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Properties

Label 891.2.a.g.1.1
Level $891$
Weight $2$
Character 891.1
Self dual yes
Analytic conductor $7.115$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [891,2,Mod(1,891)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(891, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("891.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 891 = 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 891.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: \(7.11467082010\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 99)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 891.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} +4.00000 q^{7} -3.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} +4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{10} +1.00000 q^{11} -2.00000 q^{13} +4.00000 q^{14} -1.00000 q^{16} +4.00000 q^{17} +6.00000 q^{19} -1.00000 q^{20} +1.00000 q^{22} -4.00000 q^{23} -4.00000 q^{25} -2.00000 q^{26} -4.00000 q^{28} +6.00000 q^{29} +7.00000 q^{31} +5.00000 q^{32} +4.00000 q^{34} +4.00000 q^{35} +3.00000 q^{37} +6.00000 q^{38} -3.00000 q^{40} -2.00000 q^{41} +6.00000 q^{43} -1.00000 q^{44} -4.00000 q^{46} -7.00000 q^{47} +9.00000 q^{49} -4.00000 q^{50} +2.00000 q^{52} -9.00000 q^{53} +1.00000 q^{55} -12.0000 q^{56} +6.00000 q^{58} -7.00000 q^{59} +7.00000 q^{62} +7.00000 q^{64} -2.00000 q^{65} +11.0000 q^{67} -4.00000 q^{68} +4.00000 q^{70} +9.00000 q^{71} +4.00000 q^{73} +3.00000 q^{74} -6.00000 q^{76} +4.00000 q^{77} +8.00000 q^{79} -1.00000 q^{80} -2.00000 q^{82} -12.0000 q^{83} +4.00000 q^{85} +6.00000 q^{86} -3.00000 q^{88} +6.00000 q^{89} -8.00000 q^{91} +4.00000 q^{92} -7.00000 q^{94} +6.00000 q^{95} -19.0000 q^{97} +9.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 − 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 1.00000 0.447214 0.223607 − 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) −3.00000 −1.06066
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 − 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) −4.00000 −0.755929
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 7.00000 1.25724 0.628619 − 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) 4.00000 0.685994
\(35\) 4.00000 0.676123
\(36\) 0 0
\(37\) 3.00000 0.493197 0.246598 − 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 − 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −7.00000 −1.02105 −0.510527 − 0.859861i \(-0.670550\pi\)
−0.510527 + 0.859861i \(0.670550\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) −4.00000 −0.565685
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −9.00000 −1.23625 −0.618123 − 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) −12.0000 −1.60357
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) −7.00000 −0.911322 −0.455661 − 0.890153i \(-0.650597\pi\)
−0.455661 + 0.890153i \(0.650597\pi\)
\(60\) 0 0
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 7.00000 0.889001
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 11.0000 1.34386 0.671932 − 0.740613i \(-0.265465\pi\)
0.671932 + 0.740613i \(0.265465\pi\)
\(68\) −4.00000 −0.485071
\(69\) 0 0
\(70\) 4.00000 0.478091
\(71\) 9.00000 1.06810 0.534052 − 0.845452i \(-0.320669\pi\)
0.534052 + 0.845452i \(0.320669\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 − 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 3.00000 0.348743
\(75\) 0 0
\(76\) −6.00000 −0.688247
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −2.00000 −0.220863
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 6.00000 0.646997
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) −7.00000 −0.721995
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) −19.0000 −1.92916 −0.964579 − 0.263795i \(-0.915026\pi\)
−0.964579 + 0.263795i \(0.915026\pi\)
\(98\) 9.00000 0.909137
\(99\) 0 0
\(100\) 4.00000 0.400000
\(101\) −4.00000 −0.398015 −0.199007 − 0.979998i \(-0.563772\pi\)
−0.199007 + 0.979998i \(0.563772\pi\)
\(102\) 0 0
\(103\) −13.0000 −1.28093 −0.640464 − 0.767988i \(-0.721258\pi\)
