Properties

Label 3822.2.a.h.1.1
Level $3822$
Weight $2$
Character 3822.1
Self dual yes
Analytic conductor $30.519$
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] = [3822,2,Mod(1,3822)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3822, 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("3822.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.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: \(30.5188236525\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3822.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} -2.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} +2.00000 q^{20} +4.00000 q^{22} -4.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} -2.00000 q^{29} +2.00000 q^{30} -1.00000 q^{32} +4.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -4.00000 q^{38} +1.00000 q^{39} -2.00000 q^{40} -2.00000 q^{41} +4.00000 q^{43} -4.00000 q^{44} +2.00000 q^{45} +4.00000 q^{46} +12.0000 q^{47} -1.00000 q^{48} +1.00000 q^{50} -2.00000 q^{51} -1.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} -8.00000 q^{55} -4.00000 q^{57} +2.00000 q^{58} -2.00000 q^{60} +10.0000 q^{61} +1.00000 q^{64} -2.00000 q^{65} -4.00000 q^{66} +4.00000 q^{67} +2.00000 q^{68} +4.00000 q^{69} -8.00000 q^{71} -1.00000 q^{72} +6.00000 q^{73} +2.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} -1.00000 q^{78} +8.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -8.00000 q^{83} +4.00000 q^{85} -4.00000 q^{86} +2.00000 q^{87} +4.00000 q^{88} +6.00000 q^{89} -2.00000 q^{90} -4.00000 q^{92} -12.0000 q^{94} +8.00000 q^{95} +1.00000 q^{96} -2.00000 q^{97} -4.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −1.00000 −0.235702
\(19\) 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
\(22\) 4.00000 0.852803
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 2.00000 0.365148
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.00000 0.696311
\(34\) −2.00000 −0.342997
\(35\) 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
\(39\) 1.00000 0.160128
\(40\) −2.00000 −0.316228
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −4.00000 −0.603023
\(45\) 2.00000 0.298142
\(46\) 4.00000 0.589768
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) −2.00000 −0.280056
\(52\) −1.00000 −0.138675
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 1.00000 0.136083
\(55\) −8.00000 −1.07872
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 2.00000 0.262613
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −2.00000 −0.258199
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) −4.00000 −0.492366
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 2.00000 0.242536
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −1.00000 −0.117851
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 2.00000 0.232495
\(75\) 1.00000 0.115470
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) −1.00000 −0.113228
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) −8.00000 −0.878114 −0.439057 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) −4.00000 −0.431331
\(87\) 2.00000 0.214423
\(88\) 4.00000 0.426401
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 8.00000 0.820783
\(96\) 1.00000 0.102062
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) −1.00000 −0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 2.00000 0.198030
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 8.00000 0.762770
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 4.00000 0.374634
\(115\) −8.00000 −0.746004
\(116\) −2.00000 −0.185695
