Properties

Label 8008.2.a.x.1.4
Level $8008$
Weight $2$
Character 8008.1
Self dual yes
Analytic conductor $63.944$
Analytic rank $0$
Dimension $12$
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] = [8008,2,Mod(1,8008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8008, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("8008.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8008.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: \(63.9442019386\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 17 x^{10} + 79 x^{9} + 80 x^{8} - 536 x^{7} - 4 x^{6} + 1484 x^{5} - 682 x^{4} - 1431 x^{3} + 1069 x^{2} - 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.54910\) of defining polynomial
Character \(\chi\) \(=\) 8008.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.54910 q^{3} -3.31877 q^{5} +1.00000 q^{7} -0.600281 q^{9} +O(q^{10})\) \(q-1.54910 q^{3} -3.31877 q^{5} +1.00000 q^{7} -0.600281 q^{9} +1.00000 q^{11} +1.00000 q^{13} +5.14111 q^{15} +6.21891 q^{17} -1.02163 q^{19} -1.54910 q^{21} +6.37831 q^{23} +6.01423 q^{25} +5.57720 q^{27} -7.00023 q^{29} +1.53675 q^{31} -1.54910 q^{33} -3.31877 q^{35} +0.743700 q^{37} -1.54910 q^{39} -0.0378083 q^{41} -1.82955 q^{43} +1.99219 q^{45} -1.30648 q^{47} +1.00000 q^{49} -9.63373 q^{51} -7.95351 q^{53} -3.31877 q^{55} +1.58262 q^{57} +2.85219 q^{59} +2.21055 q^{61} -0.600281 q^{63} -3.31877 q^{65} -9.93967 q^{67} -9.88066 q^{69} -0.309613 q^{71} -2.99410 q^{73} -9.31666 q^{75} +1.00000 q^{77} +13.6124 q^{79} -6.83882 q^{81} -11.3950 q^{83} -20.6391 q^{85} +10.8441 q^{87} +1.23084 q^{89} +1.00000 q^{91} -2.38058 q^{93} +3.39057 q^{95} -6.03876 q^{97} -0.600281 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9} + 12 q^{11} + 12 q^{13} - 3 q^{15} + 16 q^{17} - 2 q^{19} + 4 q^{21} + 9 q^{23} + 14 q^{25} + 7 q^{27} + 15 q^{29} + 10 q^{31} + 4 q^{33} + 6 q^{35} + 18 q^{37} + 4 q^{39} + 24 q^{41} + 15 q^{45} + 5 q^{47} + 12 q^{49} + 4 q^{51} + 15 q^{53} + 6 q^{55} - 4 q^{57} + 15 q^{59} + 17 q^{61} + 14 q^{63} + 6 q^{65} - 7 q^{67} + 9 q^{71} + 32 q^{73} - 8 q^{75} + 12 q^{77} + 20 q^{79} - 4 q^{81} - 5 q^{83} + 25 q^{85} + 19 q^{87} + 16 q^{89} + 12 q^{91} + 21 q^{93} + 8 q^{95} + 10 q^{97} + 14 q^{99}+O(q^{100}) \) 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 0
\(3\) −1.54910 −0.894375 −0.447187 0.894440i \(-0.647574\pi\)
−0.447187 + 0.894440i \(0.647574\pi\)
\(4\) 0 0
\(5\) −3.31877 −1.48420 −0.742099 0.670290i \(-0.766170\pi\)
−0.742099 + 0.670290i \(0.766170\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −0.600281 −0.200094
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 5.14111 1.32743
\(16\) 0 0
\(17\) 6.21891 1.50831 0.754153 0.656698i \(-0.228048\pi\)
0.754153 + 0.656698i \(0.228048\pi\)
\(18\) 0 0
\(19\) −1.02163 −0.234379 −0.117190 0.993110i \(-0.537388\pi\)
−0.117190 + 0.993110i \(0.537388\pi\)
\(20\) 0 0
\(21\) −1.54910 −0.338042
\(22\) 0 0
\(23\) 6.37831 1.32997 0.664985 0.746857i \(-0.268438\pi\)
0.664985 + 0.746857i \(0.268438\pi\)
\(24\) 0 0
\(25\) 6.01423 1.20285
\(26\) 0 0
\(27\) 5.57720 1.07333
\(28\) 0 0
\(29\) −7.00023 −1.29991 −0.649955 0.759973i \(-0.725212\pi\)
−0.649955 + 0.759973i \(0.725212\pi\)
\(30\) 0 0
\(31\) 1.53675 0.276008 0.138004 0.990432i \(-0.455931\pi\)
0.138004 + 0.990432i \(0.455931\pi\)
\(32\) 0 0
\(33\) −1.54910 −0.269664
\(34\) 0 0
\(35\) −3.31877 −0.560974
\(36\) 0 0
\(37\) 0.743700 0.122264 0.0611318 0.998130i \(-0.480529\pi\)
0.0611318 + 0.998130i \(0.480529\pi\)
\(38\) 0 0
\(39\) −1.54910 −0.248055
\(40\) 0 0
\(41\) −0.0378083 −0.00590466 −0.00295233 0.999996i \(-0.500940\pi\)
−0.00295233 + 0.999996i \(0.500940\pi\)
\(42\) 0 0
\(43\) −1.82955 −0.279004 −0.139502 0.990222i \(-0.544550\pi\)
−0.139502 + 0.990222i \(0.544550\pi\)
\(44\) 0 0
\(45\) 1.99219 0.296979
\(46\) 0 0
\(47\) −1.30648 −0.190570 −0.0952849 0.995450i \(-0.530376\pi\)
−0.0952849 + 0.995450i \(0.530376\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −9.63373 −1.34899
\(52\) 0 0
\(53\) −7.95351 −1.09250 −0.546250 0.837622i \(-0.683945\pi\)
−0.546250 + 0.837622i \(0.683945\pi\)
\(54\) 0 0
\(55\) −3.31877 −0.447503
\(56\) 0 0
\(57\) 1.58262 0.209623
\(58\) 0 0
\(59\) 2.85219 0.371323 0.185661 0.982614i \(-0.440557\pi\)
