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Properties

Label 1122.2.a.b.1.1
Level $1122$
Weight $2$
Character 1122.1
Self dual yes
Analytic conductor $8.959$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1122,2,Mod(1,1122)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1122, 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("1122.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1122 = 2 \cdot 3 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1122.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: \(8.95921510679\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 1122.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} +2.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} -2.00000 q^{19} +2.00000 q^{20} +2.00000 q^{21} +1.00000 q^{22} -6.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} -1.00000 q^{27} -2.00000 q^{28} +2.00000 q^{29} +2.00000 q^{30} -1.00000 q^{32} +1.00000 q^{33} +1.00000 q^{34} -4.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} +2.00000 q^{38} -2.00000 q^{40} +2.00000 q^{41} -2.00000 q^{42} -10.0000 q^{43} -1.00000 q^{44} +2.00000 q^{45} +6.00000 q^{46} +4.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} +1.00000 q^{51} +4.00000 q^{53} +1.00000 q^{54} -2.00000 q^{55} +2.00000 q^{56} +2.00000 q^{57} -2.00000 q^{58} +6.00000 q^{59} -2.00000 q^{60} -10.0000 q^{61} -2.00000 q^{63} +1.00000 q^{64} -1.00000 q^{66} -12.0000 q^{67} -1.00000 q^{68} +6.00000 q^{69} +4.00000 q^{70} +6.00000 q^{71} -1.00000 q^{72} -4.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} -2.00000 q^{76} +2.00000 q^{77} -2.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -12.0000 q^{83} +2.00000 q^{84} -2.00000 q^{85} +10.0000 q^{86} -2.00000 q^{87} +1.00000 q^{88} -2.00000 q^{89} -2.00000 q^{90} -6.00000 q^{92} -4.00000 q^{94} -4.00000 q^{95} +1.00000 q^{96} -6.00000 q^{97} +3.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
\(3\) −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
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) −1.00000 −0.301511
\(12\) −1.00000 −0.288675
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 2.00000 0.534522
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536
\(18\) −1.00000 −0.235702
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 2.00000 0.447214
\(21\) 2.00000 0.436436
\(22\) 1.00000 0.213201
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −2.00000 −0.377964
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 2.00000 0.365148
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.00000 0.174078
\(34\) 1.00000 0.171499
\(35\) −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 \(0.447432\pi\)
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) −2.00000 −0.316228
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) −2.00000 −0.308607
\(43\) −10.0000 −1.52499 −0.762493 − 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) −1.00000 −0.150756
\(45\) 2.00000 0.298142
\(46\) 6.00000 0.884652
\(47\) 4.00000 0.583460 0.291730 − 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) 1.00000 0.141421
\(51\) 1.00000 0.140028
\(52\) 0 0
\(53\) 4.00000 0.549442 0.274721 − 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 1.00000 0.136083
\(55\) −2.00000 −0.269680
\(56\) 2.00000 0.267261
\(57\) 2.00000 0.264906
\(58\) −2.00000 −0.262613
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) −2.00000 −0.258199
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.00000 −0.123091
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) −1.00000 −0.121268
\(69\) 6.00000 0.722315
\(70\) 4.00000 0.478091
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) −1.00000 −0.117851
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) −2.00000 −0.232495
\(75\) 1.00000 0.115470
\(76\) −2.00000 −0.229416
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) −2.00000 −0.225018 −0.112509 − 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 2.00000 0.218218
\(85\) −2.00000 −0.216930
\(86\) 10.0000 1.07833
\(87\) −2.00000 −0.214423
\(88\) 1.00000 0.106600
\(89\) −2.00000 −0.212000 −0.106000 − 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) 0 0
\(94\) −4.00000 −0.412568
\(95\) −4.00000 −0.410391
\(96\) 1.00000 0.102062
\(97\) −6.00000 −0.609208 −0.304604 − 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 3.00000 0.303046
\(99\) −1.00000 −0.100504
\(100\) −1.00000 −0.100000
\(101\) −10.0000 −0.995037 −0.497519 − 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) −1.00000 −0.0990148