−0.640464 + 0.767988i \(0.721258\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) −18.0000 −1.74013 −0.870063 − 0.492941i \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 − 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 1.00000 0.0953463
\(111\) 0 0
\(112\) −4.00000 −0.377964
\(113\) −15.0000 −1.41108 −0.705541 − 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) −7.00000 −0.644402
\(119\) 16.0000 1.46672
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) −7.00000 −0.628619
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 − 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) −3.00000 −0.265165
\(129\) 0 0
\(130\) −2.00000 −0.175412
\(131\) −6.00000 −0.524222 −0.262111 − 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) 24.0000 2.08106
\(134\) 11.0000 0.950255
\(135\) 0 0
\(136\) −12.0000 −1.02899
\(137\) 5.00000 0.427179 0.213589 − 0.976924i \(-0.431485\pi\)
0.213589 + 0.976924i \(0.431485\pi\)
\(138\) 0 0
\(139\) −2.00000 −0.169638 −0.0848189 − 0.996396i \(-0.527031\pi\)
−0.0848189 + 0.996396i \(0.527031\pi\)
\(140\) −4.00000 −0.338062
\(141\) 0 0
\(142\) 9.00000 0.755263
\(143\) −2.00000 −0.167248
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) 4.00000 0.331042
\(147\) 0 0
\(148\) −3.00000 −0.246598
\(149\) −4.00000 −0.327693 −0.163846 − 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 − 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) −18.0000 −1.45999
\(153\) 0 0
\(154\) 4.00000 0.322329
\(155\) 7.00000 0.562254
\(156\) 0 0
\(157\) −7.00000 −0.558661 −0.279330 − 0.960195i \(-0.590112\pi\)
−0.279330 + 0.960195i \(0.590112\pi\)
\(158\) 8.00000 0.636446
\(159\) 0 0
\(160\) 5.00000 0.395285
\(161\) −16.0000 −1.26098
\(162\) 0 0
\(163\) −11.0000 −0.861586 −0.430793 − 0.902451i \(-0.641766\pi\)
−0.430793 + 0.902451i \(0.641766\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 12.0000 0.928588 0.464294 − 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 4.00000 0.306786
\(171\) 0 0
\(172\) −6.00000 −0.457496
\(173\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) −16.0000 −1.20949
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) −9.00000 −0.672692 −0.336346 − 0.941739i \(-0.609191\pi\)
−0.336346 + 0.941739i \(0.609191\pi\)
\(180\) 0 0
\(181\) 7.00000 0.520306 0.260153 − 0.965567i \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) −8.00000 −0.592999
\(183\) 0 0
\(184\) 12.0000 0.884652
\(185\) 3.00000 0.220564
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) 7.00000 0.510527
\(189\) 0 0
\(190\) 6.00000 0.435286
\(191\) −13.0000 −0.940647 −0.470323 − 0.882494i \(-0.655863\pi\)
−0.470323 + 0.882494i \(0.655863\pi\)
\(192\) 0 0
\(193\) 4.00000 0.287926 0.143963 − 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) −19.0000 −1.36412
\(195\) 0 0
\(196\) −9.00000 −0.642857
\(197\) 16.0000 1.13995 0.569976 − 0.821661i \(-0.306952\pi\)
0.569976 + 0.821661i \(0.306952\pi\)
\(198\) 0 0
\(199\) −3.00000 −0.212664 −0.106332 − 0.994331i \(-0.533911\pi\)
−0.106332 + 0.994331i \(0.533911\pi\)
\(200\) 12.0000 0.848528
\(201\) 0 0
\(202\) −4.00000 −0.281439
\(203\) 24.0000 1.68447
\(204\) 0 0
\(205\) −2.00000 −0.139686
\(206\) −13.0000 −0.905753
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) 6.00000 0.413057 0.206529 − 0.978441i \(-0.433783\pi\)
0.206529 + 0.978441i \(0.433783\pi\)
\(212\) 9.00000 0.618123
\(213\) 0 0
\(214\) −18.0000 −1.23045
\(215\) 6.00000 0.409197
\(216\) 0 0
\(217\) 28.0000 1.90076
\(218\) −2.00000 −0.135457
\(219\) 0 0
\(220\) −1.00000 −0.0674200
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 − 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 20.0000 1.33631
\(225\) 0 0
\(226\) −15.0000 −0.997785
\(227\) −18.0000 −1.19470 −0.597351 − 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) 0 0
\(229\) −6.00000 −0.396491 −0.198246 − 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) −4.00000 −0.263752
\(231\) 0 0
\(232\) −18.0000 −1.18176
\(233\) 24.0000 1.57229 0.786146 − 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) 0 0
\(235\) −7.00000 −0.456630
\(236\) 7.00000 0.455661
