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Properties

Label 2550.2.a.f.1.1
Level $2550$
Weight $2$
Character 2550.1
Self dual yes
Analytic conductor $20.362$
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] = [2550,2,Mod(1,2550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2550, 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("2550.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2550.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: \(20.3618525154\)
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 \(\chi\) \(=\) 2550.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -5.00000 q^{11} -1.00000 q^{12} +4.00000 q^{13} -3.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} -3.00000 q^{21} +5.00000 q^{22} -4.00000 q^{23} +1.00000 q^{24} -4.00000 q^{26} -1.00000 q^{27} +3.00000 q^{28} -4.00000 q^{29} -1.00000 q^{31} -1.00000 q^{32} +5.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} -9.00000 q^{37} +1.00000 q^{38} -4.00000 q^{39} -10.0000 q^{41} +3.00000 q^{42} +11.0000 q^{43} -5.00000 q^{44} +4.00000 q^{46} +9.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} +1.00000 q^{51} +4.00000 q^{52} +3.00000 q^{53} +1.00000 q^{54} -3.00000 q^{56} +1.00000 q^{57} +4.00000 q^{58} -8.00000 q^{59} -14.0000 q^{61} +1.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} -5.00000 q^{66} -7.00000 q^{67} -1.00000 q^{68} +4.00000 q^{69} +14.0000 q^{71} -1.00000 q^{72} +2.00000 q^{73} +9.00000 q^{74} -1.00000 q^{76} -15.0000 q^{77} +4.00000 q^{78} -5.00000 q^{79} +1.00000 q^{81} +10.0000 q^{82} +8.00000 q^{83} -3.00000 q^{84} -11.0000 q^{86} +4.00000 q^{87} +5.00000 q^{88} -2.00000 q^{89} +12.0000 q^{91} -4.00000 q^{92} +1.00000 q^{93} -9.00000 q^{94} +1.00000 q^{96} -14.0000 q^{97} -2.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)

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
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 3.00000 1.13389 0.566947 − 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 − 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) −1.00000 −0.288675
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) −3.00000 −0.801784
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536
\(18\) −1.00000 −0.235702
\(19\) −1.00000 −0.229416 −0.114708 − 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 5.00000 1.06600
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) −1.00000 −0.192450
\(28\) 3.00000 0.566947
\(29\) −4.00000 −0.742781 −0.371391 − 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605 −0.0898027 − 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) −1.00000 −0.176777
\(33\) 5.00000 0.870388
\(34\) 1.00000 0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −9.00000 −1.47959 −0.739795 − 0.672832i \(-0.765078\pi\)
−0.739795 + 0.672832i \(0.765078\pi\)
\(38\) 1.00000 0.162221
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 3.00000 0.462910
\(43\) 11.0000 1.67748 0.838742 − 0.544529i \(-0.183292\pi\)
0.838742 + 0.544529i \(0.183292\pi\)
\(44\) −5.00000 −0.753778
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 9.00000 1.31278 0.656392 − 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) −1.00000 −0.144338
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 1.00000 0.140028
\(52\) 4.00000 0.554700
\(53\) 3.00000 0.412082 0.206041 − 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −3.00000 −0.400892
\(57\) 1.00000 0.132453
\(58\) 4.00000 0.525226
\(59\) −8.00000 −1.04151 −0.520756 − 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 − 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 1.00000 0.127000
\(63\) 3.00000 0.377964
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −5.00000 −0.615457
\(67\) −7.00000 −0.855186 −0.427593 − 0.903971i \(-0.640638\pi\)
−0.427593 + 0.903971i \(0.640638\pi\)
\(68\) −1.00000 −0.121268
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 14.0000 1.66149 0.830747 − 0.556650i \(-0.187914\pi\)
0.830747 + 0.556650i \(0.187914\pi\)
\(72\) −1.00000 −0.117851
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 9.00000 1.04623
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) −15.0000 −1.70941
\(78\) 4.00000 0.452911
\(79\) −5.00000 −0.562544 −0.281272 − 0.959628i \(-0.590756\pi\)
−0.281272 + 0.959628i \(0.590756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) 8.00000 0.878114 0.439057 − 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) −3.00000 −0.327327
