Properties

Label 435.2.a.d.1.1
Level $435$
Weight $2$
Character 435.1
Self dual yes
Analytic conductor $3.473$
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] = [435,2,Mod(1,435)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(435, 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("435.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 435 = 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 435.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: \(3.47349248793\)
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 \(\chi\) \(=\) 435.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} +6.00000 q^{13} +4.00000 q^{14} +1.00000 q^{15} -1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} +4.00000 q^{21} -4.00000 q^{22} -4.00000 q^{23} -3.00000 q^{24} +1.00000 q^{25} +6.00000 q^{26} +1.00000 q^{27} -4.00000 q^{28} +1.00000 q^{29} +1.00000 q^{30} -8.00000 q^{31} +5.00000 q^{32} -4.00000 q^{33} +6.00000 q^{34} +4.00000 q^{35} -1.00000 q^{36} +2.00000 q^{37} -4.00000 q^{38} +6.00000 q^{39} -3.00000 q^{40} -6.00000 q^{41} +4.00000 q^{42} +4.00000 q^{43} +4.00000 q^{44} +1.00000 q^{45} -4.00000 q^{46} -1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} +6.00000 q^{51} -6.00000 q^{52} -10.0000 q^{53} +1.00000 q^{54} -4.00000 q^{55} -12.0000 q^{56} -4.00000 q^{57} +1.00000 q^{58} -12.0000 q^{59} -1.00000 q^{60} -10.0000 q^{61} -8.00000 q^{62} +4.00000 q^{63} +7.00000 q^{64} +6.00000 q^{65} -4.00000 q^{66} +8.00000 q^{67} -6.00000 q^{68} -4.00000 q^{69} +4.00000 q^{70} -8.00000 q^{71} -3.00000 q^{72} -2.00000 q^{73} +2.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} -16.0000 q^{77} +6.00000 q^{78} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +8.00000 q^{83} -4.00000 q^{84} +6.00000 q^{85} +4.00000 q^{86} +1.00000 q^{87} +12.0000 q^{88} -6.00000 q^{89} +1.00000 q^{90} +24.0000 q^{91} +4.00000 q^{92} -8.00000 q^{93} -4.00000 q^{95} +5.00000 q^{96} -2.00000 q^{97} +9.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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 \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) 1.00000 0.447214
\(6\) 1.00000 0.408248
\(7\) 4.00000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −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 \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 4.00000 1.06904
\(15\) 1.00000 0.258199
\(16\) −1.00000 −0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −1.00000 −0.223607
\(21\) 4.00000 0.872872
\(22\) −4.00000 −0.852803
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −3.00000 −0.612372
\(25\) 1.00000 0.200000
\(26\) 6.00000 1.17670
\(27\) 1.00000 0.192450
\(28\) −4.00000 −0.755929
\(29\) 1.00000 0.185695
\(30\) 1.00000 0.182574
\(31\) −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
\(35\) 4.00000 0.676123
\(36\) −1.00000 −0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −4.00000 −0.648886
\(39\) 6.00000 0.960769
\(40\) −3.00000 −0.474342
\(41\) −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 \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 4.00000 0.603023
\(45\) 1.00000 0.149071
\(46\) −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
\(50\) 1.00000 0.141421
\(51\) 6.00000 0.840168
\(52\) −6.00000 −0.832050
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 1.00000 0.136083
\(55\) −4.00000 −0.539360
\(56\) −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 \(0.785358\pi\)
\(60\) −1.00000 −0.129099
\(61\) −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
\(65\) 6.00000 0.744208
\(66\) −4.00000 −0.492366
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −6.00000 −0.727607
\(69\) −4.00000 −0.481543
\(70\) 4.00000 0.478091