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) 0 0
\(120\) 2.00000 0.182574
\(121\) 5.00000 0.454545
\(122\) −10.0000 −0.905357
\(123\) 2.00000 0.180334
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) 2.00000 0.175412
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 4.00000 0.348155
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) −2.00000 −0.172133
\(136\) −2.00000 −0.171499
\(137\) 22.0000 1.87959 0.939793 0.341743i \(-0.111017\pi\)
0.939793 + 0.341743i \(0.111017\pi\)
\(138\) −4.00000 −0.340503
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) 8.00000 0.671345
\(143\) 4.00000 0.334497
\(144\) 1.00000 0.0833333
\(145\) −4.00000 −0.332182
\(146\) −6.00000 −0.496564
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) −4.00000 −0.324443
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 1.00000 0.0800641
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) −8.00000 −0.636446
\(159\) −6.00000 −0.475831
\(160\) −2.00000 −0.158114
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) −2.00000 −0.156174
\(165\) 8.00000 0.622799
\(166\) 8.00000 0.620920
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −4.00000 −0.306786
\(171\) 4.00000 0.305888
\(172\) 4.00000 0.304997
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 2.00000 0.149071
\(181\) 26.0000 1.93256 0.966282 0.257485i \(-0.0828937\pi\)
0.966282 + 0.257485i \(0.0828937\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 4.00000 0.294884
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) −8.00000 −0.585018
\(188\) 12.0000 0.875190
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 2.00000 0.143592
\(195\) 2.00000 0.143223
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 4.00000 0.284268
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 1.00000 0.0707107
\(201\) −4.00000 −0.282138
\(202\) −6.00000 −0.422159
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) −4.00000 −0.279372
\(206\) −8.00000 −0.557386
\(207\) −4.00000 −0.278019
\(208\) −1.00000 −0.0693375
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 6.00000 0.412082
\(213\) 8.00000 0.548151
\(214\) 0 0
\(215\) 8.00000 0.545595
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −14.0000 −0.948200
\(219\) −6.00000 −0.405442
\(220\) −8.00000 −0.539360
\(221\) −2.00000 −0.134535
\(222\) −2.00000 −0.134231
\(223\) 24.0000 1.60716 0.803579 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) −10.0000 −0.665190
\(227\) 8.00000 0.530979 0.265489 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) −4.00000 −0.264906
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) 2.00000 0.131306
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) 1.00000 0.0653720
\(235\) 24.0000 1.56559
\(236\) 0 0
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) −2.00000 −0.129099
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) −5.00000 −0.321412
\(243\) −1.00000 −0.0641500
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) −4.00000 −0.254514
\(248\) 0 0
\(249\) 8.00000 0.506979
\(250\) 12.0000 0.758947
\(251\) 4.00000 0.252478 0.126239 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) 8.00000 0.501965
\(255\) −4.00000 −0.250490
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 4.00000 0.249029
\(259\) 0 0
\(260\) −2.00000 −0.124035
\(261\) −2.00000 −0.123797
\(262\) 12.0000 0.741362
\(263\) −20.0000 −1.23325 −0.616626 0.787256i \(-0.711501\pi\)
−0.616626 + 0.787256i \(0.711501\pi\)
\(264\) −4.00000 −0.246183
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) 4.00000 0.244339