\(85\) 0 0
\(86\) −11.0000 −1.18616
\(87\) 4.00000 0.428845
\(88\) 5.00000 0.533002
\(89\) −2.00000 −0.212000 −0.106000 − 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 12.0000 1.25794
\(92\) −4.00000 −0.417029
\(93\) 1.00000 0.103695
\(94\) −9.00000 −0.928279
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −14.0000 −1.42148 −0.710742 − 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) −2.00000 −0.202031
\(99\) −5.00000 −0.502519
\(100\) 0 0
\(101\) 9.00000 0.895533 0.447767 − 0.894150i \(-0.352219\pi\)
0.447767 + 0.894150i \(0.352219\pi\)
\(102\) −1.00000 −0.0990148
\(103\) 14.0000 1.37946 0.689730 − 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) −3.00000 −0.291386
\(107\) −15.0000 −1.45010 −0.725052 − 0.688694i \(-0.758184\pi\)
−0.725052 + 0.688694i \(0.758184\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 1.00000 0.0957826 0.0478913 − 0.998853i \(-0.484750\pi\)
0.0478913 + 0.998853i \(0.484750\pi\)
\(110\) 0 0
\(111\) 9.00000 0.854242
\(112\) 3.00000 0.283473
\(113\) −3.00000 −0.282216 −0.141108 − 0.989994i \(-0.545067\pi\)
−0.141108 + 0.989994i \(0.545067\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) −4.00000 −0.371391
\(117\) 4.00000 0.369800
\(118\) 8.00000 0.736460
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 14.0000 1.26750
\(123\) 10.0000 0.901670
\(124\) −1.00000 −0.0898027
\(125\) 0 0
\(126\) −3.00000 −0.267261
\(127\) −2.00000 −0.177471 −0.0887357 − 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −11.0000 −0.968496
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 − 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 5.00000 0.435194
\(133\) −3.00000 −0.260133
\(134\) 7.00000 0.604708
\(135\) 0 0
\(136\) 1.00000 0.0857493
\(137\) −6.00000 −0.512615 −0.256307 − 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) −4.00000 −0.340503
\(139\) 8.00000 0.678551 0.339276 − 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) −9.00000 −0.757937
\(142\) −14.0000 −1.17485
\(143\) −20.0000 −1.67248
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) −2.00000 −0.164957
\(148\) −9.00000 −0.739795
\(149\) 14.0000 1.14692 0.573462 − 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 0 0
\(151\) −24.0000 −1.95309 −0.976546 − 0.215308i \(-0.930924\pi\)
−0.976546 + 0.215308i \(0.930924\pi\)
\(152\) 1.00000 0.0811107
\(153\) −1.00000 −0.0808452
\(154\) 15.0000 1.20873
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) −24.0000 −1.91541 −0.957704 − 0.287754i \(-0.907091\pi\)
−0.957704 + 0.287754i \(0.907091\pi\)
\(158\) 5.00000 0.397779
\(159\) −3.00000 −0.237915
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) −1.00000 −0.0785674
\(163\) −18.0000 −1.40987 −0.704934 − 0.709273i \(-0.749024\pi\)
−0.704934 + 0.709273i \(0.749024\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) −8.00000 −0.620920
\(167\) 18.0000 1.39288 0.696441 − 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) 3.00000 0.231455
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 11.0000 0.838742
\(173\) −14.0000 −1.06440 −0.532200 − 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) −4.00000 −0.303239
\(175\) 0 0
\(176\) −5.00000 −0.376889
\(177\) 8.00000 0.601317
\(178\) 2.00000 0.149906
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −5.00000 −0.371647 −0.185824 − 0.982583i \(-0.559495\pi\)
−0.185824 + 0.982583i \(0.559495\pi\)
\(182\) −12.0000 −0.889499
\(183\) 14.0000 1.03491
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) −1.00000 −0.0733236
\(187\) 5.00000 0.365636
\(188\) 9.00000 0.656392
\(189\) −3.00000 −0.218218
\(190\) 0 0
\(191\) −9.00000 −0.651217 −0.325609 − 0.945505i \(-0.605569\pi\)
−0.325609 + 0.945505i \(0.605569\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 6.00000 0.431889 0.215945 − 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) −12.0000 −0.854965 −0.427482 − 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 5.00000 0.355335
\(199\) −1.00000 −0.0708881 −0.0354441 − 0.999372i \(-0.511285\pi\)
−0.0354441 + 0.999372i \(0.511285\pi\)
\(200\) 0 0
\(201\) 7.00000 0.493742
\(202\) −9.00000 −0.633238
\(203\) −12.0000 −0.842235
\(204\) 1.00000 0.0700140
\(205\) 0 0
\(206\) −14.0000 −0.975426
\(207\) −4.00000 −0.278019
\(208\) 4.00000 0.277350
\(209\) 5.00000 0.345857