\(103\) −8.00000 −0.788263 −0.394132 − 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) 4.00000 0.390360
\(106\) −4.00000 −0.388514
\(107\) −12.0000 −1.16008 −0.580042 − 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −14.0000 −1.34096 −0.670478 − 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 2.00000 0.190693
\(111\) −2.00000 −0.189832
\(112\) −2.00000 −0.188982
\(113\) 20.0000 1.88144 0.940721 − 0.339182i \(-0.110150\pi\)
0.940721 + 0.339182i \(0.110150\pi\)
\(114\) −2.00000 −0.187317
\(115\) −12.0000 −1.11901
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) −6.00000 −0.552345
\(119\) 2.00000 0.183340
\(120\) 2.00000 0.182574
\(121\) 1.00000 0.0909091
\(122\) 10.0000 0.905357
\(123\) −2.00000 −0.180334
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 2.00000 0.178174
\(127\) −16.0000 −1.41977 −0.709885 − 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 10.0000 0.880451
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 − 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 1.00000 0.0870388
\(133\) 4.00000 0.346844
\(134\) 12.0000 1.03664
\(135\) −2.00000 −0.172133
\(136\) 1.00000 0.0857493
\(137\) −6.00000 −0.512615 −0.256307 − 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) −6.00000 −0.510754
\(139\) 20.0000 1.69638 0.848189 − 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) −4.00000 −0.338062
\(141\) −4.00000 −0.336861
\(142\) −6.00000 −0.503509
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 4.00000 0.332182
\(146\) 4.00000 0.331042
\(147\) 3.00000 0.247436
\(148\) 2.00000 0.164399
\(149\) −6.00000 −0.491539 −0.245770 − 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −12.0000 −0.976546 −0.488273 − 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 2.00000 0.162221
\(153\) −1.00000 −0.0808452
\(154\) −2.00000 −0.161165
\(155\) 0 0
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 − 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 2.00000 0.159111
\(159\) −4.00000 −0.317221
\(160\) −2.00000 −0.158114
\(161\) 12.0000 0.945732
\(162\) −1.00000 −0.0785674
\(163\) −12.0000 −0.939913 −0.469956 − 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 2.00000 0.156174
\(165\) 2.00000 0.155700
\(166\) 12.0000 0.931381
\(167\) −8.00000 −0.619059 −0.309529 − 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) −2.00000 −0.154303
\(169\) −13.0000 −1.00000
\(170\) 2.00000 0.153393
\(171\) −2.00000 −0.152944
\(172\) −10.0000 −0.762493
\(173\) 14.0000 1.06440 0.532200 − 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 2.00000 0.151620
\(175\) 2.00000 0.151186
\(176\) −1.00000 −0.0753778
\(177\) −6.00000 −0.450988
\(178\) 2.00000 0.149906
\(179\) 18.0000 1.34538 0.672692 − 0.739923i \(-0.265138\pi\)
0.672692 + 0.739923i \(0.265138\pi\)
\(180\) 2.00000 0.149071
\(181\) 10.0000 0.743294 0.371647 − 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) 6.00000 0.442326
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) 1.00000 0.0731272
\(188\) 4.00000 0.291730
\(189\) 2.00000 0.145479
\(190\) 4.00000 0.290191
\(191\) 8.00000 0.578860 0.289430 − 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 2.00000 0.142494 0.0712470 − 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 1.00000 0.0710669
\(199\) 8.00000 0.567105 0.283552 − 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 1.00000 0.0707107
\(201\) 12.0000 0.846415
\(202\) 10.0000 0.703598
\(203\) −4.00000 −0.280745
\(204\) 1.00000 0.0700140
\(205\) 4.00000 0.279372
\(206\) 8.00000 0.557386
\(207\) −6.00000 −0.417029
\(208\) 0 0
\(209\) 2.00000 0.138343
\(210\) −4.00000 −0.276026
\(211\) 20.0000 1.37686 0.688428 − 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 4.00000 0.274721
\(213\) −6.00000 −0.411113
\(214\) 12.0000 0.820303
\(215\) −20.0000 −1.36399
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 14.0000 0.948200
\(219\) 4.00000 0.270295
\(220\) −2.00000 −0.134840
\(221\) 0 0
\(222\) 2.00000 0.134231
\(223\) 8.00000 0.535720 0.267860 − 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 2.00000 0.133631
\(225\) −1.00000 −0.0666667
\(226\) −20.0000 −1.33038
\(227\) 4.00000 0.265489 0.132745 − 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 2.00000 0.132453
\(229\) 10.0000 0.660819 0.330409 − 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 12.0000 0.791257
\(231\) −2.00000 −0.131590
\(232\) −2.00000 −0.131306
\(233\) −6.00000 −0.393073 −0.196537 − 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 6.00000 0.390567
