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Properties

Label 99.2.a.c.1.1
Level $99$
Weight $2$
Character 99.1
Self dual yes
Analytic conductor $0.791$
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] = [99,2,Mod(1,99)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(99, 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("99.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 99.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: \(0.790518980011\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 99.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{4} +4.00000 q^{5} -2.00000 q^{7} -3.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{4} +4.00000 q^{5} -2.00000 q^{7} -3.00000 q^{8} +4.00000 q^{10} +1.00000 q^{11} -2.00000 q^{13} -2.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} -6.00000 q^{19} -4.00000 q^{20} +1.00000 q^{22} -4.00000 q^{23} +11.0000 q^{25} -2.00000 q^{26} +2.00000 q^{28} +6.00000 q^{29} +4.00000 q^{31} +5.00000 q^{32} -2.00000 q^{34} -8.00000 q^{35} -6.00000 q^{37} -6.00000 q^{38} -12.0000 q^{40} +10.0000 q^{41} +6.00000 q^{43} -1.00000 q^{44} -4.00000 q^{46} +8.00000 q^{47} -3.00000 q^{49} +11.0000 q^{50} +2.00000 q^{52} +4.00000 q^{55} +6.00000 q^{56} +6.00000 q^{58} -4.00000 q^{59} -6.00000 q^{61} +4.00000 q^{62} +7.00000 q^{64} -8.00000 q^{65} +8.00000 q^{67} +2.00000 q^{68} -8.00000 q^{70} -2.00000 q^{73} -6.00000 q^{74} +6.00000 q^{76} -2.00000 q^{77} -10.0000 q^{79} -4.00000 q^{80} +10.0000 q^{82} -12.0000 q^{83} -8.00000 q^{85} +6.00000 q^{86} -3.00000 q^{88} +4.00000 q^{91} +4.00000 q^{92} +8.00000 q^{94} -24.0000 q^{95} +2.00000 q^{97} -3.00000 q^{98} +O(q^{100})\)

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
\(5\) 4.00000 1.78885 0.894427 − 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −3.00000 −1.06066
\(9\) 0 0
\(10\) 4.00000 1.26491
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 − 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) −4.00000 −0.894427
\(21\) 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
\(25\) 11.0000 2.20000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) −8.00000 −1.35225
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) −6.00000 −0.973329
\(39\) 0 0
\(40\) −12.0000 −1.89737
\(41\) 10.0000 1.56174 0.780869 − 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 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
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 11.0000 1.55563
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 6.00000 0.801784
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 − 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) −8.00000 −0.992278
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) −8.00000 −0.956183
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 6.00000 0.688247
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 − 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) −4.00000 −0.447214
\(81\) 0 0
\(82\) 10.0000 1.10432
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −8.00000 −0.867722
\(86\) 6.00000 0.646997
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) −24.0000 −2.46235
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −3.00000 −0.303046
\(99\) 0 0
\(100\) −11.0000 −1.10000
\(101\) 14.0000 1.39305 0.696526 − 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 − 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 − 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 − 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 4.00000 0.381385
\(111\) 0 0
\(112\) 2.00000 0.188982
\(113\) −12.0000 −1.12887 −0.564433 − 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 0 0
\(115\) −16.0000 −1.49201
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) −4.00000 −0.368230
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −6.00000 −0.543214
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) 24.0000 2.14663
\(126\) 0 0