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 − 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 3.00000 0.206041
\(213\) −14.0000 −0.959264
\(214\) 15.0000 1.02538
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −3.00000 −0.203653
\(218\) −1.00000 −0.0677285
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) −9.00000 −0.604040
\(223\) 6.00000 0.401790 0.200895 − 0.979613i \(-0.435615\pi\)
0.200895 + 0.979613i \(0.435615\pi\)
\(224\) −3.00000 −0.200446
\(225\) 0 0
\(226\) 3.00000 0.199557
\(227\) −21.0000 −1.39382 −0.696909 − 0.717159i \(-0.745442\pi\)
−0.696909 + 0.717159i \(0.745442\pi\)
\(228\) 1.00000 0.0662266
\(229\) −12.0000 −0.792982 −0.396491 − 0.918039i \(-0.629772\pi\)
−0.396491 + 0.918039i \(0.629772\pi\)
\(230\) 0 0
\(231\) 15.0000 0.986928
\(232\) 4.00000 0.262613
\(233\) −6.00000 −0.393073 −0.196537 − 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) −8.00000 −0.520756
\(237\) 5.00000 0.324785
\(238\) 3.00000 0.194461
\(239\) 5.00000 0.323423 0.161712 − 0.986838i \(-0.448299\pi\)
0.161712 + 0.986838i \(0.448299\pi\)
\(240\) 0 0
\(241\) −8.00000 −0.515325 −0.257663 − 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) −14.0000 −0.899954
\(243\) −1.00000 −0.0641500
\(244\) −14.0000 −0.896258
\(245\) 0 0
\(246\) −10.0000 −0.637577
\(247\) −4.00000 −0.254514
\(248\) 1.00000 0.0635001
\(249\) −8.00000 −0.506979
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 − 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 3.00000 0.188982
\(253\) 20.0000 1.25739
\(254\) 2.00000 0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −12.0000 −0.748539 −0.374270 − 0.927320i \(-0.622107\pi\)
−0.374270 + 0.927320i \(0.622107\pi\)
\(258\) 11.0000 0.684830
\(259\) −27.0000 −1.67770
\(260\) 0 0
\(261\) −4.00000 −0.247594
\(262\) −4.00000 −0.247121
\(263\) −7.00000 −0.431638 −0.215819 − 0.976433i \(-0.569242\pi\)
−0.215819 + 0.976433i \(0.569242\pi\)
\(264\) −5.00000 −0.307729
\(265\) 0 0
\(266\) 3.00000 0.183942
\(267\) 2.00000 0.122398
\(268\) −7.00000 −0.427593
\(269\) 28.0000 1.70719 0.853595 − 0.520937i \(-0.174417\pi\)
0.853595 + 0.520937i \(0.174417\pi\)
\(270\) 0 0
\(271\) 4.00000 0.242983 0.121491 − 0.992592i \(-0.461232\pi\)
0.121491 + 0.992592i \(0.461232\pi\)
\(272\) −1.00000 −0.0606339
\(273\) −12.0000 −0.726273
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) −13.0000 −0.781094 −0.390547 − 0.920583i \(-0.627714\pi\)
−0.390547 + 0.920583i \(0.627714\pi\)
\(278\) −8.00000 −0.479808
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 − 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 9.00000 0.535942
\(283\) 22.0000 1.30776 0.653882 − 0.756596i \(-0.273139\pi\)
0.653882 + 0.756596i \(0.273139\pi\)
\(284\) 14.0000 0.830747
\(285\) 0 0
\(286\) 20.0000 1.18262
\(287\) −30.0000 −1.77084
\(288\) −1.00000 −0.0589256
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) 2.00000 0.117041
\(293\) −18.0000 −1.05157 −0.525786 − 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 2.00000 0.116642
\(295\) 0 0
\(296\) 9.00000 0.523114
\(297\) 5.00000 0.290129
\(298\) −14.0000 −0.810998
\(299\) −16.0000 −0.925304
\(300\) 0 0
\(301\) 33.0000 1.90209
\(302\) 24.0000 1.38104
\(303\) −9.00000 −0.517036
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 1.00000 0.0571662
\(307\) 8.00000 0.456584 0.228292 − 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) −15.0000 −0.854704
\(309\) −14.0000 −0.796432
\(310\) 0 0
\(311\) 10.0000 0.567048 0.283524 − 0.958965i \(-0.408496\pi\)
0.283524 + 0.958965i \(0.408496\pi\)
\(312\) 4.00000 0.226455
\(313\) −16.0000 −0.904373 −0.452187 − 0.891923i \(-0.649356\pi\)
−0.452187 + 0.891923i \(0.649356\pi\)
\(314\) 24.0000 1.35440
\(315\) 0 0
\(316\) −5.00000 −0.281272
\(317\) 4.00000 0.224662 0.112331 − 0.993671i \(-0.464168\pi\)
0.112331 + 0.993671i \(0.464168\pi\)
\(318\) 3.00000 0.168232
\(319\) 20.0000 1.11979
\(320\) 0 0
\(321\) 15.0000 0.837218
\(322\) 12.0000 0.668734
\(323\) 1.00000 0.0556415
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 18.0000 0.996928
\(327\) −1.00000 −0.0553001
\(328\) 10.0000 0.552158
\(329\) 27.0000 1.48856
\(330\) 0 0
\(331\) 5.00000 0.274825 0.137412 − 0.990514i \(-0.456121\pi\)
0.137412 + 0.990514i \(0.456121\pi\)
\(332\) 8.00000 0.439057
\(333\) −9.00000 −0.493197
\(334\) −18.0000 −0.984916
\(335\) 0 0
\(336\) −3.00000 −0.163663
\(337\) 24.0000 1.30736 0.653682 − 0.756770i \(-0.273224\pi\)
0.653682 + 0.756770i \(0.273224\pi\)