\(237\) 2.00000 0.129914
\(238\) −2.00000 −0.129641
\(239\) 16.0000 1.03495 0.517477 − 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) −2.00000 −0.129099
\(241\) 24.0000 1.54598 0.772988 − 0.634421i \(-0.218761\pi\)
0.772988 + 0.634421i \(0.218761\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −1.00000 −0.0641500
\(244\) −10.0000 −0.640184
\(245\) −6.00000 −0.383326
\(246\) 2.00000 0.127515
\(247\) 0 0
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 12.0000 0.758947
\(251\) −6.00000 −0.378717 −0.189358 − 0.981908i \(-0.560641\pi\)
−0.189358 + 0.981908i \(0.560641\pi\)
\(252\) −2.00000 −0.125988
\(253\) 6.00000 0.377217
\(254\) 16.0000 1.00393
\(255\) 2.00000 0.125245
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 − 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) −10.0000 −0.622573
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 4.00000 0.247121
\(263\) 24.0000 1.47990 0.739952 − 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) −1.00000 −0.0615457
\(265\) 8.00000 0.491436
\(266\) −4.00000 −0.245256
\(267\) 2.00000 0.122398
\(268\) −12.0000 −0.733017
\(269\) −6.00000 −0.365826 −0.182913 − 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 2.00000 0.121716
\(271\) −16.0000 −0.971931 −0.485965 − 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −1.00000 −0.0606339
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 1.00000 0.0603023
\(276\) 6.00000 0.361158
\(277\) 10.0000 0.600842 0.300421 − 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) −20.0000 −1.19952
\(279\) 0 0
\(280\) 4.00000 0.239046
\(281\) −6.00000 −0.357930 −0.178965 − 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 4.00000 0.238197
\(283\) −4.00000 −0.237775 −0.118888 − 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 6.00000 0.356034
\(285\) 4.00000 0.236940
\(286\) 0 0
\(287\) −4.00000 −0.236113
\(288\) −1.00000 −0.0589256
\(289\) 1.00000 0.0588235
\(290\) −4.00000 −0.234888
\(291\) 6.00000 0.351726
\(292\) −4.00000 −0.234082
\(293\) 2.00000 0.116841 0.0584206 − 0.998292i \(-0.481394\pi\)
0.0584206 + 0.998292i \(0.481394\pi\)
\(294\) −3.00000 −0.174964
\(295\) 12.0000 0.698667
\(296\) −2.00000 −0.116248
\(297\) 1.00000 0.0580259
\(298\) 6.00000 0.347571
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 20.0000 1.15278
\(302\) 12.0000 0.690522
\(303\) 10.0000 0.574485
\(304\) −2.00000 −0.114708
\(305\) −20.0000 −1.14520
\(306\) 1.00000 0.0571662
\(307\) −6.00000 −0.342438 −0.171219 − 0.985233i \(-0.554771\pi\)
−0.171219 + 0.985233i \(0.554771\pi\)
\(308\) 2.00000 0.113961
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) 6.00000 0.340229 0.170114 − 0.985424i \(-0.445586\pi\)
0.170114 + 0.985424i \(0.445586\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 − 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) −18.0000 −1.01580
\(315\) −4.00000 −0.225374
\(316\) −2.00000 −0.112509
\(317\) −10.0000 −0.561656 −0.280828 − 0.959758i \(-0.590609\pi\)
−0.280828 + 0.959758i \(0.590609\pi\)
\(318\) 4.00000 0.224309
\(319\) −2.00000 −0.111979
\(320\) 2.00000 0.111803
\(321\) 12.0000 0.669775
\(322\) −12.0000 −0.668734
\(323\) 2.00000 0.111283
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 12.0000 0.664619
\(327\) 14.0000 0.774202
\(328\) −2.00000 −0.110432
\(329\) −8.00000 −0.441054
\(330\) −2.00000 −0.110096
\(331\) −4.00000 −0.219860 −0.109930 − 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −12.0000 −0.658586
\(333\) 2.00000 0.109599
\(334\) 8.00000 0.437741
\(335\) −24.0000 −1.31126
\(336\) 2.00000 0.109109
\(337\) 12.0000 0.653682 0.326841 − 0.945079i \(-0.394016\pi\)
0.326841 + 0.945079i \(0.394016\pi\)
\(338\) 13.0000 0.707107
\(339\) −20.0000 −1.08625
\(340\) −2.00000 −0.108465
\(341\) 0 0
\(342\) 2.00000 0.108148
\(343\) 20.0000 1.07990
\(344\) 10.0000 0.539164
\(345\) 12.0000 0.646058
\(346\) −14.0000 −0.752645
\(347\) −4.00000 −0.214731 −0.107366 − 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) −2.00000 −0.107211
\(349\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) −2.00000 −0.106904
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 2.00000 0.106449 0.0532246 − 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) 6.00000 0.318896
\(355\) 12.0000 0.636894
\(356\) −2.00000 −0.106000
\(357\) −2.00000 −0.105851
\(358\) −18.0000 −0.951330
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −2.00000 −0.105409
\(361\) −15.0000 −0.789474
\(362\) −10.0000 −0.525588