\(237\) 0 0
\(238\) 16.0000 1.03713
\(239\) 6.00000 0.388108 0.194054 − 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) 0 0
\(245\) 9.00000 0.574989
\(246\) 0 0
\(247\) −12.0000 −0.763542
\(248\) −21.0000 −1.33350
\(249\) 0 0
\(250\) −9.00000 −0.569210
\(251\) 28.0000 1.76734 0.883672 − 0.468106i \(-0.155064\pi\)
0.883672 + 0.468106i \(0.155064\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 2.00000 0.125491
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 22.0000 1.37232 0.686161 − 0.727450i \(-0.259294\pi\)
0.686161 + 0.727450i \(0.259294\pi\)
\(258\) 0 0
\(259\) 12.0000 0.745644
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) −6.00000 −0.370681
\(263\) −16.0000 −0.986602 −0.493301 − 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 0 0
\(265\) −9.00000 −0.552866
\(266\) 24.0000 1.47153
\(267\) 0 0
\(268\) −11.0000 −0.671932
\(269\) −17.0000 −1.03651 −0.518254 − 0.855227i \(-0.673418\pi\)
−0.518254 + 0.855227i \(0.673418\pi\)
\(270\) 0 0
\(271\) −22.0000 −1.33640 −0.668202 − 0.743980i \(-0.732936\pi\)
−0.668202 + 0.743980i \(0.732936\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) 5.00000 0.302061
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 − 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) −2.00000 −0.119952
\(279\) 0 0
\(280\) −12.0000 −0.717137
\(281\) 6.00000 0.357930 0.178965 − 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 28.0000 1.66443 0.832214 − 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) −9.00000 −0.534052
\(285\) 0 0
\(286\) −2.00000 −0.118262
\(287\) −8.00000 −0.472225
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 6.00000 0.352332
\(291\) 0 0
\(292\) −4.00000 −0.234082
\(293\) 12.0000 0.701047 0.350524 − 0.936554i \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(294\) 0 0
\(295\) −7.00000 −0.407556
\(296\) −9.00000 −0.523114
\(297\) 0 0
\(298\) −4.00000 −0.231714
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) 24.0000 1.38334
\(302\) −16.0000 −0.920697
\(303\) 0 0
\(304\) −6.00000 −0.344124
\(305\) 0 0
\(306\) 0 0
\(307\) −28.0000 −1.59804 −0.799022 − 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) −4.00000 −0.227921
\(309\) 0 0
\(310\) 7.00000 0.397573
\(311\) −21.0000 −1.19080 −0.595400 − 0.803429i \(-0.703007\pi\)
−0.595400 + 0.803429i \(0.703007\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 − 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) −7.00000 −0.395033
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 22.0000 1.23564 0.617822 − 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) 7.00000 0.391312
\(321\) 0 0
\(322\) −16.0000 −0.891645
\(323\) 24.0000 1.33540
\(324\) 0 0
\(325\) 8.00000 0.443760
\(326\) −11.0000 −0.609234
\(327\) 0 0
\(328\) 6.00000 0.331295
\(329\) −28.0000 −1.54369
\(330\) 0 0
\(331\) 1.00000 0.0549650 0.0274825 − 0.999622i \(-0.491251\pi\)
0.0274825 + 0.999622i \(0.491251\pi\)
\(332\) 12.0000 0.658586
\(333\) 0 0
\(334\) 12.0000 0.656611
\(335\) 11.0000 0.600994
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 − 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) −4.00000 −0.216930
\(341\) 7.00000 0.379071
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) −18.0000 −0.970495
\(345\) 0 0
\(346\) 0 0
\(347\) 4.00000 0.214731 0.107366 − 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) 0 0
\(349\) 30.0000 1.60586 0.802932 − 0.596071i \(-0.203272\pi\)
0.802932 + 0.596071i \(0.203272\pi\)
\(350\) −16.0000 −0.855236
\(351\) 0 0
\(352\) 5.00000 0.266501
\(353\) −18.0000 −0.958043 −0.479022 − 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) 9.00000 0.477670
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) −9.00000 −0.475665
\(359\) −32.0000 −1.68890 −0.844448 − 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 7.00000 0.367912
\(363\) 0 0
\(364\) 8.00000 0.419314
\(365\) 4.00000 0.209370
\(366\) 0 0
\(367\) −11.0000 −0.574195 −0.287098 − 0.957901i \(-0.592690\pi\)
−0.287098 + 0.957901i \(0.592690\pi\)
\(368\) 4.00000 0.208514
\(369\) 0 0
\(370\) 3.00000 0.155963
\(371\) −36.0000 −1.86903
