Properties

Label 8016.2.a.z.1.1
Level $8016$
Weight $2$
Character 8016.1
Self dual yes
Analytic conductor $64.008$
Analytic rank $1$
Dimension $8$
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] = [8016,2,Mod(1,8016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8016, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.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: \(64.0080822603\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 8x^{6} + 15x^{5} + 19x^{4} - 31x^{3} - 13x^{2} + 14x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 501)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.63639\) of defining polynomial
Character \(\chi\) \(=\) 8016.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -3.62511 q^{5} +1.70865 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -3.62511 q^{5} +1.70865 q^{7} +1.00000 q^{9} +3.73929 q^{11} -5.28262 q^{13} -3.62511 q^{15} -2.55148 q^{17} -7.63986 q^{19} +1.70865 q^{21} +6.71726 q^{23} +8.14139 q^{25} +1.00000 q^{27} +2.10123 q^{29} +8.01919 q^{31} +3.73929 q^{33} -6.19405 q^{35} +5.23604 q^{37} -5.28262 q^{39} +5.40689 q^{41} -6.04679 q^{43} -3.62511 q^{45} -4.03434 q^{47} -4.08050 q^{49} -2.55148 q^{51} -3.29196 q^{53} -13.5553 q^{55} -7.63986 q^{57} -12.8952 q^{59} +11.2877 q^{61} +1.70865 q^{63} +19.1501 q^{65} +6.08146 q^{67} +6.71726 q^{69} -4.88309 q^{71} -3.70680 q^{73} +8.14139 q^{75} +6.38916 q^{77} +11.7416 q^{79} +1.00000 q^{81} -15.6696 q^{83} +9.24939 q^{85} +2.10123 q^{87} +2.46878 q^{89} -9.02617 q^{91} +8.01919 q^{93} +27.6953 q^{95} -4.76227 q^{97} +3.73929 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} + q^{5} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{3} + q^{5} + 8 q^{9} - 5 q^{11} + q^{15} - 7 q^{17} - 24 q^{19} - q^{23} + 3 q^{25} + 8 q^{27} - 11 q^{29} - 30 q^{31} - 5 q^{33} - 26 q^{35} + 11 q^{37} + 10 q^{41} - 24 q^{43} + q^{45} + 3 q^{47} + 6 q^{49} - 7 q^{51} - 25 q^{53} - 25 q^{55} - 24 q^{57} - 45 q^{59} + 16 q^{61} - 10 q^{65} - 18 q^{67} - q^{69} - 21 q^{71} - 8 q^{73} + 3 q^{75} - 18 q^{77} - 10 q^{79} + 8 q^{81} - 7 q^{83} - 11 q^{85} - 11 q^{87} + 26 q^{89} - 15 q^{91} - 30 q^{93} - q^{95} - 3 q^{97} - 5 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −3.62511 −1.62120 −0.810598 0.585603i \(-0.800858\pi\)
−0.810598 + 0.585603i \(0.800858\pi\)
\(6\) 0 0
\(7\) 1.70865 0.645810 0.322905 0.946431i \(-0.395340\pi\)
0.322905 + 0.946431i \(0.395340\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.73929 1.12744 0.563720 0.825966i \(-0.309370\pi\)
0.563720 + 0.825966i \(0.309370\pi\)
\(12\) 0 0
\(13\) −5.28262 −1.46514 −0.732568 0.680694i \(-0.761678\pi\)
−0.732568 + 0.680694i \(0.761678\pi\)
\(14\) 0 0
\(15\) −3.62511 −0.935998
\(16\) 0 0
\(17\) −2.55148 −0.618825 −0.309413 0.950928i \(-0.600132\pi\)
−0.309413 + 0.950928i \(0.600132\pi\)
\(18\) 0 0
\(19\) −7.63986 −1.75271 −0.876353 0.481670i \(-0.840030\pi\)
−0.876353 + 0.481670i \(0.840030\pi\)
\(20\) 0 0
\(21\) 1.70865 0.372859
\(22\) 0 0
\(23\) 6.71726 1.40064 0.700322 0.713827i \(-0.253040\pi\)
0.700322 + 0.713827i \(0.253040\pi\)
\(24\) 0 0
\(25\) 8.14139 1.62828
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 2.10123 0.390189 0.195095 0.980784i \(-0.437499\pi\)
0.195095 + 0.980784i \(0.437499\pi\)
\(30\) 0 0
\(31\) 8.01919 1.44029 0.720145 0.693824i \(-0.244075\pi\)
0.720145 + 0.693824i \(0.244075\pi\)
\(32\) 0 0
\(33\) 3.73929 0.650928
\(34\) 0 0
\(35\) −6.19405 −1.04699
\(36\) 0 0
\(37\) 5.23604 0.860799 0.430400 0.902638i \(-0.358373\pi\)
0.430400 + 0.902638i \(0.358373\pi\)
\(38\) 0 0
\(39\) −5.28262 −0.845896
\(40\) 0 0
\(41\) 5.40689 0.844414 0.422207 0.906499i \(-0.361256\pi\)
0.422207 + 0.906499i \(0.361256\pi\)
\(42\) 0 0
\(43\) −6.04679 −0.922127 −0.461063 0.887367i \(-0.652532\pi\)
−0.461063 + 0.887367i \(0.652532\pi\)
\(44\) 0 0
\(45\) −3.62511 −0.540399
\(46\) 0 0
\(47\) −4.03434 −0.588469 −0.294235 0.955733i \(-0.595065\pi\)
−0.294235 + 0.955733i \(0.595065\pi\)
\(48\) 0 0
\(49\) −4.08050 −0.582929
\(50\) 0 0
\(51\) −2.55148 −0.357279
\(52\) 0 0
\(53\) −3.29196 −0.452186 −0.226093 0.974106i \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(54\) 0 0
\(55\) −13.5553 −1.82780
\(56\) 0 0
