↑

Properties

Label 238.2.a.d.1.1
Level $238$
Weight $2$
Character 238.1
Self dual yes
Analytic conductor $1.900$
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] = [238,2,Mod(1,238)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(238, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("238.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 238 = 2 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 238.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: \(1.90043956811\)
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 \(\chi\) \(=\) 238.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} +2.00000 q^{10} -2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} -3.00000 q^{18} +4.00000 q^{19} +2.00000 q^{20} -1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{28} -6.00000 q^{29} +1.00000 q^{32} +1.00000 q^{34} +2.00000 q^{35} -3.00000 q^{36} -6.00000 q^{37} +4.00000 q^{38} +2.00000 q^{40} -6.00000 q^{41} -12.0000 q^{43} -6.00000 q^{45} +8.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} -2.00000 q^{53} +1.00000 q^{56} -6.00000 q^{58} +4.00000 q^{59} +2.00000 q^{61} -3.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} +12.0000 q^{67} +1.00000 q^{68} +2.00000 q^{70} -3.00000 q^{72} +2.00000 q^{73} -6.00000 q^{74} +4.00000 q^{76} -8.00000 q^{79} +2.00000 q^{80} +9.00000 q^{81} -6.00000 q^{82} +12.0000 q^{83} +2.00000 q^{85} -12.0000 q^{86} +10.0000 q^{89} -6.00000 q^{90} -2.00000 q^{91} +8.00000 q^{94} +8.00000 q^{95} -14.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) −3.00000 −1.00000
\(10\) 2.00000 0.632456
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536
\(18\) −3.00000 −0.707107
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 1.00000 0.171499
\(35\) 2.00000 0.338062
\(36\) −3.00000 −0.500000
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) 2.00000 0.316228
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −12.0000 −1.82998 −0.914991 − 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) 0 0
\(45\) −6.00000 −0.894427
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −2.00000 −0.274721 −0.137361 − 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) −3.00000 −0.377964
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 − 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 1.00000 0.121268
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −3.00000 −0.353553
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 2.00000 0.223607
\(81\) 9.00000 1.00000
\(82\) −6.00000 −0.662589
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) −12.0000 −1.29399
\(87\) 0 0
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) −6.00000 −0.632456
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 − 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) 6.00000 0.597022 0.298511 − 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 − 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) 8.00000 0.773389 0.386695 − 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 − 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 2.00000 0.188144 0.0940721 − 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 6.00000 0.554700
\(118\) 4.00000 0.368230
\(119\) 1.00000 0.0916698
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) −3.00000 −0.267261
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −4.00000 −0.350823
\(131\) 8.00000 0.698963 0.349482 − 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) 1.00000 0.0857493