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) −8.00000 −0.418739
\(366\) −10.0000 −0.522708
\(367\) −8.00000 −0.417597 −0.208798 − 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) −6.00000 −0.312772
\(369\) 2.00000 0.104116
\(370\) −4.00000 −0.207950
\(371\) −8.00000 −0.415339
\(372\) 0 0
\(373\) −8.00000 −0.414224 −0.207112 − 0.978317i \(-0.566407\pi\)
−0.207112 + 0.978317i \(0.566407\pi\)
\(374\) −1.00000 −0.0517088
\(375\) 12.0000 0.619677
\(376\) −4.00000 −0.206284
\(377\) 0 0
\(378\) −2.00000 −0.102869
\(379\) 20.0000 1.02733 0.513665 − 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) −4.00000 −0.205196
\(381\) 16.0000 0.819705
\(382\) −8.00000 −0.409316
\(383\) −32.0000 −1.63512 −0.817562 − 0.575841i \(-0.804675\pi\)
−0.817562 + 0.575841i \(0.804675\pi\)
\(384\) 1.00000 0.0510310
\(385\) 4.00000 0.203859
\(386\) 0 0
\(387\) −10.0000 −0.508329
\(388\) −6.00000 −0.304604
\(389\) 16.0000 0.811232 0.405616 − 0.914044i \(-0.367057\pi\)
0.405616 + 0.914044i \(0.367057\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) 3.00000 0.151523
\(393\) 4.00000 0.201773
\(394\) −2.00000 −0.100759
\(395\) −4.00000 −0.201262
\(396\) −1.00000 −0.0502519
\(397\) 14.0000 0.702640 0.351320 − 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) −8.00000 −0.401004
\(399\) −4.00000 −0.200250
\(400\) −1.00000 −0.0500000
\(401\) −12.0000 −0.599251 −0.299626 − 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) −12.0000 −0.598506
\(403\) 0 0
\(404\) −10.0000 −0.497519
\(405\) 2.00000 0.0993808
\(406\) 4.00000 0.198517
\(407\) −2.00000 −0.0991363
\(408\) −1.00000 −0.0495074
\(409\) 22.0000 1.08783 0.543915 − 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) −4.00000 −0.197546
\(411\) 6.00000 0.295958
\(412\) −8.00000 −0.394132
\(413\) −12.0000 −0.590481
\(414\) 6.00000 0.294884
\(415\) −24.0000 −1.17811
\(416\) 0 0
\(417\) −20.0000 −0.979404
\(418\) −2.00000 −0.0978232
\(419\) 12.0000 0.586238 0.293119 − 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 4.00000 0.195180
\(421\) −26.0000 −1.26716 −0.633581 − 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −20.0000 −0.973585
\(423\) 4.00000 0.194487
\(424\) −4.00000 −0.194257
\(425\) 1.00000 0.0485071
\(426\) 6.00000 0.290701
\(427\) 20.0000 0.967868
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 20.0000 0.964486
\(431\) −28.0000 −1.34871 −0.674356 − 0.738406i \(-0.735579\pi\)
−0.674356 + 0.738406i \(0.735579\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −18.0000 −0.865025 −0.432512 − 0.901628i \(-0.642373\pi\)
−0.432512 + 0.901628i \(0.642373\pi\)
\(434\) 0 0
\(435\) −4.00000 −0.191785
\(436\) −14.0000 −0.670478
\(437\) 12.0000 0.574038
\(438\) −4.00000 −0.191127
\(439\) −10.0000 −0.477274 −0.238637 − 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 2.00000 0.0953463
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 30.0000 1.42534 0.712672 − 0.701498i \(-0.247485\pi\)
0.712672 + 0.701498i \(0.247485\pi\)
\(444\) −2.00000 −0.0949158
\(445\) −4.00000 −0.189618
\(446\) −8.00000 −0.378811
\(447\) 6.00000 0.283790
\(448\) −2.00000 −0.0944911
\(449\) 16.0000 0.755087 0.377543 − 0.925992i \(-0.376769\pi\)
0.377543 + 0.925992i \(0.376769\pi\)
\(450\) 1.00000 0.0471405
\(451\) −2.00000 −0.0941763
\(452\) 20.0000 0.940721
\(453\) 12.0000 0.563809
\(454\) −4.00000 −0.187729
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) −2.00000 −0.0935561 −0.0467780 − 0.998905i \(-0.514895\pi\)
−0.0467780 + 0.998905i \(0.514895\pi\)
\(458\) −10.0000 −0.467269
\(459\) 1.00000 0.0466760
\(460\) −12.0000 −0.559503
\(461\) 2.00000 0.0931493 0.0465746 − 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 2.00000 0.0930484
\(463\) 32.0000 1.48717 0.743583 − 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 30.0000 1.38823 0.694117 − 0.719862i \(-0.255795\pi\)
0.694117 + 0.719862i \(0.255795\pi\)
\(468\) 0 0
\(469\) 24.0000 1.10822
\(470\) −8.00000 −0.369012
\(471\) −18.0000 −0.829396
\(472\) −6.00000 −0.276172
\(473\) 10.0000 0.459800
\(474\) −2.00000 −0.0918630
\(475\) 2.00000 0.0917663
\(476\) 2.00000 0.0916698
\(477\) 4.00000 0.183147
\(478\) −16.0000 −0.731823
\(479\) 16.0000 0.731059 0.365529 − 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 2.00000 0.0912871
\(481\) 0 0
\(482\) −24.0000 −1.09317
\(483\) −12.0000 −0.546019
\(484\) 1.00000 0.0454545
\(485\) −12.0000 −0.544892
\(486\) 1.00000 0.0453609
\(487\) 16.0000 0.725029 0.362515 − 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 10.0000 0.452679
