Properties

Label 2175.2.a.b.1.1
Level $2175$
Weight $2$
Character 2175.1
Self dual yes
Analytic conductor $17.367$
Analytic rank $2$
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] = [2175,2,Mod(1,2175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2175, 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("2175.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2175.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: \(17.3674624396\)
Analytic rank: \(2\)
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 \(\chi\) \(=\) 2175.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} -4.00000 q^{11} +1.00000 q^{12} -6.00000 q^{13} +4.00000 q^{14} -1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} +4.00000 q^{21} +4.00000 q^{22} +4.00000 q^{23} -3.00000 q^{24} +6.00000 q^{26} -1.00000 q^{27} +4.00000 q^{28} +1.00000 q^{29} -8.00000 q^{31} -5.00000 q^{32} +4.00000 q^{33} +6.00000 q^{34} -1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{38} +6.00000 q^{39} -6.00000 q^{41} -4.00000 q^{42} -4.00000 q^{43} +4.00000 q^{44} -4.00000 q^{46} +1.00000 q^{48} +9.00000 q^{49} +6.00000 q^{51} +6.00000 q^{52} +10.0000 q^{53} +1.00000 q^{54} -12.0000 q^{56} +4.00000 q^{57} -1.00000 q^{58} -12.0000 q^{59} -10.0000 q^{61} +8.00000 q^{62} -4.00000 q^{63} +7.00000 q^{64} -4.00000 q^{66} -8.00000 q^{67} +6.00000 q^{68} -4.00000 q^{69} -8.00000 q^{71} +3.00000 q^{72} +2.00000 q^{73} +2.00000 q^{74} +4.00000 q^{76} +16.0000 q^{77} -6.00000 q^{78} +1.00000 q^{81} +6.00000 q^{82} -8.00000 q^{83} -4.00000 q^{84} +4.00000 q^{86} -1.00000 q^{87} -12.0000 q^{88} -6.00000 q^{89} +24.0000 q^{91} -4.00000 q^{92} +8.00000 q^{93} +5.00000 q^{96} +2.00000 q^{97} -9.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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 \(0.615027\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000 0.288675
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 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 \(0.759367\pi\)
\(18\) −1.00000 −0.235702
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 4.00000 0.852803
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) −3.00000 −0.612372
\(25\) 0 0
\(26\) 6.00000 1.17670
\(27\) −1.00000 −0.192450
\(28\) 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
\(33\) 4.00000 0.696311
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 4.00000 0.648886
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −4.00000 −0.617213
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 6.00000 0.832050
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −12.0000 −1.60357
\(57\) 4.00000 0.529813
\(58\) −1.00000 −0.131306
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 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
\(63\) −4.00000 −0.503953
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 6.00000 0.727607
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 3.00000 0.353553
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 16.0000 1.82337
\(78\) −6.00000 −0.679366
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −8.00000 −0.878114 −0.439057 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) −4.00000 −0.436436
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) −1.00000 −0.107211
\(88\) −12.0000 −1.27920
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 24.0000 2.51588
\(92\) −4.00000 −0.417029
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) 0 0
\(96\) 5.00000 0.510310
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −9.00000 −0.909137
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) −6.00000 −0.594089
\(103\) 12.0000 1.18240 0.591198 0.806527i \(-0.298655\pi\)
0.591198 + 0.806527i \(0.298655\pi\)
\(104\) −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 \(0.626383\pi\)
\(108\) 1.00000 0.0962250
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 4.00000 0.377964