\(137\) 10.0000 0.854358 0.427179 − 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) −16.0000 −1.35710 −0.678551 − 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −3.00000 −0.250000
\(145\) −12.0000 −0.996546
\(146\) 2.00000 0.165521
\(147\) 0 0
\(148\) −6.00000 −0.493197
\(149\) 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 \(0.274337\pi\)
\(152\) 4.00000 0.324443
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 22.0000 1.75579 0.877896 − 0.478852i \(-0.158947\pi\)
0.877896 + 0.478852i \(0.158947\pi\)
\(158\) −8.00000 −0.636446
\(159\) 0 0
\(160\) 2.00000 0.158114
\(161\) 0 0
\(162\) 9.00000 0.707107
\(163\) 16.0000 1.25322 0.626608 − 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 24.0000 1.85718 0.928588 − 0.371113i \(-0.121024\pi\)
0.928588 + 0.371113i \(0.121024\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 2.00000 0.153393
\(171\) −12.0000 −0.917663
\(172\) −12.0000 −0.914991
\(173\) 10.0000 0.760286 0.380143 − 0.924928i \(-0.375875\pi\)
0.380143 + 0.924928i \(0.375875\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 0 0
\(178\) 10.0000 0.749532
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) −6.00000 −0.447214
\(181\) −22.0000 −1.63525 −0.817624 − 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) −2.00000 −0.148250
\(183\) 0 0
\(184\) 0 0
\(185\) −12.0000 −0.882258
\(186\) 0 0
\(187\) 0 0
\(188\) 8.00000 0.583460
\(189\) 0 0
\(190\) 8.00000 0.580381
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −22.0000 −1.58359 −0.791797 − 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 10.0000 0.712470 0.356235 − 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) 0 0
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 6.00000 0.422159
\(203\) −6.00000 −0.421117
\(204\) 0 0
\(205\) −12.0000 −0.838116
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 − 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) −2.00000 −0.137361
\(213\) 0 0
\(214\) 8.00000 0.546869
\(215\) −24.0000 −1.63679
\(216\) 0 0
\(217\) 0 0
\(218\) 10.0000 0.677285
\(219\) 0 0
\(220\) 0 0
\(221\) −2.00000 −0.134535
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 − 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 1.00000 0.0668153
\(225\) 3.00000 0.200000
\(226\) 2.00000 0.133038
\(227\) 8.00000 0.530979 0.265489 − 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) 0 0
\(229\) −2.00000 −0.132164 −0.0660819 − 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −6.00000 −0.393073 −0.196537 − 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 6.00000 0.392232
\(235\) 16.0000 1.04372
\(236\) 4.00000 0.260378
\(237\) 0 0
\(238\) 1.00000 0.0648204
\(239\) 16.0000 1.03495 0.517477 − 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) −30.0000 −1.93247 −0.966235 − 0.257663i \(-0.917048\pi\)
−0.966235 + 0.257663i \(0.917048\pi\)
\(242\) −11.0000 −0.707107
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) −8.00000 −0.509028
\(248\) 0 0
\(249\) 0 0
\(250\) −12.0000 −0.758947
\(251\) −12.0000 −0.757433 −0.378717 − 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) −3.00000 −0.188982
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 − 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) −4.00000 −0.248069
\(261\) 18.0000 1.11417
\(262\) 8.00000 0.494242
\(263\) −32.0000 −1.97320 −0.986602 − 0.163144i \(-0.947836\pi\)
−0.986602 + 0.163144i \(0.947836\pi\)
\(264\) 0 0
\(265\) −4.00000 −0.245718
\(266\) 4.00000 0.245256
\(267\) 0 0
\(268\) 12.0000 0.733017
\(269\) 10.0000 0.609711 0.304855 − 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 − 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 1.00000 0.0606339