\(57\) −7.63986 −1.01192
\(58\) 0 0
\(59\) −12.8952 −1.67881 −0.839403 0.543510i \(-0.817095\pi\)
−0.839403 + 0.543510i \(0.817095\pi\)
\(60\) 0 0
\(61\) 11.2877 1.44524 0.722620 0.691245i \(-0.242938\pi\)
0.722620 + 0.691245i \(0.242938\pi\)
\(62\) 0 0
\(63\) 1.70865 0.215270
\(64\) 0 0
\(65\) 19.1501 2.37527
\(66\) 0 0
\(67\) 6.08146 0.742969 0.371485 0.928439i \(-0.378849\pi\)
0.371485 + 0.928439i \(0.378849\pi\)
\(68\) 0 0
\(69\) 6.71726 0.808663
\(70\) 0 0
\(71\) −4.88309 −0.579517 −0.289758 0.957100i \(-0.593575\pi\)
−0.289758 + 0.957100i \(0.593575\pi\)
\(72\) 0 0
\(73\) −3.70680 −0.433848 −0.216924 0.976188i \(-0.569602\pi\)
−0.216924 + 0.976188i \(0.569602\pi\)
\(74\) 0 0
\(75\) 8.14139 0.940087
\(76\) 0 0
\(77\) 6.38916 0.728112
\(78\) 0 0
\(79\) 11.7416 1.32103 0.660515 0.750813i \(-0.270338\pi\)
0.660515 + 0.750813i \(0.270338\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −15.6696 −1.71996 −0.859981 0.510326i \(-0.829525\pi\)
−0.859981 + 0.510326i \(0.829525\pi\)
\(84\) 0 0
\(85\) 9.24939 1.00324
\(86\) 0 0
\(87\) 2.10123 0.225276
\(88\) 0 0
\(89\) 2.46878 0.261690 0.130845 0.991403i \(-0.458231\pi\)
0.130845 + 0.991403i \(0.458231\pi\)
\(90\) 0 0
\(91\) −9.02617 −0.946200
\(92\) 0 0
\(93\) 8.01919 0.831551
\(94\) 0 0
\(95\) 27.6953 2.84148
\(96\) 0 0
\(97\) −4.76227 −0.483536 −0.241768 0.970334i \(-0.577727\pi\)
−0.241768 + 0.970334i \(0.577727\pi\)
\(98\) 0 0
\(99\) 3.73929 0.375813
\(100\) 0 0
\(101\) −14.9847 −1.49104 −0.745518 0.666485i \(-0.767798\pi\)
−0.745518 + 0.666485i \(0.767798\pi\)
\(102\) 0 0
\(103\) −2.40203 −0.236679 −0.118340 0.992973i \(-0.537757\pi\)
−0.118340 + 0.992973i \(0.537757\pi\)
\(104\) 0 0
\(105\) −6.19405 −0.604477
\(106\) 0 0
\(107\) 4.24916 0.410782 0.205391 0.978680i \(-0.434153\pi\)
0.205391 + 0.978680i \(0.434153\pi\)
\(108\) 0 0
\(109\) −15.5656 −1.49091 −0.745457 0.666554i \(-0.767769\pi\)
−0.745457 + 0.666554i \(0.767769\pi\)
\(110\) 0 0
\(111\) 5.23604 0.496983
\(112\) 0 0
\(113\) −8.57378 −0.806553 −0.403276 0.915078i \(-0.632129\pi\)
−0.403276 + 0.915078i \(0.632129\pi\)
\(114\) 0 0
\(115\) −24.3508 −2.27072
\(116\) 0 0
\(117\) −5.28262 −0.488379
\(118\) 0 0
\(119\) −4.35960 −0.399644
\(120\) 0 0
\(121\) 2.98232 0.271120
\(122\) 0 0
\(123\) 5.40689 0.487523
\(124\) 0 0
\(125\) −11.3879 −1.01856
\(126\) 0 0
\(127\) 20.6163 1.82940 0.914702 0.404129i \(-0.132425\pi\)
0.914702 + 0.404129i \(0.132425\pi\)
\(128\) 0 0
\(129\) −6.04679 −0.532390
\(130\) 0 0
\(131\) −3.00607 −0.262642 −0.131321 0.991340i \(-0.541922\pi\)
−0.131321 + 0.991340i \(0.541922\pi\)
\(132\) 0 0
\(133\) −13.0539 −1.13192
\(134\) 0 0
\(135\) −3.62511 −0.311999
\(136\) 0 0
\(137\) −14.9358 −1.27605 −0.638025 0.770016i \(-0.720248\pi\)
−0.638025 + 0.770016i \(0.720248\pi\)
\(138\) 0 0
\(139\) −18.9035 −1.60338 −0.801689 0.597742i \(-0.796065\pi\)
−0.801689 + 0.597742i \(0.796065\pi\)
\(140\) 0 0
\(141\) −4.03434 −0.339753
\(142\) 0 0
\(143\) −19.7533 −1.65185
\(144\) 0 0
\(145\) −7.61719 −0.632573
\(146\) 0 0
\(147\) −4.08050 −0.336554
\(148\) 0 0
\(149\) −0.964909 −0.0790484 −0.0395242 0.999219i \(-0.512584\pi\)
−0.0395242 + 0.999219i \(0.512584\pi\)
\(150\) 0 0
\(151\) −18.9141 −1.53921 −0.769605 0.638520i \(-0.779547\pi\)
−0.769605 + 0.638520i \(0.779547\pi\)
\(152\) 0 0
\(153\) −2.55148 −0.206275
\(154\) 0 0
\(155\) −29.0704 −2.33499
\(156\) 0 0
\(157\) 3.36667 0.268690 0.134345 0.990935i \(-0.457107\pi\)
0.134345 + 0.990935i \(0.457107\pi\)
\(158\) 0 0
\(159\) −3.29196 −0.261070
\(160\) 0 0
\(161\) 11.4775 0.904551
\(162\) 0 0
\(163\) 0.681465 0.0533765 0.0266882 0.999644i \(-0.491504\pi\)
0.0266882 + 0.999644i \(0.491504\pi\)
\(164\) 0 0
\(165\) −13.5553 −1.05528
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) 14.9061 1.14662
\(170\) 0 0
\(171\) −7.63986 −0.584235
\(172\) 0 0
\(173\) 5.04156 0.383303 0.191651 0.981463i \(-0.438616\pi\)
0.191651 + 0.981463i \(0.438616\pi\)
\(174\) 0 0
\(175\) 13.9108 1.05156
\(176\) 0 0
\(177\) −12.8952 −0.969259
\(178\) 0 0
\(179\) 9.14897 0.683826 0.341913 0.939732i \(-0.388925\pi\)
0.341913 + 0.939732i \(0.388925\pi\)
\(180\) 0 0
\(181\) 12.2259 0.908748 0.454374 0.890811i \(-0.349863\pi\)
0.454374 + 0.890811i \(0.349863\pi\)
\(182\) 0 0
