Properties

Label 1305.2.a.b.1.1
Level $1305$
Weight $2$
Character 1305.1
Self dual yes
Analytic conductor $10.420$
Analytic rank $0$
Dimension $1$
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] = [1305,2,Mod(1,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1305.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.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: \(10.4204774638\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1305.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} +4.00000 q^{7} +3.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} +4.00000 q^{7} +3.00000 q^{8} +1.00000 q^{10} +4.00000 q^{11} +6.00000 q^{13} -4.00000 q^{14} -1.00000 q^{16} -6.00000 q^{17} -4.00000 q^{19} +1.00000 q^{20} -4.00000 q^{22} +4.00000 q^{23} +1.00000 q^{25} -6.00000 q^{26} -4.00000 q^{28} -1.00000 q^{29} -8.00000 q^{31} -5.00000 q^{32} +6.00000 q^{34} -4.00000 q^{35} +2.00000 q^{37} +4.00000 q^{38} -3.00000 q^{40} +6.00000 q^{41} +4.00000 q^{43} -4.00000 q^{44} -4.00000 q^{46} +9.00000 q^{49} -1.00000 q^{50} -6.00000 q^{52} +10.0000 q^{53} -4.00000 q^{55} +12.0000 q^{56} +1.00000 q^{58} +12.0000 q^{59} -10.0000 q^{61} +8.00000 q^{62} +7.00000 q^{64} -6.00000 q^{65} +8.00000 q^{67} +6.00000 q^{68} +4.00000 q^{70} +8.00000 q^{71} -2.00000 q^{73} -2.00000 q^{74} +4.00000 q^{76} +16.0000 q^{77} +1.00000 q^{80} -6.00000 q^{82} -8.00000 q^{83} +6.00000 q^{85} -4.00000 q^{86} +12.0000 q^{88} +6.00000 q^{89} +24.0000 q^{91} -4.00000 q^{92} +4.00000 q^{95} -2.00000 q^{97} -9.00000 q^{98} +O(q^{100})\)

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 \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) −1.00000 −0.447214
\(6\) 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
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −6.00000 −1.17670
\(27\) 0 0
\(28\) −4.00000 −0.755929
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −6.00000 −0.832050
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 12.0000 1.60357
\(57\) 0 0
\(58\) 1.00000 0.131306
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) 4.00000 0.478091
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 16.0000 1.82337
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) −8.00000 −0.878114 −0.439057 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 12.0000 1.27920
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 24.0000 2.51588
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −9.00000 −0.909137
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) −12.0000 −1.18240 −0.591198 0.806527i \(-0.701345\pi\)
−0.591198 + 0.806527i \(0.701345\pi\)
\(104\) 18.0000 1.76505
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 0 0
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 4.00000 0.381385
\(111\) 0 0
\(112\) −4.00000 −0.377964
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) 1.00000 0.0928477
\(117\) 0 0
\(118\) −12.0000 −1.10469
\(119\) −24.0000 −2.20008
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 10.0000 0.905357
\(123\) 0 0
\(124\) 8.00000 0.718421
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 3.00000 0.265165
\(129\) 0 0
\(130\) 6.00000 0.526235
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) −16.0000 −1.38738
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) −18.0000 −1.54349
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 4.00000 0.338062
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) 24.0000 2.00698
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) 2.00000 0.165521
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) −12.0000 −0.973329
\(153\) 0 0
\(154\) −16.0000 −1.28932