\(338\) −3.00000 −0.163178
\(339\) 3.00000 0.162938
\(340\) 0 0
\(341\) 5.00000 0.270765
\(342\) 1.00000 0.0540738
\(343\) −15.0000 −0.809924
\(344\) −11.0000 −0.593080
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) 11.0000 0.590511 0.295255 − 0.955418i \(-0.404595\pi\)
0.295255 + 0.955418i \(0.404595\pi\)
\(348\) 4.00000 0.214423
\(349\) −10.0000 −0.535288 −0.267644 − 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 5.00000 0.266501
\(353\) 24.0000 1.27739 0.638696 − 0.769460i \(-0.279474\pi\)
0.638696 + 0.769460i \(0.279474\pi\)
\(354\) −8.00000 −0.425195
\(355\) 0 0
\(356\) −2.00000 −0.106000
\(357\) 3.00000 0.158777
\(358\) 12.0000 0.634220
\(359\) −25.0000 −1.31945 −0.659725 − 0.751507i \(-0.729327\pi\)
−0.659725 + 0.751507i \(0.729327\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 5.00000 0.262794
\(363\) −14.0000 −0.734809
\(364\) 12.0000 0.628971
\(365\) 0 0
\(366\) −14.0000 −0.731792
\(367\) 25.0000 1.30499 0.652495 − 0.757793i \(-0.273722\pi\)
0.652495 + 0.757793i \(0.273722\pi\)
\(368\) −4.00000 −0.208514
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) 9.00000 0.467257
\(372\) 1.00000 0.0518476
\(373\) 6.00000 0.310668 0.155334 − 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) −5.00000 −0.258544
\(375\) 0 0
\(376\) −9.00000 −0.464140
\(377\) −16.0000 −0.824042
\(378\) 3.00000 0.154303
\(379\) −4.00000 −0.205466 −0.102733 − 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) 9.00000 0.460480
\(383\) −12.0000 −0.613171 −0.306586 − 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −6.00000 −0.305392
\(387\) 11.0000 0.559161
\(388\) −14.0000 −0.710742
\(389\) 21.0000 1.06474 0.532371 − 0.846511i \(-0.321301\pi\)
0.532371 + 0.846511i \(0.321301\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) −2.00000 −0.101015
\(393\) −4.00000 −0.201773
\(394\) 12.0000 0.604551
\(395\) 0 0
\(396\) −5.00000 −0.251259
\(397\) 15.0000 0.752828 0.376414 − 0.926451i \(-0.377157\pi\)
0.376414 + 0.926451i \(0.377157\pi\)
\(398\) 1.00000 0.0501255
\(399\) 3.00000 0.150188
\(400\) 0 0
\(401\) −30.0000 −1.49813 −0.749064 − 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) −7.00000 −0.349128
\(403\) −4.00000 −0.199254
\(404\) 9.00000 0.447767
\(405\) 0 0
\(406\) 12.0000 0.595550
\(407\) 45.0000 2.23057
\(408\) −1.00000 −0.0495074
\(409\) −26.0000 −1.28562 −0.642809 − 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) 14.0000 0.689730
\(413\) −24.0000 −1.18096
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) −4.00000 −0.196116
\(417\) −8.00000 −0.391762
\(418\) −5.00000 −0.244558
\(419\) 28.0000 1.36789 0.683945 − 0.729534i \(-0.260263\pi\)
0.683945 + 0.729534i \(0.260263\pi\)
\(420\) 0 0
\(421\) 24.0000 1.16969 0.584844 − 0.811146i \(-0.301156\pi\)
0.584844 + 0.811146i \(0.301156\pi\)
\(422\) 12.0000 0.584151
\(423\) 9.00000 0.437595
\(424\) −3.00000 −0.145693
\(425\) 0 0
\(426\) 14.0000 0.678302
\(427\) −42.0000 −2.03252
\(428\) −15.0000 −0.725052
\(429\) 20.0000 0.965609
\(430\) 0 0
\(431\) 14.0000 0.674356 0.337178 − 0.941441i \(-0.390528\pi\)
0.337178 + 0.941441i \(0.390528\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 27.0000 1.29754 0.648769 − 0.760986i \(-0.275284\pi\)
0.648769 + 0.760986i \(0.275284\pi\)
\(434\) 3.00000 0.144005
\(435\) 0 0
\(436\) 1.00000 0.0478913
\(437\) 4.00000 0.191346
\(438\) 2.00000 0.0955637
\(439\) 20.0000 0.954548 0.477274 − 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 4.00000 0.190261
\(443\) −38.0000 −1.80543 −0.902717 − 0.430234i \(-0.858431\pi\)
−0.902717 + 0.430234i \(0.858431\pi\)
\(444\) 9.00000 0.427121
\(445\) 0 0
\(446\) −6.00000 −0.284108
\(447\) −14.0000 −0.662177
\(448\) 3.00000 0.141737
\(449\) −41.0000 −1.93491 −0.967455 − 0.253044i \(-0.918568\pi\)
−0.967455 + 0.253044i \(0.918568\pi\)
\(450\) 0 0
\(451\) 50.0000 2.35441
\(452\) −3.00000 −0.141108
\(453\) 24.0000 1.12762
\(454\) 21.0000 0.985579
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) 37.0000 1.73079 0.865393 − 0.501093i \(-0.167069\pi\)
0.865393 + 0.501093i \(0.167069\pi\)
\(458\) 12.0000 0.560723
\(459\) 1.00000 0.0466760
\(460\) 0 0
\(461\) −5.00000 −0.232873 −0.116437 − 0.993198i \(-0.537147\pi\)
−0.116437 + 0.993198i \(0.537147\pi\)
\(462\) −15.0000 −0.697863
\(463\) 36.0000 1.67306 0.836531 − 0.547920i \(-0.184580\pi\)