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 2.00000 0.121716
\(271\) 24.0000 1.45790 0.728948 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) −22.0000 −1.32907
\(275\) 4.00000 0.241209
\(276\) 4.00000 0.240772
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) 0 0
\(281\) 22.0000 1.31241 0.656205 0.754583i \(-0.272161\pi\)
0.656205 + 0.754583i \(0.272161\pi\)
\(282\) 12.0000 0.714590
\(283\) 28.0000 1.66443 0.832214 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) −8.00000 −0.474713
\(285\) −8.00000 −0.473879
\(286\) −4.00000 −0.236525
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) 4.00000 0.234888
\(291\) 2.00000 0.117242
\(292\) 6.00000 0.351123
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2.00000 0.116248
\(297\) 4.00000 0.232104
\(298\) −18.0000 −1.04271
\(299\) 4.00000 0.231326
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) 16.0000 0.920697
\(303\) −6.00000 −0.344691
\(304\) 4.00000 0.229416
\(305\) 20.0000 1.14520
\(306\) −2.00000 −0.114332
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 22.0000 1.24153
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 6.00000 0.336463
\(319\) 8.00000 0.447914
\(320\) 2.00000 0.111803
\(321\) 0 0
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) 1.00000 0.0555556
\(325\) 1.00000 0.0554700
\(326\) −12.0000 −0.664619
\(327\) −14.0000 −0.774202
\(328\) 2.00000 0.110432
\(329\) 0 0
\(330\) −8.00000 −0.440386
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) −8.00000 −0.439057
\(333\) −2.00000 −0.109599
\(334\) 12.0000 0.656611
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −10.0000 −0.543125
\(340\) 4.00000 0.216930
\(341\) 0 0
\(342\) −4.00000 −0.216295
\(343\) 0 0
\(344\) −4.00000 −0.215666
\(345\) 8.00000 0.430706
\(346\) 2.00000 0.107521
\(347\) 16.0000 0.858925 0.429463 0.903085i \(-0.358703\pi\)
0.429463 + 0.903085i \(0.358703\pi\)
\(348\) 2.00000 0.107211
\(349\) 34.0000 1.81998 0.909989 0.414632i \(-0.136090\pi\)
0.909989 + 0.414632i \(0.136090\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 4.00000 0.213201
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 0 0
\(355\) −16.0000 −0.849192
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 24.0000 1.26844
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) −2.00000 −0.105409
\(361\) −3.00000 −0.157895
\(362\) −26.0000 −1.36653
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) 12.0000 0.628109
\(366\) 10.0000 0.522708
\(367\) 24.0000 1.25279 0.626395 0.779506i \(-0.284530\pi\)
0.626395 + 0.779506i \(0.284530\pi\)
\(368\) −4.00000 −0.208514
\(369\) −2.00000 −0.104116
\(370\) 4.00000 0.207950
\(371\) 0 0
\(372\) 0 0
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) 8.00000 0.413670
\(375\) 12.0000 0.619677
\(376\) −12.0000 −0.618853
\(377\) 2.00000 0.103005
\(378\) 0 0
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 8.00000 0.410391
\(381\) 8.00000 0.409852
\(382\) 12.0000 0.613973
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) 4.00000 0.203331
\(388\) −2.00000 −0.101535
\(389\) 14.0000 0.709828 0.354914 0.934899i \(-0.384510\pi\)
0.354914 + 0.934899i \(0.384510\pi\)
\(390\) −2.00000 −0.101274
\(391\) −8.00000 −0.404577
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 6.00000 0.302276
\(395\) 16.0000 0.805047
\(396\) −4.00000 −0.201008
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) −8.00000 −0.401004
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) 4.00000 0.199502
\(403\) 0 0
\(404\) 6.00000 0.298511