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 5.00000 0.395285
\(161\) 16.0000 1.26098
\(162\) 0 0
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 8.00000 0.620920
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) −6.00000 −0.460179
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) −24.0000 −1.77900
\(183\) 0 0
\(184\) 12.0000 0.884652
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) −24.0000 −1.75505
\(188\) 0 0
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 0 0
\(193\) 22.0000 1.58359 0.791797 0.610784i \(-0.209146\pi\)
0.791797 + 0.610784i \(0.209146\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) −9.00000 −0.642857
\(197\) −14.0000 −0.997459 −0.498729 0.866758i \(-0.666200\pi\)
−0.498729 + 0.866758i \(0.666200\pi\)
\(198\) 0 0
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 3.00000 0.212132
\(201\) 0 0
\(202\) −2.00000 −0.140720
\(203\) −4.00000 −0.280745
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) 12.0000 0.836080
\(207\) 0 0
\(208\) −6.00000 −0.416025
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) −10.0000 −0.686803
\(213\) 0 0
\(214\) 8.00000 0.546869
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) −32.0000 −2.17230
\(218\) −14.0000 −0.948200
\(219\) 0 0
\(220\) 4.00000 0.269680
\(221\) −36.0000 −2.42162
\(222\) 0 0
\(223\) 28.0000 1.87502 0.937509 0.347960i \(-0.113126\pi\)
0.937509 + 0.347960i \(0.113126\pi\)
\(224\) −20.0000 −1.33631
\(225\) 0 0
\(226\) −2.00000 −0.133038
\(227\) −24.0000 −1.59294 −0.796468 0.604681i \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(228\) 0 0
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 4.00000 0.263752
\(231\) 0 0
\(232\) −3.00000 −0.196960
\(233\) 22.0000 1.44127 0.720634 0.693316i \(-0.243851\pi\)
0.720634 + 0.693316i \(0.243851\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) 0 0
\(238\) 24.0000 1.55569
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) −5.00000 −0.321412
\(243\) 0 0
\(244\) 10.0000 0.640184
\(245\) −9.00000 −0.574989
\(246\) 0 0
\(247\) −24.0000 −1.52708
\(248\) −24.0000 −1.52400
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) 16.0000 1.00393
\(255\) 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
\(259\) 8.00000 0.497096
\(260\) 6.00000 0.372104
\(261\) 0 0
\(262\) 12.0000 0.741362
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −10.0000 −0.614295
\(266\) 16.0000 0.981023
\(267\) 0 0
\(268\) −8.00000 −0.488678
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) −26.0000 −1.56219 −0.781094 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) −4.00000 −0.239904
\(279\) 0 0
\(280\) −12.0000 −0.717137
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) 8.00000 0.475551 0.237775 0.971320i \(-0.423582\pi\)
0.237775 + 0.971320i \(0.423582\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) −24.0000 −1.41915
\(287\) 24.0000 1.41668
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) −1.00000 −0.0587220
\(291\) 0 0
\(292\) 2.00000 0.117041
\(293\) −18.0000 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 0 0
\(295\) −12.0000 −0.698667
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) −16.0000 −0.920697
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) 10.0000 0.572598
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) −16.0000 −0.911685
\(309\) 0 0
\(310\) −8.00000 −0.454369
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −30.0000 −1.69570 −0.847850 0.530236i \(-0.822103\pi\)
−0.847850 + 0.530236i \(0.822103\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) 0 0
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 0 0
\(319\) −4.00000 −0.223957
\(320\) −7.00000 −0.391312
\(321\) 0 0