0.836531 + 0.547920i \(0.184580\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) −6.00000 −0.277647 −0.138823 − 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) 4.00000 0.184900
\(469\) −21.0000 −0.969690
\(470\) 0 0
\(471\) 24.0000 1.10586
\(472\) 8.00000 0.368230
\(473\) −55.0000 −2.52890
\(474\) −5.00000 −0.229658
\(475\) 0 0
\(476\) −3.00000 −0.137505
\(477\) 3.00000 0.137361
\(478\) −5.00000 −0.228695
\(479\) 30.0000 1.37073 0.685367 − 0.728197i \(-0.259642\pi\)
0.685367 + 0.728197i \(0.259642\pi\)
\(480\) 0 0
\(481\) −36.0000 −1.64146
\(482\) 8.00000 0.364390
\(483\) 12.0000 0.546019
\(484\) 14.0000 0.636364
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −8.00000 −0.362515 −0.181257 − 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 14.0000 0.633750
\(489\) 18.0000 0.813988
\(490\) 0 0
\(491\) −30.0000 −1.35388 −0.676941 − 0.736038i \(-0.736695\pi\)
−0.676941 + 0.736038i \(0.736695\pi\)
\(492\) 10.0000 0.450835
\(493\) 4.00000 0.180151
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) −1.00000 −0.0449013
\(497\) 42.0000 1.88396
\(498\) 8.00000 0.358489
\(499\) −38.0000 −1.70111 −0.850557 − 0.525883i \(-0.823735\pi\)
−0.850557 + 0.525883i \(0.823735\pi\)
\(500\) 0 0
\(501\) −18.0000 −0.804181
\(502\) 18.0000 0.803379
\(503\) 34.0000 1.51599 0.757993 − 0.652263i \(-0.226180\pi\)
0.757993 + 0.652263i \(0.226180\pi\)
\(504\) −3.00000 −0.133631
\(505\) 0 0
\(506\) −20.0000 −0.889108
\(507\) −3.00000 −0.133235
\(508\) −2.00000 −0.0887357
\(509\) −9.00000 −0.398918 −0.199459 − 0.979906i \(-0.563918\pi\)
−0.199459 + 0.979906i \(0.563918\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) −1.00000 −0.0441942
\(513\) 1.00000 0.0441511
\(514\) 12.0000 0.529297
\(515\) 0 0
\(516\) −11.0000 −0.484248
\(517\) −45.0000 −1.97910
\(518\) 27.0000 1.18631
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) 15.0000 0.657162 0.328581 − 0.944476i \(-0.393430\pi\)
0.328581 + 0.944476i \(0.393430\pi\)
\(522\) 4.00000 0.175075
\(523\) −20.0000 −0.874539 −0.437269 − 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) 7.00000 0.305215
\(527\) 1.00000 0.0435607
\(528\) 5.00000 0.217597
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −8.00000 −0.347170
\(532\) −3.00000 −0.130066
\(533\) −40.0000 −1.73259
\(534\) −2.00000 −0.0865485
\(535\) 0 0
\(536\) 7.00000 0.302354
\(537\) 12.0000 0.517838
\(538\) −28.0000 −1.20717
\(539\) −10.0000 −0.430730
\(540\) 0 0
\(541\) 23.0000 0.988847 0.494424 − 0.869221i \(-0.335379\pi\)
0.494424 + 0.869221i \(0.335379\pi\)
\(542\) −4.00000 −0.171815
\(543\) 5.00000 0.214571
\(544\) 1.00000 0.0428746
\(545\) 0 0
\(546\) 12.0000 0.513553
\(547\) 2.00000 0.0855138 0.0427569 − 0.999086i \(-0.486386\pi\)
0.0427569 + 0.999086i \(0.486386\pi\)
\(548\) −6.00000 −0.256307
\(549\) −14.0000 −0.597505
\(550\) 0 0
\(551\) 4.00000 0.170406
\(552\) −4.00000 −0.170251
\(553\) −15.0000 −0.637865
\(554\) 13.0000 0.552317
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) 9.00000 0.381342 0.190671 − 0.981654i \(-0.438934\pi\)
0.190671 + 0.981654i \(0.438934\pi\)
\(558\) 1.00000 0.0423334
\(559\) 44.0000 1.86100
\(560\) 0 0
\(561\) −5.00000 −0.211100
\(562\) 18.0000 0.759284
\(563\) 16.0000 0.674320 0.337160 − 0.941447i \(-0.390534\pi\)
0.337160 + 0.941447i \(0.390534\pi\)
\(564\) −9.00000 −0.378968
\(565\) 0 0
\(566\) −22.0000 −0.924729
\(567\) 3.00000 0.125988
\(568\) −14.0000 −0.587427
\(569\) 14.0000 0.586911 0.293455 − 0.955973i \(-0.405195\pi\)
0.293455 + 0.955973i \(0.405195\pi\)
\(570\) 0 0
\(571\) −26.0000 −1.08807 −0.544033 − 0.839064i \(-0.683103\pi\)
−0.544033 + 0.839064i \(0.683103\pi\)
\(572\) −20.0000 −0.836242
\(573\) 9.00000 0.375980
\(574\) 30.0000 1.25218
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −45.0000 −1.87337 −0.936687 − 0.350167i \(-0.886125\pi\)
−0.936687 + 0.350167i \(0.886125\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −6.00000 −0.249351
\(580\) 0 0
\(581\) 24.0000 0.995688
\(582\) −14.0000 −0.580319
\(583\) −15.0000 −0.621237
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) 18.0000 0.743573
\(587\) 6.00000 0.247647 0.123823 − 0.992304i \(-0.460484\pi\)
0.123823 + 0.992304i \(0.460484\pi\)
\(588\) −2.00000 −0.0824786
\(589\) 1.00000 0.0412043
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) −9.00000 −0.369898
\(593\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) −5.00000 −0.205152
\(595\) 0 0
\(596\) 14.0000 0.573462
\(597\) 1.00000 0.0409273