\(405\) 2.00000 0.0993808
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 2.00000 0.0990148
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 4.00000 0.197546
\(411\) −22.0000 −1.08518
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) 4.00000 0.196589
\(415\) −16.0000 −0.785409
\(416\) 1.00000 0.0490290
\(417\) 4.00000 0.195881
\(418\) 16.0000 0.782586
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −4.00000 −0.194717
\(423\) 12.0000 0.583460
\(424\) −6.00000 −0.291386
\(425\) −2.00000 −0.0970143
\(426\) −8.00000 −0.387601
\(427\) 0 0
\(428\) 0 0
\(429\) −4.00000 −0.193122
\(430\) −8.00000 −0.385794
\(431\) −40.0000 −1.92673 −0.963366 0.268190i \(-0.913575\pi\)
−0.963366 + 0.268190i \(0.913575\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 30.0000 1.44171 0.720854 0.693087i \(-0.243750\pi\)
0.720854 + 0.693087i \(0.243750\pi\)
\(434\) 0 0
\(435\) 4.00000 0.191785
\(436\) 14.0000 0.670478
\(437\) −16.0000 −0.765384
\(438\) 6.00000 0.286691
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 8.00000 0.381385
\(441\) 0 0
\(442\) 2.00000 0.0951303
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 2.00000 0.0949158
\(445\) 12.0000 0.568855
\(446\) −24.0000 −1.13643
\(447\) −18.0000 −0.851371
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 1.00000 0.0471405
\(451\) 8.00000 0.376705
\(452\) 10.0000 0.470360
\(453\) 16.0000 0.751746
\(454\) −8.00000 −0.375459
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) −26.0000 −1.21490
\(459\) −2.00000 −0.0933520
\(460\) −8.00000 −0.373002
\(461\) 2.00000 0.0931493 0.0465746 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 0 0
\(463\) 24.0000 1.11537 0.557687 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) −26.0000 −1.20443
\(467\) −4.00000 −0.185098 −0.0925490 0.995708i \(-0.529501\pi\)
−0.0925490 + 0.995708i \(0.529501\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 0 0
\(470\) −24.0000 −1.10704
\(471\) 22.0000 1.01371
\(472\) 0 0
\(473\) −16.0000 −0.735681
\(474\) 8.00000 0.367452
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 24.0000 1.09773
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 2.00000 0.0912871
\(481\) 2.00000 0.0911922
\(482\) 2.00000 0.0910975
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) −4.00000 −0.181631
\(486\) 1.00000 0.0453609
\(487\) 32.0000 1.45006 0.725029 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) −10.0000 −0.452679
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) −8.00000 −0.361035 −0.180517 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) 2.00000 0.0901670
\(493\) −4.00000 −0.180151
\(494\) 4.00000 0.179969
\(495\) −8.00000 −0.359573
\(496\) 0 0
\(497\) 0 0
\(498\) −8.00000 −0.358489
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) −12.0000 −0.536656
\(501\) 12.0000 0.536120
\(502\) −4.00000 −0.178529
\(503\) 16.0000 0.713405 0.356702 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) −16.0000 −0.711287
\(507\) −1.00000 −0.0444116
\(508\) −8.00000 −0.354943
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 4.00000 0.177123
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −4.00000 −0.176604
\(514\) −18.0000 −0.793946
\(515\) 16.0000 0.705044
\(516\) −4.00000 −0.176090
\(517\) −48.0000 −2.11104
\(518\) 0 0
\(519\) 2.00000 0.0877903
\(520\) 2.00000 0.0877058
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 2.00000 0.0875376
\(523\) −36.0000 −1.57417 −0.787085 0.616844i \(-0.788411\pi\)
−0.787085 + 0.616844i \(0.788411\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 20.0000 0.872041