\(71\) −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 \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 2.00000 0.232495
\(75\) 1.00000 0.115470
\(76\) 4.00000 0.458831
\(77\) −16.0000 −1.82337
\(78\) 6.00000 0.679366
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) −4.00000 −0.436436
\(85\) 6.00000 0.650791
\(86\) 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 \(0.603011\pi\)
\(90\) 1.00000 0.105409
\(91\) 24.0000 2.51588
\(92\) 4.00000 0.417029
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 5.00000 0.510310
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 9.00000 0.909137
\(99\) −4.00000 −0.402015
\(100\) −1.00000 −0.100000
\(101\) −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 \(-0.701345\pi\)
−0.591198 + 0.806527i \(0.701345\pi\)
\(104\) −18.0000 −1.76505
\(105\) 4.00000 0.390360
\(106\) −10.0000 −0.971286
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) −4.00000 −0.381385
\(111\) 2.00000 0.189832
\(112\) −4.00000 −0.377964
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) −4.00000 −0.374634
\(115\) −4.00000 −0.373002
\(116\) −1.00000 −0.0928477
\(117\) 6.00000 0.554700
\(118\) −12.0000 −1.10469
\(119\) 24.0000 2.20008
\(120\) −3.00000 −0.273861
\(121\) 5.00000 0.454545
\(122\) −10.0000 −0.905357
\(123\) −6.00000 −0.541002
\(124\) 8.00000 0.718421
\(125\) 1.00000 0.0894427
\(126\) 4.00000 0.356348
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) −3.00000 −0.265165
\(129\) 4.00000 0.352180
\(130\) 6.00000 0.526235
\(131\) 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
\(135\) 1.00000 0.0860663
\(136\) −18.0000 −1.54349
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) −4.00000 −0.340503
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) −4.00000 −0.338062
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) −24.0000 −2.00698
\(144\) −1.00000 −0.0833333
\(145\) 1.00000 0.0830455
\(146\) −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 \(0.420959\pi\)
\(150\) 1.00000 0.0816497
\(151\) 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
\(155\) −8.00000 −0.642575
\(156\) −6.00000 −0.480384
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 0 0
\(159\) −10.0000 −0.793052
\(160\) 5.00000 0.395285
\(161\) −16.0000 −1.26098
\(162\) 1.00000 0.0785674
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 6.00000 0.468521
\(165\) −4.00000 −0.311400
\(166\) 8.00000 0.620920
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) −12.0000 −0.925820
\(169\) 23.0000 1.76923
\(170\) 6.00000 0.460179
\(171\) −4.00000 −0.305888
\(172\) −4.00000 −0.304997
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 1.00000 0.0758098
\(175\) 4.00000 0.302372
\(176\) 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 \(0.648028\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 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
\(185\) 2.00000 0.147043
\(186\) −8.00000 −0.586588
\(187\) −24.0000 −1.75505
\(188\) 0 0
\(189\) 4.00000 0.290957
\(190\) −4.00000 −0.290191
\(191\) −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 \(-0.209146\pi\)
0.791797 + 0.610784i \(0.209146\pi\)
\(194\) −2.00000 −0.143592
\(195\) 6.00000 0.429669
\(196\) −9.00000 −0.642857
\(197\) 14.0000 0.997459 0.498729 0.866758i \(-0.333800\pi\)
0.498729 + 0.866758i \(0.333800\pi\)
\(198\) −4.00000 −0.284268
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) −3.00000 −0.212132
\(201\) 8.00000 0.564276
\(202\) −2.00000 −0.140720
\(203\) 4.00000 0.280745
\(204\) −6.00000 −0.420084
\(205\) −6.00000 −0.419058
\(206\) −12.0000 −0.836080
\(207\) −4.00000 −0.278019
\(208\) −6.00000 −0.416025
\(209\) 16.0000 1.10674
\(210\) 4.00000 0.276026