\(598\) 16.0000 0.654289
\(599\) 39.0000 1.59350 0.796748 − 0.604311i \(-0.206552\pi\)
0.796748 + 0.604311i \(0.206552\pi\)
\(600\) 0 0
\(601\) −8.00000 −0.326327 −0.163163 − 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) −33.0000 −1.34498
\(603\) −7.00000 −0.285062
\(604\) −24.0000 −0.976546
\(605\) 0 0
\(606\) 9.00000 0.365600
\(607\) 32.0000 1.29884 0.649420 − 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) 1.00000 0.0405554
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) 36.0000 1.45640
\(612\) −1.00000 −0.0404226
\(613\) −10.0000 −0.403896 −0.201948 − 0.979396i \(-0.564727\pi\)
−0.201948 + 0.979396i \(0.564727\pi\)
\(614\) −8.00000 −0.322854
\(615\) 0 0
\(616\) 15.0000 0.604367
\(617\) 19.0000 0.764911 0.382456 − 0.923974i \(-0.375078\pi\)
0.382456 + 0.923974i \(0.375078\pi\)
\(618\) 14.0000 0.563163
\(619\) −40.0000 −1.60774 −0.803868 − 0.594808i \(-0.797228\pi\)
−0.803868 + 0.594808i \(0.797228\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) −10.0000 −0.400963
\(623\) −6.00000 −0.240385
\(624\) −4.00000 −0.160128
\(625\) 0 0
\(626\) 16.0000 0.639489
\(627\) −5.00000 −0.199681
\(628\) −24.0000 −0.957704
\(629\) 9.00000 0.358854
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 − 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 5.00000 0.198889
\(633\) 12.0000 0.476957
\(634\) −4.00000 −0.158860
\(635\) 0 0
\(636\) −3.00000 −0.118958
\(637\) 8.00000 0.316972
\(638\) −20.0000 −0.791808
\(639\) 14.0000 0.553831
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 − 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) −15.0000 −0.592003
\(643\) 44.0000 1.73519 0.867595 − 0.497271i \(-0.165665\pi\)
0.867595 + 0.497271i \(0.165665\pi\)
\(644\) −12.0000 −0.472866
\(645\) 0 0
\(646\) −1.00000 −0.0393445
\(647\) 8.00000 0.314512 0.157256 − 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 40.0000 1.57014
\(650\) 0 0
\(651\) 3.00000 0.117579
\(652\) −18.0000 −0.704934
\(653\) −10.0000 −0.391330 −0.195665 − 0.980671i \(-0.562687\pi\)
−0.195665 + 0.980671i \(0.562687\pi\)
\(654\) 1.00000 0.0391031
\(655\) 0 0
\(656\) −10.0000 −0.390434
\(657\) 2.00000 0.0780274
\(658\) −27.0000 −1.05257
\(659\) 12.0000 0.467454 0.233727 − 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) −14.0000 −0.544537 −0.272268 − 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) −5.00000 −0.194331
\(663\) 4.00000 0.155347
\(664\) −8.00000 −0.310460
\(665\) 0 0
\(666\) 9.00000 0.348743
\(667\) 16.0000 0.619522
\(668\) 18.0000 0.696441
\(669\) −6.00000 −0.231973
\(670\) 0 0
\(671\) 70.0000 2.70232
\(672\) 3.00000 0.115728
\(673\) 4.00000 0.154189 0.0770943 − 0.997024i \(-0.475436\pi\)
0.0770943 + 0.997024i \(0.475436\pi\)
\(674\) −24.0000 −0.924445
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 18.0000 0.691796 0.345898 − 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) −3.00000 −0.115214
\(679\) −42.0000 −1.61181
\(680\) 0 0
\(681\) 21.0000 0.804722
\(682\) −5.00000 −0.191460
\(683\) −8.00000 −0.306111 −0.153056 − 0.988218i \(-0.548911\pi\)
−0.153056 + 0.988218i \(0.548911\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) 15.0000 0.572703
\(687\) 12.0000 0.457829
\(688\) 11.0000 0.419371
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 6.00000 0.228251 0.114125 − 0.993466i \(-0.463593\pi\)
0.114125 + 0.993466i \(0.463593\pi\)
\(692\) −14.0000 −0.532200
\(693\) −15.0000 −0.569803
\(694\) −11.0000 −0.417554
\(695\) 0 0
\(696\) −4.00000 −0.151620
\(697\) 10.0000 0.378777
\(698\) 10.0000 0.378506
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) 22.0000 0.830929 0.415464 − 0.909610i \(-0.363619\pi\)
0.415464 + 0.909610i \(0.363619\pi\)
\(702\) 4.00000 0.150970
\(703\) 9.00000 0.339441
\(704\) −5.00000 −0.188445
\(705\) 0 0
\(706\) −24.0000 −0.903252
\(707\) 27.0000 1.01544
\(708\) 8.00000 0.300658
\(709\) 49.0000 1.84023 0.920117 − 0.391644i \(-0.128094\pi\)
0.920117 + 0.391644i \(0.128094\pi\)
\(710\) 0 0
\(711\) −5.00000 −0.187515
\(712\) 2.00000 0.0749532
\(713\) 4.00000 0.149801
\(714\) −3.00000 −0.112272
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) −5.00000 −0.186728
\(718\) 25.0000 0.932992
\(719\) −18.0000 −0.671287 −0.335643 − 0.941989i \(-0.608954\pi\)
−0.335643 + 0.941989i \(0.608954\pi\)
\(720\) 0 0
\(721\) 42.0000 1.56416
\(722\) 18.0000 0.669891
\(723\) 8.00000 0.297523
\(724\) −5.00000 −0.185824
\(725\) 0 0