\(322\) −16.0000 −0.891645
\(323\) 24.0000 1.33540
\(324\) 0 0
\(325\) 6.00000 0.332820
\(326\) 12.0000 0.664619
\(327\) 0 0
\(328\) 18.0000 0.993884
\(329\) 0 0
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 8.00000 0.439057
\(333\) 0 0
\(334\) 12.0000 0.656611
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) 30.0000 1.63420 0.817102 0.576493i \(-0.195579\pi\)
0.817102 + 0.576493i \(0.195579\pi\)
\(338\) −23.0000 −1.25104
\(339\) 0 0
\(340\) −6.00000 −0.325396
\(341\) −32.0000 −1.73290
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) −16.0000 −0.858925 −0.429463 0.903085i \(-0.641297\pi\)
−0.429463 + 0.903085i \(0.641297\pi\)
\(348\) 0 0
\(349\) −34.0000 −1.81998 −0.909989 0.414632i \(-0.863910\pi\)
−0.909989 + 0.414632i \(0.863910\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) −20.0000 −1.06600
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −22.0000 −1.15629
\(363\) 0 0
\(364\) −24.0000 −1.25794
\(365\) 2.00000 0.104685
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) 2.00000 0.103975
\(371\) 40.0000 2.07670
\(372\) 0 0
\(373\) −2.00000 −0.103556 −0.0517780 0.998659i \(-0.516489\pi\)
−0.0517780 + 0.998659i \(0.516489\pi\)
\(374\) 24.0000 1.24101
\(375\) 0 0
\(376\) 0 0
\(377\) −6.00000 −0.309016
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) −4.00000 −0.205196
\(381\) 0 0
\(382\) −16.0000 −0.818631
\(383\) −20.0000 −1.02195 −0.510976 0.859595i \(-0.670716\pi\)
−0.510976 + 0.859595i \(0.670716\pi\)
\(384\) 0 0
\(385\) −16.0000 −0.815436
\(386\) −22.0000 −1.11977
\(387\) 0 0
\(388\) 2.00000 0.101535
\(389\) 2.00000 0.101404 0.0507020 0.998714i \(-0.483854\pi\)
0.0507020 + 0.998714i \(0.483854\pi\)
\(390\) 0 0
\(391\) −24.0000 −1.21373
\(392\) 27.0000 1.36371
\(393\) 0 0
\(394\) 14.0000 0.705310
\(395\) 0 0
\(396\) 0 0
\(397\) −10.0000 −0.501886 −0.250943 0.968002i \(-0.580741\pi\)
−0.250943 + 0.968002i \(0.580741\pi\)
\(398\) 24.0000 1.20301
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −2.00000 −0.0998752 −0.0499376 0.998752i \(-0.515902\pi\)
−0.0499376 + 0.998752i \(0.515902\pi\)
\(402\) 0 0
\(403\) −48.0000 −2.39105
\(404\) −2.00000 −0.0995037
\(405\) 0 0
\(406\) 4.00000 0.198517
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 6.00000 0.296319
\(411\) 0 0
\(412\) 12.0000 0.591198
\(413\) 48.0000 2.36193
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) −30.0000 −1.47087
\(417\) 0 0
\(418\) 16.0000 0.782586
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) −18.0000 −0.877266 −0.438633 0.898666i \(-0.644537\pi\)
−0.438633 + 0.898666i \(0.644537\pi\)
\(422\) −20.0000 −0.973585
\(423\) 0 0
\(424\) 30.0000 1.45693
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) −40.0000 −1.93574
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 32.0000 1.53605
\(435\) 0 0
\(436\) −14.0000 −0.670478
\(437\) −16.0000 −0.765384
\(438\) 0 0
\(439\) −24.0000 −1.14546 −0.572729 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) −12.0000 −0.572078
\(441\) 0 0
\(442\) 36.0000 1.71235
\(443\) −20.0000 −0.950229 −0.475114 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) −28.0000 −1.32584
\(447\) 0 0
\(448\) 28.0000 1.32288
\(449\) −26.0000 −1.22702 −0.613508 0.789689i \(-0.710242\pi\)
−0.613508 + 0.789689i \(0.710242\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) −2.00000 −0.0940721
\(453\) 0 0
\(454\) 24.0000 1.12638
\(455\) −24.0000 −1.12514
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 2.00000 0.0934539
\(459\) 0 0
\(460\) 4.00000 0.186501
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) 36.0000 1.67306 0.836531 0.547920i \(-0.184580\pi\)
0.836531 + 0.547920i \(0.184580\pi\)