0.185661 + 0.982614i \(0.440557\pi\)
\(60\) 0 0
\(61\) 2.21055 0.283031 0.141516 0.989936i \(-0.454802\pi\)
0.141516 + 0.989936i \(0.454802\pi\)
\(62\) 0 0
\(63\) −0.600281 −0.0756282
\(64\) 0 0
\(65\) −3.31877 −0.411643
\(66\) 0 0
\(67\) −9.93967 −1.21432 −0.607162 0.794578i \(-0.707692\pi\)
−0.607162 + 0.794578i \(0.707692\pi\)
\(68\) 0 0
\(69\) −9.88066 −1.18949
\(70\) 0 0
\(71\) −0.309613 −0.0367443 −0.0183722 0.999831i \(-0.505848\pi\)
−0.0183722 + 0.999831i \(0.505848\pi\)
\(72\) 0 0
\(73\) −2.99410 −0.350433 −0.175216 0.984530i \(-0.556063\pi\)
−0.175216 + 0.984530i \(0.556063\pi\)
\(74\) 0 0
\(75\) −9.31666 −1.07580
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 13.6124 1.53152 0.765759 0.643127i \(-0.222363\pi\)
0.765759 + 0.643127i \(0.222363\pi\)
\(80\) 0 0
\(81\) −6.83882 −0.759869
\(82\) 0 0
\(83\) −11.3950 −1.25076 −0.625379 0.780321i \(-0.715056\pi\)
−0.625379 + 0.780321i \(0.715056\pi\)
\(84\) 0 0
\(85\) −20.6391 −2.23863
\(86\) 0 0
\(87\) 10.8441 1.16261
\(88\) 0 0
\(89\) 1.23084 0.130469 0.0652346 0.997870i \(-0.479220\pi\)
0.0652346 + 0.997870i \(0.479220\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) −2.38058 −0.246855
\(94\) 0 0
\(95\) 3.39057 0.347865
\(96\) 0 0
\(97\) −6.03876 −0.613143 −0.306571 0.951848i \(-0.599182\pi\)
−0.306571 + 0.951848i \(0.599182\pi\)
\(98\) 0 0
\(99\) −0.600281 −0.0603305
\(100\) 0 0
\(101\) 4.62869 0.460572 0.230286 0.973123i \(-0.426034\pi\)
0.230286 + 0.973123i \(0.426034\pi\)
\(102\) 0 0
\(103\) −8.85531 −0.872540 −0.436270 0.899816i \(-0.643701\pi\)
−0.436270 + 0.899816i \(0.643701\pi\)
\(104\) 0 0
\(105\) 5.14111 0.501721
\(106\) 0 0
\(107\) 15.6277 1.51079 0.755393 0.655272i \(-0.227446\pi\)
0.755393 + 0.655272i \(0.227446\pi\)
\(108\) 0 0
\(109\) −8.98139 −0.860261 −0.430130 0.902767i \(-0.641532\pi\)
−0.430130 + 0.902767i \(0.641532\pi\)
\(110\) 0 0
\(111\) −1.15207 −0.109349
\(112\) 0 0
\(113\) −8.54415 −0.803766 −0.401883 0.915691i \(-0.631644\pi\)
−0.401883 + 0.915691i \(0.631644\pi\)
\(114\) 0 0
\(115\) −21.1681 −1.97394
\(116\) 0 0
\(117\) −0.600281 −0.0554960
\(118\) 0 0
\(119\) 6.21891 0.570086
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0.0585689 0.00528098
\(124\) 0 0
\(125\) −3.36599 −0.301063
\(126\) 0 0
\(127\) −6.68676 −0.593354 −0.296677 0.954978i \(-0.595878\pi\)
−0.296677 + 0.954978i \(0.595878\pi\)
\(128\) 0 0
\(129\) 2.83417 0.249534
\(130\) 0 0
\(131\) 9.15291 0.799694 0.399847 0.916582i \(-0.369063\pi\)
0.399847 + 0.916582i \(0.369063\pi\)
\(132\) 0 0
\(133\) −1.02163 −0.0885870
\(134\) 0 0
\(135\) −18.5095 −1.59304
\(136\) 0 0
\(137\) 2.97359 0.254051 0.127026 0.991899i \(-0.459457\pi\)
0.127026 + 0.991899i \(0.459457\pi\)
\(138\) 0 0
\(139\) −10.8417 −0.919582 −0.459791 0.888027i \(-0.652076\pi\)
−0.459791 + 0.888027i \(0.652076\pi\)
\(140\) 0 0
\(141\) 2.02387 0.170441
\(142\) 0 0
\(143\) 1.00000 0.0836242
\(144\) 0 0
\(145\) 23.2321 1.92932
\(146\) 0 0
\(147\) −1.54910 −0.127768
\(148\) 0 0
\(149\) 14.2660 1.16872 0.584358 0.811496i \(-0.301346\pi\)
0.584358 + 0.811496i \(0.301346\pi\)
\(150\) 0 0
\(151\) 8.49421 0.691249 0.345625 0.938373i \(-0.387667\pi\)
0.345625 + 0.938373i \(0.387667\pi\)
\(152\) 0 0
\(153\) −3.73309 −0.301802
\(154\) 0 0
\(155\) −5.10012 −0.409651
\(156\) 0 0
\(157\) 13.2953 1.06108 0.530541 0.847659i \(-0.321989\pi\)
0.530541 + 0.847659i \(0.321989\pi\)
\(158\) 0 0
\(159\) 12.3208 0.977104
\(160\) 0 0
\(161\) 6.37831 0.502682
\(162\) 0 0
\(163\) −15.5141 −1.21516 −0.607580 0.794259i \(-0.707860\pi\)
−0.607580 + 0.794259i \(0.707860\pi\)
\(164\) 0 0
\(165\) 5.14111 0.400235
\(166\) 0 0
\(167\) 4.21029 0.325802 0.162901 0.986642i \(-0.447915\pi\)
0.162901 + 0.986642i \(0.447915\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0.613267 0.0468977
\(172\) 0 0
\(173\) 7.86188 0.597728 0.298864 0.954296i \(-0.403392\pi\)
0.298864 + 0.954296i \(0.403392\pi\)
\(174\) 0 0
\(175\) 6.01423 0.454633
\(176\) 0 0
\(177\) −4.41833 −0.332102
\(178\) 0 0
\(179\) −22.0749 −1.64995 −0.824976 0.565168i \(-0.808811\pi\)
−0.824976 + 0.565168i \(0.808811\pi\)
\(180\) 0 0
\(181\) 1.90681 0.141732 0.0708662 0.997486i \(-0.477424\pi\)
0.0708662 + 0.997486i \(0.477424\pi\)