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 − 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) 21.0000 1.08299
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 − 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) −6.00000 −0.307794
\(381\) 0 0
\(382\) −13.0000 −0.665138
\(383\) 29.0000 1.48183 0.740915 − 0.671598i \(-0.234392\pi\)
0.740915 + 0.671598i \(0.234392\pi\)
\(384\) 0 0
\(385\) 4.00000 0.203859
\(386\) 4.00000 0.203595
\(387\) 0 0
\(388\) 19.0000 0.964579
\(389\) 33.0000 1.67317 0.836583 − 0.547840i \(-0.184550\pi\)
0.836583 + 0.547840i \(0.184550\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) −27.0000 −1.36371
\(393\) 0 0
\(394\) 16.0000 0.806068
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) 13.0000 0.652451 0.326226 − 0.945292i \(-0.394223\pi\)
0.326226 + 0.945292i \(0.394223\pi\)
\(398\) −3.00000 −0.150376
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) 17.0000 0.848939 0.424470 − 0.905442i \(-0.360461\pi\)
0.424470 + 0.905442i \(0.360461\pi\)
\(402\) 0 0
\(403\) −14.0000 −0.697390
\(404\) 4.00000 0.199007
\(405\) 0 0
\(406\) 24.0000 1.19110
\(407\) 3.00000 0.148704
\(408\) 0 0
\(409\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) −2.00000 −0.0987730
\(411\) 0 0
\(412\) 13.0000 0.640464
\(413\) −28.0000 −1.37779
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) −10.0000 −0.490290
\(417\) 0 0
\(418\) 6.00000 0.293470
\(419\) 5.00000 0.244266 0.122133 − 0.992514i \(-0.461027\pi\)
0.122133 + 0.992514i \(0.461027\pi\)
\(420\) 0 0
\(421\) 13.0000 0.633581 0.316791 − 0.948495i \(-0.397395\pi\)
0.316791 + 0.948495i \(0.397395\pi\)
\(422\) 6.00000 0.292075
\(423\) 0 0
\(424\) 27.0000 1.31124
\(425\) −16.0000 −0.776114
\(426\) 0 0
\(427\) 0 0
\(428\) 18.0000 0.870063
\(429\) 0 0
\(430\) 6.00000 0.289346
\(431\) −30.0000 −1.44505 −0.722525 − 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 − 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 28.0000 1.34404
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) −24.0000 −1.14808
\(438\) 0 0
\(439\) −14.0000 −0.668184 −0.334092 − 0.942541i \(-0.608430\pi\)
−0.334092 + 0.942541i \(0.608430\pi\)
\(440\) −3.00000 −0.143019
\(441\) 0 0
\(442\) −8.00000 −0.380521
\(443\) 1.00000 0.0475114 0.0237557 − 0.999718i \(-0.492438\pi\)
0.0237557 + 0.999718i \(0.492438\pi\)
\(444\) 0 0
\(445\) 6.00000 0.284427
\(446\) 16.0000 0.757622
\(447\) 0 0
\(448\) 28.0000 1.32288
\(449\) 23.0000 1.08544 0.542719 − 0.839915i \(-0.317395\pi\)
0.542719 + 0.839915i \(0.317395\pi\)
\(450\) 0 0
\(451\) −2.00000 −0.0941763
\(452\) 15.0000 0.705541
\(453\) 0 0
\(454\) −18.0000 −0.844782
\(455\) −8.00000 −0.375046
\(456\) 0 0
\(457\) −6.00000 −0.280668 −0.140334 − 0.990104i \(-0.544818\pi\)
−0.140334 + 0.990104i \(0.544818\pi\)
\(458\) −6.00000 −0.280362
\(459\) 0 0
\(460\) 4.00000 0.186501
\(461\) −12.0000 −0.558896 −0.279448 − 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 − 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 24.0000 1.11178
\(467\) 27.0000 1.24941 0.624705 − 0.780860i \(-0.285219\pi\)
0.624705 + 0.780860i \(0.285219\pi\)
\(468\) 0 0
\(469\) 44.0000 2.03173
\(470\) −7.00000 −0.322886
\(471\) 0 0
\(472\) 21.0000 0.966603
\(473\) 6.00000 0.275880
\(474\) 0 0
\(475\) −24.0000 −1.10120
\(476\) −16.0000 −0.733359
\(477\) 0 0
\(478\) 6.00000 0.274434
\(479\) −4.00000 −0.182765 −0.0913823 − 0.995816i \(-0.529129\pi\)
−0.0913823 + 0.995816i \(0.529129\pi\)
\(480\) 0 0
\(481\) −6.00000 −0.273576
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) −1.00000 −0.0454545
\(485\) −19.0000 −0.862746
\(486\) 0 0
\(487\) −37.0000 −1.67663 −0.838315 − 0.545186i \(-0.816459\pi\)
−0.838315 + 0.545186i \(0.816459\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 9.00000 0.406579
\(491\) −8.00000 −0.361035 −0.180517 − 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) 0 0
\(493\) 24.0000 1.08091
\(494\) −12.0000 −0.539906
\(495\) 0 0
\(496\) −7.00000 −0.314309
\(497\) 36.0000 1.61482
\(498\) 0 0
\(499\) −25.0000 −1.11915 −0.559577 − 0.828778i \(-0.689036\pi\)
−0.559577 + 0.828778i \(0.689036\pi\)
\(500\) 9.00000 0.402492