\(464\) 1.00000 0.0464238
\(465\) 0 0
\(466\) −22.0000 −1.01913
\(467\) 20.0000 0.925490 0.462745 0.886492i \(-0.346865\pi\)
0.462745 + 0.886492i \(0.346865\pi\)
\(468\) 0 0
\(469\) 32.0000 1.47762
\(470\) 0 0
\(471\) 0 0
\(472\) 36.0000 1.65703
\(473\) 16.0000 0.735681
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 24.0000 1.10004
\(477\) 0 0
\(478\) −8.00000 −0.365911
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) −2.00000 −0.0910975
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) 2.00000 0.0908153
\(486\) 0 0
\(487\) −28.0000 −1.26880 −0.634401 0.773004i \(-0.718753\pi\)
−0.634401 + 0.773004i \(0.718753\pi\)
\(488\) −30.0000 −1.35804
\(489\) 0 0
\(490\) 9.00000 0.406579
\(491\) 44.0000 1.98569 0.992846 0.119401i \(-0.0380974\pi\)
0.992846 + 0.119401i \(0.0380974\pi\)
\(492\) 0 0
\(493\) 6.00000 0.270226
\(494\) 24.0000 1.07981
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 32.0000 1.43540
\(498\) 0 0
\(499\) −36.0000 −1.61158 −0.805791 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −20.0000 −0.892644
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) −2.00000 −0.0889988
\(506\) −16.0000 −0.711287
\(507\) 0 0
\(508\) 16.0000 0.709885
\(509\) 2.00000 0.0886484 0.0443242 0.999017i \(-0.485887\pi\)
0.0443242 + 0.999017i \(0.485887\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) −22.0000 −0.970378
\(515\) 12.0000 0.528783
\(516\) 0 0
\(517\) 0 0
\(518\) −8.00000 −0.351500
\(519\) 0 0
\(520\) −18.0000 −0.789352
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 0 0
\(527\) 48.0000 2.09091
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 10.0000 0.434372
\(531\) 0 0
\(532\) 16.0000 0.693688
\(533\) 36.0000 1.55933
\(534\) 0 0
\(535\) 8.00000 0.345870
\(536\) 24.0000 1.03664
\(537\) 0 0
\(538\) 6.00000 0.258678
\(539\) 36.0000 1.55063
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) −16.0000 −0.687259
\(543\) 0 0
\(544\) 30.0000 1.28624
\(545\) −14.0000 −0.599694
\(546\) 0 0
\(547\) 40.0000 1.71028 0.855138 0.518400i \(-0.173472\pi\)
0.855138 + 0.518400i \(0.173472\pi\)
\(548\) −2.00000 −0.0854358
\(549\) 0 0
\(550\) −4.00000 −0.170561
\(551\) 4.00000 0.170406
\(552\) 0 0
\(553\) 0 0
\(554\) 26.0000 1.10463
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) 4.00000 0.169031
\(561\) 0 0
\(562\) 10.0000 0.421825
\(563\) 28.0000 1.18006 0.590030 0.807382i \(-0.299116\pi\)
0.590030 + 0.807382i \(0.299116\pi\)
\(564\) 0 0
\(565\) −2.00000 −0.0841406
\(566\) −8.00000 −0.336265
\(567\) 0 0
\(568\) 24.0000 1.00702
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) −24.0000 −1.00349
\(573\) 0 0
\(574\) −24.0000 −1.00174
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) −19.0000 −0.790296
\(579\) 0 0
\(580\) −1.00000 −0.0415227
\(581\) −32.0000 −1.32758
\(582\) 0 0
\(583\) 40.0000 1.65663
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) 18.0000 0.743573
\(587\) 16.0000 0.660391 0.330195 0.943913i \(-0.392885\pi\)
0.330195 + 0.943913i \(0.392885\pi\)
\(588\) 0 0
\(589\) 32.0000 1.31854
\(590\) 12.0000 0.494032
\(591\) 0 0
\(592\) −2.00000 −0.0821995
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) 0 0
\(595\) 24.0000 0.983904
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) −24.0000 −0.981433
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) −16.0000 −0.652111
\(603\) 0 0
\(604\) −16.0000 −0.651031
\(605\) −5.00000 −0.203279
\(606\) 0 0
\(607\) 32.0000 1.29884 0.649420 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) 20.0000 0.811107
\(609\) 0 0
\(610\) −10.0000 −0.404888
\(611\) 0 0
\(612\) 0 0
\(613\) 6.00000 0.242338 0.121169 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) 48.0000 1.93398