\(182\) 0 0
\(183\) −3.42436 −0.253136
\(184\) 0 0
\(185\) −2.46817 −0.181463
\(186\) 0 0
\(187\) 6.21891 0.454772
\(188\) 0 0
\(189\) 5.57720 0.405682
\(190\) 0 0
\(191\) 5.57689 0.403530 0.201765 0.979434i \(-0.435332\pi\)
0.201765 + 0.979434i \(0.435332\pi\)
\(192\) 0 0
\(193\) 0.148948 0.0107215 0.00536076 0.999986i \(-0.498294\pi\)
0.00536076 + 0.999986i \(0.498294\pi\)
\(194\) 0 0
\(195\) 5.14111 0.368163
\(196\) 0 0
\(197\) 6.60967 0.470919 0.235460 0.971884i \(-0.424340\pi\)
0.235460 + 0.971884i \(0.424340\pi\)
\(198\) 0 0
\(199\) −9.89725 −0.701598 −0.350799 0.936451i \(-0.614090\pi\)
−0.350799 + 0.936451i \(0.614090\pi\)
\(200\) 0 0
\(201\) 15.3976 1.08606
\(202\) 0 0
\(203\) −7.00023 −0.491320
\(204\) 0 0
\(205\) 0.125477 0.00876370
\(206\) 0 0
\(207\) −3.82878 −0.266118
\(208\) 0 0
\(209\) −1.02163 −0.0706679
\(210\) 0 0
\(211\) 9.30926 0.640876 0.320438 0.947269i \(-0.396170\pi\)
0.320438 + 0.947269i \(0.396170\pi\)
\(212\) 0 0
\(213\) 0.479623 0.0328632
\(214\) 0 0
\(215\) 6.07186 0.414098
\(216\) 0 0
\(217\) 1.53675 0.104321
\(218\) 0 0
\(219\) 4.63817 0.313418
\(220\) 0 0
\(221\) 6.21891 0.418329
\(222\) 0 0
\(223\) 24.4444 1.63692 0.818459 0.574564i \(-0.194829\pi\)
0.818459 + 0.574564i \(0.194829\pi\)
\(224\) 0 0
\(225\) −3.61022 −0.240682
\(226\) 0 0
\(227\) 12.8024 0.849727 0.424863 0.905257i \(-0.360322\pi\)
0.424863 + 0.905257i \(0.360322\pi\)
\(228\) 0 0
\(229\) −9.37385 −0.619442 −0.309721 0.950828i \(-0.600236\pi\)
−0.309721 + 0.950828i \(0.600236\pi\)
\(230\) 0 0
\(231\) −1.54910 −0.101923
\(232\) 0 0
\(233\) 6.72252 0.440407 0.220203 0.975454i \(-0.429328\pi\)
0.220203 + 0.975454i \(0.429328\pi\)
\(234\) 0 0
\(235\) 4.33591 0.282843
\(236\) 0 0
\(237\) −21.0871 −1.36975
\(238\) 0 0
\(239\) 11.6728 0.755049 0.377524 0.926000i \(-0.376775\pi\)
0.377524 + 0.926000i \(0.376775\pi\)
\(240\) 0 0
\(241\) 5.42583 0.349508 0.174754 0.984612i \(-0.444087\pi\)
0.174754 + 0.984612i \(0.444087\pi\)
\(242\) 0 0
\(243\) −6.13758 −0.393726
\(244\) 0 0
\(245\) −3.31877 −0.212028
\(246\) 0 0
\(247\) −1.02163 −0.0650051
\(248\) 0 0
\(249\) 17.6520 1.11865
\(250\) 0 0
\(251\) 28.5344 1.80108 0.900539 0.434775i \(-0.143172\pi\)
0.900539 + 0.434775i \(0.143172\pi\)
\(252\) 0 0
\(253\) 6.37831 0.401001
\(254\) 0 0
\(255\) 31.9721 2.00217
\(256\) 0 0
\(257\) 16.3385 1.01916 0.509582 0.860422i \(-0.329800\pi\)
0.509582 + 0.860422i \(0.329800\pi\)
\(258\) 0 0
\(259\) 0.743700 0.0462113
\(260\) 0 0
\(261\) 4.20210 0.260104
\(262\) 0 0
\(263\) −10.5084 −0.647977 −0.323989 0.946061i \(-0.605024\pi\)
−0.323989 + 0.946061i \(0.605024\pi\)
\(264\) 0 0
\(265\) 26.3959 1.62149
\(266\) 0 0
\(267\) −1.90670 −0.116688
\(268\) 0 0
\(269\) 26.8012 1.63410 0.817048 0.576570i \(-0.195609\pi\)
0.817048 + 0.576570i \(0.195609\pi\)
\(270\) 0 0
\(271\) −7.10670 −0.431701 −0.215851 0.976426i \(-0.569252\pi\)
−0.215851 + 0.976426i \(0.569252\pi\)
\(272\) 0 0
\(273\) −1.54910 −0.0937560
\(274\) 0 0
\(275\) 6.01423 0.362672
\(276\) 0 0
\(277\) −8.13705 −0.488908 −0.244454 0.969661i \(-0.578609\pi\)
−0.244454 + 0.969661i \(0.578609\pi\)
\(278\) 0 0
\(279\) −0.922481 −0.0552275
\(280\) 0 0
\(281\) −30.9310 −1.84519 −0.922594 0.385771i \(-0.873935\pi\)
−0.922594 + 0.385771i \(0.873935\pi\)
\(282\) 0 0
\(283\) −17.2695 −1.02657 −0.513284 0.858219i \(-0.671571\pi\)
−0.513284 + 0.858219i \(0.671571\pi\)
\(284\) 0 0
\(285\) −5.25234 −0.311122
\(286\) 0 0
\(287\) −0.0378083 −0.00223175
\(288\) 0 0
\(289\) 21.6748 1.27499
\(290\) 0 0
\(291\) 9.35466 0.548380
\(292\) 0 0
\(293\) −7.47772 −0.436853 −0.218427 0.975853i \(-0.570092\pi\)
−0.218427 + 0.975853i \(0.570092\pi\)
\(294\) 0 0
\(295\) −9.46574 −0.551117
\(296\) 0 0
\(297\) 5.57720 0.323622
\(298\) 0 0
\(299\) 6.37831 0.368867
\(300\) 0 0
\(301\) −1.82955 −0.105454
\(302\) 0 0
\(303\) −7.17032 −0.411924
\(304\) 0 0
\(305\) −7.33629 −0.420075
\(306\) 0 0
\(307\) −15.4902 −0.884070 −0.442035 0.896998i \(-0.645743\pi\)
−0.442035 + 0.896998i \(0.645743\pi\)
\(308\) 0 0
\(309\) 13.7178 0.780377
\(310\) 0 0
\(311\) 15.4377 0.875393 0.437696 0.899123i \(-0.355795\pi\)
0.437696 + 0.899123i \(0.355795\pi\)
\(312\) 0 0
\(313\) 35.0431 1.98075 0.990377 0.138395i \(-0.0441943\pi\)