\(726\) 14.0000 0.519589
\(727\) 32.0000 1.18681 0.593407 − 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) −12.0000 −0.444750
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −11.0000 −0.406850
\(732\) 14.0000 0.517455
\(733\) −2.00000 −0.0738717 −0.0369358 − 0.999318i \(-0.511760\pi\)
−0.0369358 + 0.999318i \(0.511760\pi\)
\(734\) −25.0000 −0.922767
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) 35.0000 1.28924
\(738\) 10.0000 0.368105
\(739\) −17.0000 −0.625355 −0.312678 − 0.949859i \(-0.601226\pi\)
−0.312678 + 0.949859i \(0.601226\pi\)
\(740\) 0 0
\(741\) 4.00000 0.146944
\(742\) −9.00000 −0.330400
\(743\) 38.0000 1.39408 0.697042 − 0.717030i \(-0.254499\pi\)
0.697042 + 0.717030i \(0.254499\pi\)
\(744\) −1.00000 −0.0366618
\(745\) 0 0
\(746\) −6.00000 −0.219676
\(747\) 8.00000 0.292705
\(748\) 5.00000 0.182818
\(749\) −45.0000 −1.64426
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 − 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 9.00000 0.328196
\(753\) 18.0000 0.655956
\(754\) 16.0000 0.582686
\(755\) 0 0
\(756\) −3.00000 −0.109109
\(757\) −46.0000 −1.67190 −0.835949 − 0.548807i \(-0.815082\pi\)
−0.835949 + 0.548807i \(0.815082\pi\)
\(758\) 4.00000 0.145287
\(759\) −20.0000 −0.725954
\(760\) 0 0
\(761\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) −2.00000 −0.0724524
\(763\) 3.00000 0.108607
\(764\) −9.00000 −0.325609
\(765\) 0 0
\(766\) 12.0000 0.433578
\(767\) −32.0000 −1.15545
\(768\) −1.00000 −0.0360844
\(769\) 11.0000 0.396670 0.198335 − 0.980134i \(-0.436447\pi\)
0.198335 + 0.980134i \(0.436447\pi\)
\(770\) 0 0
\(771\) 12.0000 0.432169
\(772\) 6.00000 0.215945
\(773\) 22.0000 0.791285 0.395643 − 0.918405i \(-0.370522\pi\)
0.395643 + 0.918405i \(0.370522\pi\)
\(774\) −11.0000 −0.395387
\(775\) 0 0
\(776\) 14.0000 0.502571
\(777\) 27.0000 0.968620
\(778\) −21.0000 −0.752886
\(779\) 10.0000 0.358287
\(780\) 0 0
\(781\) −70.0000 −2.50480
\(782\) −4.00000 −0.143040
\(783\) 4.00000 0.142948
\(784\) 2.00000 0.0714286
\(785\) 0 0
\(786\) 4.00000 0.142675
\(787\) 50.0000 1.78231 0.891154 − 0.453701i \(-0.149897\pi\)
0.891154 + 0.453701i \(0.149897\pi\)
\(788\) −12.0000 −0.427482
\(789\) 7.00000 0.249207
\(790\) 0 0
\(791\) −9.00000 −0.320003
\(792\) 5.00000 0.177667
\(793\) −56.0000 −1.98862
\(794\) −15.0000 −0.532330
\(795\) 0 0
\(796\) −1.00000 −0.0354441
\(797\) −39.0000 −1.38145 −0.690725 − 0.723117i \(-0.742709\pi\)
−0.690725 + 0.723117i \(0.742709\pi\)
\(798\) −3.00000 −0.106199
\(799\) −9.00000 −0.318397
\(800\) 0 0
\(801\) −2.00000 −0.0706665
\(802\) 30.0000 1.05934
\(803\) −10.0000 −0.352892
\(804\) 7.00000 0.246871
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) −28.0000 −0.985647
\(808\) −9.00000 −0.316619
\(809\) 39.0000 1.37117 0.685583 − 0.727994i \(-0.259547\pi\)
0.685583 + 0.727994i \(0.259547\pi\)
\(810\) 0 0
\(811\) −22.0000 −0.772524 −0.386262 − 0.922389i \(-0.626234\pi\)
−0.386262 + 0.922389i \(0.626234\pi\)
\(812\) −12.0000 −0.421117
\(813\) −4.00000 −0.140286
\(814\) −45.0000 −1.57725
\(815\) 0 0
\(816\) 1.00000 0.0350070
\(817\) −11.0000 −0.384841
\(818\) 26.0000 0.909069
\(819\) 12.0000 0.419314
\(820\) 0 0
\(821\) −30.0000 −1.04701 −0.523504 − 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) −6.00000 −0.209274
\(823\) 8.00000 0.278862 0.139431 − 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) −14.0000 −0.487713
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) −39.0000 −1.35616 −0.678081 − 0.734987i \(-0.737188\pi\)
−0.678081 + 0.734987i \(0.737188\pi\)
\(828\) −4.00000 −0.139010
\(829\) −24.0000 −0.833554 −0.416777 − 0.909009i \(-0.636840\pi\)
−0.416777 + 0.909009i \(0.636840\pi\)
\(830\) 0 0
\(831\) 13.0000 0.450965
\(832\) 4.00000 0.138675
\(833\) −2.00000 −0.0692959
\(834\) 8.00000 0.277017
\(835\) 0 0
\(836\) 5.00000 0.172929
\(837\) 1.00000 0.0345651
\(838\) −28.0000 −0.967244
\(839\) 10.0000 0.345238 0.172619 − 0.984989i \(-0.444777\pi\)
0.172619 + 0.984989i \(0.444777\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) −24.0000 −0.827095
\(843\) 18.0000 0.619953
\(844\) −12.0000 −0.413057
\(845\) 0 0
\(846\) −9.00000 −0.309426
\(847\) 42.0000 1.44314
\(848\) 3.00000 0.103020
\(849\) −22.0000 −0.755038
\(850\) 0 0
\(851\) 36.0000 1.23406
\(852\) −14.0000 −0.479632
\(853\) 9.00000 0.308154 0.154077 − 0.988059i \(-0.450760\pi\)
0.154077 + 0.988059i \(0.450760\pi\)
\(854\) 42.0000 1.43721