\(211\) 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
\(215\) 4.00000 0.272798
\(216\) −3.00000 −0.204124
\(217\) −32.0000 −2.17230
\(218\) 14.0000 0.948200
\(219\) −2.00000 −0.135147
\(220\) 4.00000 0.269680
\(221\) 36.0000 2.42162
\(222\) 2.00000 0.134231
\(223\) 28.0000 1.87502 0.937509 0.347960i \(-0.113126\pi\)
0.937509 + 0.347960i \(0.113126\pi\)
\(224\) 20.0000 1.33631
\(225\) 1.00000 0.0666667
\(226\) −2.00000 −0.133038
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) 4.00000 0.264906
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) −4.00000 −0.263752
\(231\) −16.0000 −1.05272
\(232\) −3.00000 −0.196960
\(233\) −22.0000 −1.44127 −0.720634 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(234\) 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 \(0.583307\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 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
\(245\) 9.00000 0.574989
\(246\) −6.00000 −0.382546
\(247\) −24.0000 −1.52708
\(248\) 24.0000 1.52400
\(249\) 8.00000 0.506979
\(250\) 1.00000 0.0632456
\(251\) −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
\(255\) 6.00000 0.375735
\(256\) −17.0000 −1.06250
\(257\) −22.0000 −1.37232 −0.686161 0.727450i \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) 4.00000 0.249029
\(259\) 8.00000 0.497096
\(260\) −6.00000 −0.372104
\(261\) 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
\(265\) −10.0000 −0.614295
\(266\) −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 \(0.441447\pi\)
\(270\) 1.00000 0.0608581
\(271\) 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
\(275\) −4.00000 −0.241209
\(276\) 4.00000 0.240772
\(277\) −26.0000 −1.56219 −0.781094 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) 4.00000 0.239904
\(279\) −8.00000 −0.478947
\(280\) −12.0000 −0.717137
\(281\) 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 \(-0.423582\pi\)
0.237775 + 0.971320i \(0.423582\pi\)
\(284\) 8.00000 0.474713
\(285\) −4.00000 −0.236940
\(286\) −24.0000 −1.41915
\(287\) −24.0000 −1.41668
\(288\) 5.00000 0.294628
\(289\) 19.0000 1.11765
\(290\) 1.00000 0.0587220
\(291\) −2.00000 −0.117242
\(292\) 2.00000 0.117041
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 9.00000 0.524891
\(295\) −12.0000 −0.698667
\(296\) −6.00000 −0.348743
\(297\) −4.00000 −0.232104
\(298\) 6.00000 0.347571
\(299\) −24.0000 −1.38796
\(300\) −1.00000 −0.0577350
\(301\) 16.0000 0.922225
\(302\) 16.0000 0.920697
\(303\) −2.00000 −0.114897
\(304\) 4.00000 0.229416
\(305\) −10.0000 −0.572598
\(306\) 6.00000 0.342997
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 16.0000 0.911685
\(309\) −12.0000 −0.682656
\(310\) −8.00000 −0.454369
\(311\) 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 \(-0.822103\pi\)
−0.847850 + 0.530236i \(0.822103\pi\)
\(314\) 18.0000 1.01580
\(315\) 4.00000 0.225374
\(316\) 0 0
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) −10.0000 −0.560772
\(319\) −4.00000 −0.223957
\(320\) 7.00000 0.391312
\(321\) 8.00000 0.446516
\(322\) −16.0000 −0.891645
\(323\) −24.0000 −1.33540
\(324\) −1.00000 −0.0555556
\(325\) 6.00000 0.332820
\(326\) −12.0000 −0.664619
\(327\) 14.0000 0.774202
\(328\) 18.0000 0.993884
\(329\) 0 0
\(330\) −4.00000 −0.220193
\(331\) 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
\(335\) 8.00000 0.437087
\(336\) −4.00000 −0.218218
\(337\) 30.0000 1.63420 0.817102 0.576493i \(-0.195579\pi\)
0.817102 + 0.576493i \(0.195579\pi\)
\(338\) 23.0000 1.25104
\(339\) −2.00000 −0.108625
\(340\) −6.00000 −0.325396
\(341\) 32.0000 1.73290
\(342\) −4.00000 −0.216295
\(343\) 8.00000 0.431959
\(344\) −12.0000 −0.646997
\(345\) −4.00000 −0.215353
\(346\) 6.00000 0.322562