0.990377 + 0.138395i \(0.0441943\pi\)
\(314\) 0 0
\(315\) 1.99219 0.112247
\(316\) 0 0
\(317\) −14.4070 −0.809175 −0.404587 0.914499i \(-0.632585\pi\)
−0.404587 + 0.914499i \(0.632585\pi\)
\(318\) 0 0
\(319\) −7.00023 −0.391938
\(320\) 0 0
\(321\) −24.2089 −1.35121
\(322\) 0 0
\(323\) −6.35345 −0.353515
\(324\) 0 0
\(325\) 6.01423 0.333609
\(326\) 0 0
\(327\) 13.9131 0.769396
\(328\) 0 0
\(329\) −1.30648 −0.0720286
\(330\) 0 0
\(331\) −11.2853 −0.620294 −0.310147 0.950689i \(-0.600378\pi\)
−0.310147 + 0.950689i \(0.600378\pi\)
\(332\) 0 0
\(333\) −0.446429 −0.0244641
\(334\) 0 0
\(335\) 32.9875 1.80230
\(336\) 0 0
\(337\) −21.6167 −1.17754 −0.588769 0.808301i \(-0.700387\pi\)
−0.588769 + 0.808301i \(0.700387\pi\)
\(338\) 0 0
\(339\) 13.2358 0.718868
\(340\) 0 0
\(341\) 1.53675 0.0832197
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 32.7916 1.76544
\(346\) 0 0
\(347\) −3.96522 −0.212864 −0.106432 0.994320i \(-0.533943\pi\)
−0.106432 + 0.994320i \(0.533943\pi\)
\(348\) 0 0
\(349\) 27.0076 1.44569 0.722843 0.691013i \(-0.242835\pi\)
0.722843 + 0.691013i \(0.242835\pi\)
\(350\) 0 0
\(351\) 5.57720 0.297689
\(352\) 0 0
\(353\) 18.5211 0.985779 0.492890 0.870092i \(-0.335941\pi\)
0.492890 + 0.870092i \(0.335941\pi\)
\(354\) 0 0
\(355\) 1.02754 0.0545359
\(356\) 0 0
\(357\) −9.63373 −0.509871
\(358\) 0 0
\(359\) −21.2727 −1.12273 −0.561364 0.827569i \(-0.689723\pi\)
−0.561364 + 0.827569i \(0.689723\pi\)
\(360\) 0 0
\(361\) −17.9563 −0.945066
\(362\) 0 0
\(363\) −1.54910 −0.0813068
\(364\) 0 0
\(365\) 9.93672 0.520112
\(366\) 0 0
\(367\) −2.65798 −0.138745 −0.0693727 0.997591i \(-0.522100\pi\)
−0.0693727 + 0.997591i \(0.522100\pi\)
\(368\) 0 0
\(369\) 0.0226956 0.00118149
\(370\) 0 0
\(371\) −7.95351 −0.412926
\(372\) 0 0
\(373\) 26.1344 1.35319 0.676593 0.736357i \(-0.263456\pi\)
0.676593 + 0.736357i \(0.263456\pi\)
\(374\) 0 0
\(375\) 5.21427 0.269264
\(376\) 0 0
\(377\) −7.00023 −0.360530
\(378\) 0 0
\(379\) −2.00213 −0.102843 −0.0514213 0.998677i \(-0.516375\pi\)
−0.0514213 + 0.998677i \(0.516375\pi\)
\(380\) 0 0
\(381\) 10.3585 0.530681
\(382\) 0 0
\(383\) −10.3948 −0.531149 −0.265574 0.964090i \(-0.585562\pi\)
−0.265574 + 0.964090i \(0.585562\pi\)
\(384\) 0 0
\(385\) −3.31877 −0.169140
\(386\) 0 0
\(387\) 1.09825 0.0558269
\(388\) 0 0
\(389\) −31.3009 −1.58702 −0.793510 0.608558i \(-0.791748\pi\)
−0.793510 + 0.608558i \(0.791748\pi\)
\(390\) 0 0
\(391\) 39.6661 2.00600
\(392\) 0 0
\(393\) −14.1788 −0.715226
\(394\) 0 0
\(395\) −45.1765 −2.27308
\(396\) 0 0
\(397\) 23.3595 1.17238 0.586190 0.810174i \(-0.300627\pi\)
0.586190 + 0.810174i \(0.300627\pi\)
\(398\) 0 0
\(399\) 1.58262 0.0792299
\(400\) 0 0
\(401\) 33.3811 1.66697 0.833485 0.552542i \(-0.186342\pi\)
0.833485 + 0.552542i \(0.186342\pi\)
\(402\) 0 0
\(403\) 1.53675 0.0765510
\(404\) 0 0
\(405\) 22.6965 1.12780
\(406\) 0 0
\(407\) 0.743700 0.0368638
\(408\) 0 0
\(409\) 25.5848 1.26509 0.632543 0.774525i \(-0.282011\pi\)
0.632543 + 0.774525i \(0.282011\pi\)
\(410\) 0 0
\(411\) −4.60640 −0.227217
\(412\) 0 0
\(413\) 2.85219 0.140347
\(414\) 0 0
\(415\) 37.8172 1.85637
\(416\) 0 0
\(417\) 16.7949 0.822451
\(418\) 0 0
\(419\) −7.09521 −0.346624 −0.173312 0.984867i \(-0.555447\pi\)
−0.173312 + 0.984867i \(0.555447\pi\)
\(420\) 0 0
\(421\) 20.1596 0.982519 0.491260 0.871013i \(-0.336537\pi\)
0.491260 + 0.871013i \(0.336537\pi\)
\(422\) 0 0
\(423\) 0.784255 0.0381318
\(424\) 0 0
\(425\) 37.4019 1.81426
\(426\) 0 0
\(427\) 2.21055 0.106976
\(428\) 0 0
\(429\) −1.54910 −0.0747914
\(430\) 0 0
\(431\) 0.245405 0.0118208 0.00591038 0.999983i \(-0.498119\pi\)
0.00591038 + 0.999983i \(0.498119\pi\)
\(432\) 0 0
\(433\) 12.7873 0.614519 0.307260 0.951626i \(-0.400588\pi\)
0.307260 + 0.951626i \(0.400588\pi\)
\(434\) 0 0
\(435\) −35.9890 −1.72554
\(436\) 0 0
\(437\) −6.51631 −0.311717
\(438\) 0 0
\(439\) 5.63337 0.268866 0.134433 0.990923i \(-0.457079\pi\)
0.134433 + 0.990923i \(0.457079\pi\)
\(440\) 0 0
\(441\) −0.600281 −0.0285848
\(442\) 0 0
\(443\) 9.46289 0.449596 0.224798 0.974405i \(-0.427828\pi\)
0.224798 + 0.974405i \(0.427828\pi\)
\(444\) 0 0
\(445\) −4.08489 −0.193642
\(446\) 0 0
\(447\) −22.0995 −1.04527
\(448\) 0 0