\(527\) 0 0
\(528\) 4.00000 0.174078
\(529\) −7.00000 −0.304348
\(530\) −12.0000 −0.521247
\(531\) 0 0
\(532\) 0 0
\(533\) 2.00000 0.0866296
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 24.0000 1.03568
\(538\) −6.00000 −0.258678
\(539\) 0 0
\(540\) −2.00000 −0.0860663
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) −24.0000 −1.03089
\(543\) −26.0000 −1.11577
\(544\) −2.00000 −0.0857493
\(545\) 28.0000 1.19939
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 22.0000 0.939793
\(549\) 10.0000 0.426790
\(550\) −4.00000 −0.170561
\(551\) −8.00000 −0.340811
\(552\) −4.00000 −0.170251
\(553\) 0 0
\(554\) −22.0000 −0.934690
\(555\) 4.00000 0.169791
\(556\) −4.00000 −0.169638
\(557\) −38.0000 −1.61011 −0.805056 0.593199i \(-0.797865\pi\)
−0.805056 + 0.593199i \(0.797865\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) −22.0000 −0.928014
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) −12.0000 −0.505291
\(565\) 20.0000 0.841406
\(566\) −28.0000 −1.17693
\(567\) 0 0
\(568\) 8.00000 0.335673
\(569\) 26.0000 1.08998 0.544988 0.838444i \(-0.316534\pi\)
0.544988 + 0.838444i \(0.316534\pi\)
\(570\) 8.00000 0.335083
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 4.00000 0.167248
\(573\) 12.0000 0.501307
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 1.00000 0.0416667
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) 13.0000 0.540729
\(579\) −2.00000 −0.0831172
\(580\) −4.00000 −0.166091
\(581\) 0 0
\(582\) −2.00000 −0.0829027
\(583\) −24.0000 −0.993978
\(584\) −6.00000 −0.248282
\(585\) −2.00000 −0.0826898
\(586\) 14.0000 0.578335
\(587\) 24.0000 0.990586 0.495293 0.868726i \(-0.335061\pi\)
0.495293 + 0.868726i \(0.335061\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) −2.00000 −0.0821995
\(593\) 22.0000 0.903432 0.451716 0.892162i \(-0.350812\pi\)
0.451716 + 0.892162i \(0.350812\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) −8.00000 −0.327418
\(598\) −4.00000 −0.163572
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) −16.0000 −0.651031
\(605\) 10.0000 0.406558
\(606\) 6.00000 0.243733
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) −20.0000 −0.809776
\(611\) −12.0000 −0.485468
\(612\) 2.00000 0.0808452
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) −12.0000 −0.484281
\(615\) 4.00000 0.161296
\(616\) 0 0
\(617\) 22.0000 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) 8.00000 0.321807
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 24.0000 0.962312
\(623\) 0 0
\(624\) 1.00000 0.0400320
\(625\) −19.0000 −0.760000
\(626\) −6.00000 −0.239808
\(627\) 16.0000 0.638978
\(628\) −22.0000 −0.877896
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) −8.00000 −0.318223
\(633\) −4.00000 −0.158986
\(634\) −2.00000 −0.0794301
\(635\) −16.0000 −0.634941
\(636\) −6.00000 −0.237915
\(637\) 0 0
\(638\) −8.00000 −0.316723
\(639\) −8.00000 −0.316475
\(640\) −2.00000 −0.0790569
\(641\) −22.0000 −0.868948 −0.434474 0.900684i \(-0.643066\pi\)
−0.434474 + 0.900684i \(0.643066\pi\)
\(642\) 0 0
\(643\) −20.0000 −0.788723 −0.394362 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) −8.00000 −0.314756
\(647\) 48.0000 1.88707 0.943537 0.331266i \(-0.107476\pi\)
0.943537 + 0.331266i \(0.107476\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) −1.00000 −0.0392232
\(651\) 0 0
\(652\) 12.0000 0.469956
\(653\) 30.0000 1.17399 0.586995 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) 14.0000 0.547443
\(655\) −24.0000 −0.937758
\(656\) −2.00000 −0.0780869