\(183\) 11.2877 0.834410
\(184\) 0 0
\(185\) −18.9812 −1.39552
\(186\) 0 0
\(187\) −9.54074 −0.697688
\(188\) 0 0
\(189\) 1.70865 0.124286
\(190\) 0 0
\(191\) −16.6142 −1.20216 −0.601080 0.799189i \(-0.705263\pi\)
−0.601080 + 0.799189i \(0.705263\pi\)
\(192\) 0 0
\(193\) 7.78696 0.560518 0.280259 0.959924i \(-0.409580\pi\)
0.280259 + 0.959924i \(0.409580\pi\)
\(194\) 0 0
\(195\) 19.1501 1.37136
\(196\) 0 0
\(197\) 11.7525 0.837328 0.418664 0.908141i \(-0.362499\pi\)
0.418664 + 0.908141i \(0.362499\pi\)
\(198\) 0 0
\(199\) 6.51873 0.462101 0.231050 0.972942i \(-0.425784\pi\)
0.231050 + 0.972942i \(0.425784\pi\)
\(200\) 0 0
\(201\) 6.08146 0.428953
\(202\) 0 0
\(203\) 3.59028 0.251988
\(204\) 0 0
\(205\) −19.6005 −1.36896
\(206\) 0 0
\(207\) 6.71726 0.466882
\(208\) 0 0
\(209\) −28.5677 −1.97607
\(210\) 0 0
\(211\) 13.2143 0.909711 0.454856 0.890565i \(-0.349691\pi\)
0.454856 + 0.890565i \(0.349691\pi\)
\(212\) 0 0
\(213\) −4.88309 −0.334584
\(214\) 0 0
\(215\) 21.9202 1.49495
\(216\) 0 0
\(217\) 13.7020 0.930154
\(218\) 0 0
\(219\) −3.70680 −0.250482
\(220\) 0 0
\(221\) 13.4785 0.906663
\(222\) 0 0
\(223\) −15.9483 −1.06797 −0.533987 0.845493i \(-0.679307\pi\)
−0.533987 + 0.845493i \(0.679307\pi\)
\(224\) 0 0
\(225\) 8.14139 0.542759
\(226\) 0 0
\(227\) 1.27727 0.0847754 0.0423877 0.999101i \(-0.486504\pi\)
0.0423877 + 0.999101i \(0.486504\pi\)
\(228\) 0 0
\(229\) 2.75137 0.181816 0.0909078 0.995859i \(-0.471023\pi\)
0.0909078 + 0.995859i \(0.471023\pi\)
\(230\) 0 0
\(231\) 6.38916 0.420376
\(232\) 0 0
\(233\) −2.73207 −0.178984 −0.0894921 0.995988i \(-0.528524\pi\)
−0.0894921 + 0.995988i \(0.528524\pi\)
\(234\) 0 0
\(235\) 14.6249 0.954025
\(236\) 0 0
\(237\) 11.7416 0.762697
\(238\) 0 0
\(239\) −27.1609 −1.75689 −0.878445 0.477843i \(-0.841419\pi\)
−0.878445 + 0.477843i \(0.841419\pi\)
\(240\) 0 0
\(241\) −9.73684 −0.627205 −0.313603 0.949554i \(-0.601536\pi\)
−0.313603 + 0.949554i \(0.601536\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 14.7923 0.945042
\(246\) 0 0
\(247\) 40.3585 2.56795
\(248\) 0 0
\(249\) −15.6696 −0.993020
\(250\) 0 0
\(251\) −18.8215 −1.18800 −0.594001 0.804464i \(-0.702452\pi\)
−0.594001 + 0.804464i \(0.702452\pi\)
\(252\) 0 0
\(253\) 25.1178 1.57914
\(254\) 0 0
\(255\) 9.24939 0.579219
\(256\) 0 0
\(257\) 2.66522 0.166252 0.0831259 0.996539i \(-0.473510\pi\)
0.0831259 + 0.996539i \(0.473510\pi\)
\(258\) 0 0
\(259\) 8.94658 0.555913
\(260\) 0 0
\(261\) 2.10123 0.130063
\(262\) 0 0
\(263\) −9.38514 −0.578712 −0.289356 0.957221i \(-0.593441\pi\)
−0.289356 + 0.957221i \(0.593441\pi\)
\(264\) 0 0
\(265\) 11.9337 0.733082
\(266\) 0 0
\(267\) 2.46878 0.151087
\(268\) 0 0
\(269\) 11.2658 0.686890 0.343445 0.939173i \(-0.388406\pi\)
0.343445 + 0.939173i \(0.388406\pi\)
\(270\) 0 0
\(271\) 19.4755 1.18305 0.591527 0.806285i \(-0.298525\pi\)
0.591527 + 0.806285i \(0.298525\pi\)
\(272\) 0 0
\(273\) −9.02617 −0.546289
\(274\) 0 0
\(275\) 30.4430 1.83578
\(276\) 0 0
\(277\) 7.75098 0.465711 0.232856 0.972511i \(-0.425193\pi\)
0.232856 + 0.972511i \(0.425193\pi\)
\(278\) 0 0
\(279\) 8.01919 0.480096
\(280\) 0 0
\(281\) −19.9976 −1.19295 −0.596477 0.802630i \(-0.703433\pi\)
−0.596477 + 0.802630i \(0.703433\pi\)
\(282\) 0 0
\(283\) −1.99842 −0.118793 −0.0593967 0.998234i \(-0.518918\pi\)
−0.0593967 + 0.998234i \(0.518918\pi\)
\(284\) 0 0
\(285\) 27.6953 1.64053
\(286\) 0 0
\(287\) 9.23850 0.545331
\(288\) 0 0
\(289\) −10.4899 −0.617055
\(290\) 0 0
\(291\) −4.76227 −0.279169
\(292\) 0 0
\(293\) −13.1462 −0.768008 −0.384004 0.923331i \(-0.625455\pi\)
−0.384004 + 0.923331i \(0.625455\pi\)
\(294\) 0 0
\(295\) 46.7463 2.72167
\(296\) 0 0
\(297\) 3.73929 0.216976
\(298\) 0 0
\(299\) −35.4847 −2.05213
\(300\) 0 0
\(301\) −10.3319 −0.595519
\(302\) 0 0
\(303\) −14.9847 −0.860851
\(304\) 0 0
\(305\) −40.9191 −2.34302
\(306\) 0 0
\(307\) −19.0378 −1.08654 −0.543272 0.839557i \(-0.682815\pi\)
−0.543272 + 0.839557i \(0.682815\pi\)
\(308\) 0 0
\(309\) −2.40203 −0.136647
\(310\) 0 0
\(311\) −7.31774 −0.414951 −0.207475 0.978240i \(-0.566525\pi\)
−0.207475 + 0.978240i \(0.566525\pi\)
\(312\) 0 0
\(313\) −21.5818 −1.21987 −0.609937 0.792450i \(-0.708805\pi\)
−0.609937 + 0.792450i \(0.708805\pi\)