\(489\) 12.0000 0.542659
\(490\) 6.00000 0.271052
\(491\) 40.0000 1.80517 0.902587 − 0.430507i \(-0.141665\pi\)
0.902587 + 0.430507i \(0.141665\pi\)
\(492\) −2.00000 −0.0901670
\(493\) −2.00000 −0.0900755
\(494\) 0 0
\(495\) −2.00000 −0.0898933
\(496\) 0 0
\(497\) −12.0000 −0.538274
\(498\) −12.0000 −0.537733
\(499\) 20.0000 0.895323 0.447661 − 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) −12.0000 −0.536656
\(501\) 8.00000 0.357414
\(502\) 6.00000 0.267793
\(503\) −20.0000 −0.891756 −0.445878 − 0.895094i \(-0.647108\pi\)
−0.445878 + 0.895094i \(0.647108\pi\)
\(504\) 2.00000 0.0890871
\(505\) −20.0000 −0.889988
\(506\) −6.00000 −0.266733
\(507\) 13.0000 0.577350
\(508\) −16.0000 −0.709885
\(509\) 8.00000 0.354594 0.177297 − 0.984157i \(-0.443265\pi\)
0.177297 + 0.984157i \(0.443265\pi\)
\(510\) −2.00000 −0.0885615
\(511\) 8.00000 0.353899
\(512\) −1.00000 −0.0441942
\(513\) 2.00000 0.0883022
\(514\) 6.00000 0.264649
\(515\) −16.0000 −0.705044
\(516\) 10.0000 0.440225
\(517\) −4.00000 −0.175920
\(518\) 4.00000 0.175750
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) −4.00000 −0.175243 −0.0876216 − 0.996154i \(-0.527927\pi\)
−0.0876216 + 0.996154i \(0.527927\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −38.0000 −1.66162 −0.830812 − 0.556553i \(-0.812124\pi\)
−0.830812 + 0.556553i \(0.812124\pi\)
\(524\) −4.00000 −0.174741
\(525\) −2.00000 −0.0872872
\(526\) −24.0000 −1.04645
\(527\) 0 0
\(528\) 1.00000 0.0435194
\(529\) 13.0000 0.565217
\(530\) −8.00000 −0.347498
\(531\) 6.00000 0.260378
\(532\) 4.00000 0.173422
\(533\) 0 0
\(534\) −2.00000 −0.0865485
\(535\) −24.0000 −1.03761
\(536\) 12.0000 0.518321
\(537\) −18.0000 −0.776757
\(538\) 6.00000 0.258678
\(539\) 3.00000 0.129219
\(540\) −2.00000 −0.0860663
\(541\) 30.0000 1.28980 0.644900 − 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 16.0000 0.687259
\(543\) −10.0000 −0.429141
\(544\) 1.00000 0.0428746
\(545\) −28.0000 −1.19939
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 − 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) −6.00000 −0.256307
\(549\) −10.0000 −0.426790
\(550\) −1.00000 −0.0426401
\(551\) −4.00000 −0.170406
\(552\) −6.00000 −0.255377
\(553\) 4.00000 0.170097
\(554\) −10.0000 −0.424859
\(555\) −4.00000 −0.169791
\(556\) 20.0000 0.848189
\(557\) −22.0000 −0.932170 −0.466085 − 0.884740i \(-0.654336\pi\)
−0.466085 + 0.884740i \(0.654336\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −4.00000 −0.169031
\(561\) −1.00000 −0.0422200
\(562\) 6.00000 0.253095
\(563\) −28.0000 −1.18006 −0.590030 − 0.807382i \(-0.700884\pi\)
−0.590030 + 0.807382i \(0.700884\pi\)
\(564\) −4.00000 −0.168430
\(565\) 40.0000 1.68281
\(566\) 4.00000 0.168133
\(567\) −2.00000 −0.0839921
\(568\) −6.00000 −0.251754
\(569\) 6.00000 0.251533 0.125767 − 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) −4.00000 −0.167542
\(571\) −28.0000 −1.17176 −0.585882 − 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 0 0
\(573\) −8.00000 −0.334205
\(574\) 4.00000 0.166957
\(575\) 6.00000 0.250217
\(576\) 1.00000 0.0416667
\(577\) −18.0000 −0.749350 −0.374675 − 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 0 0
\(580\) 4.00000 0.166091
\(581\) 24.0000 0.995688
\(582\) −6.00000 −0.248708
\(583\) −4.00000 −0.165663
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) −2.00000 −0.0826192
\(587\) −18.0000 −0.742940 −0.371470 − 0.928445i \(-0.621146\pi\)
−0.371470 + 0.928445i \(0.621146\pi\)
\(588\) 3.00000 0.123718
\(589\) 0 0
\(590\) −12.0000 −0.494032
\(591\) −2.00000 −0.0822690
\(592\) 2.00000 0.0821995
\(593\) 18.0000 0.739171 0.369586 − 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 4.00000 0.163984
\(596\) −6.00000 −0.245770
\(597\) −8.00000 −0.327418
\(598\) 0 0
\(599\) −40.0000 −1.63436 −0.817178 − 0.576386i \(-0.804463\pi\)
−0.817178 + 0.576386i \(0.804463\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 4.00000 0.163163 0.0815817 − 0.996667i \(-0.474003\pi\)
0.0815817 + 0.996667i \(0.474003\pi\)
\(602\) −20.0000 −0.815139
\(603\) −12.0000 −0.488678
\(604\) −12.0000 −0.488273
\(605\) 2.00000 0.0813116
\(606\) −10.0000 −0.406222
\(607\) 22.0000 0.892952 0.446476 − 0.894795i \(-0.352679\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(608\) 2.00000 0.0811107
\(609\) 4.00000 0.162088
\(610\) 20.0000 0.809776
\(611\) 0 0
\(612\) −1.00000 −0.0404226
\(613\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(614\) 6.00000 0.242140
\(615\) −4.00000 −0.161296
\(616\) −2.00000 −0.0805823