\(127\) −10.0000 −0.887357 −0.443678 − 0.896186i \(-0.646327\pi\)
−0.443678 + 0.896186i \(0.646327\pi\)
\(128\) −3.00000 −0.265165
\(129\) 0 0
\(130\) −8.00000 −0.701646
\(131\) 12.0000 1.04844 0.524222 − 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 12.0000 1.04053
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) −4.00000 −0.341743 −0.170872 − 0.985293i \(-0.554658\pi\)
−0.170872 + 0.985293i \(0.554658\pi\)
\(138\) 0 0
\(139\) 10.0000 0.848189 0.424094 − 0.905618i \(-0.360592\pi\)
0.424094 + 0.905618i \(0.360592\pi\)
\(140\) 8.00000 0.676123
\(141\) 0 0
\(142\) 0 0
\(143\) −2.00000 −0.167248
\(144\) 0 0
\(145\) 24.0000 1.99309
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) 6.00000 0.493197
\(149\) 2.00000 0.163846 0.0819232 − 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) 0 0
\(151\) 14.0000 1.13930 0.569652 − 0.821886i \(-0.307078\pi\)
0.569652 + 0.821886i \(0.307078\pi\)
\(152\) 18.0000 1.45999
\(153\) 0 0
\(154\) −2.00000 −0.161165
\(155\) 16.0000 1.28515
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 − 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) −10.0000 −0.795557
\(159\) 0 0
\(160\) 20.0000 1.58114
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) −20.0000 −1.56652 −0.783260 − 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −8.00000 −0.613572
\(171\) 0 0
\(172\) −6.00000 −0.457496
\(173\) 6.00000 0.456172 0.228086 − 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) −22.0000 −1.66304
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 0 0
\(179\) 24.0000 1.79384 0.896922 − 0.442189i \(-0.145798\pi\)
0.896922 + 0.442189i \(0.145798\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 − 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 4.00000 0.296500
\(183\) 0 0
\(184\) 12.0000 0.884652
\(185\) −24.0000 −1.76452
\(186\) 0 0
\(187\) −2.00000 −0.146254
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) −24.0000 −1.74114
\(191\) −16.0000 −1.15772 −0.578860 − 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) −14.0000 −1.00774 −0.503871 − 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) −2.00000 −0.142494 −0.0712470 − 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) 12.0000 0.850657 0.425329 − 0.905039i \(-0.360158\pi\)
0.425329 + 0.905039i \(0.360158\pi\)
\(200\) −33.0000 −2.33345
\(201\) 0 0
\(202\) 14.0000 0.985037
\(203\) −12.0000 −0.842235
\(204\) 0 0
\(205\) 40.0000 2.79372
\(206\) 8.00000 0.557386
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) −6.00000 −0.413057 −0.206529 − 0.978441i \(-0.566217\pi\)
−0.206529 + 0.978441i \(0.566217\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 24.0000 1.63679
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) −2.00000 −0.135457
\(219\) 0 0
\(220\) −4.00000 −0.269680
\(221\) 4.00000 0.269069
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 − 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) −10.0000 −0.668153
\(225\) 0 0
\(226\) −12.0000 −0.798228
\(227\) 12.0000 0.796468 0.398234 − 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 − 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) −16.0000 −1.05501
\(231\) 0 0
\(232\) −18.0000 −1.18176
\(233\) −6.00000 −0.393073 −0.196537 − 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 32.0000 2.08745
\(236\) 4.00000 0.260378
\(237\) 0 0
\(238\) 4.00000 0.259281
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −26.0000 −1.67481 −0.837404 − 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) 6.00000 0.384111
\(245\) −12.0000 −0.766652
\(246\) 0 0
\(247\) 12.0000 0.763542
\(248\) −12.0000 −0.762001
\(249\) 0 0
\(250\) 24.0000 1.51789
\(251\) −8.00000 −0.504956 −0.252478 − 0.967603i \(-0.581245\pi\)
−0.252478 + 0.967603i \(0.581245\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) −10.0000 −0.627456
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −8.00000 −0.499026 −0.249513 − 0.968371i \(-0.580271\pi\)
−0.249513 + 0.968371i \(0.580271\pi\)
\(258\) 0 0
\(259\) 12.0000 0.745644