\(501\) 0 0
\(502\) 28.0000 1.24970
\(503\) −14.0000 −0.624229 −0.312115 − 0.950044i \(-0.601037\pi\)
−0.312115 + 0.950044i \(0.601037\pi\)
\(504\) 0 0
\(505\) −4.00000 −0.177998
\(506\) −4.00000 −0.177822
\(507\) 0 0
\(508\) −2.00000 −0.0887357
\(509\) −6.00000 −0.265945 −0.132973 − 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 16.0000 0.707798
\(512\) −11.0000 −0.486136
\(513\) 0 0
\(514\) 22.0000 0.970378
\(515\) −13.0000 −0.572848
\(516\) 0 0
\(517\) −7.00000 −0.307860
\(518\) 12.0000 0.527250
\(519\) 0 0
\(520\) 6.00000 0.263117
\(521\) −3.00000 −0.131432 −0.0657162 − 0.997838i \(-0.520933\pi\)
−0.0657162 + 0.997838i \(0.520933\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 − 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 6.00000 0.262111
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) 28.0000 1.21970
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) −9.00000 −0.390935
\(531\) 0 0
\(532\) −24.0000 −1.04053
\(533\) 4.00000 0.173259
\(534\) 0 0
\(535\) −18.0000 −0.778208
\(536\) −33.0000 −1.42538
\(537\) 0 0
\(538\) −17.0000 −0.732922
\(539\) 9.00000 0.387657
\(540\) 0 0
\(541\) 10.0000 0.429934 0.214967 − 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) −22.0000 −0.944981
\(543\) 0 0
\(544\) 20.0000 0.857493
\(545\) −2.00000 −0.0856706
\(546\) 0 0
\(547\) −22.0000 −0.940652 −0.470326 − 0.882493i \(-0.655864\pi\)
−0.470326 + 0.882493i \(0.655864\pi\)
\(548\) −5.00000 −0.213589
\(549\) 0 0
\(550\) −4.00000 −0.170561
\(551\) 36.0000 1.53365
\(552\) 0 0
\(553\) 32.0000 1.36078
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) 2.00000 0.0848189
\(557\) 40.0000 1.69485 0.847427 − 0.530912i \(-0.178150\pi\)
0.847427 + 0.530912i \(0.178150\pi\)
\(558\) 0 0
\(559\) −12.0000 −0.507546
\(560\) −4.00000 −0.169031
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) −14.0000 −0.590030 −0.295015 − 0.955493i \(-0.595325\pi\)
−0.295015 + 0.955493i \(0.595325\pi\)
\(564\) 0 0
\(565\) −15.0000 −0.631055
\(566\) 28.0000 1.17693
\(567\) 0 0
\(568\) −27.0000 −1.13289
\(569\) −6.00000 −0.251533 −0.125767 − 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 − 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 2.00000 0.0836242
\(573\) 0 0
\(574\) −8.00000 −0.333914
\(575\) 16.0000 0.667246
\(576\) 0 0
\(577\) −3.00000 −0.124892 −0.0624458 − 0.998048i \(-0.519890\pi\)
−0.0624458 + 0.998048i \(0.519890\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 0 0
\(580\) −6.00000 −0.249136
\(581\) −48.0000 −1.99138
\(582\) 0 0
\(583\) −9.00000 −0.372742
\(584\) −12.0000 −0.496564
\(585\) 0 0
\(586\) 12.0000 0.495715
\(587\) −5.00000 −0.206372 −0.103186 − 0.994662i \(-0.532904\pi\)
−0.103186 + 0.994662i \(0.532904\pi\)
\(588\) 0 0
\(589\) 42.0000 1.73058
\(590\) −7.00000 −0.288185
\(591\) 0 0
\(592\) −3.00000 −0.123299
\(593\) −22.0000 −0.903432 −0.451716 − 0.892162i \(-0.649188\pi\)
−0.451716 + 0.892162i \(0.649188\pi\)
\(594\) 0 0
\(595\) 16.0000 0.655936
\(596\) 4.00000 0.163846
\(597\) 0 0
\(598\) 8.00000 0.327144
\(599\) 4.00000 0.163436 0.0817178 − 0.996656i \(-0.473959\pi\)
0.0817178 + 0.996656i \(0.473959\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 − 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 24.0000 0.978167
\(603\) 0 0
\(604\) 16.0000 0.651031
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) 8.00000 0.324710 0.162355 − 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 30.0000 1.21666
\(609\) 0 0
\(610\) 0 0
\(611\) 14.0000 0.566379
\(612\) 0 0
\(613\) −16.0000 −0.646234 −0.323117 − 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) −28.0000 −1.12999
\(615\) 0 0
\(616\) −12.0000 −0.483494
\(617\) −15.0000 −0.603877 −0.301939 − 0.953327i \(-0.597634\pi\)
−0.301939 + 0.953327i \(0.597634\pi\)
\(618\) 0 0
\(619\) −25.0000 −1.00483 −0.502417 − 0.864625i \(-0.667556\pi\)
−0.502417 + 0.864625i \(0.667556\pi\)
\(620\) −7.00000 −0.281127
\(621\) 0 0
\(622\) −21.0000 −0.842023
\(623\) 24.0000 0.961540
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 14.0000 0.559553
\(627\) 0 0
\(628\) 7.00000 0.279330