\(449\) 17.0017 0.802359 0.401180 0.915999i \(-0.368600\pi\)
0.401180 + 0.915999i \(0.368600\pi\)
\(450\) 0 0
\(451\) −0.0378083 −0.00178032
\(452\) 0 0
\(453\) −13.1584 −0.618236
\(454\) 0 0
\(455\) −3.31877 −0.155586
\(456\) 0 0
\(457\) 6.86138 0.320962 0.160481 0.987039i \(-0.448696\pi\)
0.160481 + 0.987039i \(0.448696\pi\)
\(458\) 0 0
\(459\) 34.6841 1.61892
\(460\) 0 0
\(461\) −5.45887 −0.254245 −0.127123 0.991887i \(-0.540574\pi\)
−0.127123 + 0.991887i \(0.540574\pi\)
\(462\) 0 0
\(463\) −35.8728 −1.66715 −0.833575 0.552406i \(-0.813710\pi\)
−0.833575 + 0.552406i \(0.813710\pi\)
\(464\) 0 0
\(465\) 7.90061 0.366382
\(466\) 0 0
\(467\) 7.35921 0.340544 0.170272 0.985397i \(-0.445535\pi\)
0.170272 + 0.985397i \(0.445535\pi\)
\(468\) 0 0
\(469\) −9.93967 −0.458971
\(470\) 0 0
\(471\) −20.5958 −0.949005
\(472\) 0 0
\(473\) −1.82955 −0.0841229
\(474\) 0 0
\(475\) −6.14434 −0.281922
\(476\) 0 0
\(477\) 4.77434 0.218602
\(478\) 0 0
\(479\) −21.2874 −0.972646 −0.486323 0.873779i \(-0.661662\pi\)
−0.486323 + 0.873779i \(0.661662\pi\)
\(480\) 0 0
\(481\) 0.743700 0.0339098
\(482\) 0 0
\(483\) −9.88066 −0.449586
\(484\) 0 0
\(485\) 20.0412 0.910026
\(486\) 0 0
\(487\) −8.86941 −0.401911 −0.200956 0.979600i \(-0.564405\pi\)
−0.200956 + 0.979600i \(0.564405\pi\)
\(488\) 0 0
\(489\) 24.0330 1.08681
\(490\) 0 0
\(491\) 6.85321 0.309281 0.154640 0.987971i \(-0.450578\pi\)
0.154640 + 0.987971i \(0.450578\pi\)
\(492\) 0 0
\(493\) −43.5338 −1.96066
\(494\) 0 0
\(495\) 1.99219 0.0895424
\(496\) 0 0
\(497\) −0.309613 −0.0138881
\(498\) 0 0
\(499\) −10.1193 −0.453002 −0.226501 0.974011i \(-0.572729\pi\)
−0.226501 + 0.974011i \(0.572729\pi\)
\(500\) 0 0
\(501\) −6.52217 −0.291389
\(502\) 0 0
\(503\) 37.4867 1.67145 0.835726 0.549147i \(-0.185047\pi\)
0.835726 + 0.549147i \(0.185047\pi\)
\(504\) 0 0
\(505\) −15.3616 −0.683580
\(506\) 0 0
\(507\) −1.54910 −0.0687981
\(508\) 0 0
\(509\) 40.1767 1.78080 0.890401 0.455176i \(-0.150424\pi\)
0.890401 + 0.455176i \(0.150424\pi\)
\(510\) 0 0
\(511\) −2.99410 −0.132451
\(512\) 0 0
\(513\) −5.69787 −0.251567
\(514\) 0 0
\(515\) 29.3887 1.29502
\(516\) 0 0
\(517\) −1.30648 −0.0574589
\(518\) 0 0
\(519\) −12.1789 −0.534593
\(520\) 0 0
\(521\) 28.2066 1.23575 0.617877 0.786274i \(-0.287993\pi\)
0.617877 + 0.786274i \(0.287993\pi\)
\(522\) 0 0
\(523\) 7.17407 0.313700 0.156850 0.987622i \(-0.449866\pi\)
0.156850 + 0.987622i \(0.449866\pi\)
\(524\) 0 0
\(525\) −9.31666 −0.406612
\(526\) 0 0
\(527\) 9.55691 0.416305
\(528\) 0 0
\(529\) 17.6829 0.768821
\(530\) 0 0
\(531\) −1.71211 −0.0742993
\(532\) 0 0
\(533\) −0.0378083 −0.00163766
\(534\) 0 0
\(535\) −51.8647 −2.24231
\(536\) 0 0
\(537\) 34.1962 1.47568
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −9.07954 −0.390360 −0.195180 0.980767i \(-0.562529\pi\)
−0.195180 + 0.980767i \(0.562529\pi\)
\(542\) 0 0
\(543\) −2.95385 −0.126762
\(544\) 0 0
\(545\) 29.8072 1.27680
\(546\) 0 0
\(547\) 31.3830 1.34184 0.670919 0.741530i \(-0.265900\pi\)
0.670919 + 0.741530i \(0.265900\pi\)
\(548\) 0 0
\(549\) −1.32695 −0.0566328
\(550\) 0 0
\(551\) 7.15168 0.304672
\(552\) 0 0
\(553\) 13.6124 0.578860
\(554\) 0 0
\(555\) 3.82345 0.162296
\(556\) 0 0
\(557\) 19.5763 0.829474 0.414737 0.909941i \(-0.363874\pi\)
0.414737 + 0.909941i \(0.363874\pi\)
\(558\) 0 0
\(559\) −1.82955 −0.0773818
\(560\) 0 0
\(561\) −9.63373 −0.406736
\(562\) 0 0
\(563\) −41.3845 −1.74415 −0.872074 0.489373i \(-0.837226\pi\)
−0.872074 + 0.489373i \(0.837226\pi\)
\(564\) 0 0
\(565\) 28.3560 1.19295
\(566\) 0 0
\(567\) −6.83882 −0.287204
\(568\) 0 0
\(569\) 18.1712 0.761774 0.380887 0.924621i \(-0.375619\pi\)
0.380887 + 0.924621i \(0.375619\pi\)
\(570\) 0 0
\(571\) −8.45424 −0.353799 −0.176900 0.984229i \(-0.556607\pi\)
−0.176900 + 0.984229i \(0.556607\pi\)
\(572\) 0 0
\(573\) −8.63918 −0.360907
\(574\) 0 0
\(575\) 38.3606 1.59975
\(576\) 0 0
\(577\) 14.9679 0.623121 0.311561 0.950226i \(-0.399148\pi\)
0.311561 + 0.950226i \(0.399148\pi\)
\(578\) 0 0
\(579\) −0.230736 −0.00958906
\(580\) 0 0
\(581\) −11.3950 −0.472742
\(582\) 0 0
\(583\) −7.95351 −0.329401
\(584\) 0 0
\(585\) 1.99219 0.0823670
\(586\) 0 0
\(587\) −15.1400 −0.624894 −0.312447 0.949935i \(-0.601149\pi\)
−0.312447 + 0.949935i \(0.601149\pi\)