\(617\) 24.0000 0.966204 0.483102 − 0.875564i \(-0.339510\pi\)
0.483102 + 0.875564i \(0.339510\pi\)
\(618\) −8.00000 −0.321807
\(619\) −20.0000 −0.803868 −0.401934 − 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 6.00000 0.240772
\(622\) −6.00000 −0.240578
\(623\) 4.00000 0.160257
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 14.0000 0.559553
\(627\) −2.00000 −0.0798723
\(628\) 18.0000 0.718278
\(629\) −2.00000 −0.0797452
\(630\) 4.00000 0.159364
\(631\) 8.00000 0.318475 0.159237 − 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 2.00000 0.0795557
\(633\) −20.0000 −0.794929
\(634\) 10.0000 0.397151
\(635\) −32.0000 −1.26988
\(636\) −4.00000 −0.158610
\(637\) 0 0
\(638\) 2.00000 0.0791808
\(639\) 6.00000 0.237356
\(640\) −2.00000 −0.0790569
\(641\) −20.0000 −0.789953 −0.394976 − 0.918691i \(-0.629247\pi\)
−0.394976 + 0.918691i \(0.629247\pi\)
\(642\) −12.0000 −0.473602
\(643\) −20.0000 −0.788723 −0.394362 − 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) 12.0000 0.472866
\(645\) 20.0000 0.787499
\(646\) −2.00000 −0.0786889
\(647\) −20.0000 −0.786281 −0.393141 − 0.919478i \(-0.628611\pi\)
−0.393141 + 0.919478i \(0.628611\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −6.00000 −0.235521
\(650\) 0 0
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) −46.0000 −1.80012 −0.900060 − 0.435767i \(-0.856477\pi\)
−0.900060 + 0.435767i \(0.856477\pi\)
\(654\) −14.0000 −0.547443
\(655\) −8.00000 −0.312586
\(656\) 2.00000 0.0780869
\(657\) −4.00000 −0.156055
\(658\) 8.00000 0.311872
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 2.00000 0.0778499
\(661\) −14.0000 −0.544537 −0.272268 − 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) 4.00000 0.155464
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) 8.00000 0.310227
\(666\) −2.00000 −0.0774984
\(667\) −12.0000 −0.464642
\(668\) −8.00000 −0.309529
\(669\) −8.00000 −0.309298
\(670\) 24.0000 0.927201
\(671\) 10.0000 0.386046
\(672\) −2.00000 −0.0771517
\(673\) 16.0000 0.616755 0.308377 − 0.951264i \(-0.400214\pi\)
0.308377 + 0.951264i \(0.400214\pi\)
\(674\) −12.0000 −0.462223
\(675\) 1.00000 0.0384900
\(676\) −13.0000 −0.500000
\(677\) 2.00000 0.0768662 0.0384331 − 0.999261i \(-0.487763\pi\)
0.0384331 + 0.999261i \(0.487763\pi\)
\(678\) 20.0000 0.768095
\(679\) 12.0000 0.460518
\(680\) 2.00000 0.0766965
\(681\) −4.00000 −0.153280
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 − 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) −2.00000 −0.0764719
\(685\) −12.0000 −0.458496
\(686\) −20.0000 −0.763604
\(687\) −10.0000 −0.381524
\(688\) −10.0000 −0.381246
\(689\) 0 0
\(690\) −12.0000 −0.456832
\(691\) −20.0000 −0.760836 −0.380418 − 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 14.0000 0.532200
\(693\) 2.00000 0.0759737
\(694\) 4.00000 0.151838
\(695\) 40.0000 1.51729
\(696\) 2.00000 0.0758098
\(697\) −2.00000 −0.0757554
\(698\) 0 0
\(699\) 6.00000 0.226941
\(700\) 2.00000 0.0755929
\(701\) −6.00000 −0.226617 −0.113308 − 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 0 0
\(703\) −4.00000 −0.150863
\(704\) −1.00000 −0.0376889
\(705\) −8.00000 −0.301297
\(706\) −2.00000 −0.0752710
\(707\) 20.0000 0.752177
\(708\) −6.00000 −0.225494
\(709\) 14.0000 0.525781 0.262891 − 0.964826i \(-0.415324\pi\)
0.262891 + 0.964826i \(0.415324\pi\)
\(710\) −12.0000 −0.450352
\(711\) −2.00000 −0.0750059
\(712\) 2.00000 0.0749532
\(713\) 0 0
\(714\) 2.00000 0.0748481
\(715\) 0 0
\(716\) 18.0000 0.672692
\(717\) −16.0000 −0.597531
\(718\) 0 0
\(719\) 18.0000 0.671287 0.335643 − 0.941989i \(-0.391046\pi\)
0.335643 + 0.941989i \(0.391046\pi\)
\(720\) 2.00000 0.0745356
\(721\) 16.0000 0.595871
\(722\) 15.0000 0.558242
\(723\) −24.0000 −0.892570
\(724\) 10.0000 0.371647
\(725\) −2.00000 −0.0742781
\(726\) 1.00000 0.0371135
\(727\) −32.0000 −1.18681 −0.593407 − 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 8.00000 0.296093
\(731\) 10.0000 0.369863
\(732\) 10.0000 0.369611
\(733\) 24.0000 0.886460 0.443230 − 0.896408i \(-0.353832\pi\)
0.443230 + 0.896408i \(0.353832\pi\)
\(734\) 8.00000 0.295285
\(735\) 6.00000 0.221313
\(736\) 6.00000 0.221163
\(737\) 12.0000 0.442026
\(738\) −2.00000 −0.0736210
\(739\) 6.00000 0.220714 0.110357 − 0.993892i \(-0.464801\pi\)
0.110357 + 0.993892i \(0.464801\pi\)
\(740\) 4.00000 0.147043
\(741\) 0 0
\(742\) 8.00000 0.293689
\(743\) −28.0000 −1.02722 −0.513610 − 0.858024i \(-0.671692\pi\)
−0.513610 + 0.858024i \(0.671692\pi\)
\(744\) 0 0
\(745\) −12.0000 −0.439646