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) −1.00000 −0.0928477
\(117\) −6.00000 −0.554700
\(118\) 12.0000 1.10469
\(119\) 24.0000 2.20008
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 10.0000 0.905357
\(123\) 6.00000 0.541002
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) 4.00000 0.356348
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 3.00000 0.265165
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) −4.00000 −0.348155
\(133\) 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 \(0.472772\pi\)
\(138\) 4.00000 0.340503
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) 24.0000 2.00698
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) −9.00000 −0.742307
\(148\) 2.00000 0.164399
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 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
\(153\) −6.00000 −0.485071
\(154\) −16.0000 −1.28932
\(155\) 0 0
\(156\) −6.00000 −0.480384
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 0 0
\(159\) −10.0000 −0.793052
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) −1.00000 −0.0785674
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 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 \(0.653692\pi\)
\(168\) 12.0000 0.925820
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 4.00000 0.304997
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 1.00000 0.0758098
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 12.0000 0.901975
\(178\) 6.00000 0.449719
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 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
\(183\) 10.0000 0.739221
\(184\) 12.0000 0.884652
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) 24.0000 1.75505
\(188\) 0 0
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) −7.00000 −0.505181
\(193\) −22.0000 −1.58359 −0.791797 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) −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 \(0.666200\pi\)
\(198\) 4.00000 0.284268
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 2.00000 0.140720
\(203\) −4.00000 −0.280745
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) −12.0000 −0.836080
\(207\) 4.00000 0.278019
\(208\) 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
\(213\) 8.00000 0.548151
\(214\) 8.00000 0.546869
\(215\) 0 0
\(216\) −3.00000 −0.204124
\(217\) 32.0000 2.17230
\(218\) −14.0000 −0.948200
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) 36.0000 2.42162
\(222\) −2.00000 −0.134231
\(223\) −28.0000 −1.87502 −0.937509 0.347960i \(-0.886874\pi\)
−0.937509 + 0.347960i \(0.886874\pi\)
\(224\) 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 \(0.793301\pi\)
\(228\) −4.00000 −0.264906
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) −16.0000 −1.05272
\(232\) 3.00000 0.196960
\(233\) 22.0000 1.44127 0.720634 0.693316i \(-0.243851\pi\)
0.720634 + 0.693316i \(0.243851\pi\)
\(234\) 6.00000 0.392232
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) −24.0000 −1.55569
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 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
\(243\) −1.00000 −0.0641500
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 24.0000 1.52708
\(248\) −24.0000 −1.52400
\(249\) 8.00000 0.506979
\(250\) 0 0
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 4.00000 0.251976
\(253\) −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 \(0.259294\pi\)
\(258\) −4.00000 −0.249029
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) −12.0000 −0.741362
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 12.0000 0.738549
\(265\) 0 0
\(266\) −16.0000 −0.981023
\(267\) 6.00000 0.367194
\(268\) 8.00000 0.488678
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 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
\(273\) −24.0000 −1.45255
\(274\) −2.00000 −0.120824