\(347\) 16.0000 0.858925 0.429463 0.903085i \(-0.358703\pi\)
0.429463 + 0.903085i \(0.358703\pi\)
\(348\) −1.00000 −0.0536056
\(349\) −34.0000 −1.81998 −0.909989 0.414632i \(-0.863910\pi\)
−0.909989 + 0.414632i \(0.863910\pi\)
\(350\) 4.00000 0.213809
\(351\) 6.00000 0.320256
\(352\) −20.0000 −1.06600
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) −12.0000 −0.637793
\(355\) −8.00000 −0.424596
\(356\) 6.00000 0.317999
\(357\) 24.0000 1.27021
\(358\) −12.0000 −0.634220
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −3.00000 −0.158114
\(361\) −3.00000 −0.157895
\(362\) 22.0000 1.15629
\(363\) 5.00000 0.262432
\(364\) −24.0000 −1.25794
\(365\) −2.00000 −0.104685
\(366\) −10.0000 −0.522708
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 4.00000 0.208514
\(369\) −6.00000 −0.312348
\(370\) 2.00000 0.103975
\(371\) −40.0000 −2.07670
\(372\) 8.00000 0.414781
\(373\) −2.00000 −0.103556 −0.0517780 0.998659i \(-0.516489\pi\)
−0.0517780 + 0.998659i \(0.516489\pi\)
\(374\) −24.0000 −1.24101
\(375\) 1.00000 0.0516398
\(376\) 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 \(0.671713\pi\)
\(380\) 4.00000 0.205196
\(381\) −16.0000 −0.819705
\(382\) −16.0000 −0.818631
\(383\) 20.0000 1.02195 0.510976 0.859595i \(-0.329284\pi\)
0.510976 + 0.859595i \(0.329284\pi\)
\(384\) −3.00000 −0.153093
\(385\) −16.0000 −0.815436
\(386\) 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 \(0.516146\pi\)
\(390\) 6.00000 0.303822
\(391\) −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 \(-0.580741\pi\)
−0.250943 + 0.968002i \(0.580741\pi\)
\(398\) −24.0000 −1.20301
\(399\) −16.0000 −0.801002
\(400\) −1.00000 −0.0500000
\(401\) 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
\(405\) 1.00000 0.0496904
\(406\) 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 \(0.683059\pi\)
\(410\) −6.00000 −0.296319
\(411\) −2.00000 −0.0986527
\(412\) 12.0000 0.591198
\(413\) −48.0000 −2.36193
\(414\) −4.00000 −0.196589
\(415\) 8.00000 0.392705
\(416\) 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 \(0.531151\pi\)
\(420\) −4.00000 −0.195180
\(421\) −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
\(426\) −8.00000 −0.387601
\(427\) −40.0000 −1.93574
\(428\) −8.00000 −0.386695
\(429\) −24.0000 −1.15873
\(430\) 4.00000 0.192897
\(431\) 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 \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) −32.0000 −1.53605
\(435\) 1.00000 0.0479463
\(436\) −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 \(0.694115\pi\)
\(440\) 12.0000 0.572078
\(441\) 9.00000 0.428571
\(442\) 36.0000 1.71235
\(443\) 20.0000 0.950229 0.475114 0.879924i \(-0.342407\pi\)
0.475114 + 0.879924i \(0.342407\pi\)
\(444\) −2.00000 −0.0949158
\(445\) −6.00000 −0.284427
\(446\) 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 \(0.289758\pi\)
\(450\) 1.00000 0.0471405
\(451\) 24.0000 1.13012
\(452\) 2.00000 0.0940721
\(453\) 16.0000 0.751746
\(454\) 24.0000 1.12638
\(455\) 24.0000 1.12514
\(456\) 12.0000 0.561951
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) −2.00000 −0.0934539
\(459\) 6.00000 0.280056
\(460\) 4.00000 0.186501
\(461\) 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 \(-0.184580\pi\)
0.836531 + 0.547920i \(0.184580\pi\)
\(464\) −1.00000 −0.0464238
\(465\) −8.00000 −0.370991
\(466\) −22.0000 −1.01913
\(467\) −20.0000 −0.925490 −0.462745 0.886492i \(-0.653135\pi\)
−0.462745 + 0.886492i \(0.653135\pi\)
\(468\) −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
\(475\) −4.00000 −0.183533
\(476\) −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 \(0.619112\pi\)
\(480\) 5.00000 0.228218
\(481\) 12.0000 0.547153
\(482\) 2.00000 0.0910975
\(483\) −16.0000 −0.728025