\(746\) 8.00000 0.292901
\(747\) −12.0000 −0.439057
\(748\) 1.00000 0.0365636
\(749\) 24.0000 0.876941
\(750\) −12.0000 −0.438178
\(751\) −28.0000 −1.02173 −0.510867 − 0.859660i \(-0.670676\pi\)
−0.510867 + 0.859660i \(0.670676\pi\)
\(752\) 4.00000 0.145865
\(753\) 6.00000 0.218652
\(754\) 0 0
\(755\) −24.0000 −0.873449
\(756\) 2.00000 0.0727393
\(757\) 14.0000 0.508839 0.254419 − 0.967094i \(-0.418116\pi\)
0.254419 + 0.967094i \(0.418116\pi\)
\(758\) −20.0000 −0.726433
\(759\) −6.00000 −0.217786
\(760\) 4.00000 0.145095
\(761\) 10.0000 0.362500 0.181250 − 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) −16.0000 −0.579619
\(763\) 28.0000 1.01367
\(764\) 8.00000 0.289430
\(765\) −2.00000 −0.0723102
\(766\) 32.0000 1.15621
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −22.0000 −0.793340 −0.396670 − 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) −4.00000 −0.144150
\(771\) 6.00000 0.216085
\(772\) 0 0
\(773\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 10.0000 0.359443
\(775\) 0 0
\(776\) 6.00000 0.215387
\(777\) 4.00000 0.143499
\(778\) −16.0000 −0.573628
\(779\) −4.00000 −0.143315
\(780\) 0 0
\(781\) −6.00000 −0.214697
\(782\) −6.00000 −0.214560
\(783\) −2.00000 −0.0714742
\(784\) −3.00000 −0.107143
\(785\) 36.0000 1.28490
\(786\) −4.00000 −0.142675
\(787\) 48.0000 1.71102 0.855508 − 0.517790i \(-0.173245\pi\)
0.855508 + 0.517790i \(0.173245\pi\)
\(788\) 2.00000 0.0712470
\(789\) −24.0000 −0.854423
\(790\) 4.00000 0.142314
\(791\) −40.0000 −1.42224
\(792\) 1.00000 0.0355335
\(793\) 0 0
\(794\) −14.0000 −0.496841
\(795\) −8.00000 −0.283731
\(796\) 8.00000 0.283552
\(797\) −44.0000 −1.55856 −0.779280 − 0.626676i \(-0.784415\pi\)
−0.779280 + 0.626676i \(0.784415\pi\)
\(798\) 4.00000 0.141598
\(799\) −4.00000 −0.141510
\(800\) 1.00000 0.0353553
\(801\) −2.00000 −0.0706665
\(802\) 12.0000 0.423735
\(803\) 4.00000 0.141157
\(804\) 12.0000 0.423207
\(805\) 24.0000 0.845889
\(806\) 0 0
\(807\) 6.00000 0.211210
\(808\) 10.0000 0.351799
\(809\) 22.0000 0.773479 0.386739 − 0.922189i \(-0.373601\pi\)
0.386739 + 0.922189i \(0.373601\pi\)
\(810\) −2.00000 −0.0702728
\(811\) −24.0000 −0.842754 −0.421377 − 0.906886i \(-0.638453\pi\)
−0.421377 + 0.906886i \(0.638453\pi\)
\(812\) −4.00000 −0.140372
\(813\) 16.0000 0.561144
\(814\) 2.00000 0.0701000
\(815\) −24.0000 −0.840683
\(816\) 1.00000 0.0350070
\(817\) 20.0000 0.699711
\(818\) −22.0000 −0.769212
\(819\) 0 0
\(820\) 4.00000 0.139686
\(821\) −42.0000 −1.46581 −0.732905 − 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) −6.00000 −0.209274
\(823\) 12.0000 0.418294 0.209147 − 0.977884i \(-0.432931\pi\)
0.209147 + 0.977884i \(0.432931\pi\)
\(824\) 8.00000 0.278693
\(825\) −1.00000 −0.0348155
\(826\) 12.0000 0.417533
\(827\) 52.0000 1.80822 0.904109 − 0.427303i \(-0.140536\pi\)
0.904109 + 0.427303i \(0.140536\pi\)
\(828\) −6.00000 −0.208514
\(829\) 6.00000 0.208389 0.104194 − 0.994557i \(-0.466774\pi\)
0.104194 + 0.994557i \(0.466774\pi\)
\(830\) 24.0000 0.833052
\(831\) −10.0000 −0.346896
\(832\) 0 0
\(833\) 3.00000 0.103944
\(834\) 20.0000 0.692543
\(835\) −16.0000 −0.553703
\(836\) 2.00000 0.0691714
\(837\) 0 0
\(838\) −12.0000 −0.414533
\(839\) 54.0000 1.86429 0.932144 − 0.362089i \(-0.117936\pi\)
0.932144 + 0.362089i \(0.117936\pi\)
\(840\) −4.00000 −0.138013
\(841\) −25.0000 −0.862069
\(842\) 26.0000 0.896019
\(843\) 6.00000 0.206651
\(844\) 20.0000 0.688428
\(845\) −26.0000 −0.894427
\(846\) −4.00000 −0.137523
\(847\) −2.00000 −0.0687208
\(848\) 4.00000 0.137361
\(849\) 4.00000 0.137280
\(850\) −1.00000 −0.0342997
\(851\) −12.0000 −0.411355
\(852\) −6.00000 −0.205557
\(853\) 50.0000 1.71197 0.855984 − 0.517003i \(-0.172952\pi\)
0.855984 + 0.517003i \(0.172952\pi\)
\(854\) −20.0000 −0.684386
\(855\) −4.00000 −0.136797
\(856\) 12.0000 0.410152
\(857\) −34.0000 −1.16142 −0.580709 − 0.814111i \(-0.697225\pi\)
−0.580709 + 0.814111i \(0.697225\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 − 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) −20.0000 −0.681994
\(861\) 4.00000 0.136320
\(862\) 28.0000 0.953684
\(863\) −24.0000 −0.816970 −0.408485 − 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 1.00000 0.0340207
\(865\) 28.0000 0.952029
\(866\) 18.0000 0.611665
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) 2.00000 0.0678454
\(870\) 4.00000 0.135613
\(871\) 0 0
\(872\) 14.0000 0.474100
\(873\) −6.00000 −0.203069
\(874\) −12.0000 −0.405906
\(875\) 24.0000 0.811348
\(876\) 4.00000 0.135147