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) 0 0
\(276\) 0 0
\(277\) 18.0000 1.08152 0.540758 − 0.841178i \(-0.318138\pi\)
0.540758 + 0.841178i \(0.318138\pi\)
\(278\) −16.0000 −0.959616
\(279\) 0 0
\(280\) 2.00000 0.119523
\(281\) −22.0000 −1.31241 −0.656205 − 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) 0 0
\(283\) −24.0000 −1.42665 −0.713326 − 0.700832i \(-0.752812\pi\)
−0.713326 + 0.700832i \(0.752812\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) −3.00000 −0.176777
\(289\) 1.00000 0.0588235
\(290\) −12.0000 −0.704664
\(291\) 0 0
\(292\) 2.00000 0.117041
\(293\) 30.0000 1.75262 0.876309 − 0.481749i \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) −6.00000 −0.348743
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) 0 0
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) 16.0000 0.920697
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) 4.00000 0.229039
\(306\) −3.00000 −0.171499
\(307\) 4.00000 0.228292 0.114146 − 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 − 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 − 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 22.0000 1.24153
\(315\) −6.00000 −0.338062
\(316\) −8.00000 −0.450035
\(317\) 18.0000 1.01098 0.505490 − 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 2.00000 0.111803
\(321\) 0 0
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) 9.00000 0.500000
\(325\) 2.00000 0.110940
\(326\) 16.0000 0.886158
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 − 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 12.0000 0.658586
\(333\) 18.0000 0.986394
\(334\) 24.0000 1.31322
\(335\) 24.0000 1.31126
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 − 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) 2.00000 0.108465
\(341\) 0 0
\(342\) −12.0000 −0.648886
\(343\) 1.00000 0.0539949
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) 10.0000 0.537603
\(347\) −24.0000 −1.28839 −0.644194 − 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) 0 0
\(349\) 30.0000 1.60586 0.802932 − 0.596071i \(-0.203272\pi\)
0.802932 + 0.596071i \(0.203272\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) 0 0
\(353\) 18.0000 0.958043 0.479022 − 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 24.0000 1.26667 0.633336 − 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) −6.00000 −0.316228
\(361\) −3.00000 −0.157895
\(362\) −22.0000 −1.15629
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) 4.00000 0.209370
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 − 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) 18.0000 0.937043
\(370\) −12.0000 −0.623850
\(371\) −2.00000 −0.103835
\(372\) 0 0
\(373\) −2.00000 −0.103556 −0.0517780 − 0.998659i \(-0.516489\pi\)
−0.0517780 + 0.998659i \(0.516489\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 8.00000 0.410391
\(381\) 0 0
\(382\) 0 0
\(383\) −16.0000 −0.817562 −0.408781 − 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −22.0000 −1.11977
\(387\) 36.0000 1.82998
\(388\) −14.0000 −0.710742
\(389\) −10.0000 −0.507020 −0.253510 − 0.967333i \(-0.581585\pi\)
−0.253510 + 0.967333i \(0.581585\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 10.0000 0.503793
\(395\) −16.0000 −0.805047
\(396\) 0 0
\(397\) 34.0000 1.70641 0.853206 − 0.521575i \(-0.174655\pi\)
0.853206 + 0.521575i \(0.174655\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 10.0000 0.499376 0.249688 − 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 6.00000 0.298511
\(405\) 18.0000 0.894427
\(406\) −6.00000 −0.297775
\(407\) 0 0