\(588\) 0 0
\(589\) −1.57000 −0.0646906
\(590\) 0 0
\(591\) −10.2391 −0.421178
\(592\) 0 0
\(593\) −21.5490 −0.884912 −0.442456 0.896790i \(-0.645893\pi\)
−0.442456 + 0.896790i \(0.645893\pi\)
\(594\) 0 0
\(595\) −20.6391 −0.846121
\(596\) 0 0
\(597\) 15.3319 0.627491
\(598\) 0 0
\(599\) 33.7469 1.37886 0.689430 0.724352i \(-0.257861\pi\)
0.689430 + 0.724352i \(0.257861\pi\)
\(600\) 0 0
\(601\) −15.5524 −0.634398 −0.317199 0.948359i \(-0.602742\pi\)
−0.317199 + 0.948359i \(0.602742\pi\)
\(602\) 0 0
\(603\) 5.96659 0.242978
\(604\) 0 0
\(605\) −3.31877 −0.134927
\(606\) 0 0
\(607\) −39.5930 −1.60703 −0.803514 0.595285i \(-0.797039\pi\)
−0.803514 + 0.595285i \(0.797039\pi\)
\(608\) 0 0
\(609\) 10.8441 0.439424
\(610\) 0 0
\(611\) −1.30648 −0.0528545
\(612\) 0 0
\(613\) 10.6653 0.430767 0.215383 0.976530i \(-0.430900\pi\)
0.215383 + 0.976530i \(0.430900\pi\)
\(614\) 0 0
\(615\) −0.194377 −0.00783803
\(616\) 0 0
\(617\) −4.94294 −0.198995 −0.0994975 0.995038i \(-0.531724\pi\)
−0.0994975 + 0.995038i \(0.531724\pi\)
\(618\) 0 0
\(619\) 26.9675 1.08392 0.541958 0.840406i \(-0.317683\pi\)
0.541958 + 0.840406i \(0.317683\pi\)
\(620\) 0 0
\(621\) 35.5732 1.42750
\(622\) 0 0
\(623\) 1.23084 0.0493127
\(624\) 0 0
\(625\) −18.9002 −0.756008
\(626\) 0 0
\(627\) 1.58262 0.0632036
\(628\) 0 0
\(629\) 4.62500 0.184411
\(630\) 0 0
\(631\) −38.6565 −1.53889 −0.769446 0.638712i \(-0.779468\pi\)
−0.769446 + 0.638712i \(0.779468\pi\)
\(632\) 0 0
\(633\) −14.4210 −0.573184
\(634\) 0 0
\(635\) 22.1918 0.880655
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) 0.185855 0.00735231
\(640\) 0 0
\(641\) 6.02450 0.237953 0.118977 0.992897i \(-0.462039\pi\)
0.118977 + 0.992897i \(0.462039\pi\)
\(642\) 0 0
\(643\) −20.6890 −0.815896 −0.407948 0.913005i \(-0.633756\pi\)
−0.407948 + 0.913005i \(0.633756\pi\)
\(644\) 0 0
\(645\) −9.40594 −0.370359
\(646\) 0 0
\(647\) 26.7240 1.05063 0.525315 0.850908i \(-0.323948\pi\)
0.525315 + 0.850908i \(0.323948\pi\)
\(648\) 0 0
\(649\) 2.85219 0.111958
\(650\) 0 0
\(651\) −2.38058 −0.0933024
\(652\) 0 0
\(653\) 24.5007 0.958788 0.479394 0.877600i \(-0.340857\pi\)
0.479394 + 0.877600i \(0.340857\pi\)
\(654\) 0 0
\(655\) −30.3764 −1.18690
\(656\) 0 0
\(657\) 1.79730 0.0701193
\(658\) 0 0
\(659\) −17.9495 −0.699214 −0.349607 0.936896i \(-0.613685\pi\)
−0.349607 + 0.936896i \(0.613685\pi\)
\(660\) 0 0
\(661\) 42.8531 1.66679 0.833397 0.552675i \(-0.186393\pi\)
0.833397 + 0.552675i \(0.186393\pi\)
\(662\) 0 0
\(663\) −9.63373 −0.374143
\(664\) 0 0
\(665\) 3.39057 0.131481
\(666\) 0 0
\(667\) −44.6497 −1.72884
\(668\) 0 0
\(669\) −37.8669 −1.46402
\(670\) 0 0
\(671\) 2.21055 0.0853372
\(672\) 0 0
\(673\) 20.9187 0.806358 0.403179 0.915121i \(-0.367905\pi\)
0.403179 + 0.915121i \(0.367905\pi\)
\(674\) 0 0
\(675\) 33.5426 1.29105
\(676\) 0 0
\(677\) 37.5378 1.44270 0.721348 0.692573i \(-0.243523\pi\)
0.721348 + 0.692573i \(0.243523\pi\)
\(678\) 0 0
\(679\) −6.03876 −0.231746
\(680\) 0 0
\(681\) −19.8323 −0.759974
\(682\) 0 0
\(683\) −21.7111 −0.830753 −0.415376 0.909650i \(-0.636350\pi\)
−0.415376 + 0.909650i \(0.636350\pi\)
\(684\) 0 0
\(685\) −9.86866 −0.377062
\(686\) 0 0
\(687\) 14.5211 0.554013
\(688\) 0 0
\(689\) −7.95351 −0.303005
\(690\) 0 0
\(691\) 13.4383 0.511218 0.255609 0.966780i \(-0.417724\pi\)
0.255609 + 0.966780i \(0.417724\pi\)
\(692\) 0 0
\(693\) −0.600281 −0.0228028
\(694\) 0 0
\(695\) 35.9811 1.36484
\(696\) 0 0
\(697\) −0.235126 −0.00890604
\(698\) 0 0
\(699\) −10.4139 −0.393889
\(700\) 0 0
\(701\) −37.1966 −1.40489 −0.702447 0.711736i \(-0.747909\pi\)
−0.702447 + 0.711736i \(0.747909\pi\)
\(702\) 0 0
\(703\) −0.759790 −0.0286560
\(704\) 0 0
\(705\) −6.71676 −0.252968
\(706\) 0 0
\(707\) 4.62869 0.174080
\(708\) 0 0
\(709\) 10.4426 0.392179 0.196089 0.980586i \(-0.437176\pi\)
0.196089 + 0.980586i \(0.437176\pi\)
\(710\) 0 0
\(711\) −8.17128 −0.306447
\(712\) 0 0
\(713\) 9.80187 0.367083
\(714\) 0 0
\(715\) −3.31877 −0.124115
\(716\) 0 0
\(717\) −18.0823 −0.675296
\(718\) 0 0
\(719\) 33.4983 1.24928 0.624638 0.780914i \(-0.285247\pi\)
0.624638 + 0.780914i \(0.285247\pi\)
\(720\) 0 0
\(721\) −8.85531 −0.329789
\(722\) 0 0
\(723\) −8.40517 −0.312592
\(724\) 0 0