\(877\) 18.0000 0.607817 0.303908 − 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) 10.0000 0.337484
\(879\) −2.00000 −0.0674583
\(880\) −2.00000 −0.0674200
\(881\) −40.0000 −1.34763 −0.673817 − 0.738898i \(-0.735346\pi\)
−0.673817 + 0.738898i \(0.735346\pi\)
\(882\) 3.00000 0.101015
\(883\) 32.0000 1.07689 0.538443 − 0.842662i \(-0.319013\pi\)
0.538443 + 0.842662i \(0.319013\pi\)
\(884\) 0 0
\(885\) −12.0000 −0.403376
\(886\) −30.0000 −1.00787
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 2.00000 0.0671156
\(889\) 32.0000 1.07325
\(890\) 4.00000 0.134080
\(891\) −1.00000 −0.0335013
\(892\) 8.00000 0.267860
\(893\) −8.00000 −0.267710
\(894\) −6.00000 −0.200670
\(895\) 36.0000 1.20335
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) −16.0000 −0.533927
\(899\) 0 0
\(900\) −1.00000 −0.0333333
\(901\) −4.00000 −0.133259
\(902\) 2.00000 0.0665927
\(903\) −20.0000 −0.665558
\(904\) −20.0000 −0.665190
\(905\) 20.0000 0.664822
\(906\) −12.0000 −0.398673
\(907\) 12.0000 0.398453 0.199227 − 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) 4.00000 0.132745
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) 38.0000 1.25900 0.629498 − 0.777002i \(-0.283261\pi\)
0.629498 + 0.777002i \(0.283261\pi\)
\(912\) 2.00000 0.0662266
\(913\) 12.0000 0.397142
\(914\) 2.00000 0.0661541
\(915\) 20.0000 0.661180
\(916\) 10.0000 0.330409
\(917\) 8.00000 0.264183
\(918\) −1.00000 −0.0330049
\(919\) −20.0000 −0.659739 −0.329870 − 0.944027i \(-0.607005\pi\)
−0.329870 + 0.944027i \(0.607005\pi\)
\(920\) 12.0000 0.395628
\(921\) 6.00000 0.197707
\(922\) −2.00000 −0.0658665
\(923\) 0 0
\(924\) −2.00000 −0.0657952
\(925\) −2.00000 −0.0657596
\(926\) −32.0000 −1.05159
\(927\) −8.00000 −0.262754
\(928\) −2.00000 −0.0656532
\(929\) 32.0000 1.04989 0.524943 − 0.851137i \(-0.324087\pi\)
0.524943 + 0.851137i \(0.324087\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) −6.00000 −0.196537
\(933\) −6.00000 −0.196431
\(934\) −30.0000 −0.981630
\(935\) 2.00000 0.0654070
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 − 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) −24.0000 −0.783628
\(939\) 14.0000 0.456873
\(940\) 8.00000 0.260931
\(941\) −18.0000 −0.586783 −0.293392 − 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) 18.0000 0.586472
\(943\) −12.0000 −0.390774
\(944\) 6.00000 0.195283
\(945\) 4.00000 0.130120
\(946\) −10.0000 −0.325128
\(947\) −24.0000 −0.779895 −0.389948 − 0.920837i \(-0.627507\pi\)
−0.389948 + 0.920837i \(0.627507\pi\)
\(948\) 2.00000 0.0649570
\(949\) 0 0
\(950\) −2.00000 −0.0648886
\(951\) 10.0000 0.324272
\(952\) −2.00000 −0.0648204
\(953\) −38.0000 −1.23094 −0.615470 − 0.788160i \(-0.711034\pi\)
−0.615470 + 0.788160i \(0.711034\pi\)
\(954\) −4.00000 −0.129505
\(955\) 16.0000 0.517748
\(956\) 16.0000 0.517477
\(957\) 2.00000 0.0646508
\(958\) −16.0000 −0.516937
\(959\) 12.0000 0.387500
\(960\) −2.00000 −0.0645497
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) 24.0000 0.772988
\(965\) 0 0
\(966\) 12.0000 0.386094
\(967\) −8.00000 −0.257263 −0.128631 − 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) −1.00000 −0.0321412
\(969\) −2.00000 −0.0642493
\(970\) 12.0000 0.385297
\(971\) 14.0000 0.449281 0.224641 − 0.974442i \(-0.427879\pi\)
0.224641 + 0.974442i \(0.427879\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −40.0000 −1.28234
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) −22.0000 −0.703842 −0.351921 − 0.936030i \(-0.614471\pi\)
−0.351921 + 0.936030i \(0.614471\pi\)
\(978\) −12.0000 −0.383718
\(979\) 2.00000 0.0639203
\(980\) −6.00000 −0.191663
\(981\) −14.0000 −0.446986
\(982\) −40.0000 −1.27645
\(983\) −14.0000 −0.446531 −0.223265 − 0.974758i \(-0.571672\pi\)
−0.223265 + 0.974758i \(0.571672\pi\)
\(984\) 2.00000 0.0637577
\(985\) 4.00000 0.127451
\(986\) 2.00000 0.0636930
\(987\) 8.00000 0.254643
\(988\) 0 0
\(989\) 60.0000 1.90789
\(990\) 2.00000 0.0635642
\(991\) −4.00000 −0.127064 −0.0635321 − 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) 0 0
\(993\) 4.00000 0.126936
\(994\) 12.0000 0.380617
\(995\) 16.0000 0.507234
\(996\) 12.0000 0.380235
\(997\) 46.0000 1.45683 0.728417 − 0.685134i \(-0.240256\pi\)
0.728417 + 0.685134i \(0.240256\pi\)
\(998\) −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 1122.2.a.b.1.1 ✓ 1
3.2 odd 2 3366.2.a.l.1.1 1
4.3 odd 2 8976.2.a.bc.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1122.2.a.b.1.1 ✓ 1 1.1 even 1 trivial
3366.2.a.l.1.1 1 3.2 odd 2
8976.2.a.bc.1.1 1 4.3 odd 2