\(260\) 8.00000 0.496139
\(261\) 0 0
\(262\) 12.0000 0.741362
\(263\) 32.0000 1.97320 0.986602 − 0.163144i \(-0.0521635\pi\)
0.986602 + 0.163144i \(0.0521635\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 12.0000 0.735767
\(267\) 0 0
\(268\) −8.00000 −0.488678
\(269\) 16.0000 0.975537 0.487769 − 0.872973i \(-0.337811\pi\)
0.487769 + 0.872973i \(0.337811\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 − 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) −4.00000 −0.241649
\(275\) 11.0000 0.663325
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 − 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 10.0000 0.599760
\(279\) 0 0
\(280\) 24.0000 1.43427
\(281\) −6.00000 −0.357930 −0.178965 − 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) −2.00000 −0.118888 −0.0594438 − 0.998232i \(-0.518933\pi\)
−0.0594438 + 0.998232i \(0.518933\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −2.00000 −0.118262
\(287\) −20.0000 −1.18056
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 24.0000 1.40933
\(291\) 0 0
\(292\) 2.00000 0.117041
\(293\) 18.0000 1.05157 0.525786 − 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 0 0
\(295\) −16.0000 −0.931556
\(296\) 18.0000 1.04623
\(297\) 0 0
\(298\) 2.00000 0.115857
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) 14.0000 0.805609
\(303\) 0 0
\(304\) 6.00000 0.344124
\(305\) −24.0000 −1.37424
\(306\) 0 0
\(307\) −22.0000 −1.25561 −0.627803 − 0.778372i \(-0.716046\pi\)
−0.627803 + 0.778372i \(0.716046\pi\)
\(308\) 2.00000 0.113961
\(309\) 0 0
\(310\) 16.0000 0.908739
\(311\) −12.0000 −0.680458 −0.340229 − 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) −34.0000 −1.92179 −0.960897 − 0.276907i \(-0.910691\pi\)
−0.960897 + 0.276907i \(0.910691\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) 4.00000 0.224662 0.112331 − 0.993671i \(-0.464168\pi\)
0.112331 + 0.993671i \(0.464168\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) 28.0000 1.56525
\(321\) 0 0
\(322\) 8.00000 0.445823
\(323\) 12.0000 0.667698
\(324\) 0 0
\(325\) −22.0000 −1.22034
\(326\) −20.0000 −1.10770
\(327\) 0 0
\(328\) −30.0000 −1.65647
\(329\) −16.0000 −0.882109
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 − 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 12.0000 0.658586
\(333\) 0 0
\(334\) 0 0
\(335\) 32.0000 1.74835
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 − 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) 8.00000 0.433861
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) −18.0000 −0.970495
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) −20.0000 −1.07366 −0.536828 − 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) 0 0
\(349\) 18.0000 0.963518 0.481759 − 0.876304i \(-0.339998\pi\)
0.481759 + 0.876304i \(0.339998\pi\)
\(350\) −22.0000 −1.17595
\(351\) 0 0
\(352\) 5.00000 0.266501
\(353\) −24.0000 −1.27739 −0.638696 − 0.769460i \(-0.720526\pi\)
−0.638696 + 0.769460i \(0.720526\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 24.0000 1.26844
\(359\) −8.00000 −0.422224 −0.211112 − 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 10.0000 0.525588
\(363\) 0 0
\(364\) −4.00000 −0.209657
\(365\) −8.00000 −0.418739
\(366\) 0 0
\(367\) 16.0000 0.835193 0.417597 − 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 4.00000 0.208514
\(369\) 0 0
\(370\) −24.0000 −1.24770
\(371\) 0 0
\(372\) 0 0
\(373\) 34.0000 1.76045 0.880227 − 0.474554i \(-0.157390\pi\)
0.880227 + 0.474554i \(0.157390\pi\)
\(374\) −2.00000 −0.103418
\(375\) 0 0
\(376\) −24.0000 −1.23771
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 − 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 24.0000 1.23117
\(381\) 0 0
\(382\) −16.0000 −0.818631
\(383\) 20.0000 1.02195 0.510976 − 0.859595i \(-0.329284\pi\)
0.510976 + 0.859595i \(0.329284\pi\)
\(384\) 0 0
\(385\) −8.00000 −0.407718
\(386\) −14.0000 −0.712581
\(387\) 0 0
\(388\) −2.00000 −0.101535
\(389\) −36.0000 −1.82527 −0.912636 − 0.408773i \(-0.865957\pi\)