\(314\) 0 0
\(315\) −6.19405 −0.348995
\(316\) 0 0
\(317\) −19.2702 −1.08232 −0.541162 0.840918i \(-0.682016\pi\)
−0.541162 + 0.840918i \(0.682016\pi\)
\(318\) 0 0
\(319\) 7.85713 0.439915
\(320\) 0 0
\(321\) 4.24916 0.237165
\(322\) 0 0
\(323\) 19.4930 1.08462
\(324\) 0 0
\(325\) −43.0079 −2.38565
\(326\) 0 0
\(327\) −15.5656 −0.860780
\(328\) 0 0
\(329\) −6.89330 −0.380040
\(330\) 0 0
\(331\) −27.0258 −1.48547 −0.742737 0.669583i \(-0.766473\pi\)
−0.742737 + 0.669583i \(0.766473\pi\)
\(332\) 0 0
\(333\) 5.23604 0.286933
\(334\) 0 0
\(335\) −22.0459 −1.20450
\(336\) 0 0
\(337\) 27.0867 1.47550 0.737752 0.675072i \(-0.235887\pi\)
0.737752 + 0.675072i \(0.235887\pi\)
\(338\) 0 0
\(339\) −8.57378 −0.465664
\(340\) 0 0
\(341\) 29.9861 1.62384
\(342\) 0 0
\(343\) −18.9327 −1.02227
\(344\) 0 0
\(345\) −24.3508 −1.31100
\(346\) 0 0
\(347\) −1.04990 −0.0563618 −0.0281809 0.999603i \(-0.508971\pi\)
−0.0281809 + 0.999603i \(0.508971\pi\)
\(348\) 0 0
\(349\) −4.79660 −0.256756 −0.128378 0.991725i \(-0.540977\pi\)
−0.128378 + 0.991725i \(0.540977\pi\)
\(350\) 0 0
\(351\) −5.28262 −0.281965
\(352\) 0 0
\(353\) −10.1349 −0.539425 −0.269713 0.962941i \(-0.586929\pi\)
−0.269713 + 0.962941i \(0.586929\pi\)
\(354\) 0 0
\(355\) 17.7017 0.939510
\(356\) 0 0
\(357\) −4.35960 −0.230734
\(358\) 0 0
\(359\) 6.68943 0.353055 0.176527 0.984296i \(-0.443514\pi\)
0.176527 + 0.984296i \(0.443514\pi\)
\(360\) 0 0
\(361\) 39.3675 2.07197
\(362\) 0 0
\(363\) 2.98232 0.156531
\(364\) 0 0
\(365\) 13.4375 0.703353
\(366\) 0 0
\(367\) 5.32088 0.277748 0.138874 0.990310i \(-0.455652\pi\)
0.138874 + 0.990310i \(0.455652\pi\)
\(368\) 0 0
\(369\) 5.40689 0.281471
\(370\) 0 0
\(371\) −5.62483 −0.292026
\(372\) 0 0
\(373\) −21.9039 −1.13414 −0.567071 0.823669i \(-0.691924\pi\)
−0.567071 + 0.823669i \(0.691924\pi\)
\(374\) 0 0
\(375\) −11.3879 −0.588067
\(376\) 0 0
\(377\) −11.1000 −0.571680
\(378\) 0 0
\(379\) −19.1233 −0.982300 −0.491150 0.871075i \(-0.663423\pi\)
−0.491150 + 0.871075i \(0.663423\pi\)
\(380\) 0 0
\(381\) 20.6163 1.05621
\(382\) 0 0
\(383\) −31.4789 −1.60850 −0.804248 0.594293i \(-0.797432\pi\)
−0.804248 + 0.594293i \(0.797432\pi\)
\(384\) 0 0
\(385\) −23.1614 −1.18041
\(386\) 0 0
\(387\) −6.04679 −0.307376
\(388\) 0 0
\(389\) −34.5326 −1.75087 −0.875435 0.483335i \(-0.839425\pi\)
−0.875435 + 0.483335i \(0.839425\pi\)
\(390\) 0 0
\(391\) −17.1390 −0.866755
\(392\) 0 0
\(393\) −3.00607 −0.151636
\(394\) 0 0
\(395\) −42.5644 −2.14165
\(396\) 0 0
\(397\) 30.0959 1.51047 0.755234 0.655455i \(-0.227523\pi\)
0.755234 + 0.655455i \(0.227523\pi\)
\(398\) 0 0
\(399\) −13.0539 −0.653512
\(400\) 0 0
\(401\) 13.9203 0.695146 0.347573 0.937653i \(-0.387006\pi\)
0.347573 + 0.937653i \(0.387006\pi\)
\(402\) 0 0
\(403\) −42.3624 −2.11022
\(404\) 0 0
\(405\) −3.62511 −0.180133
\(406\) 0 0
\(407\) 19.5791 0.970499
\(408\) 0 0
\(409\) −0.684344 −0.0338386 −0.0169193 0.999857i \(-0.505386\pi\)
−0.0169193 + 0.999857i \(0.505386\pi\)
\(410\) 0 0
\(411\) −14.9358 −0.736728
\(412\) 0 0
\(413\) −22.0333 −1.08419
\(414\) 0 0
\(415\) 56.8039 2.78840
\(416\) 0 0
\(417\) −18.9035 −0.925710
\(418\) 0 0
\(419\) −12.0874 −0.590508 −0.295254 0.955419i \(-0.595404\pi\)
−0.295254 + 0.955419i \(0.595404\pi\)
\(420\) 0 0
\(421\) 25.7142 1.25324 0.626618 0.779327i \(-0.284439\pi\)
0.626618 + 0.779327i \(0.284439\pi\)
\(422\) 0 0
\(423\) −4.03434 −0.196156
\(424\) 0 0
\(425\) −20.7726 −1.00762
\(426\) 0 0
\(427\) 19.2868 0.933351
\(428\) 0 0
\(429\) −19.7533 −0.953697
\(430\) 0 0
\(431\) −25.2017 −1.21392 −0.606961 0.794731i \(-0.707612\pi\)
−0.606961 + 0.794731i \(0.707612\pi\)
\(432\) 0 0
\(433\) 6.89916 0.331553 0.165776 0.986163i \(-0.446987\pi\)
0.165776 + 0.986163i \(0.446987\pi\)
\(434\) 0 0
\(435\) −7.61719 −0.365216
\(436\) 0 0
\(437\) −51.3189 −2.45492
\(438\) 0 0
\(439\) 39.6783 1.89374 0.946871 0.321615i \(-0.104226\pi\)
0.946871 + 0.321615i \(0.104226\pi\)
\(440\) 0 0
\(441\) −4.08050 −0.194310
\(442\) 0 0
\(443\) 9.33342 0.443444 0.221722 0.975110i \(-0.428832\pi\)
0.221722 + 0.975110i \(0.428832\pi\)
\(444\) 0 0
\(445\) −8.94958 −0.424251
\(446\) 0 0
\(447\) −0.964909 −0.0456386
\(448\) 0 0
\(449\) −12.2858 −0.579805 −0.289902 0.957056i \(-0.593623\pi\)