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) −4.00000 −0.239904
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) −8.00000 −0.475551 −0.237775 0.971320i \(-0.576418\pi\)
−0.237775 + 0.971320i \(0.576418\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) −24.0000 −1.41915
\(287\) 24.0000 1.41668
\(288\) −5.00000 −0.294628
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) −2.00000 −0.117041
\(293\) −18.0000 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 9.00000 0.524891
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) 4.00000 0.232104
\(298\) −6.00000 −0.347571
\(299\) −24.0000 −1.38796
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) −16.0000 −0.920697
\(303\) 2.00000 0.114897
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) −16.0000 −0.911685
\(309\) −12.0000 −0.682656
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 18.0000 1.01905
\(313\) 30.0000 1.69570 0.847850 0.530236i \(-0.177897\pi\)
0.847850 + 0.530236i \(0.177897\pi\)
\(314\) 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 \(0.517887\pi\)
\(318\) 10.0000 0.560772
\(319\) −4.00000 −0.223957
\(320\) 0 0
\(321\) 8.00000 0.446516
\(322\) 16.0000 0.891645
\(323\) 24.0000 1.33540
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −12.0000 −0.664619
\(327\) −14.0000 −0.774202
\(328\) −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
\(333\) −2.00000 −0.109599
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) −30.0000 −1.63420 −0.817102 0.576493i \(-0.804421\pi\)
−0.817102 + 0.576493i \(0.804421\pi\)
\(338\) −23.0000 −1.25104
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) 32.0000 1.73290
\(342\) 4.00000 0.216295
\(343\) −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 \(0.641297\pi\)
\(348\) 1.00000 0.0536056
\(349\) −34.0000 −1.81998 −0.909989 0.414632i \(-0.863910\pi\)
−0.909989 + 0.414632i \(0.863910\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) 20.0000 1.06600
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) −24.0000 −1.27021
\(358\) 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
\(363\) −5.00000 −0.262432
\(364\) −24.0000 −1.25794
\(365\) 0 0
\(366\) −10.0000 −0.522708
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) −4.00000 −0.208514
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) −40.0000 −2.07670
\(372\) −8.00000 −0.414781
\(373\) 2.00000 0.103556 0.0517780 0.998659i \(-0.483511\pi\)
0.0517780 + 0.998659i \(0.483511\pi\)
\(374\) −24.0000 −1.24101
\(375\) 0 0
\(376\) 0 0
\(377\) −6.00000 −0.309016
\(378\) −4.00000 −0.205738
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 16.0000 0.818631
\(383\) −20.0000 −1.02195 −0.510976 0.859595i \(-0.670716\pi\)
−0.510976 + 0.859595i \(0.670716\pi\)
\(384\) −3.00000 −0.153093
\(385\) 0 0
\(386\) 22.0000 1.11977
\(387\) −4.00000 −0.203331
\(388\) −2.00000 −0.101535
\(389\) −2.00000 −0.101404 −0.0507020 0.998714i \(-0.516146\pi\)
−0.0507020 + 0.998714i \(0.516146\pi\)
\(390\) 0 0
\(391\) −24.0000 −1.21373
\(392\) 27.0000 1.36371
\(393\) −12.0000 −0.605320
\(394\) 14.0000 0.705310
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) 10.0000 0.501886 0.250943 0.968002i \(-0.419259\pi\)
0.250943 + 0.968002i \(0.419259\pi\)
\(398\) 24.0000 1.20301
\(399\) −16.0000 −0.801002
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) −8.00000 −0.399004
\(403\) 48.0000 2.39105
\(404\) 2.00000 0.0995037
\(405\) 0 0
\(406\) 4.00000 0.198517
\(407\) 8.00000 0.396545
\(408\) 18.0000 0.891133
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) −12.0000 −0.591198
\(413\) 48.0000 2.36193
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) 30.0000 1.47087
\(417\) −4.00000 −0.195881
\(418\) −16.0000 −0.782586
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) 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
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) 40.0000 1.93574
\(428\) 8.00000 0.386695