\(484\) −5.00000 −0.227273
\(485\) −2.00000 −0.0908153
\(486\) 1.00000 0.0453609
\(487\) −28.0000 −1.26880 −0.634401 0.773004i \(-0.718753\pi\)
−0.634401 + 0.773004i \(0.718753\pi\)
\(488\) 30.0000 1.35804
\(489\) −12.0000 −0.542659
\(490\) 9.00000 0.406579
\(491\) −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
\(495\) −4.00000 −0.179787
\(496\) 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 \(0.798259\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 12.0000 0.536120
\(502\) −20.0000 −0.892644
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) −12.0000 −0.534522
\(505\) −2.00000 −0.0889988
\(506\) 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 \(0.514113\pi\)
\(510\) 6.00000 0.265684
\(511\) −8.00000 −0.353899
\(512\) −11.0000 −0.486136
\(513\) −4.00000 −0.176604
\(514\) −22.0000 −0.970378
\(515\) −12.0000 −0.528783
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 8.00000 0.351500
\(519\) 6.00000 0.263371
\(520\) −18.0000 −0.789352
\(521\) −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
\(525\) 4.00000 0.174574
\(526\) 0 0
\(527\) −48.0000 −2.09091
\(528\) 4.00000 0.174078
\(529\) −7.00000 −0.304348
\(530\) −10.0000 −0.434372
\(531\) −12.0000 −0.520756
\(532\) 16.0000 0.693688
\(533\) −36.0000 −1.55933
\(534\) −6.00000 −0.259645
\(535\) 8.00000 0.345870
\(536\) −24.0000 −1.03664
\(537\) −12.0000 −0.517838
\(538\) 6.00000 0.258678
\(539\) −36.0000 −1.55063
\(540\) −1.00000 −0.0430331
\(541\) −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
\(545\) 14.0000 0.599694
\(546\) 24.0000 1.02711
\(547\) 40.0000 1.71028 0.855138 0.518400i \(-0.173472\pi\)
0.855138 + 0.518400i \(0.173472\pi\)
\(548\) 2.00000 0.0854358
\(549\) −10.0000 −0.426790
\(550\) −4.00000 −0.170561
\(551\) −4.00000 −0.170406
\(552\) 12.0000 0.510754
\(553\) 0 0
\(554\) −26.0000 −1.10463
\(555\) 2.00000 0.0848953
\(556\) −4.00000 −0.169638
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) −8.00000 −0.338667
\(559\) 24.0000 1.01509
\(560\) −4.00000 −0.169031
\(561\) −24.0000 −1.01328
\(562\) 10.0000 0.421825
\(563\) −28.0000 −1.18006 −0.590030 0.807382i \(-0.700884\pi\)
−0.590030 + 0.807382i \(0.700884\pi\)
\(564\) 0 0
\(565\) −2.00000 −0.0841406
\(566\) 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 \(0.716467\pi\)
\(570\) −4.00000 −0.167542
\(571\) 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
\(575\) −4.00000 −0.166812
\(576\) 7.00000 0.291667
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) 19.0000 0.790296
\(579\) 22.0000 0.914289
\(580\) −1.00000 −0.0415227
\(581\) 32.0000 1.32758
\(582\) −2.00000 −0.0829027
\(583\) 40.0000 1.65663
\(584\) 6.00000 0.248282
\(585\) 6.00000 0.248069
\(586\) 18.0000 0.743573
\(587\) −16.0000 −0.660391 −0.330195 0.943913i \(-0.607115\pi\)
−0.330195 + 0.943913i \(0.607115\pi\)
\(588\) −9.00000 −0.371154
\(589\) 32.0000 1.31854
\(590\) −12.0000 −0.494032
\(591\) 14.0000 0.575883
\(592\) −2.00000 −0.0821995
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) −4.00000 −0.164122
\(595\) 24.0000 0.983904
\(596\) −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 \(0.336885\pi\)
\(600\) −3.00000 −0.122474
\(601\) −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
\(605\) 5.00000 0.203279
\(606\) −2.00000 −0.0812444
\(607\) 32.0000 1.29884 0.649420 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) −20.0000 −0.811107
\(609\) 4.00000 0.162088
\(610\) −10.0000 −0.404888
\(611\) 0 0
\(612\) −6.00000 −0.242536
\(613\) 6.00000 0.242338 0.121169 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) 12.0000 0.484281
\(615\) −6.00000 −0.241943
\(616\) 48.0000 1.93398
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) −12.0000 −0.482711