\(408\) 0 0
\(409\) −6.00000 −0.296681 −0.148340 − 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) −12.0000 −0.592638
\(411\) 0 0
\(412\) −8.00000 −0.394132
\(413\) 4.00000 0.196827
\(414\) 0 0
\(415\) 24.0000 1.17811
\(416\) −2.00000 −0.0980581
\(417\) 0 0
\(418\) 0 0
\(419\) −8.00000 −0.390826 −0.195413 − 0.980721i \(-0.562605\pi\)
−0.195413 + 0.980721i \(0.562605\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 − 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) −8.00000 −0.389434
\(423\) −24.0000 −1.16692
\(424\) −2.00000 −0.0971286
\(425\) −1.00000 −0.0485071
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) −24.0000 −1.15738
\(431\) −24.0000 −1.15604 −0.578020 − 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 − 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) 0 0
\(438\) 0 0
\(439\) −16.0000 −0.763638 −0.381819 − 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) −2.00000 −0.0951303
\(443\) 20.0000 0.950229 0.475114 − 0.879924i \(-0.342407\pi\)
0.475114 + 0.879924i \(0.342407\pi\)
\(444\) 0 0
\(445\) 20.0000 0.948091
\(446\) −8.00000 −0.378811
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −38.0000 −1.79333 −0.896665 − 0.442709i \(-0.854018\pi\)
−0.896665 + 0.442709i \(0.854018\pi\)
\(450\) 3.00000 0.141421
\(451\) 0 0
\(452\) 2.00000 0.0940721
\(453\) 0 0
\(454\) 8.00000 0.375459
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 − 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) −2.00000 −0.0934539
\(459\) 0 0
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 − 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) 24.0000 1.11537 0.557687 − 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) −12.0000 −0.555294 −0.277647 − 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 6.00000 0.277350
\(469\) 12.0000 0.554109
\(470\) 16.0000 0.738025
\(471\) 0 0
\(472\) 4.00000 0.184115
\(473\) 0 0
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 1.00000 0.0458349
\(477\) 6.00000 0.274721
\(478\) 16.0000 0.731823
\(479\) −40.0000 −1.82765 −0.913823 − 0.406112i \(-0.866884\pi\)
−0.913823 + 0.406112i \(0.866884\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) −30.0000 −1.36646
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) −28.0000 −1.27141
\(486\) 0 0
\(487\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 2.00000 0.0905357
\(489\) 0 0
\(490\) 2.00000 0.0903508
\(491\) −20.0000 −0.902587 −0.451294 − 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 0 0
\(493\) −6.00000 −0.270226
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 24.0000 1.07439 0.537194 − 0.843459i \(-0.319484\pi\)
0.537194 + 0.843459i \(0.319484\pi\)
\(500\) −12.0000 −0.536656
\(501\) 0 0
\(502\) −12.0000 −0.535586
\(503\) −16.0000 −0.713405 −0.356702 − 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) −3.00000 −0.133631
\(505\) 12.0000 0.533993
\(506\) 0 0
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) −10.0000 −0.443242 −0.221621 − 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −14.0000 −0.617514
\(515\) −16.0000 −0.705044
\(516\) 0 0
\(517\) 0 0
\(518\) −6.00000 −0.263625
\(519\) 0 0
\(520\) −4.00000 −0.175412
\(521\) 42.0000 1.84005 0.920027 − 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) 18.0000 0.787839
\(523\) 4.00000 0.174908 0.0874539 − 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 8.00000 0.349482
\(525\) 0 0
\(526\) −32.0000 −1.39527
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) −4.00000 −0.173749
\(531\) −12.0000 −0.520756
\(532\) 4.00000 0.173422