\(429\) −24.0000 −1.15873
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 1.00000 0.0481125
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) −32.0000 −1.53605
\(435\) 0 0
\(436\) −14.0000 −0.670478
\(437\) −16.0000 −0.765384
\(438\) 2.00000 0.0955637
\(439\) −24.0000 −1.14546 −0.572729 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) −36.0000 −1.71235
\(443\) −20.0000 −0.950229 −0.475114 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) 28.0000 1.32584
\(447\) −6.00000 −0.283790
\(448\) −28.0000 −1.32288
\(449\) 26.0000 1.22702 0.613508 0.789689i \(-0.289758\pi\)
0.613508 + 0.789689i \(0.289758\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) −2.00000 −0.0940721
\(453\) −16.0000 −0.751746
\(454\) 24.0000 1.12638
\(455\) 0 0
\(456\) 12.0000 0.561951
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 2.00000 0.0934539
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 16.0000 0.744387
\(463\) −36.0000 −1.67306 −0.836531 0.547920i \(-0.815420\pi\)
−0.836531 + 0.547920i \(0.815420\pi\)
\(464\) −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 \(0.346865\pi\)
\(468\) 6.00000 0.277350
\(469\) 32.0000 1.47762
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) −36.0000 −1.65703
\(473\) 16.0000 0.735681
\(474\) 0 0
\(475\) 0 0
\(476\) −24.0000 −1.10004
\(477\) 10.0000 0.457869
\(478\) 8.00000 0.365911
\(479\) −16.0000 −0.731059 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) −2.00000 −0.0910975
\(483\) 16.0000 0.728025
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 28.0000 1.26880 0.634401 0.773004i \(-0.281247\pi\)
0.634401 + 0.773004i \(0.281247\pi\)
\(488\) −30.0000 −1.35804
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) −44.0000 −1.98569 −0.992846 0.119401i \(-0.961903\pi\)
−0.992846 + 0.119401i \(0.961903\pi\)
\(492\) −6.00000 −0.270501
\(493\) −6.00000 −0.270226
\(494\) −24.0000 −1.07981
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 32.0000 1.43540
\(498\) −8.00000 −0.358489
\(499\) −36.0000 −1.61158 −0.805791 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 20.0000 0.892644
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) −12.0000 −0.534522
\(505\) 0 0
\(506\) 16.0000 0.711287
\(507\) −23.0000 −1.02147
\(508\) −16.0000 −0.709885
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) 11.0000 0.486136
\(513\) 4.00000 0.176604
\(514\) −22.0000 −0.970378
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) −8.00000 −0.351500
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −22.0000 −0.963837 −0.481919 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) −1.00000 −0.0437688
\(523\) 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
\(528\) −4.00000 −0.174078
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) −16.0000 −0.693688
\(533\) 36.0000 1.55933
\(534\) −6.00000 −0.259645
\(535\) 0 0
\(536\) −24.0000 −1.03664
\(537\) 12.0000 0.517838
\(538\) −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
\(543\) −22.0000 −0.944110
\(544\) 30.0000 1.28624
\(545\) 0 0
\(546\) 24.0000 1.02711
\(547\) −40.0000 −1.71028 −0.855138 0.518400i \(-0.826528\pi\)
−0.855138 + 0.518400i \(0.826528\pi\)
\(548\) −2.00000 −0.0854358
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) −4.00000 −0.170406
\(552\) −12.0000 −0.510754
\(553\) 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 \(0.719235\pi\)
\(558\) 8.00000 0.338667
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) −24.0000 −1.01328
\(562\) −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
\(565\) 0 0
\(566\) 8.00000 0.336265
\(567\) −4.00000 −0.167984
\(568\) −24.0000 −1.00702
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 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
\(573\) 16.0000 0.668410
\(574\) −24.0000 −1.00174
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) 10.0000 0.416305 0.208153 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) −19.0000 −0.790296