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) 37.0000 1.47295 0.736473 − 0.676467i \(-0.236490\pi\)
0.736473 + 0.676467i \(0.236490\pi\)
\(632\) −24.0000 −0.954669
\(633\) 0 0
\(634\) 22.0000 0.873732
\(635\) 2.00000 0.0793676
\(636\) 0 0
\(637\) −18.0000 −0.713186
\(638\) 6.00000 0.237542
\(639\) 0 0
\(640\) −3.00000 −0.118585
\(641\) −30.0000 −1.18493 −0.592464 − 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) −16.0000 −0.630978 −0.315489 − 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) 16.0000 0.630488
\(645\) 0 0
\(646\) 24.0000 0.944267
\(647\) 32.0000 1.25805 0.629025 − 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) 0 0
\(649\) −7.00000 −0.274774
\(650\) 8.00000 0.313786
\(651\) 0 0
\(652\) 11.0000 0.430793
\(653\) 31.0000 1.21312 0.606562 − 0.795036i \(-0.292548\pi\)
0.606562 + 0.795036i \(0.292548\pi\)
\(654\) 0 0
\(655\) −6.00000 −0.234439
\(656\) 2.00000 0.0780869
\(657\) 0 0
\(658\) −28.0000 −1.09155
\(659\) 34.0000 1.32445 0.662226 − 0.749304i \(-0.269612\pi\)
0.662226 + 0.749304i \(0.269612\pi\)
\(660\) 0 0
\(661\) 49.0000 1.90588 0.952940 − 0.303160i \(-0.0980418\pi\)
0.952940 + 0.303160i \(0.0980418\pi\)
\(662\) 1.00000 0.0388661
\(663\) 0 0
\(664\) 36.0000 1.39707
\(665\) 24.0000 0.930680
\(666\) 0 0
\(667\) −24.0000 −0.929284
\(668\) −12.0000 −0.464294
\(669\) 0 0
\(670\) 11.0000 0.424967
\(671\) 0 0
\(672\) 0 0
\(673\) −22.0000 −0.848038 −0.424019 − 0.905653i \(-0.639381\pi\)
−0.424019 + 0.905653i \(0.639381\pi\)
\(674\) −22.0000 −0.847408
\(675\) 0 0
\(676\) 9.00000 0.346154
\(677\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) −76.0000 −2.91661
\(680\) −12.0000 −0.460179
\(681\) 0 0
\(682\) 7.00000 0.268044
\(683\) 5.00000 0.191320 0.0956598 − 0.995414i \(-0.469504\pi\)
0.0956598 + 0.995414i \(0.469504\pi\)
\(684\) 0 0
\(685\) 5.00000 0.191040
\(686\) 8.00000 0.305441
\(687\) 0 0
\(688\) −6.00000 −0.228748
\(689\) 18.0000 0.685745
\(690\) 0 0
\(691\) 35.0000 1.33146 0.665731 − 0.746191i \(-0.268120\pi\)
0.665731 + 0.746191i \(0.268120\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 4.00000 0.151838
\(695\) −2.00000 −0.0758643
\(696\) 0 0
\(697\) −8.00000 −0.303022
\(698\) 30.0000 1.13552
\(699\) 0 0
\(700\) 16.0000 0.604743
\(701\) 2.00000 0.0755390 0.0377695 − 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 0 0
\(703\) 18.0000 0.678883
\(704\) 7.00000 0.263822
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) −16.0000 −0.601742
\(708\) 0 0
\(709\) −13.0000 −0.488225 −0.244113 − 0.969747i \(-0.578497\pi\)
−0.244113 + 0.969747i \(0.578497\pi\)
\(710\) 9.00000 0.337764
\(711\) 0 0
\(712\) −18.0000 −0.674579
\(713\) −28.0000 −1.04861
\(714\) 0 0
\(715\) −2.00000 −0.0747958
\(716\) 9.00000 0.336346
\(717\) 0 0
\(718\) −32.0000 −1.19423
\(719\) −15.0000 −0.559406 −0.279703 − 0.960087i \(-0.590236\pi\)
−0.279703 + 0.960087i \(0.590236\pi\)
\(720\) 0 0
\(721\) −52.0000 −1.93658
\(722\) 17.0000 0.632674
\(723\) 0 0
\(724\) −7.00000 −0.260153
\(725\) −24.0000 −0.891338
\(726\) 0 0
\(727\) −33.0000 −1.22390 −0.611951 − 0.790896i \(-0.709615\pi\)
−0.611951 + 0.790896i \(0.709615\pi\)
\(728\) 24.0000 0.889499
\(729\) 0 0
\(730\) 4.00000 0.148047
\(731\) 24.0000 0.887672
\(732\) 0 0
\(733\) −6.00000 −0.221615 −0.110808 − 0.993842i \(-0.535344\pi\)
−0.110808 + 0.993842i \(0.535344\pi\)
\(734\) −11.0000 −0.406017
\(735\) 0 0
\(736\) −20.0000 −0.737210
\(737\) 11.0000 0.405190
\(738\) 0 0
\(739\) −16.0000 −0.588570 −0.294285 − 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) −3.00000 −0.110282
\(741\) 0 0
\(742\) −36.0000 −1.32160
\(743\) −50.0000 −1.83432 −0.917161 − 0.398517i \(-0.869525\pi\)
−0.917161 + 0.398517i \(0.869525\pi\)
\(744\) 0 0
\(745\) −4.00000 −0.146549
\(746\) 10.0000 0.366126
\(747\) 0 0
\(748\) −4.00000 −0.146254
\(749\) −72.0000 −2.63082
\(750\) 0 0
\(751\) −11.0000 −0.401396 −0.200698 − 0.979653i \(-0.564321\pi\)
−0.200698 + 0.979653i \(0.564321\pi\)
\(752\) 7.00000 0.255264
\(753\) 0 0
\(754\) −12.0000 −0.437014
\(755\) −16.0000 −0.582300
\(756\) 0 0
\(757\) −19.0000 −0.690567 −0.345283 − 0.938498i \(-0.612217\pi\)