−0.912636 + 0.408773i \(0.865957\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 9.00000 0.454569
\(393\) 0 0
\(394\) −2.00000 −0.100759
\(395\) −40.0000 −2.01262
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 − 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 12.0000 0.601506
\(399\) 0 0
\(400\) −11.0000 −0.550000
\(401\) −28.0000 −1.39825 −0.699127 − 0.714998i \(-0.746428\pi\)
−0.699127 + 0.714998i \(0.746428\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) −14.0000 −0.696526
\(405\) 0 0
\(406\) −12.0000 −0.595550
\(407\) −6.00000 −0.297409
\(408\) 0 0
\(409\) 6.00000 0.296681 0.148340 − 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 40.0000 1.97546
\(411\) 0 0
\(412\) −8.00000 −0.394132
\(413\) 8.00000 0.393654
\(414\) 0 0
\(415\) −48.0000 −2.35623
\(416\) −10.0000 −0.490290
\(417\) 0 0
\(418\) −6.00000 −0.293470
\(419\) 32.0000 1.56330 0.781651 − 0.623716i \(-0.214378\pi\)
0.781651 + 0.623716i \(0.214378\pi\)
\(420\) 0 0
\(421\) 34.0000 1.65706 0.828529 − 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) −6.00000 −0.292075
\(423\) 0 0
\(424\) 0 0
\(425\) −22.0000 −1.06716
\(426\) 0 0
\(427\) 12.0000 0.580721
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 24.0000 1.15738
\(431\) −24.0000 −1.15604 −0.578020 − 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 − 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) −8.00000 −0.384012
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 24.0000 1.14808
\(438\) 0 0
\(439\) 10.0000 0.477274 0.238637 − 0.971109i \(-0.423299\pi\)
0.238637 + 0.971109i \(0.423299\pi\)
\(440\) −12.0000 −0.572078
\(441\) 0 0
\(442\) 4.00000 0.190261
\(443\) 4.00000 0.190046 0.0950229 − 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) 0 0
\(448\) −14.0000 −0.661438
\(449\) 32.0000 1.51017 0.755087 − 0.655625i \(-0.227595\pi\)
0.755087 + 0.655625i \(0.227595\pi\)
\(450\) 0 0
\(451\) 10.0000 0.470882
\(452\) 12.0000 0.564433
\(453\) 0 0
\(454\) 12.0000 0.563188
\(455\) 16.0000 0.750092
\(456\) 0 0
\(457\) −18.0000 −0.842004 −0.421002 − 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) 6.00000 0.280362
\(459\) 0 0
\(460\) 16.0000 0.746004
\(461\) −6.00000 −0.279448 −0.139724 − 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 − 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) −36.0000 −1.66588 −0.832941 − 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) 0 0
\(469\) −16.0000 −0.738811
\(470\) 32.0000 1.47605
\(471\) 0 0
\(472\) 12.0000 0.552345
\(473\) 6.00000 0.275880
\(474\) 0 0
\(475\) −66.0000 −3.02829
\(476\) −4.00000 −0.183340
\(477\) 0 0
\(478\) 0 0
\(479\) 32.0000 1.46212 0.731059 − 0.682315i \(-0.239027\pi\)
0.731059 + 0.682315i \(0.239027\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) −26.0000 −1.18427
\(483\) 0 0
\(484\) −1.00000 −0.0454545
\(485\) 8.00000 0.363261
\(486\) 0 0
\(487\) −4.00000 −0.181257 −0.0906287 − 0.995885i \(-0.528888\pi\)
−0.0906287 + 0.995885i \(0.528888\pi\)
\(488\) 18.0000 0.814822
\(489\) 0 0
\(490\) −12.0000 −0.542105
\(491\) 28.0000 1.26362 0.631811 − 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) 12.0000 0.539906
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) 0 0
\(499\) −16.0000 −0.716258 −0.358129 − 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) −24.0000 −1.07331
\(501\) 0 0
\(502\) −8.00000 −0.357057
\(503\) 40.0000 1.78351 0.891756 − 0.452517i \(-0.149474\pi\)
0.891756 + 0.452517i \(0.149474\pi\)
\(504\) 0 0
\(505\) 56.0000 2.49197
\(506\) −4.00000 −0.177822
\(507\) 0 0
\(508\) 10.0000 0.443678
\(509\) 24.0000 1.06378 0.531891 − 0.846813i \(-0.321482\pi\)
0.531891 + 0.846813i \(0.321482\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) −11.0000 −0.486136
\(513\) 0 0
\(514\) −8.00000 −0.352865
\(515\) 32.0000 1.41009
\(516\) 0 0
\(517\) 8.00000 0.351840
\(518\) 12.0000 0.527250
\(519\) 0 0
\(520\) 24.0000 1.05247
\(521\) 12.0000 0.525730 0.262865 − 0.964833i \(-0.415333\pi\)