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) −8.00000 −0.321288
\(621\) 0 0
\(622\) 0 0
\(623\) 24.0000 0.961540
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 30.0000 1.19904
\(627\) 0 0
\(628\) −18.0000 −0.718278
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 24.0000 0.955425 0.477712 0.878516i \(-0.341466\pi\)
0.477712 + 0.878516i \(0.341466\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 2.00000 0.0794301
\(635\) 16.0000 0.634941
\(636\) 0 0
\(637\) 54.0000 2.13956
\(638\) 4.00000 0.158362
\(639\) 0 0
\(640\) −3.00000 −0.118585
\(641\) −34.0000 −1.34292 −0.671460 0.741041i \(-0.734332\pi\)
−0.671460 + 0.741041i \(0.734332\pi\)
\(642\) 0 0
\(643\) 16.0000 0.630978 0.315489 0.948929i \(-0.397831\pi\)
0.315489 + 0.948929i \(0.397831\pi\)
\(644\) −16.0000 −0.630488
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) 28.0000 1.10079 0.550397 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) 0 0
\(649\) 48.0000 1.88416
\(650\) −6.00000 −0.235339
\(651\) 0 0
\(652\) 12.0000 0.469956
\(653\) −50.0000 −1.95665 −0.978326 0.207072i \(-0.933606\pi\)
−0.978326 + 0.207072i \(0.933606\pi\)
\(654\) 0 0
\(655\) 12.0000 0.468879
\(656\) −6.00000 −0.234261
\(657\) 0 0
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) −24.0000 −0.931381
\(665\) 16.0000 0.620453
\(666\) 0 0
\(667\) −4.00000 −0.154881
\(668\) 12.0000 0.464294
\(669\) 0 0
\(670\) 8.00000 0.309067
\(671\) −40.0000 −1.54418
\(672\) 0 0
\(673\) −6.00000 −0.231283 −0.115642 0.993291i \(-0.536892\pi\)
−0.115642 + 0.993291i \(0.536892\pi\)
\(674\) −30.0000 −1.15556
\(675\) 0 0
\(676\) −23.0000 −0.884615
\(677\) −50.0000 −1.92166 −0.960828 0.277145i \(-0.910612\pi\)
−0.960828 + 0.277145i \(0.910612\pi\)
\(678\) 0 0
\(679\) −8.00000 −0.307012
\(680\) 18.0000 0.690268
\(681\) 0 0
\(682\) 32.0000 1.22534
\(683\) −40.0000 −1.53056 −0.765279 0.643699i \(-0.777399\pi\)
−0.765279 + 0.643699i \(0.777399\pi\)
\(684\) 0 0
\(685\) −2.00000 −0.0764161
\(686\) −8.00000 −0.305441
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) 60.0000 2.28582
\(690\) 0 0
\(691\) 44.0000 1.67384 0.836919 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) 16.0000 0.607352
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) −36.0000 −1.36360
\(698\) 34.0000 1.28692
\(699\) 0 0
\(700\) −4.00000 −0.151186
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 28.0000 1.05529
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) 8.00000 0.300871
\(708\) 0 0
\(709\) 22.0000 0.826227 0.413114 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 8.00000 0.300235
\(711\) 0 0
\(712\) 18.0000 0.674579
\(713\) −32.0000 −1.19841
\(714\) 0 0
\(715\) −24.0000 −0.897549
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) 0 0
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) −48.0000 −1.78761
\(722\) 3.00000 0.111648
\(723\) 0 0
\(724\) −22.0000 −0.817624
\(725\) −1.00000 −0.0371391
\(726\) 0 0
\(727\) 16.0000 0.593407 0.296704 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 72.0000 2.66850
\(729\) 0 0
\(730\) −2.00000 −0.0740233
\(731\) −24.0000 −0.887672
\(732\) 0 0
\(733\) 2.00000 0.0738717 0.0369358 0.999318i \(-0.488240\pi\)
0.0369358 + 0.999318i \(0.488240\pi\)
\(734\) −8.00000 −0.295285
\(735\) 0 0
\(736\) −20.0000 −0.737210
\(737\) 32.0000 1.17874
\(738\) 0 0
\(739\) −36.0000 −1.32428 −0.662141 0.749380i \(-0.730352\pi\)
−0.662141 + 0.749380i \(0.730352\pi\)
\(740\) 2.00000 0.0735215
\(741\) 0 0
\(742\) −40.0000 −1.46845
\(743\) 8.00000 0.293492 0.146746 0.989174i \(-0.453120\pi\)
0.146746 + 0.989174i \(0.453120\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 2.00000 0.0732252
\(747\) 0 0