−0.345283 + 0.938498i \(0.612217\pi\)
\(758\) 16.0000 0.581146
\(759\) 0 0
\(760\) −18.0000 −0.652929
\(761\) 36.0000 1.30500 0.652499 − 0.757789i \(-0.273720\pi\)
0.652499 + 0.757789i \(0.273720\pi\)
\(762\) 0 0
\(763\) −8.00000 −0.289619
\(764\) 13.0000 0.470323
\(765\) 0 0
\(766\) 29.0000 1.04781
\(767\) 14.0000 0.505511
\(768\) 0 0
\(769\) −16.0000 −0.576975 −0.288487 − 0.957484i \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) 4.00000 0.144150
\(771\) 0 0
\(772\) −4.00000 −0.143963
\(773\) 18.0000 0.647415 0.323708 − 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) 0 0
\(775\) −28.0000 −1.00579
\(776\) 57.0000 2.04618
\(777\) 0 0
\(778\) 33.0000 1.18311
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) 9.00000 0.322045
\(782\) −16.0000 −0.572159
\(783\) 0 0
\(784\) −9.00000 −0.321429
\(785\) −7.00000 −0.249841
\(786\) 0 0
\(787\) −14.0000 −0.499046 −0.249523 − 0.968369i \(-0.580274\pi\)
−0.249523 + 0.968369i \(0.580274\pi\)
\(788\) −16.0000 −0.569976
\(789\) 0 0
\(790\) 8.00000 0.284627
\(791\) −60.0000 −2.13335
\(792\) 0 0
\(793\) 0 0
\(794\) 13.0000 0.461353
\(795\) 0 0
\(796\) 3.00000 0.106332
\(797\) 5.00000 0.177109 0.0885545 − 0.996071i \(-0.471775\pi\)
0.0885545 + 0.996071i \(0.471775\pi\)
\(798\) 0 0
\(799\) −28.0000 −0.990569
\(800\) −20.0000 −0.707107
\(801\) 0 0
\(802\) 17.0000 0.600291
\(803\) 4.00000 0.141157
\(804\) 0 0
\(805\) −16.0000 −0.563926
\(806\) −14.0000 −0.493129
\(807\) 0 0
\(808\) 12.0000 0.422159
\(809\) −30.0000 −1.05474 −0.527372 − 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) −14.0000 −0.491606 −0.245803 − 0.969320i \(-0.579052\pi\)
−0.245803 + 0.969320i \(0.579052\pi\)
\(812\) −24.0000 −0.842235
\(813\) 0 0
\(814\) 3.00000 0.105150
\(815\) −11.0000 −0.385313
\(816\) 0 0
\(817\) 36.0000 1.25948
\(818\) 0 0
\(819\) 0 0
\(820\) 2.00000 0.0698430
\(821\) −20.0000 −0.698005 −0.349002 − 0.937122i \(-0.613479\pi\)
−0.349002 + 0.937122i \(0.613479\pi\)
\(822\) 0 0
\(823\) −24.0000 −0.836587 −0.418294 − 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) 39.0000 1.35863
\(825\) 0 0
\(826\) −28.0000 −0.974245
\(827\) −40.0000 −1.39094 −0.695468 − 0.718557i \(-0.744803\pi\)
−0.695468 + 0.718557i \(0.744803\pi\)
\(828\) 0 0
\(829\) 25.0000 0.868286 0.434143 − 0.900844i \(-0.357051\pi\)
0.434143 + 0.900844i \(0.357051\pi\)
\(830\) −12.0000 −0.416526
\(831\) 0 0
\(832\) −14.0000 −0.485363
\(833\) 36.0000 1.24733
\(834\) 0 0
\(835\) 12.0000 0.415277
\(836\) −6.00000 −0.207514
\(837\) 0 0
\(838\) 5.00000 0.172722
\(839\) 16.0000 0.552381 0.276191 − 0.961103i \(-0.410928\pi\)
0.276191 + 0.961103i \(0.410928\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 13.0000 0.448010
\(843\) 0 0
\(844\) −6.00000 −0.206529
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 4.00000 0.137442
\(848\) 9.00000 0.309061
\(849\) 0 0
\(850\) −16.0000 −0.548795
\(851\) −12.0000 −0.411355
\(852\) 0 0
\(853\) 2.00000 0.0684787 0.0342393 − 0.999414i \(-0.489099\pi\)
0.0342393 + 0.999414i \(0.489099\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 54.0000 1.84568
\(857\) 38.0000 1.29806 0.649028 − 0.760765i \(-0.275176\pi\)
0.649028 + 0.760765i \(0.275176\pi\)
\(858\) 0 0
\(859\) 15.0000 0.511793 0.255897 − 0.966704i \(-0.417629\pi\)
0.255897 + 0.966704i \(0.417629\pi\)
\(860\) −6.00000 −0.204598
\(861\) 0 0
\(862\) −30.0000 −1.02180
\(863\) 24.0000 0.816970 0.408485 − 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −14.0000 −0.475739
\(867\) 0 0
\(868\) −28.0000 −0.950382
\(869\) 8.00000 0.271381
\(870\) 0 0
\(871\) −22.0000 −0.745442
\(872\) 6.00000 0.203186
\(873\) 0 0
\(874\) −24.0000 −0.811812
\(875\) −36.0000 −1.21702
\(876\) 0 0
\(877\) −18.0000 −0.607817 −0.303908 − 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) −14.0000 −0.472477
\(879\) 0 0
\(880\) −1.00000 −0.0337100
\(881\) −7.00000 −0.235836 −0.117918 − 0.993023i \(-0.537622\pi\)
−0.117918 + 0.993023i \(0.537622\pi\)
\(882\) 0 0
\(883\) −5.00000 −0.168263 −0.0841317 − 0.996455i \(-0.526812\pi\)
−0.0841317 + 0.996455i \(0.526812\pi\)
\(884\) 8.00000 0.269069
\(885\) 0 0