0.262865 + 0.964833i \(0.415333\pi\)
\(522\) 0 0
\(523\) 38.0000 1.66162 0.830812 − 0.556553i \(-0.187876\pi\)
0.830812 + 0.556553i \(0.187876\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 32.0000 1.39527
\(527\) −8.00000 −0.348485
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) −12.0000 −0.520266
\(533\) −20.0000 −0.866296
\(534\) 0 0
\(535\) −48.0000 −2.07522
\(536\) −24.0000 −1.03664
\(537\) 0 0
\(538\) 16.0000 0.689809
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 22.0000 0.945854 0.472927 − 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 2.00000 0.0859074
\(543\) 0 0
\(544\) −10.0000 −0.428746
\(545\) −8.00000 −0.342682
\(546\) 0 0
\(547\) 38.0000 1.62476 0.812381 − 0.583127i \(-0.198171\pi\)
0.812381 + 0.583127i \(0.198171\pi\)
\(548\) 4.00000 0.170872
\(549\) 0 0
\(550\) 11.0000 0.469042
\(551\) −36.0000 −1.53365
\(552\) 0 0
\(553\) 20.0000 0.850487
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) −10.0000 −0.424094
\(557\) −38.0000 −1.61011 −0.805056 − 0.593199i \(-0.797865\pi\)
−0.805056 + 0.593199i \(0.797865\pi\)
\(558\) 0 0
\(559\) −12.0000 −0.507546
\(560\) 8.00000 0.338062
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) 28.0000 1.18006 0.590030 − 0.807382i \(-0.299116\pi\)
0.590030 + 0.807382i \(0.299116\pi\)
\(564\) 0 0
\(565\) −48.0000 −2.01938
\(566\) −2.00000 −0.0840663
\(567\) 0 0
\(568\) 0 0
\(569\) −18.0000 −0.754599 −0.377300 − 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) −34.0000 −1.42286 −0.711428 − 0.702759i \(-0.751951\pi\)
−0.711428 + 0.702759i \(0.751951\pi\)
\(572\) 2.00000 0.0836242
\(573\) 0 0
\(574\) −20.0000 −0.834784
\(575\) −44.0000 −1.83493
\(576\) 0 0
\(577\) 18.0000 0.749350 0.374675 − 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) −13.0000 −0.540729
\(579\) 0 0
\(580\) −24.0000 −0.996546
\(581\) 24.0000 0.995688
\(582\) 0 0
\(583\) 0 0
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) 18.0000 0.743573
\(587\) 4.00000 0.165098 0.0825488 − 0.996587i \(-0.473694\pi\)
0.0825488 + 0.996587i \(0.473694\pi\)
\(588\) 0 0
\(589\) −24.0000 −0.988903
\(590\) −16.0000 −0.658710
\(591\) 0 0
\(592\) 6.00000 0.246598
\(593\) 26.0000 1.06769 0.533846 − 0.845582i \(-0.320746\pi\)
0.533846 + 0.845582i \(0.320746\pi\)
\(594\) 0 0
\(595\) 16.0000 0.655936
\(596\) −2.00000 −0.0819232
\(597\) 0 0
\(598\) 8.00000 0.327144
\(599\) 28.0000 1.14405 0.572024 − 0.820237i \(-0.306158\pi\)
0.572024 + 0.820237i \(0.306158\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 − 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) −12.0000 −0.489083
\(603\) 0 0
\(604\) −14.0000 −0.569652
\(605\) 4.00000 0.162623
\(606\) 0 0
\(607\) −10.0000 −0.405887 −0.202944 − 0.979190i \(-0.565051\pi\)
−0.202944 + 0.979190i \(0.565051\pi\)
\(608\) −30.0000 −1.21666
\(609\) 0 0
\(610\) −24.0000 −0.971732
\(611\) −16.0000 −0.647291
\(612\) 0 0
\(613\) −46.0000 −1.85792 −0.928961 − 0.370177i \(-0.879297\pi\)
−0.928961 + 0.370177i \(0.879297\pi\)
\(614\) −22.0000 −0.887848
\(615\) 0 0
\(616\) 6.00000 0.241747
\(617\) 12.0000 0.483102 0.241551 − 0.970388i \(-0.422344\pi\)
0.241551 + 0.970388i \(0.422344\pi\)
\(618\) 0 0
\(619\) 44.0000 1.76851 0.884255 − 0.467005i \(-0.154667\pi\)
0.884255 + 0.467005i \(0.154667\pi\)
\(620\) −16.0000 −0.642575
\(621\) 0 0
\(622\) −12.0000 −0.481156
\(623\) 0 0
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) −34.0000 −1.35891
\(627\) 0 0
\(628\) 10.0000 0.399043
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) 4.00000 0.159237 0.0796187 − 0.996825i \(-0.474630\pi\)
0.0796187 + 0.996825i \(0.474630\pi\)
\(632\) 30.0000 1.19334
\(633\) 0 0
\(634\) 4.00000 0.158860
\(635\) −40.0000 −1.58735
\(636\) 0 0
\(637\) 6.00000 0.237729
\(638\) 6.00000 0.237542
\(639\) 0 0
\(640\) −12.0000 −0.474342
\(641\) 24.0000 0.947943 0.473972 − 0.880540i \(-0.342820\pi\)
0.473972 + 0.880540i \(0.342820\pi\)
\(642\) 0 0
\(643\) −4.00000 −0.157745 −0.0788723 − 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) −8.00000 −0.315244