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) 16.0000 0.691740
\(536\) 12.0000 0.518321
\(537\) 0 0
\(538\) 10.0000 0.431131
\(539\) 0 0
\(540\) 0 0
\(541\) −14.0000 −0.601907 −0.300954 − 0.953639i \(-0.597305\pi\)
−0.300954 + 0.953639i \(0.597305\pi\)
\(542\) −8.00000 −0.343629
\(543\) 0 0
\(544\) 1.00000 0.0428746
\(545\) 20.0000 0.856706
\(546\) 0 0
\(547\) 16.0000 0.684111 0.342055 − 0.939680i \(-0.388877\pi\)
0.342055 + 0.939680i \(0.388877\pi\)
\(548\) 10.0000 0.427179
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) 18.0000 0.764747
\(555\) 0 0
\(556\) −16.0000 −0.678551
\(557\) −42.0000 −1.77960 −0.889799 − 0.456354i \(-0.849155\pi\)
−0.889799 + 0.456354i \(0.849155\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) 2.00000 0.0845154
\(561\) 0 0
\(562\) −22.0000 −0.928014
\(563\) 12.0000 0.505740 0.252870 − 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) 4.00000 0.168281
\(566\) −24.0000 −1.00880
\(567\) 9.00000 0.377964
\(568\) 0 0
\(569\) −22.0000 −0.922288 −0.461144 − 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) 0 0
\(571\) −8.00000 −0.334790 −0.167395 − 0.985890i \(-0.553535\pi\)
−0.167395 + 0.985890i \(0.553535\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −6.00000 −0.250435
\(575\) 0 0
\(576\) −3.00000 −0.125000
\(577\) 34.0000 1.41544 0.707719 − 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) 1.00000 0.0415945
\(579\) 0 0
\(580\) −12.0000 −0.498273
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 0 0
\(584\) 2.00000 0.0827606
\(585\) 12.0000 0.496139
\(586\) 30.0000 1.23929
\(587\) 28.0000 1.15568 0.577842 − 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 8.00000 0.329355
\(591\) 0 0
\(592\) −6.00000 −0.246598
\(593\) −30.0000 −1.23195 −0.615976 − 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) 2.00000 0.0819920
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) 0 0
\(599\) −16.0000 −0.653742 −0.326871 − 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 0 0
\(601\) −46.0000 −1.87638 −0.938190 − 0.346122i \(-0.887498\pi\)
−0.938190 + 0.346122i \(0.887498\pi\)
\(602\) −12.0000 −0.489083
\(603\) −36.0000 −1.46603
\(604\) 16.0000 0.651031
\(605\) −22.0000 −0.894427
\(606\) 0 0
\(607\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) 4.00000 0.161955
\(611\) −16.0000 −0.647291
\(612\) −3.00000 −0.121268
\(613\) 22.0000 0.888572 0.444286 − 0.895885i \(-0.353457\pi\)
0.444286 + 0.895885i \(0.353457\pi\)
\(614\) 4.00000 0.161427
\(615\) 0 0
\(616\) 0 0
\(617\) 42.0000 1.69086 0.845428 − 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) 0 0
\(619\) 32.0000 1.28619 0.643094 − 0.765787i \(-0.277650\pi\)
0.643094 + 0.765787i \(0.277650\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −8.00000 −0.320771
\(623\) 10.0000 0.400642
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) 22.0000 0.877896
\(629\) −6.00000 −0.239236
\(630\) −6.00000 −0.239046
\(631\) 40.0000 1.59237 0.796187 − 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) −8.00000 −0.318223
\(633\) 0 0
\(634\) 18.0000 0.714871
\(635\) 16.0000 0.634941
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) 0 0
\(640\) 2.00000 0.0790569
\(641\) 26.0000 1.02694 0.513469 − 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) 0 0
\(643\) 16.0000 0.630978 0.315489 − 0.948929i \(-0.397831\pi\)
0.315489 + 0.948929i \(0.397831\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 4.00000 0.157378
\(647\) 32.0000 1.25805 0.629025 − 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) 9.00000 0.353553
\(649\) 0 0
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) 16.0000 0.626608