\(725\) −42.1010 −1.56359
\(726\) 0 0
\(727\) 24.7916 0.919469 0.459735 0.888056i \(-0.347945\pi\)
0.459735 + 0.888056i \(0.347945\pi\)
\(728\) 0 0
\(729\) 30.0242 1.11201
\(730\) 0 0
\(731\) −11.3778 −0.420824
\(732\) 0 0
\(733\) 4.51313 0.166696 0.0833481 0.996520i \(-0.473439\pi\)
0.0833481 + 0.996520i \(0.473439\pi\)
\(734\) 0 0
\(735\) 5.14111 0.189633
\(736\) 0 0
\(737\) −9.93967 −0.366132
\(738\) 0 0
\(739\) 19.2604 0.708505 0.354252 0.935150i \(-0.384735\pi\)
0.354252 + 0.935150i \(0.384735\pi\)
\(740\) 0 0
\(741\) 1.58262 0.0581389
\(742\) 0 0
\(743\) 11.1762 0.410016 0.205008 0.978760i \(-0.434278\pi\)
0.205008 + 0.978760i \(0.434278\pi\)
\(744\) 0 0
\(745\) −47.3455 −1.73461
\(746\) 0 0
\(747\) 6.84017 0.250269
\(748\) 0 0
\(749\) 15.6277 0.571023
\(750\) 0 0
\(751\) 4.11214 0.150054 0.0750270 0.997181i \(-0.476096\pi\)
0.0750270 + 0.997181i \(0.476096\pi\)
\(752\) 0 0
\(753\) −44.2028 −1.61084
\(754\) 0 0
\(755\) −28.1903 −1.02595
\(756\) 0 0
\(757\) 45.8571 1.66670 0.833352 0.552742i \(-0.186419\pi\)
0.833352 + 0.552742i \(0.186419\pi\)
\(758\) 0 0
\(759\) −9.88066 −0.358645
\(760\) 0 0
\(761\) 21.0542 0.763213 0.381606 0.924325i \(-0.375371\pi\)
0.381606 + 0.924325i \(0.375371\pi\)
\(762\) 0 0
\(763\) −8.98139 −0.325148
\(764\) 0 0
\(765\) 12.3893 0.447935
\(766\) 0 0
\(767\) 2.85219 0.102986
\(768\) 0 0
\(769\) 5.97553 0.215483 0.107742 0.994179i \(-0.465638\pi\)
0.107742 + 0.994179i \(0.465638\pi\)
\(770\) 0 0
\(771\) −25.3099 −0.911516
\(772\) 0 0
\(773\) 23.3308 0.839152 0.419576 0.907720i \(-0.362179\pi\)
0.419576 + 0.907720i \(0.362179\pi\)
\(774\) 0 0
\(775\) 9.24237 0.331996
\(776\) 0 0
\(777\) −1.15207 −0.0413302
\(778\) 0 0
\(779\) 0.0386263 0.00138393
\(780\) 0 0
\(781\) −0.309613 −0.0110788
\(782\) 0 0
\(783\) −39.0417 −1.39524
\(784\) 0 0
\(785\) −44.1241 −1.57486
\(786\) 0 0
\(787\) −16.1562 −0.575905 −0.287953 0.957645i \(-0.592975\pi\)
−0.287953 + 0.957645i \(0.592975\pi\)
\(788\) 0 0
\(789\) 16.2786 0.579535
\(790\) 0 0
\(791\) −8.54415 −0.303795
\(792\) 0 0
\(793\) 2.21055 0.0784988
\(794\) 0 0
\(795\) −40.8899 −1.45022
\(796\) 0 0
\(797\) 16.5884 0.587590 0.293795 0.955869i \(-0.405082\pi\)
0.293795 + 0.955869i \(0.405082\pi\)
\(798\) 0 0
\(799\) −8.12488 −0.287438
\(800\) 0 0
\(801\) −0.738852 −0.0261061
\(802\) 0 0
\(803\) −2.99410 −0.105659
\(804\) 0 0
\(805\) −21.1681 −0.746079
\(806\) 0 0
\(807\) −41.5177 −1.46149
\(808\) 0 0
\(809\) −13.0829 −0.459969 −0.229984 0.973194i \(-0.573868\pi\)
−0.229984 + 0.973194i \(0.573868\pi\)
\(810\) 0 0
\(811\) −7.99860 −0.280869 −0.140434 0.990090i \(-0.544850\pi\)
−0.140434 + 0.990090i \(0.544850\pi\)
\(812\) 0 0
\(813\) 11.0090 0.386103
\(814\) 0 0
\(815\) 51.4878 1.80354
\(816\) 0 0
\(817\) 1.86913 0.0653927
\(818\) 0 0
\(819\) −0.600281 −0.0209755
\(820\) 0 0
\(821\) 21.3754 0.746008 0.373004 0.927830i \(-0.378328\pi\)
0.373004 + 0.927830i \(0.378328\pi\)
\(822\) 0 0
\(823\) −25.1332 −0.876088 −0.438044 0.898954i \(-0.644329\pi\)
−0.438044 + 0.898954i \(0.644329\pi\)
\(824\) 0 0
\(825\) −9.31666 −0.324364
\(826\) 0 0
\(827\) 51.5180 1.79145 0.895727 0.444604i \(-0.146655\pi\)
0.895727 + 0.444604i \(0.146655\pi\)
\(828\) 0 0
\(829\) −52.9584 −1.83932 −0.919661 0.392713i \(-0.871536\pi\)
−0.919661 + 0.392713i \(0.871536\pi\)
\(830\) 0 0
\(831\) 12.6051 0.437267
\(832\) 0 0
\(833\) 6.21891 0.215472
\(834\) 0 0
\(835\) −13.9730 −0.483555
\(836\) 0 0
\(837\) 8.57077 0.296249
\(838\) 0 0
\(839\) 15.6430 0.540056 0.270028 0.962852i \(-0.412967\pi\)
0.270028 + 0.962852i \(0.412967\pi\)
\(840\) 0 0
\(841\) 20.0032 0.689766
\(842\) 0 0
\(843\) 47.9153 1.65029
\(844\) 0 0
\(845\) −3.31877 −0.114169
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0 0
\(849\) 26.7523 0.918136
\(850\) 0 0
\(851\) 4.74355 0.162607
\(852\) 0 0
\(853\) −11.4578 −0.392308 −0.196154 0.980573i \(-0.562845\pi\)
−0.196154 + 0.980573i \(0.562845\pi\)
\(854\) 0 0
\(855\) −2.03529 −0.0696056
\(856\) 0 0
\(857\) 27.2015 0.929187 0.464594 0.885524i \(-0.346200\pi\)
0.464594 + 0.885524i \(0.346200\pi\)
\(858\) 0 0
\(859\) 0.794861 0.0271203 0.0135602 0.999908i \(-0.495684\pi\)
0.0135602 + 0.999908i \(0.495684\pi\)
\(860\) 0 0
\(861\) 0.0585689 0.00199602
\(862\) 0 0