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) −28.0000 −1.10079 −0.550397 − 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) 0 0
\(649\) −4.00000 −0.157014
\(650\) −22.0000 −0.862911
\(651\) 0 0
\(652\) 20.0000 0.783260
\(653\) 16.0000 0.626128 0.313064 − 0.949732i \(-0.398644\pi\)
0.313064 + 0.949732i \(0.398644\pi\)
\(654\) 0 0
\(655\) 48.0000 1.87552
\(656\) −10.0000 −0.390434
\(657\) 0 0
\(658\) −16.0000 −0.623745
\(659\) −44.0000 −1.71400 −0.856998 − 0.515319i \(-0.827673\pi\)
−0.856998 + 0.515319i \(0.827673\pi\)
\(660\) 0 0
\(661\) −50.0000 −1.94477 −0.972387 − 0.233373i \(-0.925024\pi\)
−0.972387 + 0.233373i \(0.925024\pi\)
\(662\) 4.00000 0.155464
\(663\) 0 0
\(664\) 36.0000 1.39707
\(665\) 48.0000 1.86136
\(666\) 0 0
\(667\) −24.0000 −0.929284
\(668\) 0 0
\(669\) 0 0
\(670\) 32.0000 1.23627
\(671\) −6.00000 −0.231627
\(672\) 0 0
\(673\) 26.0000 1.00223 0.501113 − 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) 14.0000 0.539260
\(675\) 0 0
\(676\) 9.00000 0.346154
\(677\) 6.00000 0.230599 0.115299 − 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) −4.00000 −0.153506
\(680\) 24.0000 0.920358
\(681\) 0 0
\(682\) 4.00000 0.153168
\(683\) 32.0000 1.22445 0.612223 − 0.790685i \(-0.290275\pi\)
0.612223 + 0.790685i \(0.290275\pi\)
\(684\) 0 0
\(685\) −16.0000 −0.611329
\(686\) 20.0000 0.763604
\(687\) 0 0
\(688\) −6.00000 −0.228748
\(689\) 0 0
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 − 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) −20.0000 −0.759190
\(695\) 40.0000 1.51729
\(696\) 0 0
\(697\) −20.0000 −0.757554
\(698\) 18.0000 0.681310
\(699\) 0 0
\(700\) 22.0000 0.831522
\(701\) −22.0000 −0.830929 −0.415464 − 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) 0 0
\(703\) 36.0000 1.35777
\(704\) 7.00000 0.263822
\(705\) 0 0
\(706\) −24.0000 −0.903252
\(707\) −28.0000 −1.05305
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 − 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −16.0000 −0.599205
\(714\) 0 0
\(715\) −8.00000 −0.299183
\(716\) −24.0000 −0.896922
\(717\) 0 0
\(718\) −8.00000 −0.298557
\(719\) −24.0000 −0.895049 −0.447524 − 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) 17.0000 0.632674
\(723\) 0 0
\(724\) −10.0000 −0.371647
\(725\) 66.0000 2.45118
\(726\) 0 0
\(727\) 12.0000 0.445055 0.222528 − 0.974926i \(-0.428569\pi\)
0.222528 + 0.974926i \(0.428569\pi\)
\(728\) −12.0000 −0.444750
\(729\) 0 0
\(730\) −8.00000 −0.296093
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) 6.00000 0.221615 0.110808 − 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) 16.0000 0.590571
\(735\) 0 0
\(736\) −20.0000 −0.737210
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) −34.0000 −1.25071 −0.625355 − 0.780340i \(-0.715046\pi\)
−0.625355 + 0.780340i \(0.715046\pi\)
\(740\) 24.0000 0.882258
\(741\) 0 0
\(742\) 0 0
\(743\) 16.0000 0.586983 0.293492 − 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 0 0
\(745\) 8.00000 0.293097
\(746\) 34.0000 1.24483
\(747\) 0 0
\(748\) 2.00000 0.0731272
\(749\) 24.0000 0.876941
\(750\) 0 0
\(751\) −20.0000 −0.729810 −0.364905 − 0.931045i \(-0.618899\pi\)
−0.364905 + 0.931045i \(0.618899\pi\)
\(752\) −8.00000 −0.291730
\(753\) 0 0
\(754\) −12.0000 −0.437014
\(755\) 56.0000 2.03805
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 − 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) 72.0000 2.61171
\(761\) −42.0000 −1.52250 −0.761249 − 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) 0 0
\(763\) 4.00000 0.144810
\(764\) 16.0000 0.578860
\(765\) 0 0
\(766\) 20.0000 0.722629
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) 2.00000 0.0721218 0.0360609 − 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) −8.00000 −0.288300
\(771\) 0 0
\(772\) 14.0000 0.503871
\(773\) 24.0000 0.863220 0.431610 − 0.902060i \(-0.357946\pi\)
0.431610 + 0.902060i \(0.357946\pi\)
\(774\) 0 0
\(775\) 44.0000 1.58053
\(776\) −6.00000 −0.215387