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 8.00000 0.321288
\(621\) −4.00000 −0.160514
\(622\) 0 0
\(623\) −24.0000 −0.961540
\(624\) −6.00000 −0.240192
\(625\) 1.00000 0.0400000
\(626\) −30.0000 −1.19904
\(627\) 16.0000 0.638978
\(628\) −18.0000 −0.718278
\(629\) 12.0000 0.478471
\(630\) 4.00000 0.159364
\(631\) 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
\(635\) −16.0000 −0.634941
\(636\) 10.0000 0.396526
\(637\) 54.0000 2.13956
\(638\) −4.00000 −0.158362
\(639\) −8.00000 −0.316475
\(640\) −3.00000 −0.118585
\(641\) 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 \(-0.397831\pi\)
0.315489 + 0.948929i \(0.397831\pi\)
\(644\) 16.0000 0.630488
\(645\) 4.00000 0.157500
\(646\) −24.0000 −0.944267
\(647\) −28.0000 −1.10079 −0.550397 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) −3.00000 −0.117851
\(649\) 48.0000 1.88416
\(650\) 6.00000 0.235339
\(651\) −32.0000 −1.25418
\(652\) 12.0000 0.469956
\(653\) 50.0000 1.95665 0.978326 0.207072i \(-0.0663936\pi\)
0.978326 + 0.207072i \(0.0663936\pi\)
\(654\) 14.0000 0.547443
\(655\) 12.0000 0.468879
\(656\) 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 \(0.424908\pi\)
\(660\) 4.00000 0.155700
\(661\) 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
\(665\) −16.0000 −0.620453
\(666\) 2.00000 0.0774984
\(667\) −4.00000 −0.154881
\(668\) −12.0000 −0.464294
\(669\) 28.0000 1.08254
\(670\) 8.00000 0.309067
\(671\) 40.0000 1.54418
\(672\) 20.0000 0.771517
\(673\) −6.00000 −0.231283 −0.115642 0.993291i \(-0.536892\pi\)
−0.115642 + 0.993291i \(0.536892\pi\)
\(674\) 30.0000 1.15556
\(675\) 1.00000 0.0384900
\(676\) −23.0000 −0.884615
\(677\) 50.0000 1.92166 0.960828 0.277145i \(-0.0893883\pi\)
0.960828 + 0.277145i \(0.0893883\pi\)
\(678\) −2.00000 −0.0768095
\(679\) −8.00000 −0.307012
\(680\) −18.0000 −0.690268
\(681\) 24.0000 0.919682
\(682\) 32.0000 1.22534
\(683\) 40.0000 1.53056 0.765279 0.643699i \(-0.222601\pi\)
0.765279 + 0.643699i \(0.222601\pi\)
\(684\) 4.00000 0.152944
\(685\) −2.00000 −0.0764161
\(686\) 8.00000 0.305441
\(687\) −2.00000 −0.0763048
\(688\) −4.00000 −0.152499
\(689\) −60.0000 −2.28582
\(690\) −4.00000 −0.152277
\(691\) 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
\(695\) 4.00000 0.151729
\(696\) −3.00000 −0.113715
\(697\) −36.0000 −1.36360
\(698\) −34.0000 −1.28692
\(699\) −22.0000 −0.832116
\(700\) −4.00000 −0.151186
\(701\) 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 \(0.364441\pi\)
\(710\) −8.00000 −0.300235
\(711\) 0 0
\(712\) 18.0000 0.674579
\(713\) 32.0000 1.19841
\(714\) 24.0000 0.898177
\(715\) −24.0000 −0.897549
\(716\) 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 \(0.647694\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −48.0000 −1.78761
\(722\) −3.00000 −0.111648
\(723\) 2.00000 0.0743808
\(724\) −22.0000 −0.817624
\(725\) 1.00000 0.0371391
\(726\) 5.00000 0.185567
\(727\) 16.0000 0.593407 0.296704 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) −72.0000 −2.66850
\(729\) 1.00000 0.0370370
\(730\) −2.00000 −0.0740233
\(731\) 24.0000 0.887672
\(732\) 10.0000 0.369611
\(733\) 2.00000 0.0738717 0.0369358 0.999318i \(-0.488240\pi\)
0.0369358 + 0.999318i \(0.488240\pi\)
\(734\) 8.00000 0.295285
\(735\) 9.00000 0.331970
\(736\) −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 \(0.730352\pi\)
\(740\) −2.00000 −0.0735215
\(741\) −24.0000 −0.881662
\(742\) −40.0000 −1.46845
\(743\) −8.00000 −0.293492 −0.146746 0.989174i \(-0.546880\pi\)
−0.146746 + 0.989174i \(0.546880\pi\)
\(744\) 24.0000 0.879883
\(745\) 6.00000 0.219823
\(746\) −2.00000 −0.0732252
\(747\) 8.00000 0.292705
\(748\) 24.0000 0.877527
\(749\) 32.0000 1.16925
\(750\) 1.00000 0.0365148