\(653\) 26.0000 1.01746 0.508729 − 0.860927i \(-0.330115\pi\)
0.508729 + 0.860927i \(0.330115\pi\)
\(654\) 0 0
\(655\) 16.0000 0.625172
\(656\) −6.00000 −0.234261
\(657\) −6.00000 −0.234082
\(658\) 8.00000 0.311872
\(659\) −44.0000 −1.71400 −0.856998 − 0.515319i \(-0.827673\pi\)
−0.856998 + 0.515319i \(0.827673\pi\)
\(660\) 0 0
\(661\) 38.0000 1.47803 0.739014 − 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 4.00000 0.155464
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) 8.00000 0.310227
\(666\) 18.0000 0.697486
\(667\) 0 0
\(668\) 24.0000 0.928588
\(669\) 0 0
\(670\) 24.0000 0.927201
\(671\) 0 0
\(672\) 0 0
\(673\) −22.0000 −0.848038 −0.424019 − 0.905653i \(-0.639381\pi\)
−0.424019 + 0.905653i \(0.639381\pi\)
\(674\) 2.00000 0.0770371
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −22.0000 −0.845529 −0.422764 − 0.906240i \(-0.638940\pi\)
−0.422764 + 0.906240i \(0.638940\pi\)
\(678\) 0 0
\(679\) −14.0000 −0.537271
\(680\) 2.00000 0.0766965
\(681\) 0 0
\(682\) 0 0
\(683\) −48.0000 −1.83667 −0.918334 − 0.395805i \(-0.870466\pi\)
−0.918334 + 0.395805i \(0.870466\pi\)
\(684\) −12.0000 −0.458831
\(685\) 20.0000 0.764161
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) −12.0000 −0.457496
\(689\) 4.00000 0.152388
\(690\) 0 0
\(691\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 10.0000 0.380143
\(693\) 0 0
\(694\) −24.0000 −0.911028
\(695\) −32.0000 −1.21383
\(696\) 0 0
\(697\) −6.00000 −0.227266
\(698\) 30.0000 1.13552
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) −10.0000 −0.377695 −0.188847 − 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) 0 0
\(703\) −24.0000 −0.905177
\(704\) 0 0
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) 6.00000 0.225653
\(708\) 0 0
\(709\) −46.0000 −1.72757 −0.863783 − 0.503864i \(-0.831911\pi\)
−0.863783 + 0.503864i \(0.831911\pi\)
\(710\) 0 0
\(711\) 24.0000 0.900070
\(712\) 10.0000 0.374766
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) 24.0000 0.895672
\(719\) 8.00000 0.298350 0.149175 − 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) −6.00000 −0.223607
\(721\) −8.00000 −0.297936
\(722\) −3.00000 −0.111648
\(723\) 0 0
\(724\) −22.0000 −0.817624
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) −48.0000 −1.78022 −0.890111 − 0.455744i \(-0.849373\pi\)
−0.890111 + 0.455744i \(0.849373\pi\)
\(728\) −2.00000 −0.0741249
\(729\) −27.0000 −1.00000
\(730\) 4.00000 0.148047
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) 30.0000 1.10808 0.554038 − 0.832492i \(-0.313086\pi\)
0.554038 + 0.832492i \(0.313086\pi\)
\(734\) −8.00000 −0.295285
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 18.0000 0.662589
\(739\) 20.0000 0.735712 0.367856 − 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) −12.0000 −0.441129
\(741\) 0 0
\(742\) −2.00000 −0.0734223
\(743\) 24.0000 0.880475 0.440237 − 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 0 0
\(745\) 12.0000 0.439646
\(746\) −2.00000 −0.0732252
\(747\) −36.0000 −1.31717
\(748\) 0 0
\(749\) 8.00000 0.292314
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 − 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 8.00000 0.291730
\(753\) 0 0
\(754\) 12.0000 0.437014
\(755\) 32.0000 1.16460
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 − 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 8.00000 0.290191
\(761\) −6.00000 −0.217500 −0.108750 − 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 0 0
\(763\) 10.0000 0.362024
\(764\) 0 0
\(765\) −6.00000 −0.216930
\(766\) −16.0000 −0.578103
\(767\) −8.00000 −0.288863