\(777\) 0 0
\(778\) −36.0000 −1.29066
\(779\) −60.0000 −2.14972
\(780\) 0 0
\(781\) 0 0
\(782\) 8.00000 0.286079
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) −40.0000 −1.42766
\(786\) 0 0
\(787\) 22.0000 0.784215 0.392108 − 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) 2.00000 0.0712470
\(789\) 0 0
\(790\) −40.0000 −1.42314
\(791\) 24.0000 0.853342
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) −12.0000 −0.425329
\(797\) −28.0000 −0.991811 −0.495905 − 0.868377i \(-0.665164\pi\)
−0.495905 + 0.868377i \(0.665164\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 55.0000 1.94454
\(801\) 0 0
\(802\) −28.0000 −0.988714
\(803\) −2.00000 −0.0705785
\(804\) 0 0
\(805\) 32.0000 1.12785
\(806\) −8.00000 −0.281788
\(807\) 0 0
\(808\) −42.0000 −1.47755
\(809\) 30.0000 1.05474 0.527372 − 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) 34.0000 1.19390 0.596951 − 0.802278i \(-0.296379\pi\)
0.596951 + 0.802278i \(0.296379\pi\)
\(812\) 12.0000 0.421117
\(813\) 0 0
\(814\) −6.00000 −0.210300
\(815\) −80.0000 −2.80228
\(816\) 0 0
\(817\) −36.0000 −1.25948
\(818\) 6.00000 0.209785
\(819\) 0 0
\(820\) −40.0000 −1.39686
\(821\) 22.0000 0.767805 0.383903 − 0.923374i \(-0.374580\pi\)
0.383903 + 0.923374i \(0.374580\pi\)
\(822\) 0 0
\(823\) −48.0000 −1.67317 −0.836587 − 0.547833i \(-0.815453\pi\)
−0.836587 + 0.547833i \(0.815453\pi\)
\(824\) −24.0000 −0.836080
\(825\) 0 0
\(826\) 8.00000 0.278356
\(827\) −28.0000 −0.973655 −0.486828 − 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 − 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) −48.0000 −1.66610
\(831\) 0 0
\(832\) −14.0000 −0.485363
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) 0 0
\(836\) 6.00000 0.207514
\(837\) 0 0
\(838\) 32.0000 1.10542
\(839\) −20.0000 −0.690477 −0.345238 − 0.938515i \(-0.612202\pi\)
−0.345238 + 0.938515i \(0.612202\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 34.0000 1.17172
\(843\) 0 0
\(844\) 6.00000 0.206529
\(845\) −36.0000 −1.23844
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) 0 0
\(849\) 0 0
\(850\) −22.0000 −0.754594
\(851\) 24.0000 0.822709
\(852\) 0 0
\(853\) 50.0000 1.71197 0.855984 − 0.517003i \(-0.172952\pi\)
0.855984 + 0.517003i \(0.172952\pi\)
\(854\) 12.0000 0.410632
\(855\) 0 0
\(856\) 36.0000 1.23045
\(857\) 14.0000 0.478231 0.239115 − 0.970991i \(-0.423143\pi\)
0.239115 + 0.970991i \(0.423143\pi\)
\(858\) 0 0
\(859\) −36.0000 −1.22830 −0.614152 − 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) −24.0000 −0.818393
\(861\) 0 0
\(862\) −24.0000 −0.817443
\(863\) −36.0000 −1.22545 −0.612727 − 0.790295i \(-0.709928\pi\)
−0.612727 + 0.790295i \(0.709928\pi\)
\(864\) 0 0
\(865\) 24.0000 0.816024
\(866\) −2.00000 −0.0679628
\(867\) 0 0
\(868\) 8.00000 0.271538
\(869\) −10.0000 −0.339227
\(870\) 0 0
\(871\) −16.0000 −0.542139
\(872\) 6.00000 0.203186
\(873\) 0 0
\(874\) 24.0000 0.811812
\(875\) −48.0000 −1.62270
\(876\) 0 0
\(877\) 42.0000 1.41824 0.709120 − 0.705088i \(-0.249093\pi\)
0.709120 + 0.705088i \(0.249093\pi\)
\(878\) 10.0000 0.337484
\(879\) 0 0
\(880\) −4.00000 −0.134840
\(881\) 20.0000 0.673817 0.336909 − 0.941537i \(-0.390619\pi\)
0.336909 + 0.941537i \(0.390619\pi\)
\(882\) 0 0
\(883\) −32.0000 −1.07689 −0.538443 − 0.842662i \(-0.680987\pi\)
−0.538443 + 0.842662i \(0.680987\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) −16.0000 −0.537227 −0.268614 − 0.963248i \(-0.586566\pi\)
−0.268614 + 0.963248i \(0.586566\pi\)
\(888\) 0 0
\(889\) 20.0000 0.670778
\(890\) 0 0
\(891\) 0 0
\(892\) 8.00000 0.267860
\(893\) −48.0000 −1.60626
\(894\) 0 0
\(895\) 96.0000 3.20893
\(896\) 6.00000 0.200446
\(897\) 0 0
\(898\) 32.0000 1.06785
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) 0 0
\(902\) 10.0000 0.332964
\(903\) 0 0
\(904\) 36.0000 1.19734
\(905\) 40.0000 1.32964
\(906\) 0 0
\(907\) −12.0000 −0.398453 −0.199227 − 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) −12.0000 −0.398234
\(909\) 0 0
\(910\) 16.0000 0.530395