−0.289902 + 0.957056i \(0.593623\pi\)
\(450\) 0 0
\(451\) 20.2179 0.952026
\(452\) 0 0
\(453\) −18.9141 −0.888664
\(454\) 0 0
\(455\) 32.7208 1.53398
\(456\) 0 0
\(457\) 28.3360 1.32550 0.662751 0.748840i \(-0.269389\pi\)
0.662751 + 0.748840i \(0.269389\pi\)
\(458\) 0 0
\(459\) −2.55148 −0.119093
\(460\) 0 0
\(461\) −24.1768 −1.12603 −0.563014 0.826447i \(-0.690358\pi\)
−0.563014 + 0.826447i \(0.690358\pi\)
\(462\) 0 0
\(463\) −3.08154 −0.143211 −0.0716057 0.997433i \(-0.522812\pi\)
−0.0716057 + 0.997433i \(0.522812\pi\)
\(464\) 0 0
\(465\) −29.0704 −1.34811
\(466\) 0 0
\(467\) 7.43216 0.343919 0.171960 0.985104i \(-0.444990\pi\)
0.171960 + 0.985104i \(0.444990\pi\)
\(468\) 0 0
\(469\) 10.3911 0.479817
\(470\) 0 0
\(471\) 3.36667 0.155128
\(472\) 0 0
\(473\) −22.6107 −1.03964
\(474\) 0 0
\(475\) −62.1991 −2.85389
\(476\) 0 0
\(477\) −3.29196 −0.150729
\(478\) 0 0
\(479\) 16.6251 0.759619 0.379810 0.925065i \(-0.375989\pi\)
0.379810 + 0.925065i \(0.375989\pi\)
\(480\) 0 0
\(481\) −27.6600 −1.26119
\(482\) 0 0
\(483\) 11.4775 0.522243
\(484\) 0 0
\(485\) 17.2637 0.783906
\(486\) 0 0
\(487\) −30.5765 −1.38555 −0.692777 0.721152i \(-0.743613\pi\)
−0.692777 + 0.721152i \(0.743613\pi\)
\(488\) 0 0
\(489\) 0.681465 0.0308169
\(490\) 0 0
\(491\) 4.79196 0.216258 0.108129 0.994137i \(-0.465514\pi\)
0.108129 + 0.994137i \(0.465514\pi\)
\(492\) 0 0
\(493\) −5.36126 −0.241459
\(494\) 0 0
\(495\) −13.5553 −0.609267
\(496\) 0 0
\(497\) −8.34352 −0.374258
\(498\) 0 0
\(499\) 3.15628 0.141294 0.0706472 0.997501i \(-0.477494\pi\)
0.0706472 + 0.997501i \(0.477494\pi\)
\(500\) 0 0
\(501\) −1.00000 −0.0446767
\(502\) 0 0
\(503\) −30.0941 −1.34183 −0.670916 0.741534i \(-0.734099\pi\)
−0.670916 + 0.741534i \(0.734099\pi\)
\(504\) 0 0
\(505\) 54.3212 2.41726
\(506\) 0 0
\(507\) 14.9061 0.662003
\(508\) 0 0
\(509\) −19.8120 −0.878152 −0.439076 0.898450i \(-0.644694\pi\)
−0.439076 + 0.898450i \(0.644694\pi\)
\(510\) 0 0
\(511\) −6.33364 −0.280184
\(512\) 0 0
\(513\) −7.63986 −0.337308
\(514\) 0 0
\(515\) 8.70761 0.383703
\(516\) 0 0
\(517\) −15.0856 −0.663464
\(518\) 0 0
\(519\) 5.04156 0.221300
\(520\) 0 0
\(521\) −19.3463 −0.847579 −0.423789 0.905761i \(-0.639300\pi\)
−0.423789 + 0.905761i \(0.639300\pi\)
\(522\) 0 0
\(523\) −28.9423 −1.26556 −0.632778 0.774333i \(-0.718086\pi\)
−0.632778 + 0.774333i \(0.718086\pi\)
\(524\) 0 0
\(525\) 13.9108 0.607118
\(526\) 0 0
\(527\) −20.4608 −0.891287
\(528\) 0 0
\(529\) 22.1215 0.961806
\(530\) 0 0
\(531\) −12.8952 −0.559602
\(532\) 0 0
\(533\) −28.5625 −1.23718
\(534\) 0 0
\(535\) −15.4037 −0.665959
\(536\) 0 0
\(537\) 9.14897 0.394807
\(538\) 0 0
\(539\) −15.2582 −0.657217
\(540\) 0 0
\(541\) −29.6947 −1.27667 −0.638336 0.769757i \(-0.720377\pi\)
−0.638336 + 0.769757i \(0.720377\pi\)
\(542\) 0 0
\(543\) 12.2259 0.524666
\(544\) 0 0
\(545\) 56.4269 2.41706
\(546\) 0 0
\(547\) 3.38735 0.144833 0.0724163 0.997374i \(-0.476929\pi\)
0.0724163 + 0.997374i \(0.476929\pi\)
\(548\) 0 0
\(549\) 11.2877 0.481747
\(550\) 0 0
\(551\) −16.0531 −0.683887
\(552\) 0 0
\(553\) 20.0623 0.853135
\(554\) 0 0
\(555\) −18.9812 −0.805707
\(556\) 0 0
\(557\) 5.24723 0.222332 0.111166 0.993802i \(-0.464541\pi\)
0.111166 + 0.993802i \(0.464541\pi\)
\(558\) 0 0
\(559\) 31.9429 1.35104
\(560\) 0 0
\(561\) −9.54074 −0.402810
\(562\) 0 0
\(563\) −13.1413 −0.553840 −0.276920 0.960893i \(-0.589314\pi\)
−0.276920 + 0.960893i \(0.589314\pi\)
\(564\) 0 0
\(565\) 31.0808 1.30758
\(566\) 0 0
\(567\) 1.70865 0.0717567
\(568\) 0 0
\(569\) 27.1300 1.13735 0.568675 0.822562i \(-0.307456\pi\)
0.568675 + 0.822562i \(0.307456\pi\)
\(570\) 0 0
\(571\) 8.07458 0.337911 0.168955 0.985624i \(-0.445961\pi\)
0.168955 + 0.985624i \(0.445961\pi\)
\(572\) 0 0
\(573\) −16.6142 −0.694068
\(574\) 0 0
\(575\) 54.6878 2.28064
\(576\) 0 0
\(577\) 16.7649 0.697931 0.348965 0.937136i \(-0.386533\pi\)
0.348965 + 0.937136i \(0.386533\pi\)
\(578\) 0 0
\(579\) 7.78696 0.323615
\(580\) 0 0
\(581\) −26.7739 −1.11077
\(582\) 0 0
\(583\) −12.3096 −0.509812
\(584\) 0 0
\(585\) 19.1501 0.791758
\(586\) 0 0
\(587\) 36.1028 1.49012 0.745061 0.666996i \(-0.232420\pi\)
0.745061 + 0.666996i \(0.232420\pi\)
\(588\) 0 0
\(589\) −61.2655 −2.52440