\(768\) 0 0
\(769\) 2.00000 0.0721218 0.0360609 − 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −22.0000 −0.791797
\(773\) 30.0000 1.07903 0.539513 − 0.841978i \(-0.318609\pi\)
0.539513 + 0.841978i \(0.318609\pi\)
\(774\) 36.0000 1.29399
\(775\) 0 0
\(776\) −14.0000 −0.502571
\(777\) 0 0
\(778\) −10.0000 −0.358517
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 44.0000 1.57043
\(786\) 0 0
\(787\) −8.00000 −0.285169 −0.142585 − 0.989783i \(-0.545541\pi\)
−0.142585 + 0.989783i \(0.545541\pi\)
\(788\) 10.0000 0.356235
\(789\) 0 0
\(790\) −16.0000 −0.569254
\(791\) 2.00000 0.0711118
\(792\) 0 0
\(793\) −4.00000 −0.142044
\(794\) 34.0000 1.20661
\(795\) 0 0
\(796\) 0 0
\(797\) 14.0000 0.495905 0.247953 − 0.968772i \(-0.420242\pi\)
0.247953 + 0.968772i \(0.420242\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) −1.00000 −0.0353553
\(801\) −30.0000 −1.06000
\(802\) 10.0000 0.353112
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 6.00000 0.211079
\(809\) −46.0000 −1.61727 −0.808637 − 0.588308i \(-0.799794\pi\)
−0.808637 + 0.588308i \(0.799794\pi\)
\(810\) 18.0000 0.632456
\(811\) 56.0000 1.96643 0.983213 − 0.182462i \(-0.0584065\pi\)
0.983213 + 0.182462i \(0.0584065\pi\)
\(812\) −6.00000 −0.210559
\(813\) 0 0
\(814\) 0 0
\(815\) 32.0000 1.12091
\(816\) 0 0
\(817\) −48.0000 −1.67931
\(818\) −6.00000 −0.209785
\(819\) 6.00000 0.209657
\(820\) −12.0000 −0.419058
\(821\) −38.0000 −1.32621 −0.663105 − 0.748527i \(-0.730762\pi\)
−0.663105 + 0.748527i \(0.730762\pi\)
\(822\) 0 0
\(823\) 16.0000 0.557725 0.278862 − 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) 4.00000 0.139178
\(827\) 48.0000 1.66912 0.834562 − 0.550914i \(-0.185721\pi\)
0.834562 + 0.550914i \(0.185721\pi\)
\(828\) 0 0
\(829\) 6.00000 0.208389 0.104194 − 0.994557i \(-0.466774\pi\)
0.104194 + 0.994557i \(0.466774\pi\)
\(830\) 24.0000 0.833052
\(831\) 0 0
\(832\) −2.00000 −0.0693375
\(833\) 1.00000 0.0346479
\(834\) 0 0
\(835\) 48.0000 1.66111
\(836\) 0 0
\(837\) 0 0
\(838\) −8.00000 −0.276355
\(839\) −40.0000 −1.38095 −0.690477 − 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −2.00000 −0.0689246
\(843\) 0 0
\(844\) −8.00000 −0.275371
\(845\) −18.0000 −0.619219
\(846\) −24.0000 −0.825137
\(847\) −11.0000 −0.377964
\(848\) −2.00000 −0.0686803
\(849\) 0 0
\(850\) −1.00000 −0.0342997
\(851\) 0 0
\(852\) 0 0
\(853\) −30.0000 −1.02718 −0.513590 − 0.858036i \(-0.671685\pi\)
−0.513590 + 0.858036i \(0.671685\pi\)
\(854\) 2.00000 0.0684386
\(855\) −24.0000 −0.820783
\(856\) 8.00000 0.273434
\(857\) 42.0000 1.43469 0.717346 − 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 − 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) −24.0000 −0.818393
\(861\) 0 0
\(862\) −24.0000 −0.817443
\(863\) 24.0000 0.816970 0.408485 − 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) 20.0000 0.680020
\(866\) 2.00000 0.0679628
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 10.0000 0.338643
\(873\) 42.0000 1.42148
\(874\) 0 0
\(875\) −12.0000 −0.405674
\(876\) 0 0
\(877\) −54.0000 −1.82345 −0.911725 − 0.410801i \(-0.865249\pi\)
−0.911725 + 0.410801i \(0.865249\pi\)
\(878\) −16.0000 −0.539974
\(879\) 0 0
\(880\) 0 0
\(881\) 10.0000 0.336909 0.168454 − 0.985709i \(-0.446122\pi\)
0.168454 + 0.985709i \(0.446122\pi\)
\(882\) −3.00000 −0.101015
\(883\) −20.0000 −0.673054 −0.336527 − 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) −2.00000 −0.0672673
\(885\) 0 0
\(886\) 20.0000 0.671913
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 20.0000 0.670402
\(891\) 0 0
\(892\) −8.00000 −0.267860