\(911\) −48.0000 −1.59031 −0.795155 − 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) −18.0000 −0.595387
\(915\) 0 0
\(916\) −6.00000 −0.198246
\(917\) −24.0000 −0.792550
\(918\) 0 0
\(919\) −14.0000 −0.461817 −0.230909 − 0.972975i \(-0.574170\pi\)
−0.230909 + 0.972975i \(0.574170\pi\)
\(920\) 48.0000 1.58251
\(921\) 0 0
\(922\) −6.00000 −0.197599
\(923\) 0 0
\(924\) 0 0
\(925\) −66.0000 −2.17007
\(926\) 16.0000 0.525793
\(927\) 0 0
\(928\) 30.0000 0.984798
\(929\) 12.0000 0.393707 0.196854 − 0.980433i \(-0.436928\pi\)
0.196854 + 0.980433i \(0.436928\pi\)
\(930\) 0 0
\(931\) 18.0000 0.589926
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) −36.0000 −1.17796
\(935\) −8.00000 −0.261628
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 − 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) −16.0000 −0.522419
\(939\) 0 0
\(940\) −32.0000 −1.04372
\(941\) 30.0000 0.977972 0.488986 − 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 0 0
\(943\) −40.0000 −1.30258
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 6.00000 0.195077
\(947\) 48.0000 1.55979 0.779895 − 0.625910i \(-0.215272\pi\)
0.779895 + 0.625910i \(0.215272\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) −66.0000 −2.14132
\(951\) 0 0
\(952\) −12.0000 −0.388922
\(953\) −26.0000 −0.842223 −0.421111 − 0.907009i \(-0.638360\pi\)
−0.421111 + 0.907009i \(0.638360\pi\)
\(954\) 0 0
\(955\) −64.0000 −2.07099
\(956\) 0 0
\(957\) 0 0
\(958\) 32.0000 1.03387
\(959\) 8.00000 0.258333
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 12.0000 0.386896
\(963\) 0 0
\(964\) 26.0000 0.837404
\(965\) −56.0000 −1.80270
\(966\) 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
\(970\) 8.00000 0.256865
\(971\) −4.00000 −0.128366 −0.0641831 − 0.997938i \(-0.520444\pi\)
−0.0641831 + 0.997938i \(0.520444\pi\)
\(972\) 0 0
\(973\) −20.0000 −0.641171
\(974\) −4.00000 −0.128168
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) 36.0000 1.15174 0.575871 − 0.817541i \(-0.304663\pi\)
0.575871 + 0.817541i \(0.304663\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 12.0000 0.383326
\(981\) 0 0
\(982\) 28.0000 0.893516
\(983\) 36.0000 1.14822 0.574111 − 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) 0 0
\(985\) −8.00000 −0.254901
\(986\) −12.0000 −0.382158
\(987\) 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 \(0.746306\pi\)
\(992\) 20.0000 0.635001
\(993\) 0 0
\(994\) 0 0
\(995\) 48.0000 1.52170
\(996\) 0 0
\(997\) 50.0000 1.58352 0.791758 − 0.610835i \(-0.209166\pi\)
0.791758 + 0.610835i \(0.209166\pi\)
\(998\) −16.0000 −0.506471
\(999\) 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 99.2.a.c.1.1 yes 1
3.2 odd 2 99.2.a.a.1.1 ✓ 1
4.3 odd 2 1584.2.a.r.1.1 1
5.2 odd 4 2475.2.c.g.199.2 2
5.3 odd 4 2475.2.c.g.199.1 2
5.4 even 2 2475.2.a.c.1.1 1
7.6 odd 2 4851.2.a.o.1.1 1
8.3 odd 2 6336.2.a.f.1.1 1
8.5 even 2 6336.2.a.b.1.1 1
9.2 odd 6 891.2.e.j.595.1 2
9.4 even 3 891.2.e.c.298.1 2
9.5 odd 6 891.2.e.j.298.1 2
9.7 even 3 891.2.e.c.595.1 2
11.10 odd 2 1089.2.a.d.1.1 1
12.11 even 2 1584.2.a.b.1.1 1
15.2 even 4 2475.2.c.b.199.1 2
15.8 even 4 2475.2.c.b.199.2 2
15.14 odd 2 2475.2.a.j.1.1 1
21.20 even 2 4851.2.a.g.1.1 1
24.5 odd 2 6336.2.a.cl.1.1 1
24.11 even 2 6336.2.a.cm.1.1 1
33.32 even 2 1089.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.2.a.a.1.1 ✓ 1 3.2 odd 2
99.2.a.c.1.1 yes 1 1.1 even 1 trivial
891.2.e.c.298.1 2 9.4 even 3
891.2.e.c.595.1 2 9.7 even 3
891.2.e.j.298.1 2 9.5 odd 6
891.2.e.j.595.1 2 9.2 odd 6
1089.2.a.d.1.1 1 11.10 odd 2
1089.2.a.h.1.1 1 33.32 even 2
1584.2.a.b.1.1 1 12.11 even 2
1584.2.a.r.1.1 1 4.3 odd 2
2475.2.a.c.1.1 1 5.4 even 2
2475.2.a.j.1.1 1 15.14 odd 2
2475.2.c.b.199.1 2 15.2 even 4
2475.2.c.b.199.2 2 15.8 even 4
2475.2.c.g.199.1 2 5.3 odd 4
2475.2.c.g.199.2 2 5.2 odd 4
4851.2.a.g.1.1 1 21.20 even 2
4851.2.a.o.1.1 1 7.6 odd 2
6336.2.a.b.1.1 1 8.5 even 2
6336.2.a.f.1.1 1 8.3 odd 2
6336.2.a.cl.1.1 1 24.5 odd 2
6336.2.a.cm.1.1 1 24.11 even 2