\(590\) 0 0
\(591\) 11.7525 0.483431
\(592\) 0 0
\(593\) 28.6934 1.17830 0.589148 0.808025i \(-0.299463\pi\)
0.589148 + 0.808025i \(0.299463\pi\)
\(594\) 0 0
\(595\) 15.8040 0.647901
\(596\) 0 0
\(597\) 6.51873 0.266794
\(598\) 0 0
\(599\) −24.2332 −0.990141 −0.495071 0.868853i \(-0.664858\pi\)
−0.495071 + 0.868853i \(0.664858\pi\)
\(600\) 0 0
\(601\) −29.5123 −1.20383 −0.601917 0.798559i \(-0.705596\pi\)
−0.601917 + 0.798559i \(0.705596\pi\)
\(602\) 0 0
\(603\) 6.08146 0.247656
\(604\) 0 0
\(605\) −10.8112 −0.439539
\(606\) 0 0
\(607\) −32.1543 −1.30510 −0.652552 0.757744i \(-0.726302\pi\)
−0.652552 + 0.757744i \(0.726302\pi\)
\(608\) 0 0
\(609\) 3.59028 0.145486
\(610\) 0 0
\(611\) 21.3119 0.862188
\(612\) 0 0
\(613\) 15.7926 0.637857 0.318928 0.947779i \(-0.396677\pi\)
0.318928 + 0.947779i \(0.396677\pi\)
\(614\) 0 0
\(615\) −19.6005 −0.790370
\(616\) 0 0
\(617\) −17.6167 −0.709220 −0.354610 0.935014i \(-0.615386\pi\)
−0.354610 + 0.935014i \(0.615386\pi\)
\(618\) 0 0
\(619\) 2.58374 0.103849 0.0519246 0.998651i \(-0.483464\pi\)
0.0519246 + 0.998651i \(0.483464\pi\)
\(620\) 0 0
\(621\) 6.71726 0.269554
\(622\) 0 0
\(623\) 4.21829 0.169002
\(624\) 0 0
\(625\) 0.575264 0.0230105
\(626\) 0 0
\(627\) −28.5677 −1.14088
\(628\) 0 0
\(629\) −13.3597 −0.532684
\(630\) 0 0
\(631\) −20.0988 −0.800120 −0.400060 0.916489i \(-0.631011\pi\)
−0.400060 + 0.916489i \(0.631011\pi\)
\(632\) 0 0
\(633\) 13.2143 0.525222
\(634\) 0 0
\(635\) −74.7364 −2.96582
\(636\) 0 0
\(637\) 21.5558 0.854070
\(638\) 0 0
\(639\) −4.88309 −0.193172
\(640\) 0 0
\(641\) −35.2904 −1.39389 −0.696943 0.717126i \(-0.745457\pi\)
−0.696943 + 0.717126i \(0.745457\pi\)
\(642\) 0 0
\(643\) −12.1897 −0.480715 −0.240357 0.970684i \(-0.577265\pi\)
−0.240357 + 0.970684i \(0.577265\pi\)
\(644\) 0 0
\(645\) 21.9202 0.863109
\(646\) 0 0
\(647\) −9.13850 −0.359272 −0.179636 0.983733i \(-0.557492\pi\)
−0.179636 + 0.983733i \(0.557492\pi\)
\(648\) 0 0
\(649\) −48.2188 −1.89275
\(650\) 0 0
\(651\) 13.7020 0.537025
\(652\) 0 0
\(653\) 10.3695 0.405792 0.202896 0.979200i \(-0.434965\pi\)
0.202896 + 0.979200i \(0.434965\pi\)
\(654\) 0 0
\(655\) 10.8973 0.425794
\(656\) 0 0
\(657\) −3.70680 −0.144616
\(658\) 0 0
\(659\) 41.5378 1.61808 0.809042 0.587751i \(-0.199987\pi\)
0.809042 + 0.587751i \(0.199987\pi\)
\(660\) 0 0
\(661\) 45.3751 1.76489 0.882443 0.470419i \(-0.155897\pi\)
0.882443 + 0.470419i \(0.155897\pi\)
\(662\) 0 0
\(663\) 13.4785 0.523462
\(664\) 0 0
\(665\) 47.3217 1.83506
\(666\) 0 0
\(667\) 14.1145 0.546517
\(668\) 0 0
\(669\) −15.9483 −0.616595
\(670\) 0 0
\(671\) 42.2080 1.62942
\(672\) 0 0
\(673\) −0.658734 −0.0253923 −0.0126962 0.999919i \(-0.504041\pi\)
−0.0126962 + 0.999919i \(0.504041\pi\)
\(674\) 0 0
\(675\) 8.14139 0.313362
\(676\) 0 0
\(677\) 1.87746 0.0721566 0.0360783 0.999349i \(-0.488513\pi\)
0.0360783 + 0.999349i \(0.488513\pi\)
\(678\) 0 0
\(679\) −8.13708 −0.312272
\(680\) 0 0
\(681\) 1.27727 0.0489451
\(682\) 0 0
\(683\) 39.0783 1.49529 0.747644 0.664099i \(-0.231185\pi\)
0.747644 + 0.664099i \(0.231185\pi\)
\(684\) 0 0
\(685\) 54.1438 2.06873
\(686\) 0 0
\(687\) 2.75137 0.104971
\(688\) 0 0
\(689\) 17.3902 0.662514
\(690\) 0 0
\(691\) −18.5443 −0.705458 −0.352729 0.935725i \(-0.614746\pi\)
−0.352729 + 0.935725i \(0.614746\pi\)
\(692\) 0 0
\(693\) 6.38916 0.242704
\(694\) 0 0
\(695\) 68.5273 2.59939
\(696\) 0 0
\(697\) −13.7956 −0.522545
\(698\) 0 0
\(699\) −2.73207 −0.103337
\(700\) 0 0
\(701\) 7.92644 0.299378 0.149689 0.988733i \(-0.452173\pi\)
0.149689 + 0.988733i \(0.452173\pi\)
\(702\) 0 0
\(703\) −40.0026 −1.50873
\(704\) 0 0
\(705\) 14.6249 0.550806
\(706\) 0 0
\(707\) −25.6037 −0.962927
\(708\) 0 0
\(709\) −48.5390 −1.82292 −0.911461 0.411387i \(-0.865044\pi\)
−0.911461 + 0.411387i \(0.865044\pi\)
\(710\) 0 0
\(711\) 11.7416 0.440343
\(712\) 0 0
\(713\) 53.8670 2.01733
\(714\) 0 0
\(715\) 71.6077 2.67798
\(716\) 0 0
\(717\) −27.1609 −1.01434
\(718\) 0 0
\(719\) 5.98110 0.223057 0.111529 0.993761i \(-0.464425\pi\)
0.111529 + 0.993761i \(0.464425\pi\)
\(720\) 0 0
\(721\) −4.10424 −0.152850
\(722\) 0 0
\(723\) −9.73684 −0.362117
\(724\) 0 0
\(725\) 17.1070 0.635336
\(726\) 0 0
\(727\) −0.978393 −0.0362866 −0.0181433 0.999835i \(-0.505776\pi\)