\(579\) 22.0000 0.914289
\(580\) 0 0
\(581\) 32.0000 1.32758
\(582\) 2.00000 0.0829027
\(583\) −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 \(0.392885\pi\)
\(588\) 9.00000 0.371154
\(589\) 32.0000 1.31854
\(590\) 0 0
\(591\) 14.0000 0.575883
\(592\) 2.00000 0.0821995
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 24.0000 0.982255
\(598\) 24.0000 0.981433
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 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
\(603\) −8.00000 −0.325785
\(604\) −16.0000 −0.651031
\(605\) 0 0
\(606\) −2.00000 −0.0812444
\(607\) −32.0000 −1.29884 −0.649420 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) 20.0000 0.811107
\(609\) 4.00000 0.162088
\(610\) 0 0
\(611\) 0 0
\(612\) 6.00000 0.242536
\(613\) −6.00000 −0.242338 −0.121169 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) 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 \(0.381983\pi\)
\(618\) 12.0000 0.482711
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 0 0
\(623\) 24.0000 0.961540
\(624\) −6.00000 −0.240192
\(625\) 0 0
\(626\) −30.0000 −1.19904
\(627\) −16.0000 −0.638978
\(628\) 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
\(633\) −20.0000 −0.794929
\(634\) 2.00000 0.0794301
\(635\) 0 0
\(636\) 10.0000 0.396526
\(637\) −54.0000 −2.13956
\(638\) 4.00000 0.158362
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) 34.0000 1.34292 0.671460 0.741041i \(-0.265668\pi\)
0.671460 + 0.741041i \(0.265668\pi\)
\(642\) −8.00000 −0.315735
\(643\) −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
\(647\) 28.0000 1.10079 0.550397 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) 3.00000 0.117851
\(649\) 48.0000 1.88416
\(650\) 0 0
\(651\) −32.0000 −1.25418
\(652\) −12.0000 −0.469956
\(653\) −50.0000 −1.95665 −0.978326 0.207072i \(-0.933606\pi\)
−0.978326 + 0.207072i \(0.933606\pi\)
\(654\) 14.0000 0.547443
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 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
\(663\) −36.0000 −1.39812
\(664\) −24.0000 −0.931381
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) 4.00000 0.154881
\(668\) 12.0000 0.464294
\(669\) 28.0000 1.08254
\(670\) 0 0
\(671\) 40.0000 1.54418
\(672\) −20.0000 −0.771517
\(673\) 6.00000 0.231283 0.115642 0.993291i \(-0.463108\pi\)
0.115642 + 0.993291i \(0.463108\pi\)
\(674\) 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 \(0.910612\pi\)
\(678\) 2.00000 0.0768095
\(679\) −8.00000 −0.307012
\(680\) 0 0
\(681\) 24.0000 0.919682
\(682\) −32.0000 −1.22534
\(683\) −40.0000 −1.53056 −0.765279 0.643699i \(-0.777399\pi\)
−0.765279 + 0.643699i \(0.777399\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) 8.00000 0.305441
\(687\) 2.00000 0.0763048
\(688\) 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
\(693\) 16.0000 0.607790
\(694\) 16.0000 0.607352
\(695\) 0 0
\(696\) −3.00000 −0.113715
\(697\) 36.0000 1.36360
\(698\) 34.0000 1.28692
\(699\) −22.0000 −0.832116
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) −6.00000 −0.226455
\(703\) 8.00000 0.301726
\(704\) −28.0000 −1.05529
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) 8.00000 0.300871
\(708\) −12.0000 −0.450988
\(709\) 22.0000 0.826227 0.413114 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −18.0000 −0.674579
\(713\) −32.0000 −1.19841
\(714\) 24.0000 0.898177
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 8.00000 0.298765
\(718\) 0 0
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) −48.0000 −1.78761
\(722\) 3.00000 0.111648
\(723\) −2.00000 −0.0743808
\(724\) −22.0000 −0.817624
\(725\) 0 0
\(726\) 5.00000 0.185567
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 72.0000 2.66850
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) −10.0000 −0.369611
\(733\) −2.00000 −0.0738717 −0.0369358 0.999318i \(-0.511760\pi\)