\(751\) −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
\(755\) 16.0000 0.582300
\(756\) −4.00000 −0.145479
\(757\) −14.0000 −0.508839 −0.254419 0.967094i \(-0.581884\pi\)
−0.254419 + 0.967094i \(0.581884\pi\)
\(758\) −20.0000 −0.726433
\(759\) 16.0000 0.580763
\(760\) 12.0000 0.435286
\(761\) 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
\(765\) 6.00000 0.216930
\(766\) 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 \(0.226528\pi\)
\(770\) −16.0000 −0.576600
\(771\) −22.0000 −0.792311
\(772\) −22.0000 −0.791797
\(773\) −38.0000 −1.36677 −0.683383 0.730061i \(-0.739492\pi\)
−0.683383 + 0.730061i \(0.739492\pi\)
\(774\) 4.00000 0.143777
\(775\) −8.00000 −0.287368
\(776\) 6.00000 0.215387
\(777\) 8.00000 0.286998
\(778\) −2.00000 −0.0717035
\(779\) 24.0000 0.859889
\(780\) −6.00000 −0.214834
\(781\) 32.0000 1.14505
\(782\) −24.0000 −0.858238
\(783\) 1.00000 0.0357371
\(784\) −9.00000 −0.321429
\(785\) 18.0000 0.642448
\(786\) 12.0000 0.428026
\(787\) −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
\(792\) 12.0000 0.426401
\(793\) −60.0000 −2.13066
\(794\) −10.0000 −0.354887
\(795\) −10.0000 −0.354663
\(796\) 24.0000 0.850657
\(797\) 50.0000 1.77109 0.885545 0.464553i \(-0.153785\pi\)
0.885545 + 0.464553i \(0.153785\pi\)
\(798\) −16.0000 −0.566394
\(799\) 0 0
\(800\) 5.00000 0.176777
\(801\) −6.00000 −0.212000
\(802\) 2.00000 0.0706225
\(803\) 8.00000 0.282314
\(804\) −8.00000 −0.282138
\(805\) −16.0000 −0.563926
\(806\) −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 \(0.533636\pi\)
\(810\) 1.00000 0.0351364
\(811\) −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
\(815\) −12.0000 −0.420342
\(816\) −6.00000 −0.210042
\(817\) −16.0000 −0.559769
\(818\) −22.0000 −0.769212
\(819\) 24.0000 0.838628
\(820\) 6.00000 0.209529
\(821\) −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 \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 36.0000 1.25412
\(825\) −4.00000 −0.139262
\(826\) −48.0000 −1.67013
\(827\) 20.0000 0.695468 0.347734 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) 4.00000 0.139010
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 8.00000 0.277684
\(831\) −26.0000 −0.901930
\(832\) 42.0000 1.45609
\(833\) 54.0000 1.87099
\(834\) 4.00000 0.138509
\(835\) 12.0000 0.415277
\(836\) −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 \(0.635969\pi\)
\(840\) −12.0000 −0.414039
\(841\) 1.00000 0.0344828
\(842\) −18.0000 −0.620321
\(843\) 10.0000 0.344418
\(844\) −20.0000 −0.688428
\(845\) 23.0000 0.791224
\(846\) 0 0
\(847\) 20.0000 0.687208
\(848\) 10.0000 0.343401
\(849\) 8.00000 0.274559
\(850\) 6.00000 0.205798
\(851\) −8.00000 −0.274236
\(852\) 8.00000 0.274075
\(853\) −38.0000 −1.30110 −0.650548 0.759465i \(-0.725461\pi\)
−0.650548 + 0.759465i \(0.725461\pi\)
\(854\) −40.0000 −1.36877
\(855\) −4.00000 −0.136797
\(856\) −24.0000 −0.820303
\(857\) 2.00000 0.0683187 0.0341593 0.999416i \(-0.489125\pi\)
0.0341593 + 0.999416i \(0.489125\pi\)
\(858\) −24.0000 −0.819346
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) −4.00000 −0.136399
\(861\) −24.0000 −0.817918
\(862\) 24.0000 0.817443
\(863\) 4.00000 0.136162 0.0680808 0.997680i \(-0.478312\pi\)
0.0680808 + 0.997680i \(0.478312\pi\)
\(864\) 5.00000 0.170103
\(865\) 6.00000 0.204006
\(866\) −34.0000 −1.15537
\(867\) 19.0000 0.645274
\(868\) 32.0000 1.08615
\(869\) 0 0
\(870\) 1.00000 0.0339032
\(871\) 48.0000 1.62642
\(872\) −42.0000 −1.42230
\(873\) −2.00000 −0.0676897
\(874\) 16.0000 0.541208
\(875\) 4.00000 0.135225
\(876\) 2.00000 0.0675737
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) −24.0000 −0.809961
\(879\) 18.0000 0.607125
\(880\) 4.00000 0.134840