\(748\) 24.0000 0.877527
\(749\) −32.0000 −1.16925
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 6.00000 0.218507
\(755\) −16.0000 −0.582300
\(756\) 0 0
\(757\) −14.0000 −0.508839 −0.254419 0.967094i \(-0.581884\pi\)
−0.254419 + 0.967094i \(0.581884\pi\)
\(758\) 20.0000 0.726433
\(759\) 0 0
\(760\) 12.0000 0.435286
\(761\) −26.0000 −0.942499 −0.471250 0.882000i \(-0.656197\pi\)
−0.471250 + 0.882000i \(0.656197\pi\)
\(762\) 0 0
\(763\) 56.0000 2.02734
\(764\) −16.0000 −0.578860
\(765\) 0 0
\(766\) 20.0000 0.722629
\(767\) 72.0000 2.59977
\(768\) 0 0
\(769\) 42.0000 1.51456 0.757279 0.653091i \(-0.226528\pi\)
0.757279 + 0.653091i \(0.226528\pi\)
\(770\) 16.0000 0.576600
\(771\) 0 0
\(772\) −22.0000 −0.791797
\(773\) 38.0000 1.36677 0.683383 0.730061i \(-0.260508\pi\)
0.683383 + 0.730061i \(0.260508\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) −6.00000 −0.215387
\(777\) 0 0
\(778\) −2.00000 −0.0717035
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) 24.0000 0.858238
\(783\) 0 0
\(784\) −9.00000 −0.321429
\(785\) −18.0000 −0.642448
\(786\) 0 0
\(787\) −48.0000 −1.71102 −0.855508 0.517790i \(-0.826755\pi\)
−0.855508 + 0.517790i \(0.826755\pi\)
\(788\) 14.0000 0.498729
\(789\) 0 0
\(790\) 0 0
\(791\) 8.00000 0.284447
\(792\) 0 0
\(793\) −60.0000 −2.13066
\(794\) 10.0000 0.354887
\(795\) 0 0
\(796\) 24.0000 0.850657
\(797\) −50.0000 −1.77109 −0.885545 0.464553i \(-0.846215\pi\)
−0.885545 + 0.464553i \(0.846215\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −5.00000 −0.176777
\(801\) 0 0
\(802\) 2.00000 0.0706225
\(803\) −8.00000 −0.282314
\(804\) 0 0
\(805\) −16.0000 −0.563926
\(806\) 48.0000 1.69073
\(807\) 0 0
\(808\) 6.00000 0.211079
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) −20.0000 −0.702295 −0.351147 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 4.00000 0.140372
\(813\) 0 0
\(814\) −8.00000 −0.280400
\(815\) 12.0000 0.420342
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) 22.0000 0.769212
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) 10.0000 0.349002 0.174501 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 0 0
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) −36.0000 −1.25412
\(825\) 0 0
\(826\) −48.0000 −1.67013
\(827\) −20.0000 −0.695468 −0.347734 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) −8.00000 −0.277684
\(831\) 0 0
\(832\) 42.0000 1.45609
\(833\) −54.0000 −1.87099
\(834\) 0 0
\(835\) 12.0000 0.415277
\(836\) 16.0000 0.553372
\(837\) 0 0
\(838\) −4.00000 −0.138178
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 18.0000 0.620321
\(843\) 0 0
\(844\) −20.0000 −0.688428
\(845\) −23.0000 −0.791224
\(846\) 0 0
\(847\) 20.0000 0.687208
\(848\) −10.0000 −0.343401
\(849\) 0 0
\(850\) 6.00000 0.205798
\(851\) 8.00000 0.274236
\(852\) 0 0
\(853\) −38.0000 −1.30110 −0.650548 0.759465i \(-0.725461\pi\)
−0.650548 + 0.759465i \(0.725461\pi\)
\(854\) 40.0000 1.36877
\(855\) 0 0
\(856\) −24.0000 −0.820303
\(857\) −2.00000 −0.0683187 −0.0341593 0.999416i \(-0.510875\pi\)
−0.0341593 + 0.999416i \(0.510875\pi\)
\(858\) 0 0
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) 24.0000 0.817443
\(863\) −4.00000 −0.136162 −0.0680808 0.997680i \(-0.521688\pi\)
−0.0680808 + 0.997680i \(0.521688\pi\)
\(864\) 0 0
\(865\) 6.00000 0.204006
\(866\) 34.0000 1.15537
\(867\) 0 0
\(868\) 32.0000 1.08615
\(869\) 0 0
\(870\) 0 0
\(871\) 48.0000 1.62642
\(872\) 42.0000 1.42230
\(873\) 0 0
\(874\) 16.0000 0.541208
\(875\) −4.00000 −0.135225
\(876\) 0 0
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) 24.0000 0.809961
\(879\) 0 0
\(880\) 4.00000 0.134840
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) −56.0000 −1.88455 −0.942275 0.334840i \(-0.891318\pi\)