\(893\) 32.0000 1.07084
\(894\) 0 0
\(895\) −24.0000 −0.802232
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −38.0000 −1.26808
\(899\) 0 0
\(900\) 3.00000 0.100000
\(901\) −2.00000 −0.0666297
\(902\) 0 0
\(903\) 0 0
\(904\) 2.00000 0.0665190
\(905\) −44.0000 −1.46261
\(906\) 0 0
\(907\) −32.0000 −1.06254 −0.531271 − 0.847202i \(-0.678286\pi\)
−0.531271 + 0.847202i \(0.678286\pi\)
\(908\) 8.00000 0.265489
\(909\) −18.0000 −0.597022
\(910\) −4.00000 −0.132599
\(911\) 48.0000 1.59031 0.795155 − 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −22.0000 −0.727695
\(915\) 0 0
\(916\) −2.00000 −0.0660819
\(917\) 8.00000 0.264183
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 − 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −18.0000 −0.592798
\(923\) 0 0
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 24.0000 0.788689
\(927\) 24.0000 0.788263
\(928\) −6.00000 −0.196960
\(929\) 34.0000 1.11550 0.557752 − 0.830008i \(-0.311664\pi\)
0.557752 + 0.830008i \(0.311664\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) −6.00000 −0.196537
\(933\) 0 0
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) 6.00000 0.196116
\(937\) −54.0000 −1.76410 −0.882052 − 0.471153i \(-0.843838\pi\)
−0.882052 + 0.471153i \(0.843838\pi\)
\(938\) 12.0000 0.391814
\(939\) 0 0
\(940\) 16.0000 0.521862
\(941\) 18.0000 0.586783 0.293392 − 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 0 0
\(947\) −32.0000 −1.03986 −0.519930 − 0.854209i \(-0.674042\pi\)
−0.519930 + 0.854209i \(0.674042\pi\)
\(948\) 0 0
\(949\) −4.00000 −0.129845
\(950\) −4.00000 −0.129777
\(951\) 0 0
\(952\) 1.00000 0.0324102
\(953\) −22.0000 −0.712650 −0.356325 − 0.934362i \(-0.615970\pi\)
−0.356325 + 0.934362i \(0.615970\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) 0 0
\(958\) −40.0000 −1.29234
\(959\) 10.0000 0.322917
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 12.0000 0.386896
\(963\) −24.0000 −0.773389
\(964\) −30.0000 −0.966235
\(965\) −44.0000 −1.41641
\(966\) 0 0
\(967\) 40.0000 1.28631 0.643157 − 0.765735i \(-0.277624\pi\)
0.643157 + 0.765735i \(0.277624\pi\)
\(968\) −11.0000 −0.353553
\(969\) 0 0
\(970\) −28.0000 −0.899026
\(971\) 52.0000 1.66876 0.834380 − 0.551190i \(-0.185826\pi\)
0.834380 + 0.551190i \(0.185826\pi\)
\(972\) 0 0
\(973\) −16.0000 −0.512936
\(974\) 0 0
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 34.0000 1.08776 0.543878 − 0.839164i \(-0.316955\pi\)
0.543878 + 0.839164i \(0.316955\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 2.00000 0.0638877
\(981\) −30.0000 −0.957826
\(982\) −20.0000 −0.638226
\(983\) −48.0000 −1.53096 −0.765481 − 0.643458i \(-0.777499\pi\)
−0.765481 + 0.643458i \(0.777499\pi\)
\(984\) 0 0
\(985\) 20.0000 0.637253
\(986\) −6.00000 −0.191079
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 0 0
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 − 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 10.0000 0.316703 0.158352 − 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) 24.0000 0.759707
\(999\) 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 238.2.a.d.1.1 ✓ 1
3.2 odd 2 2142.2.a.c.1.1 1
4.3 odd 2 1904.2.a.d.1.1 1
5.4 even 2 5950.2.a.f.1.1 1
7.6 odd 2 1666.2.a.j.1.1 1
8.3 odd 2 7616.2.a.e.1.1 1
8.5 even 2 7616.2.a.f.1.1 1
17.16 even 2 4046.2.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
238.2.a.d.1.1 ✓ 1 1.1 even 1 trivial
1666.2.a.j.1.1 1 7.6 odd 2
1904.2.a.d.1.1 1 4.3 odd 2
2142.2.a.c.1.1 1 3.2 odd 2
4046.2.a.m.1.1 1 17.16 even 2
5950.2.a.f.1.1 1 5.4 even 2
7616.2.a.e.1.1 1 8.3 odd 2
7616.2.a.f.1.1 1 8.5 even 2