi\)
0.346141 + 0.938182i \(0.387492\pi\)
\(920\) 8.72024 0.287498
\(921\) 0 0
\(922\) −17.2882 −0.569358
\(923\) −2.11491 −0.0696131
\(924\) 0 0
\(925\) −2.41596 −0.0794363
\(926\) −10.5600 −0.347024
\(927\) 0 0
\(928\) 25.9723 0.852581
\(929\) 42.3000 1.38782 0.693910 0.720062i \(-0.255887\pi\)
0.693910 + 0.720062i \(0.255887\pi\)
\(930\) 0 0
\(931\) 4.39995 0.144203
\(932\) −35.5500 −1.16448
\(933\) 0 0
\(934\) −1.19054 −0.0389555
\(935\) 0.687597 0.0224868
\(936\) 0 0
\(937\) 32.8141 1.07199 0.535996 0.844221i \(-0.319936\pi\)
0.535996 + 0.844221i \(0.319936\pi\)
\(938\) −2.57106 −0.0839480
\(939\) 0 0
\(940\) −41.8523 −1.36507
\(941\) −13.6694 −0.445609 −0.222805 0.974863i \(-0.571521\pi\)
−0.222805 + 0.974863i \(0.571521\pi\)
\(942\) 0 0
\(943\) 9.01427 0.293545
\(944\) −23.0636 −0.750656
\(945\) 0 0
\(946\) −0.393113 −0.0127812
\(947\) 38.3611 1.24657 0.623283 0.781996i \(-0.285798\pi\)
0.623283 + 0.781996i \(0.285798\pi\)
\(948\) 0 0
\(949\) −49.7253 −1.61415
\(950\) 2.21176 0.0717590
\(951\) 0 0
\(952\) −4.82456 −0.156365
\(953\) −26.1473 −0.846993 −0.423497 0.905898i \(-0.639197\pi\)
−0.423497 + 0.905898i \(0.639197\pi\)
\(954\) 0 0
\(955\) 17.6765 0.571998
\(956\) −19.9169 −0.644159
\(957\) 0 0
\(958\) −7.53759 −0.243528
\(959\) −5.11663 −0.165225
\(960\) 0 0
\(961\) −26.7545 −0.863047
\(962\) −5.65420 −0.182299
\(963\) 0 0
\(964\) −40.0941 −1.29134
\(965\) −13.3850 −0.430877
\(966\) 0 0
\(967\) −0.229274 −0.00737297 −0.00368648 0.999993i \(-0.501173\pi\)
−0.00368648 + 0.999993i \(0.501173\pi\)
\(968\) −19.7657 −0.635293
\(969\) 0 0
\(970\) −2.16430 −0.0694914
\(971\) 35.4274 1.13692 0.568460 0.822711i \(-0.307539\pi\)
0.568460 + 0.822711i \(0.307539\pi\)
\(972\) 0 0
\(973\) 10.1869 0.326576
\(974\) 14.4340 0.462495
\(975\) 0 0
\(976\) 8.27853 0.264989
\(977\) −45.0167 −1.44021 −0.720105 0.693865i \(-0.755907\pi\)
−0.720105 + 0.693865i \(0.755907\pi\)
\(978\) 0 0
\(979\) −0.107553 −0.00343741
\(980\) −3.52097 −0.112473
\(981\) 0 0
\(982\) 2.76414 0.0882072
\(983\) 36.6478 1.16888 0.584441 0.811436i \(-0.301314\pi\)
0.584441 + 0.811436i \(0.301314\pi\)
\(984\) 0 0
\(985\) 35.9609 1.14581
\(986\) 6.80553 0.216732
\(987\) 0 0
\(988\) −40.3159 −1.28262
\(989\) 15.5689 0.495062
\(990\) 0 0
\(991\) 16.3587 0.519650 0.259825 0.965656i \(-0.416335\pi\)
0.259825 + 0.965656i \(0.416335\pi\)
\(992\) −10.0566 −0.319297
\(993\) 0 0
\(994\) −0.195159 −0.00619007
\(995\) 37.0344 1.17407
\(996\) 0 0
\(997\) −33.2285 −1.05236 −0.526178 0.850374i \(-0.676376\pi\)
−0.526178 + 0.850374i \(0.676376\pi\)
\(998\) −15.8927 −0.503074
\(999\) 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.p.1.6 14
3.2 odd 2 2667.2.a.m.1.9 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2667.2.a.m.1.9 14 3.2 odd 2
8001.2.a.p.1.6 14 1.1 even 1 trivial