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 8.00000 0.311400
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) 12.0000 0.466393
\(663\) 2.00000 0.0776736
\(664\) 8.00000 0.310460
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) 8.00000 0.309761
\(668\) −12.0000 −0.464294
\(669\) −24.0000 −0.927894
\(670\) −8.00000 −0.309067
\(671\) −40.0000 −1.54418
\(672\) 0 0
\(673\) 18.0000 0.693849 0.346925 0.937893i \(-0.387226\pi\)
0.346925 + 0.937893i \(0.387226\pi\)
\(674\) −18.0000 −0.693334
\(675\) 1.00000 0.0384900
\(676\) 1.00000 0.0384615
\(677\) −34.0000 −1.30673 −0.653363 0.757045i \(-0.726642\pi\)
−0.653363 + 0.757045i \(0.726642\pi\)
\(678\) 10.0000 0.384048
\(679\) 0 0
\(680\) −4.00000 −0.153393
\(681\) −8.00000 −0.306561
\(682\) 0 0
\(683\) −44.0000 −1.68361 −0.841807 0.539779i \(-0.818508\pi\)
−0.841807 + 0.539779i \(0.818508\pi\)
\(684\) 4.00000 0.152944
\(685\) 44.0000 1.68115
\(686\) 0 0
\(687\) −26.0000 −0.991962
\(688\) 4.00000 0.152499
\(689\) −6.00000 −0.228582
\(690\) −8.00000 −0.304555
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) −2.00000 −0.0760286
\(693\) 0 0
\(694\) −16.0000 −0.607352
\(695\) −8.00000 −0.303457
\(696\) −2.00000 −0.0758098
\(697\) −4.00000 −0.151511
\(698\) −34.0000 −1.28692
\(699\) −26.0000 −0.983410
\(700\) 0 0
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −8.00000 −0.301726
\(704\) −4.00000 −0.150756
\(705\) −24.0000 −0.903892
\(706\) −14.0000 −0.526897
\(707\) 0 0
\(708\) 0 0
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) 16.0000 0.600469
\(711\) 8.00000 0.300023
\(712\) −6.00000 −0.224860
\(713\) 0 0
\(714\) 0 0
\(715\) 8.00000 0.299183
\(716\) −24.0000 −0.896922
\(717\) 24.0000 0.896296
\(718\) −24.0000 −0.895672
\(719\) −8.00000 −0.298350 −0.149175 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) 2.00000 0.0745356
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) 2.00000 0.0743808
\(724\) 26.0000 0.966282
\(725\) 2.00000 0.0742781
\(726\) 5.00000 0.185567
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −12.0000 −0.444140
\(731\) 8.00000 0.295891
\(732\) −10.0000 −0.369611
\(733\) −30.0000 −1.10808 −0.554038 0.832492i \(-0.686914\pi\)
−0.554038 + 0.832492i \(0.686914\pi\)
\(734\) −24.0000 −0.885856
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) −16.0000 −0.589368
\(738\) 2.00000 0.0736210
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) −4.00000 −0.147043
\(741\) 4.00000 0.146944
\(742\) 0 0
\(743\) 40.0000 1.46746 0.733729 0.679442i \(-0.237778\pi\)
0.733729 + 0.679442i \(0.237778\pi\)
\(744\) 0 0
\(745\) 36.0000 1.31894
\(746\) 26.0000 0.951928
\(747\) −8.00000 −0.292705
\(748\) −8.00000 −0.292509
\(749\) 0 0
\(750\) −12.0000 −0.438178
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 12.0000 0.437595
\(753\) −4.00000 −0.145768
\(754\) −2.00000 −0.0728357
\(755\) −32.0000 −1.16460
\(756\) 0 0
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) −28.0000 −1.01701
\(759\) −16.0000 −0.580763
\(760\) −8.00000 −0.290191
\(761\) 14.0000 0.507500 0.253750 0.967270i \(-0.418336\pi\)
0.253750 + 0.967270i \(0.418336\pi\)
\(762\) −8.00000 −0.289809
\(763\) 0 0
\(764\) −12.0000 −0.434145
\(765\) 4.00000 0.144620
\(766\) −12.0000 −0.433578
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −34.0000 −1.22607 −0.613036 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 2.00000 0.0719816
\(773\) 34.0000 1.22290 0.611448 0.791285i \(-0.290588\pi\)
0.611448 + 0.791285i \(0.290588\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) −14.0000 −0.501924