−0.0181433 + 0.999835i \(0.505776\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 15.4283 0.570635
\(732\) 0 0
\(733\) 48.7386 1.80020 0.900101 0.435681i \(-0.143492\pi\)
0.900101 + 0.435681i \(0.143492\pi\)
\(734\) 0 0
\(735\) 14.7923 0.545620
\(736\) 0 0
\(737\) 22.7404 0.837653
\(738\) 0 0
\(739\) −18.8810 −0.694549 −0.347275 0.937763i \(-0.612893\pi\)
−0.347275 + 0.937763i \(0.612893\pi\)
\(740\) 0 0
\(741\) 40.3585 1.48261
\(742\) 0 0
\(743\) 9.54298 0.350098 0.175049 0.984560i \(-0.443992\pi\)
0.175049 + 0.984560i \(0.443992\pi\)
\(744\) 0 0
\(745\) 3.49790 0.128153
\(746\) 0 0
\(747\) −15.6696 −0.573321
\(748\) 0 0
\(749\) 7.26035 0.265287
\(750\) 0 0
\(751\) −38.7014 −1.41223 −0.706117 0.708096i \(-0.749555\pi\)
−0.706117 + 0.708096i \(0.749555\pi\)
\(752\) 0 0
\(753\) −18.8215 −0.685893
\(754\) 0 0
\(755\) 68.5657 2.49536
\(756\) 0 0
\(757\) 50.4031 1.83193 0.915965 0.401257i \(-0.131427\pi\)
0.915965 + 0.401257i \(0.131427\pi\)
\(758\) 0 0
\(759\) 25.1178 0.911718
\(760\) 0 0
\(761\) 23.1763 0.840141 0.420070 0.907492i \(-0.362005\pi\)
0.420070 + 0.907492i \(0.362005\pi\)
\(762\) 0 0
\(763\) −26.5962 −0.962848
\(764\) 0 0
\(765\) 9.24939 0.334412
\(766\) 0 0
\(767\) 68.1202 2.45968
\(768\) 0 0
\(769\) 27.0045 0.973808 0.486904 0.873455i \(-0.338126\pi\)
0.486904 + 0.873455i \(0.338126\pi\)
\(770\) 0 0
\(771\) 2.66522 0.0959855
\(772\) 0 0
\(773\) 42.4965 1.52849 0.764246 0.644925i \(-0.223112\pi\)
0.764246 + 0.644925i \(0.223112\pi\)
\(774\) 0 0
\(775\) 65.2873 2.34519
\(776\) 0 0
\(777\) 8.94658 0.320957
\(778\) 0 0
\(779\) −41.3079 −1.48001
\(780\) 0 0
\(781\) −18.2593 −0.653370
\(782\) 0 0
\(783\) 2.10123 0.0750920
\(784\) 0 0
\(785\) −12.2046 −0.435599
\(786\) 0 0
\(787\) 45.2673 1.61361 0.806803 0.590821i \(-0.201196\pi\)
0.806803 + 0.590821i \(0.201196\pi\)
\(788\) 0 0
\(789\) −9.38514 −0.334120
\(790\) 0 0
\(791\) −14.6496 −0.520880
\(792\) 0 0
\(793\) −59.6286 −2.11747
\(794\) 0 0
\(795\) 11.9337 0.423245
\(796\) 0 0
\(797\) −10.7500 −0.380783 −0.190392 0.981708i \(-0.560976\pi\)
−0.190392 + 0.981708i \(0.560976\pi\)
\(798\) 0 0
\(799\) 10.2936 0.364160
\(800\) 0 0
\(801\) 2.46878 0.0872300
\(802\) 0 0
\(803\) −13.8608 −0.489137
\(804\) 0 0
\(805\) −41.6070 −1.46645
\(806\) 0 0
\(807\) 11.2658 0.396576
\(808\) 0 0
\(809\) 21.5809 0.758743 0.379371 0.925244i \(-0.376140\pi\)
0.379371 + 0.925244i \(0.376140\pi\)
\(810\) 0 0
\(811\) 28.3625 0.995943 0.497972 0.867193i \(-0.334078\pi\)
0.497972 + 0.867193i \(0.334078\pi\)
\(812\) 0 0
\(813\) 19.4755 0.683037
\(814\) 0 0
\(815\) −2.47038 −0.0865338
\(816\) 0 0
\(817\) 46.1966 1.61622
\(818\) 0 0
\(819\) −9.02617 −0.315400
\(820\) 0 0
\(821\) −20.3162 −0.709039 −0.354519 0.935049i \(-0.615356\pi\)
−0.354519 + 0.935049i \(0.615356\pi\)
\(822\) 0 0
\(823\) −14.9449 −0.520948 −0.260474 0.965481i \(-0.583879\pi\)
−0.260474 + 0.965481i \(0.583879\pi\)
\(824\) 0 0
\(825\) 30.4430 1.05989
\(826\) 0 0
\(827\) −19.9501 −0.693734 −0.346867 0.937914i \(-0.612754\pi\)
−0.346867 + 0.937914i \(0.612754\pi\)
\(828\) 0 0
\(829\) −52.0430 −1.80753 −0.903765 0.428029i \(-0.859208\pi\)
−0.903765 + 0.428029i \(0.859208\pi\)
\(830\) 0 0
\(831\) 7.75098 0.268878
\(832\) 0 0
\(833\) 10.4113 0.360731
\(834\) 0 0
\(835\) 3.62511 0.125452
\(836\) 0 0
\(837\) 8.01919 0.277184
\(838\) 0 0
\(839\) −16.3093 −0.563058 −0.281529 0.959553i \(-0.590842\pi\)
−0.281529 + 0.959553i \(0.590842\pi\)
\(840\) 0 0
\(841\) −24.5848 −0.847752
\(842\) 0 0
\(843\) −19.9976 −0.688753
\(844\) 0 0
\(845\) −54.0362 −1.85890
\(846\) 0 0
\(847\) 5.09575 0.175092
\(848\) 0 0
\(849\) −1.99842 −0.0685855
\(850\) 0 0
\(851\) 35.1718 1.20567
\(852\) 0 0
\(853\) −0.499221 −0.0170930 −0.00854649 0.999963i \(-0.502720\pi\)
−0.00854649 + 0.999963i \(0.502720\pi\)
\(854\) 0 0
\(855\) 27.6953 0.947160
\(856\) 0 0
\(857\) 19.5417 0.667531 0.333765 0.942656i \(-0.391681\pi\)
0.333765 + 0.942656i \(0.391681\pi\)
\(858\) 0 0
\(859\) −5.17233 −0.176478 −0.0882389 0.996099i \(-0.528124\pi\)
−0.0882389 + 0.996099i \(0.528124\pi\)
\(860\) 0 0
\(861\) 9.23850 0.314847
\(862\) 0 0
\(863\) −51.4457 −1.75123 −0.875615 0.483009i \(-0.839544\pi\)
−0.875615 + 0.483009i \(0.839544\pi\)
\(864\) 0 0