\(881\) 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 \(-0.891318\pi\)
−0.942275 + 0.334840i \(0.891318\pi\)
\(884\) −36.0000 −1.21081
\(885\) −12.0000 −0.403376
\(886\) 20.0000 0.671913
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) −6.00000 −0.201347
\(889\) −64.0000 −2.14649
\(890\) −6.00000 −0.201120
\(891\) −4.00000 −0.134005
\(892\) −28.0000 −0.937509
\(893\) 0 0
\(894\) 6.00000 0.200670
\(895\) −12.0000 −0.401116
\(896\) −12.0000 −0.400892
\(897\) −24.0000 −0.801337
\(898\) 26.0000 0.867631
\(899\) −8.00000 −0.266815
\(900\) −1.00000 −0.0333333
\(901\) −60.0000 −1.99889
\(902\) 24.0000 0.799113
\(903\) 16.0000 0.532447
\(904\) 6.00000 0.199557
\(905\) 22.0000 0.731305
\(906\) 16.0000 0.531564
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) −24.0000 −0.796468
\(909\) −2.00000 −0.0663358
\(910\) 24.0000 0.795592
\(911\) −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
\(915\) −10.0000 −0.330590
\(916\) 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 \(0.457877\pi\)
\(920\) 12.0000 0.395628
\(921\) 12.0000 0.395413
\(922\) 6.00000 0.197599
\(923\) −48.0000 −1.57994
\(924\) 16.0000 0.526361
\(925\) 2.00000 0.0657596
\(926\) 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 \(0.311664\pi\)
\(930\) −8.00000 −0.262330
\(931\) −36.0000 −1.17985
\(932\) 22.0000 0.720634
\(933\) 0 0
\(934\) −20.0000 −0.654420
\(935\) −24.0000 −0.784884
\(936\) −18.0000 −0.588348
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 32.0000 1.04484
\(939\) −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
\(945\) 4.00000 0.130120
\(946\) −16.0000 −0.520205
\(947\) 28.0000 0.909878 0.454939 0.890523i \(-0.349661\pi\)
0.454939 + 0.890523i \(0.349661\pi\)
\(948\) 0 0
\(949\) −12.0000 −0.389536
\(950\) −4.00000 −0.129777
\(951\) 2.00000 0.0648544
\(952\) −72.0000 −2.33353
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) −10.0000 −0.323762
\(955\) −16.0000 −0.517748
\(956\) 8.00000 0.258738
\(957\) −4.00000 −0.129302
\(958\) −16.0000 −0.516937
\(959\) −8.00000 −0.258333
\(960\) 7.00000 0.225924
\(961\) 33.0000 1.06452
\(962\) 12.0000 0.386896
\(963\) 8.00000 0.257796
\(964\) −2.00000 −0.0644157
\(965\) 22.0000 0.708205
\(966\) −16.0000 −0.514792
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) −15.0000 −0.482118
\(969\) −24.0000 −0.770991
\(970\) −2.00000 −0.0642161
\(971\) 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
\(975\) 6.00000 0.192154
\(976\) 10.0000 0.320092
\(977\) −6.00000 −0.191957 −0.0959785 0.995383i \(-0.530598\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(978\) −12.0000 −0.383718
\(979\) 24.0000 0.767043
\(980\) −9.00000 −0.287494
\(981\) 14.0000 0.446986
\(982\) −44.0000 −1.40410
\(983\) 40.0000 1.27580 0.637901 0.770118i \(-0.279803\pi\)
0.637901 + 0.770118i \(0.279803\pi\)
\(984\) 18.0000 0.573819
\(985\) 14.0000 0.446077
\(986\) 6.00000 0.191079
\(987\) 0 0
\(988\) 24.0000 0.763542
\(989\) −16.0000 −0.508770
\(990\) −4.00000 −0.127128
\(991\) −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
\(995\) −24.0000 −0.760851
\(996\) −8.00000 −0.253490
\(997\) 50.0000 1.58352 0.791758 0.610835i \(-0.209166\pi\)
0.791758 + 0.610835i \(0.209166\pi\)
\(998\) −36.0000 −1.13956
\(999\) 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 435.2.a.d.1.1 1
3.2 odd 2 1305.2.a.b.1.1 1
4.3 odd 2 6960.2.a.l.1.1 1
5.2 odd 4 2175.2.c.b.349.2 2
5.3 odd 4 2175.2.c.b.349.1 2
5.4 even 2 2175.2.a.b.1.1 1
15.14 odd 2 6525.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.d.1.1 1 1.1 even 1 trivial
1305.2.a.b.1.1 1 3.2 odd 2
2175.2.a.b.1.1 1 5.4 even 2
2175.2.c.b.349.1 2 5.3 odd 4
2175.2.c.b.349.2 2 5.2 odd 4
6525.2.a.j.1.1 1 15.14 odd 2
6960.2.a.l.1.1 1 4.3 odd 2