\(779\) −8.00000 −0.286630
\(780\) 2.00000 0.0716115
\(781\) 32.0000 1.14505
\(782\) 8.00000 0.286079
\(783\) 2.00000 0.0714742
\(784\) 0 0
\(785\) −44.0000 −1.57043
\(786\) −12.0000 −0.428026
\(787\) −44.0000 −1.56843 −0.784215 0.620489i \(-0.786934\pi\)
−0.784215 + 0.620489i \(0.786934\pi\)
\(788\) −6.00000 −0.213741
\(789\) 20.0000 0.712019
\(790\) −16.0000 −0.569254
\(791\) 0 0
\(792\) 4.00000 0.142134
\(793\) −10.0000 −0.355110
\(794\) 14.0000 0.496841
\(795\) −12.0000 −0.425596
\(796\) 8.00000 0.283552
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 1.00000 0.0353553
\(801\) 6.00000 0.212000
\(802\) 10.0000 0.353112
\(803\) −24.0000 −0.846942
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 0 0
\(807\) −6.00000 −0.211210
\(808\) −6.00000 −0.211079
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) −2.00000 −0.0702728
\(811\) 4.00000 0.140459 0.0702295 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) 0 0
\(813\) −24.0000 −0.841717
\(814\) −8.00000 −0.280400
\(815\) 24.0000 0.840683
\(816\) −2.00000 −0.0700140
\(817\) 16.0000 0.559769
\(818\) 26.0000 0.909069
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) 34.0000 1.18661 0.593304 0.804978i \(-0.297823\pi\)
0.593304 + 0.804978i \(0.297823\pi\)
\(822\) 22.0000 0.767338
\(823\) −8.00000 −0.278862 −0.139431 0.990232i \(-0.544527\pi\)
−0.139431 + 0.990232i \(0.544527\pi\)
\(824\) −8.00000 −0.278693
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) 28.0000 0.973655 0.486828 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) −4.00000 −0.139010
\(829\) −30.0000 −1.04194 −0.520972 0.853574i \(-0.674430\pi\)
−0.520972 + 0.853574i \(0.674430\pi\)
\(830\) 16.0000 0.555368
\(831\) −22.0000 −0.763172
\(832\) −1.00000 −0.0346688
\(833\) 0 0
\(834\) −4.00000 −0.138509
\(835\) −24.0000 −0.830554
\(836\) −16.0000 −0.553372
\(837\) 0 0
\(838\) 20.0000 0.690889
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 26.0000 0.896019
\(843\) −22.0000 −0.757720
\(844\) 4.00000 0.137686
\(845\) 2.00000 0.0688021
\(846\) −12.0000 −0.412568
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) −28.0000 −0.960958
\(850\) 2.00000 0.0685994
\(851\) 8.00000 0.274236
\(852\) 8.00000 0.274075
\(853\) 26.0000 0.890223 0.445112 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 0 0
\(855\) 8.00000 0.273594
\(856\) 0 0
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 4.00000 0.136558
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) 40.0000 1.36241
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 1.00000 0.0340207
\(865\) −4.00000 −0.136004
\(866\) −30.0000 −1.01944
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) −32.0000 −1.08553
\(870\) −4.00000 −0.135613
\(871\) −4.00000 −0.135535
\(872\) −14.0000 −0.474100
\(873\) −2.00000 −0.0676897
\(874\) 16.0000 0.541208
\(875\) 0 0
\(876\) −6.00000 −0.202721
\(877\) −10.0000 −0.337676 −0.168838 0.985644i \(-0.554001\pi\)
−0.168838 + 0.985644i \(0.554001\pi\)
\(878\) −32.0000 −1.07995
\(879\) 14.0000 0.472208
\(880\) −8.00000 −0.269680
\(881\) −22.0000 −0.741199 −0.370599 0.928793i \(-0.620848\pi\)
−0.370599 + 0.928793i \(0.620848\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) −2.00000 −0.0672673
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) 16.0000 0.537227 0.268614 0.963248i \(-0.413434\pi\)
0.268614 + 0.963248i \(0.413434\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 0 0
\(890\) −12.0000 −0.402241
\(891\) −4.00000 −0.134005
\(892\) 24.0000 0.803579
\(893\) 48.0000 1.60626
\(894\) 18.0000 0.602010