−0.0369358 + 0.999318i \(0.511760\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) −20.0000 −0.737210
\(737\) 32.0000 1.17874
\(738\) 6.00000 0.220863
\(739\) −36.0000 −1.32428 −0.662141 0.749380i \(-0.730352\pi\)
−0.662141 + 0.749380i \(0.730352\pi\)
\(740\) 0 0
\(741\) −24.0000 −0.881662
\(742\) 40.0000 1.46845
\(743\) 8.00000 0.293492 0.146746 0.989174i \(-0.453120\pi\)
0.146746 + 0.989174i \(0.453120\pi\)
\(744\) 24.0000 0.879883
\(745\) 0 0
\(746\) −2.00000 −0.0732252
\(747\) −8.00000 −0.292705
\(748\) −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
\(753\) 20.0000 0.728841
\(754\) 6.00000 0.218507
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) 14.0000 0.508839 0.254419 0.967094i \(-0.418116\pi\)
0.254419 + 0.967094i \(0.418116\pi\)
\(758\) 20.0000 0.726433
\(759\) 16.0000 0.580763
\(760\) 0 0
\(761\) 26.0000 0.942499 0.471250 0.882000i \(-0.343803\pi\)
0.471250 + 0.882000i \(0.343803\pi\)
\(762\) 16.0000 0.579619
\(763\) −56.0000 −2.02734
\(764\) 16.0000 0.578860
\(765\) 0 0
\(766\) 20.0000 0.722629
\(767\) 72.0000 2.59977
\(768\) 17.0000 0.613435
\(769\) 42.0000 1.51456 0.757279 0.653091i \(-0.226528\pi\)
0.757279 + 0.653091i \(0.226528\pi\)
\(770\) 0 0
\(771\) −22.0000 −0.792311
\(772\) 22.0000 0.791797
\(773\) 38.0000 1.36677 0.683383 0.730061i \(-0.260508\pi\)
0.683383 + 0.730061i \(0.260508\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) 6.00000 0.215387
\(777\) −8.00000 −0.286998
\(778\) 2.00000 0.0717035
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) 24.0000 0.858238
\(783\) −1.00000 −0.0357371
\(784\) −9.00000 −0.321429
\(785\) 0 0
\(786\) 12.0000 0.428026
\(787\) 48.0000 1.71102 0.855508 0.517790i \(-0.173245\pi\)
0.855508 + 0.517790i \(0.173245\pi\)
\(788\) 14.0000 0.498729
\(789\) 0 0
\(790\) 0 0
\(791\) −8.00000 −0.284447
\(792\) −12.0000 −0.426401
\(793\) 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 \(0.846215\pi\)
\(798\) 16.0000 0.566394
\(799\) 0 0
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) −2.00000 −0.0706225
\(803\) −8.00000 −0.282314
\(804\) −8.00000 −0.282138
\(805\) 0 0
\(806\) −48.0000 −1.69073
\(807\) −6.00000 −0.211210
\(808\) −6.00000 −0.211079
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 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
\(813\) −16.0000 −0.561144
\(814\) −8.00000 −0.280400
\(815\) 0 0
\(816\) −6.00000 −0.210042
\(817\) 16.0000 0.559769
\(818\) 22.0000 0.769212
\(819\) 24.0000 0.838628
\(820\) 0 0
\(821\) −10.0000 −0.349002 −0.174501 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) 2.00000 0.0697580
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 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 \(0.613049\pi\)
\(828\) −4.00000 −0.139010
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 0 0
\(831\) −26.0000 −0.901930
\(832\) −42.0000 −1.45609
\(833\) −54.0000 −1.87099
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) −16.0000 −0.553372
\(837\) 8.00000 0.276520
\(838\) 4.00000 0.138178
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 18.0000 0.620321
\(843\) −10.0000 −0.344418
\(844\) −20.0000 −0.688428
\(845\) 0 0
\(846\) 0 0
\(847\) −20.0000 −0.687208
\(848\) −10.0000 −0.343401
\(849\) 8.00000 0.274559
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) −8.00000 −0.274075
\(853\) 38.0000 1.30110 0.650548 0.759465i \(-0.274539\pi\)
0.650548 + 0.759465i \(0.274539\pi\)
\(854\) −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 \(0.510875\pi\)
\(858\) 24.0000 0.819346
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 0 0
\(861\) −24.0000 −0.817918
\(862\) −24.0000 −0.817443
\(863\) −4.00000 −0.136162 −0.0680808 0.997680i \(-0.521688\pi\)
−0.0680808 + 0.997680i \(0.521688\pi\)
\(864\) 5.00000 0.170103
\(865\) 0 0
\(866\) −34.0000 −1.15537
\(867\) −19.0000 −0.645274
\(868\) −32.0000 −1.08615
\(869\) 0 0
\(870\) 0 0
\(871\) 48.0000 1.62642
\(872\) 42.0000 1.42230