\(865\) −18.2762 −0.621409
\(866\) 0 0
\(867\) −10.4899 −0.356257
\(868\) 0 0
\(869\) 43.9052 1.48938
\(870\) 0 0
\(871\) −32.1261 −1.08855
\(872\) 0 0
\(873\) −4.76227 −0.161179
\(874\) 0 0
\(875\) −19.4579 −0.657798
\(876\) 0 0
\(877\) −24.5013 −0.827350 −0.413675 0.910425i \(-0.635755\pi\)
−0.413675 + 0.910425i \(0.635755\pi\)
\(878\) 0 0
\(879\) −13.1462 −0.443409
\(880\) 0 0
\(881\) −14.3995 −0.485132 −0.242566 0.970135i \(-0.577989\pi\)
−0.242566 + 0.970135i \(0.577989\pi\)
\(882\) 0 0
\(883\) −5.38774 −0.181312 −0.0906559 0.995882i \(-0.528896\pi\)
−0.0906559 + 0.995882i \(0.528896\pi\)
\(884\) 0 0
\(885\) 46.7463 1.57136
\(886\) 0 0
\(887\) −7.97543 −0.267789 −0.133894 0.990996i \(-0.542748\pi\)
−0.133894 + 0.990996i \(0.542748\pi\)
\(888\) 0 0
\(889\) 35.2262 1.18145
\(890\) 0 0
\(891\) 3.73929 0.125271
\(892\) 0 0
\(893\) 30.8218 1.03141
\(894\) 0 0
\(895\) −33.1660 −1.10862
\(896\) 0 0
\(897\) −35.4847 −1.18480
\(898\) 0 0
\(899\) 16.8502 0.561985
\(900\) 0 0
\(901\) 8.39939 0.279824
\(902\) 0 0
\(903\) −10.3319 −0.343823
\(904\) 0 0
\(905\) −44.3204 −1.47326
\(906\) 0 0
\(907\) −47.9048 −1.59065 −0.795326 0.606182i \(-0.792700\pi\)
−0.795326 + 0.606182i \(0.792700\pi\)
\(908\) 0 0
\(909\) −14.9847 −0.497012
\(910\) 0 0
\(911\) −33.3844 −1.10607 −0.553037 0.833157i \(-0.686531\pi\)
−0.553037 + 0.833157i \(0.686531\pi\)
\(912\) 0 0
\(913\) −58.5932 −1.93915
\(914\) 0 0
\(915\) −40.9191 −1.35274
\(916\) 0 0
\(917\) −5.13634 −0.169617
\(918\) 0 0
\(919\) 4.31358 0.142292 0.0711459 0.997466i \(-0.477334\pi\)
0.0711459 + 0.997466i \(0.477334\pi\)
\(920\) 0 0
\(921\) −19.0378 −0.627317
\(922\) 0 0
\(923\) 25.7955 0.849070
\(924\) 0 0
\(925\) 42.6286 1.40162
\(926\) 0 0
\(927\) −2.40203 −0.0788930
\(928\) 0 0
\(929\) 39.8838 1.30854 0.654272 0.756259i \(-0.272975\pi\)
0.654272 + 0.756259i \(0.272975\pi\)
\(930\) 0 0
\(931\) 31.1745 1.02170
\(932\) 0 0
\(933\) −7.31774 −0.239572
\(934\) 0 0
\(935\) 34.5862 1.13109
\(936\) 0 0
\(937\) −24.4960 −0.800250 −0.400125 0.916461i \(-0.631033\pi\)
−0.400125 + 0.916461i \(0.631033\pi\)
\(938\) 0 0
\(939\) −21.5818 −0.704294
\(940\) 0 0
\(941\) 20.5676 0.670486 0.335243 0.942132i \(-0.391182\pi\)
0.335243 + 0.942132i \(0.391182\pi\)
\(942\) 0 0
\(943\) 36.3195 1.18272
\(944\) 0 0
\(945\) −6.19405 −0.201492
\(946\) 0 0
\(947\) −10.7521 −0.349395 −0.174697 0.984622i \(-0.555895\pi\)
−0.174697 + 0.984622i \(0.555895\pi\)
\(948\) 0 0
\(949\) 19.5816 0.635646
\(950\) 0 0
\(951\) −19.2702 −0.624880
\(952\) 0 0
\(953\) 32.8772 1.06500 0.532499 0.846431i \(-0.321253\pi\)
0.532499 + 0.846431i \(0.321253\pi\)
\(954\) 0 0
\(955\) 60.2282 1.94894
\(956\) 0 0
\(957\) 7.85713 0.253985
\(958\) 0 0
\(959\) −25.5201 −0.824086
\(960\) 0 0
\(961\) 33.3074 1.07443
\(962\) 0 0
\(963\) 4.24916 0.136927
\(964\) 0 0
\(965\) −28.2286 −0.908710
\(966\) 0 0
\(967\) 50.0286 1.60881 0.804405 0.594081i \(-0.202484\pi\)
0.804405 + 0.594081i \(0.202484\pi\)
\(968\) 0 0
\(969\) 19.4930 0.626205
\(970\) 0 0
\(971\) −6.37248 −0.204503 −0.102251 0.994759i \(-0.532605\pi\)
−0.102251 + 0.994759i \(0.532605\pi\)
\(972\) 0 0
\(973\) −32.2996 −1.03548
\(974\) 0 0
\(975\) −43.0079 −1.37735
\(976\) 0 0
\(977\) 5.12270 0.163890 0.0819448 0.996637i \(-0.473887\pi\)
0.0819448 + 0.996637i \(0.473887\pi\)
\(978\) 0 0
\(979\) 9.23148 0.295040
\(980\) 0 0
\(981\) −15.5656 −0.496971
\(982\) 0 0
\(983\) −0.608262 −0.0194006 −0.00970028 0.999953i \(-0.503088\pi\)
−0.00970028 + 0.999953i \(0.503088\pi\)
\(984\) 0 0
\(985\) −42.6039 −1.35747
\(986\) 0 0
\(987\) −6.89330 −0.219416
\(988\) 0 0
\(989\) −40.6178 −1.29157
\(990\) 0 0
\(991\) 36.9745 1.17453 0.587267 0.809393i \(-0.300204\pi\)
0.587267 + 0.809393i \(0.300204\pi\)
\(992\) 0 0
\(993\) −27.0258 −0.857639
\(994\) 0 0
\(995\) −23.6311 −0.749156
\(996\) 0 0
\(997\) 59.1927 1.87465 0.937326 0.348454i \(-0.113293\pi\)
0.937326 + 0.348454i \(0.113293\pi\)
\(998\) 0 0
\(999\) 5.23604 0.165661
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 8016.2.a.z.1.1 8
4.3 odd 2 501.2.a.d.1.2 8
12.11 even 2 1503.2.a.f.1.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
501.2.a.d.1.2 8 4.3 odd 2
1503.2.a.f.1.7 8 12.11 even 2
8016.2.a.z.1.1 8 1.1 even 1 trivial