\(886\) 1.00000 0.0335957
\(887\) 8.00000 0.268614 0.134307 − 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 6.00000 0.201120
\(891\) 0 0
\(892\) −16.0000 −0.535720
\(893\) −42.0000 −1.40548
\(894\) 0 0
\(895\) −9.00000 −0.300837
\(896\) −12.0000 −0.400892
\(897\) 0 0
\(898\) 23.0000 0.767520
\(899\) 42.0000 1.40078
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) −2.00000 −0.0665927
\(903\) 0 0
\(904\) 45.0000 1.49668
\(905\) 7.00000 0.232688
\(906\) 0 0
\(907\) 12.0000 0.398453 0.199227 − 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) 18.0000 0.597351
\(909\) 0 0
\(910\) −8.00000 −0.265197
\(911\) 27.0000 0.894550 0.447275 − 0.894397i \(-0.352395\pi\)
0.447275 + 0.894397i \(0.352395\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) −6.00000 −0.198462
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) −24.0000 −0.792550
\(918\) 0 0
\(919\) 52.0000 1.71532 0.857661 − 0.514216i \(-0.171917\pi\)
0.857661 + 0.514216i \(0.171917\pi\)
\(920\) 12.0000 0.395628
\(921\) 0 0
\(922\) −12.0000 −0.395199
\(923\) −18.0000 −0.592477
\(924\) 0 0
\(925\) −12.0000 −0.394558
\(926\) −8.00000 −0.262896
\(927\) 0 0
\(928\) 30.0000 0.984798
\(929\) 21.0000 0.688988 0.344494 − 0.938789i \(-0.388051\pi\)
0.344494 + 0.938789i \(0.388051\pi\)
\(930\) 0 0
\(931\) 54.0000 1.76978
\(932\) −24.0000 −0.786146
\(933\) 0 0
\(934\) 27.0000 0.883467
\(935\) 4.00000 0.130814
\(936\) 0 0
\(937\) 14.0000 0.457360 0.228680 − 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) 44.0000 1.43665
\(939\) 0 0
\(940\) 7.00000 0.228315
\(941\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 8.00000 0.260516
\(944\) 7.00000 0.227831
\(945\) 0 0
\(946\) 6.00000 0.195077
\(947\) 15.0000 0.487435 0.243717 − 0.969846i \(-0.421633\pi\)
0.243717 + 0.969846i \(0.421633\pi\)
\(948\) 0 0
\(949\) −8.00000 −0.259691
\(950\) −24.0000 −0.778663
\(951\) 0 0
\(952\) −48.0000 −1.55569
\(953\) −26.0000 −0.842223 −0.421111 − 0.907009i \(-0.638360\pi\)
−0.421111 + 0.907009i \(0.638360\pi\)
\(954\) 0 0
\(955\) −13.0000 −0.420670
\(956\) −6.00000 −0.194054
\(957\) 0 0
\(958\) −4.00000 −0.129234
\(959\) 20.0000 0.645834
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) −6.00000 −0.193448
\(963\) 0 0
\(964\) −10.0000 −0.322078
\(965\) 4.00000 0.128765
\(966\) 0 0
\(967\) −50.0000 −1.60789 −0.803946 − 0.594703i \(-0.797270\pi\)
−0.803946 + 0.594703i \(0.797270\pi\)
\(968\) −3.00000 −0.0964237
\(969\) 0 0
\(970\) −19.0000 −0.610053
\(971\) 20.0000 0.641831 0.320915 − 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) 0 0
\(973\) −8.00000 −0.256468
\(974\) −37.0000 −1.18556
\(975\) 0 0
\(976\) 0 0
\(977\) 18.0000 0.575871 0.287936 − 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 0 0
\(979\) 6.00000 0.191761
\(980\) −9.00000 −0.287494
\(981\) 0 0
\(982\) −8.00000 −0.255290
\(983\) 3.00000 0.0956851 0.0478426 − 0.998855i \(-0.484765\pi\)
0.0478426 + 0.998855i \(0.484765\pi\)
\(984\) 0 0
\(985\) 16.0000 0.509802
\(986\) 24.0000 0.764316
\(987\) 0 0
\(988\) 12.0000 0.381771
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) −44.0000 −1.39771 −0.698853 − 0.715265i \(-0.746306\pi\)
−0.698853 + 0.715265i \(0.746306\pi\)
\(992\) 35.0000 1.11125
\(993\) 0 0
\(994\) 36.0000 1.14185
\(995\) −3.00000 −0.0951064
\(996\) 0 0
\(997\) −4.00000 −0.126681 −0.0633406 − 0.997992i \(-0.520175\pi\)
−0.0633406 + 0.997992i \(0.520175\pi\)
\(998\) −25.0000 −0.791361
\(999\) 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 891.2.a.g.1.1 1
3.2 odd 2 891.2.a.c.1.1 1
9.2 odd 6 297.2.e.c.199.1 2
9.4 even 3 99.2.e.a.34.1 ✓ 2
9.5 odd 6 297.2.e.c.100.1 2
9.7 even 3 99.2.e.a.67.1 yes 2
11.10 odd 2 9801.2.a.c.1.1 1
33.32 even 2 9801.2.a.i.1.1 1
99.43 odd 6 1089.2.e.c.364.1 2
99.76 odd 6 1089.2.e.c.727.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.2.e.a.34.1 ✓ 2 9.4 even 3
99.2.e.a.67.1 yes 2 9.7 even 3
297.2.e.c.100.1 2 9.5 odd 6
297.2.e.c.199.1 2 9.2 odd 6
891.2.a.c.1.1 1 3.2 odd 2
891.2.a.g.1.1 1 1.1 even 1 trivial
1089.2.e.c.364.1 2 99.43 odd 6
1089.2.e.c.727.1 2 99.76 odd 6
9801.2.a.c.1.1 1 11.10 odd 2
9801.2.a.i.1.1 1 33.32 even 2