\(873\) 2.00000 0.0676897
\(874\) 16.0000 0.541208
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) −14.0000 −0.472746 −0.236373 0.971662i \(-0.575959\pi\)
−0.236373 + 0.971662i \(0.575959\pi\)
\(878\) 24.0000 0.809961
\(879\) 18.0000 0.607125
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) −9.00000 −0.303046
\(883\) 56.0000 1.88455 0.942275 0.334840i \(-0.108682\pi\)
0.942275 + 0.334840i \(0.108682\pi\)
\(884\) −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 \(0.542881\pi\)
\(888\) 6.00000 0.201347
\(889\) −64.0000 −2.14649
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 28.0000 0.937509
\(893\) 0 0
\(894\) 6.00000 0.200670
\(895\) 0 0
\(896\) −12.0000 −0.400892
\(897\) 24.0000 0.801337
\(898\) −26.0000 −0.867631
\(899\) −8.00000 −0.266815
\(900\) 0 0
\(901\) −60.0000 −1.99889
\(902\) −24.0000 −0.799113
\(903\) −16.0000 −0.532447
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) 16.0000 0.531564
\(907\) 44.0000 1.46100 0.730498 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) 24.0000 0.796468
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) −56.0000 −1.85536 −0.927681 0.373373i \(-0.878201\pi\)
−0.927681 + 0.373373i \(0.878201\pi\)
\(912\) −4.00000 −0.132453
\(913\) 32.0000 1.05905
\(914\) 10.0000 0.330771
\(915\) 0 0
\(916\) 2.00000 0.0660819
\(917\) −48.0000 −1.58510
\(918\) −6.00000 −0.198030
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) −6.00000 −0.197599
\(923\) 48.0000 1.57994
\(924\) 16.0000 0.526361
\(925\) 0 0
\(926\) 36.0000 1.18303
\(927\) 12.0000 0.394132
\(928\) −5.00000 −0.164133
\(929\) 34.0000 1.11550 0.557752 0.830008i \(-0.311664\pi\)
0.557752 + 0.830008i \(0.311664\pi\)
\(930\) 0 0
\(931\) −36.0000 −1.17985
\(932\) −22.0000 −0.720634
\(933\) 0 0
\(934\) −20.0000 −0.654420
\(935\) 0 0
\(936\) −18.0000 −0.588348
\(937\) −10.0000 −0.326686 −0.163343 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(938\) −32.0000 −1.04484
\(939\) −30.0000 −0.979013
\(940\) 0 0
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) −18.0000 −0.586472
\(943\) −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
\(950\) 0 0
\(951\) 2.00000 0.0648544
\(952\) 72.0000 2.33353
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) −10.0000 −0.323762
\(955\) 0 0
\(956\) 8.00000 0.258738
\(957\) 4.00000 0.129302
\(958\) 16.0000 0.516937
\(959\) −8.00000 −0.258333
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −12.0000 −0.386896
\(963\) −8.00000 −0.257796
\(964\) −2.00000 −0.0644157
\(965\) 0 0
\(966\) −16.0000 −0.514792
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 15.0000 0.482118
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) 20.0000 0.641831 0.320915 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) 1.00000 0.0320750
\(973\) −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 \(0.469402\pi\)
\(978\) 12.0000 0.383718
\(979\) 24.0000 0.767043
\(980\) 0 0
\(981\) 14.0000 0.446986
\(982\) 44.0000 1.40410
\(983\) −40.0000 −1.27580 −0.637901 0.770118i \(-0.720197\pi\)
−0.637901 + 0.770118i \(0.720197\pi\)
\(984\) 18.0000 0.573819
\(985\) 0 0
\(986\) 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
\(993\) −4.00000 −0.126936
\(994\) −32.0000 −1.01498
\(995\) 0 0
\(996\) −8.00000 −0.253490
\(997\) −50.0000 −1.58352 −0.791758 0.610835i \(-0.790834\pi\)
−0.791758 + 0.610835i \(0.790834\pi\)
\(998\) 36.0000 1.13956
\(999\) 2.00000 0.0632772
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 2175.2.a.b.1.1 1
3.2 odd 2 6525.2.a.j.1.1 1
5.2 odd 4 2175.2.c.b.349.1 2
5.3 odd 4 2175.2.c.b.349.2 2
5.4 even 2 435.2.a.d.1.1 1
15.14 odd 2 1305.2.a.b.1.1 1
20.19 odd 2 6960.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.d.1.1 1 5.4 even 2
1305.2.a.b.1.1 1 15.14 odd 2
2175.2.a.b.1.1 1 1.1 even 1 trivial
2175.2.c.b.349.1 2 5.2 odd 4
2175.2.c.b.349.2 2 5.3 odd 4
6525.2.a.j.1.1 1 3.2 odd 2
6960.2.a.l.1.1 1 20.19 odd 2