−0.942275 + 0.334840i \(0.891318\pi\)
\(884\) 36.0000 1.21081
\(885\) 0 0
\(886\) 20.0000 0.671913
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) 0 0
\(889\) −64.0000 −2.14649
\(890\) 6.00000 0.201120
\(891\) 0 0
\(892\) −28.0000 −0.937509
\(893\) 0 0
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) 12.0000 0.400892
\(897\) 0 0
\(898\) 26.0000 0.867631
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) −60.0000 −1.99889
\(902\) −24.0000 −0.799113
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) −22.0000 −0.731305
\(906\) 0 0
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) 24.0000 0.796468
\(909\) 0 0
\(910\) 24.0000 0.795592
\(911\) 56.0000 1.85536 0.927681 0.373373i \(-0.121799\pi\)
0.927681 + 0.373373i \(0.121799\pi\)
\(912\) 0 0
\(913\) −32.0000 −1.05905
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) 2.00000 0.0660819
\(917\) −48.0000 −1.58510
\(918\) 0 0
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) −12.0000 −0.395628
\(921\) 0 0
\(922\) 6.00000 0.197599
\(923\) 48.0000 1.57994
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) −36.0000 −1.18303
\(927\) 0 0
\(928\) 5.00000 0.164133
\(929\) −34.0000 −1.11550 −0.557752 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 0 0
\(931\) −36.0000 −1.17985
\(932\) −22.0000 −0.720634
\(933\) 0 0
\(934\) −20.0000 −0.654420
\(935\) 24.0000 0.784884
\(936\) 0 0
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) −32.0000 −1.04484
\(939\) 0 0
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 0 0
\(943\) 24.0000 0.781548
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) −28.0000 −0.909878 −0.454939 0.890523i \(-0.650339\pi\)
−0.454939 + 0.890523i \(0.650339\pi\)
\(948\) 0 0
\(949\) −12.0000 −0.389536
\(950\) 4.00000 0.129777
\(951\) 0 0
\(952\) −72.0000 −2.33353
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) −16.0000 −0.517748
\(956\) −8.00000 −0.258738
\(957\) 0 0
\(958\) −16.0000 −0.516937
\(959\) 8.00000 0.258333
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −12.0000 −0.386896
\(963\) 0 0
\(964\) −2.00000 −0.0644157
\(965\) −22.0000 −0.708205
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 15.0000 0.482118
\(969\) 0 0
\(970\) −2.00000 −0.0642161
\(971\) −20.0000 −0.641831 −0.320915 0.947108i \(-0.603990\pi\)
−0.320915 + 0.947108i \(0.603990\pi\)
\(972\) 0 0
\(973\) 16.0000 0.512936
\(974\) 28.0000 0.897178
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) 6.00000 0.191957 0.0959785 0.995383i \(-0.469402\pi\)
0.0959785 + 0.995383i \(0.469402\pi\)
\(978\) 0 0
\(979\) 24.0000 0.767043
\(980\) 9.00000 0.287494
\(981\) 0 0
\(982\) −44.0000 −1.40410
\(983\) −40.0000 −1.27580 −0.637901 0.770118i \(-0.720197\pi\)
−0.637901 + 0.770118i \(0.720197\pi\)
\(984\) 0 0
\(985\) 14.0000 0.446077
\(986\) −6.00000 −0.191079
\(987\) 0 0
\(988\) 24.0000 0.763542
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 40.0000 1.27000
\(993\) 0 0
\(994\) −32.0000 −1.01498
\(995\) 24.0000 0.760851
\(996\) 0 0
\(997\) 50.0000 1.58352 0.791758 0.610835i \(-0.209166\pi\)
0.791758 + 0.610835i \(0.209166\pi\)
\(998\) 36.0000 1.13956
\(999\) 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 1305.2.a.b.1.1 1
3.2 odd 2 435.2.a.d.1.1 1
5.4 even 2 6525.2.a.j.1.1 1
12.11 even 2 6960.2.a.l.1.1 1
15.2 even 4 2175.2.c.b.349.2 2
15.8 even 4 2175.2.c.b.349.1 2
15.14 odd 2 2175.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.d.1.1 1 3.2 odd 2
1305.2.a.b.1.1 1 1.1 even 1 trivial
2175.2.a.b.1.1 1 15.14 odd 2
2175.2.c.b.349.1 2 15.8 even 4
2175.2.c.b.349.2 2 15.2 even 4
6525.2.a.j.1.1 1 5.4 even 2
6960.2.a.l.1.1 1 12.11 even 2