\(895\) −48.0000 −1.60446
\(896\) 0 0
\(897\) −4.00000 −0.133556
\(898\) 18.0000 0.600668
\(899\) 0 0
\(900\) −1.00000 −0.0333333
\(901\) 12.0000 0.399778
\(902\) −8.00000 −0.266371
\(903\) 0 0
\(904\) −10.0000 −0.332595
\(905\) 52.0000 1.72854
\(906\) −16.0000 −0.531564
\(907\) −36.0000 −1.19536 −0.597680 0.801735i \(-0.703911\pi\)
−0.597680 + 0.801735i \(0.703911\pi\)
\(908\) 8.00000 0.265489
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) −4.00000 −0.132453
\(913\) 32.0000 1.05905
\(914\) −18.0000 −0.595387
\(915\) −20.0000 −0.661180
\(916\) 26.0000 0.859064
\(917\) 0 0
\(918\) 2.00000 0.0660098
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 8.00000 0.263752
\(921\) −12.0000 −0.395413
\(922\) −2.00000 −0.0658665
\(923\) 8.00000 0.263323
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) −24.0000 −0.788689
\(927\) 8.00000 0.262754
\(928\) 2.00000 0.0656532
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 26.0000 0.851658
\(933\) 24.0000 0.785725
\(934\) 4.00000 0.130884
\(935\) −16.0000 −0.523256
\(936\) 1.00000 0.0326860
\(937\) −42.0000 −1.37208 −0.686040 0.727564i \(-0.740653\pi\)
−0.686040 + 0.727564i \(0.740653\pi\)
\(938\) 0 0
\(939\) −6.00000 −0.195803
\(940\) 24.0000 0.782794
\(941\) 34.0000 1.10837 0.554184 0.832394i \(-0.313030\pi\)
0.554184 + 0.832394i \(0.313030\pi\)
\(942\) −22.0000 −0.716799
\(943\) 8.00000 0.260516
\(944\) 0 0
\(945\) 0 0
\(946\) 16.0000 0.520205
\(947\) −20.0000 −0.649913 −0.324956 0.945729i \(-0.605350\pi\)
−0.324956 + 0.945729i \(0.605350\pi\)
\(948\) −8.00000 −0.259828
\(949\) −6.00000 −0.194768
\(950\) 4.00000 0.129777
\(951\) −2.00000 −0.0648544
\(952\) 0 0
\(953\) −46.0000 −1.49009 −0.745043 0.667016i \(-0.767571\pi\)
−0.745043 + 0.667016i \(0.767571\pi\)
\(954\) −6.00000 −0.194257
\(955\) −24.0000 −0.776622
\(956\) −24.0000 −0.776215
\(957\) −8.00000 −0.258603
\(958\) 12.0000 0.387702
\(959\) 0 0
\(960\) −2.00000 −0.0645497
\(961\) −31.0000 −1.00000
\(962\) −2.00000 −0.0644826
\(963\) 0 0
\(964\) −2.00000 −0.0644157
\(965\) 4.00000 0.128765
\(966\) 0 0
\(967\) −24.0000 −0.771788 −0.385894 0.922543i \(-0.626107\pi\)
−0.385894 + 0.922543i \(0.626107\pi\)
\(968\) −5.00000 −0.160706
\(969\) −8.00000 −0.256997
\(970\) 4.00000 0.128432
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −32.0000 −1.02535
\(975\) −1.00000 −0.0320256
\(976\) 10.0000 0.320092
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) 12.0000 0.383718
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) 14.0000 0.446986
\(982\) 8.00000 0.255290
\(983\) 4.00000 0.127580 0.0637901 0.997963i \(-0.479681\pi\)
0.0637901 + 0.997963i \(0.479681\pi\)
\(984\) −2.00000 −0.0637577
\(985\) −12.0000 −0.382352
\(986\) 4.00000 0.127386
\(987\) 0 0
\(988\) −4.00000 −0.127257
\(989\) −16.0000 −0.508770
\(990\) 8.00000 0.254257
\(991\) 24.0000 0.762385 0.381193 0.924496i \(-0.375513\pi\)
0.381193 + 0.924496i \(0.375513\pi\)
\(992\) 0 0
\(993\) 12.0000 0.380808
\(994\) 0 0
\(995\) 16.0000 0.507234
\(996\) 8.00000 0.253490
\(997\) −46.0000 −1.45683 −0.728417 0.685134i \(-0.759744\pi\)
−0.728417 + 0.685134i \(0.759744\pi\)
\(998\) 20.0000 0.633089
\(999\) 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 3822.2.a.h.1.1 1
7.6 odd 2 546.2.a.b.1.1 1
21.20 even 2 1638.2.a.s.1.1 1
28.27 even 2 4368.2.a.d.1.1 1
91.90 odd 2 7098.2.a.be.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.b.1.1 1 7.6 odd 2
1638.2.a.s.1.1 1 21.20 even 2
3822.2.a.h.1.1 1 1.1 even 1 trivial
4368.2.a.d.1.1 1 28.27 even 2
7098.2.a.be.1.1 1 91.90 odd 2