\(863\) −40.1721 −1.36747 −0.683737 0.729729i \(-0.739646\pi\)
−0.683737 + 0.729729i \(0.739646\pi\)
\(864\) 0 0
\(865\) −26.0918 −0.887146
\(866\) 0 0
\(867\) −33.5765 −1.14032
\(868\) 0 0
\(869\) 13.6124 0.461770
\(870\) 0 0
\(871\) −9.93967 −0.336793
\(872\) 0 0
\(873\) 3.62495 0.122686
\(874\) 0 0
\(875\) −3.36599 −0.113791
\(876\) 0 0
\(877\) −1.07543 −0.0363147 −0.0181573 0.999835i \(-0.505780\pi\)
−0.0181573 + 0.999835i \(0.505780\pi\)
\(878\) 0 0
\(879\) 11.5838 0.390710
\(880\) 0 0
\(881\) 5.83898 0.196720 0.0983600 0.995151i \(-0.468640\pi\)
0.0983600 + 0.995151i \(0.468640\pi\)
\(882\) 0 0
\(883\) 23.8634 0.803066 0.401533 0.915845i \(-0.368478\pi\)
0.401533 + 0.915845i \(0.368478\pi\)
\(884\) 0 0
\(885\) 14.6634 0.492905
\(886\) 0 0
\(887\) 7.83140 0.262953 0.131476 0.991319i \(-0.458028\pi\)
0.131476 + 0.991319i \(0.458028\pi\)
\(888\) 0 0
\(889\) −6.68676 −0.224267
\(890\) 0 0
\(891\) −6.83882 −0.229109
\(892\) 0 0
\(893\) 1.33475 0.0446656
\(894\) 0 0
\(895\) 73.2613 2.44886
\(896\) 0 0
\(897\) −9.88066 −0.329906
\(898\) 0 0
\(899\) −10.7576 −0.358786
\(900\) 0 0
\(901\) −49.4622 −1.64782
\(902\) 0 0
\(903\) 2.83417 0.0943151
\(904\) 0 0
\(905\) −6.32828 −0.210359
\(906\) 0 0
\(907\) 34.0608 1.13097 0.565485 0.824758i \(-0.308689\pi\)
0.565485 + 0.824758i \(0.308689\pi\)
\(908\) 0 0
\(909\) −2.77851 −0.0921574
\(910\) 0 0
\(911\) 3.58092 0.118641 0.0593205 0.998239i \(-0.481107\pi\)
0.0593205 + 0.998239i \(0.481107\pi\)
\(912\) 0 0
\(913\) −11.3950 −0.377118
\(914\) 0 0
\(915\) 11.3647 0.375704
\(916\) 0 0
\(917\) 9.15291 0.302256
\(918\) 0 0
\(919\) 17.7970 0.587070 0.293535 0.955948i \(-0.405168\pi\)
0.293535 + 0.955948i \(0.405168\pi\)
\(920\) 0 0
\(921\) 23.9958 0.790690
\(922\) 0 0
\(923\) −0.309613 −0.0101910
\(924\) 0 0
\(925\) 4.47278 0.147064
\(926\) 0 0
\(927\) 5.31567 0.174590
\(928\) 0 0
\(929\) −41.8452 −1.37290 −0.686448 0.727179i \(-0.740831\pi\)
−0.686448 + 0.727179i \(0.740831\pi\)
\(930\) 0 0
\(931\) −1.02163 −0.0334827
\(932\) 0 0
\(933\) −23.9146 −0.782929
\(934\) 0 0
\(935\) −20.6391 −0.674971
\(936\) 0 0
\(937\) −45.8519 −1.49792 −0.748959 0.662617i \(-0.769446\pi\)
−0.748959 + 0.662617i \(0.769446\pi\)
\(938\) 0 0
\(939\) −54.2854 −1.77154
\(940\) 0 0
\(941\) 54.3692 1.77238 0.886192 0.463318i \(-0.153341\pi\)
0.886192 + 0.463318i \(0.153341\pi\)
\(942\) 0 0
\(943\) −0.241153 −0.00785303
\(944\) 0 0
\(945\) −18.5095 −0.602113
\(946\) 0 0
\(947\) −6.36608 −0.206870 −0.103435 0.994636i \(-0.532983\pi\)
−0.103435 + 0.994636i \(0.532983\pi\)
\(948\) 0 0
\(949\) −2.99410 −0.0971926
\(950\) 0 0
\(951\) 22.3178 0.723706
\(952\) 0 0
\(953\) 40.7296 1.31936 0.659681 0.751545i \(-0.270691\pi\)
0.659681 + 0.751545i \(0.270691\pi\)
\(954\) 0 0
\(955\) −18.5084 −0.598918
\(956\) 0 0
\(957\) 10.8441 0.350539
\(958\) 0 0
\(959\) 2.97359 0.0960223
\(960\) 0 0
\(961\) −28.6384 −0.923819
\(962\) 0 0
\(963\) −9.38100 −0.302298
\(964\) 0 0
\(965\) −0.494325 −0.0159129
\(966\) 0 0
\(967\) −46.0953 −1.48232 −0.741162 0.671326i \(-0.765725\pi\)
−0.741162 + 0.671326i \(0.765725\pi\)
\(968\) 0 0
\(969\) 9.84215 0.316175
\(970\) 0 0
\(971\) −6.02712 −0.193419 −0.0967097 0.995313i \(-0.530832\pi\)
−0.0967097 + 0.995313i \(0.530832\pi\)
\(972\) 0 0
\(973\) −10.8417 −0.347569
\(974\) 0 0
\(975\) −9.31666 −0.298372
\(976\) 0 0
\(977\) −24.6234 −0.787771 −0.393885 0.919160i \(-0.628869\pi\)
−0.393885 + 0.919160i \(0.628869\pi\)
\(978\) 0 0
\(979\) 1.23084 0.0393380
\(980\) 0 0
\(981\) 5.39135 0.172133
\(982\) 0 0
\(983\) 37.3295 1.19063 0.595313 0.803494i \(-0.297028\pi\)
0.595313 + 0.803494i \(0.297028\pi\)
\(984\) 0 0
\(985\) −21.9360 −0.698938
\(986\) 0 0
\(987\) 2.02387 0.0644206
\(988\) 0 0
\(989\) −11.6695 −0.371067
\(990\) 0 0
\(991\) 37.2932 1.18466 0.592329 0.805696i \(-0.298209\pi\)
0.592329 + 0.805696i \(0.298209\pi\)
\(992\) 0 0
\(993\) 17.4820 0.554775
\(994\) 0 0
\(995\) 32.8467 1.04131
\(996\) 0 0
\(997\) 6.93327 0.219579 0.109789 0.993955i \(-0.464982\pi\)
0.109789 + 0.993955i \(0.464982\pi\)
\(998\) 0 0
\(999\) 4.14777 0.131230
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 8008.2.a.x.1.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8008.2.a.x.1.4 12 1.1 even 1 trivial