\(855\) 0 0
\(856\) 15.0000 0.512689
\(857\) −5.00000 −0.170797 −0.0853984 − 0.996347i \(-0.527216\pi\)
−0.0853984 + 0.996347i \(0.527216\pi\)
\(858\) −20.0000 −0.682789
\(859\) −25.0000 −0.852989 −0.426494 − 0.904490i \(-0.640252\pi\)
−0.426494 + 0.904490i \(0.640252\pi\)
\(860\) 0 0
\(861\) 30.0000 1.02240
\(862\) −14.0000 −0.476842
\(863\) 17.0000 0.578687 0.289343 − 0.957225i \(-0.406563\pi\)
0.289343 + 0.957225i \(0.406563\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −27.0000 −0.917497
\(867\) −1.00000 −0.0339618
\(868\) −3.00000 −0.101827
\(869\) 25.0000 0.848067
\(870\) 0 0
\(871\) −28.0000 −0.948744
\(872\) −1.00000 −0.0338643
\(873\) −14.0000 −0.473828
\(874\) −4.00000 −0.135302
\(875\) 0 0
\(876\) −2.00000 −0.0675737
\(877\) 46.0000 1.55331 0.776655 − 0.629926i \(-0.216915\pi\)
0.776655 + 0.629926i \(0.216915\pi\)
\(878\) −20.0000 −0.674967
\(879\) 18.0000 0.607125
\(880\) 0 0
\(881\) −35.0000 −1.17918 −0.589590 − 0.807703i \(-0.700711\pi\)
−0.589590 + 0.807703i \(0.700711\pi\)
\(882\) −2.00000 −0.0673435
\(883\) 48.0000 1.61533 0.807664 − 0.589643i \(-0.200731\pi\)
0.807664 + 0.589643i \(0.200731\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) 38.0000 1.27663
\(887\) −12.0000 −0.402921 −0.201460 − 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) −9.00000 −0.302020
\(889\) −6.00000 −0.201234
\(890\) 0 0
\(891\) −5.00000 −0.167506
\(892\) 6.00000 0.200895
\(893\) −9.00000 −0.301174
\(894\) 14.0000 0.468230
\(895\) 0 0
\(896\) −3.00000 −0.100223
\(897\) 16.0000 0.534224
\(898\) 41.0000 1.36819
\(899\) 4.00000 0.133407
\(900\) 0 0
\(901\) −3.00000 −0.0999445
\(902\) −50.0000 −1.66482
\(903\) −33.0000 −1.09817
\(904\) 3.00000 0.0997785
\(905\) 0 0
\(906\) −24.0000 −0.797347
\(907\) 24.0000 0.796907 0.398453 − 0.917189i \(-0.369547\pi\)
0.398453 + 0.917189i \(0.369547\pi\)
\(908\) −21.0000 −0.696909
\(909\) 9.00000 0.298511
\(910\) 0 0
\(911\) 36.0000 1.19273 0.596367 − 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 1.00000 0.0331133
\(913\) −40.0000 −1.32381
\(914\) −37.0000 −1.22385
\(915\) 0 0
\(916\) −12.0000 −0.396491
\(917\) 12.0000 0.396275
\(918\) −1.00000 −0.0330049
\(919\) −14.0000 −0.461817 −0.230909 − 0.972975i \(-0.574170\pi\)
−0.230909 + 0.972975i \(0.574170\pi\)
\(920\) 0 0
\(921\) −8.00000 −0.263609
\(922\) 5.00000 0.164666
\(923\) 56.0000 1.84326
\(924\) 15.0000 0.493464
\(925\) 0 0
\(926\) −36.0000 −1.18303
\(927\) 14.0000 0.459820
\(928\) 4.00000 0.131306
\(929\) 1.00000 0.0328089 0.0164045 − 0.999865i \(-0.494778\pi\)
0.0164045 + 0.999865i \(0.494778\pi\)
\(930\) 0 0
\(931\) −2.00000 −0.0655474
\(932\) −6.00000 −0.196537
\(933\) −10.0000 −0.327385
\(934\) 6.00000 0.196326
\(935\) 0 0
\(936\) −4.00000 −0.130744
\(937\) −38.0000 −1.24141 −0.620703 − 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) 21.0000 0.685674
\(939\) 16.0000 0.522140
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 − 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) −24.0000 −0.781962
\(943\) 40.0000 1.30258
\(944\) −8.00000 −0.260378
\(945\) 0 0
\(946\) 55.0000 1.78820
\(947\) −29.0000 −0.942373 −0.471187 − 0.882034i \(-0.656174\pi\)
−0.471187 + 0.882034i \(0.656174\pi\)
\(948\) 5.00000 0.162392
\(949\) 8.00000 0.259691
\(950\) 0 0
\(951\) −4.00000 −0.129709
\(952\) 3.00000 0.0972306
\(953\) 12.0000 0.388718 0.194359 − 0.980930i \(-0.437737\pi\)
0.194359 + 0.980930i \(0.437737\pi\)
\(954\) −3.00000 −0.0971286
\(955\) 0 0
\(956\) 5.00000 0.161712
\(957\) −20.0000 −0.646508
\(958\) −30.0000 −0.969256
\(959\) −18.0000 −0.581250
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 36.0000 1.16069
\(963\) −15.0000 −0.483368
\(964\) −8.00000 −0.257663
\(965\) 0 0
\(966\) −12.0000 −0.386094
\(967\) −24.0000 −0.771788 −0.385894 − 0.922543i \(-0.626107\pi\)
−0.385894 + 0.922543i \(0.626107\pi\)
\(968\) −14.0000 −0.449977
\(969\) −1.00000 −0.0321246
\(970\) 0 0
\(971\) −10.0000 −0.320915 −0.160458 − 0.987043i \(-0.551297\pi\)
−0.160458 + 0.987043i \(0.551297\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 24.0000 0.769405
\(974\) 8.00000 0.256337
\(975\) 0 0
\(976\) −14.0000 −0.448129
\(977\) −48.0000 −1.53566 −0.767828 − 0.640656i \(-0.778662\pi\)
−0.767828 + 0.640656i \(0.778662\pi\)
\(978\) −18.0000 −0.575577
\(979\) 10.0000 0.319601
\(980\) 0 0
\(981\) 1.00000 0.0319275
\(982\) 30.0000 0.957338
\(983\) −28.0000 −0.893061 −0.446531 − 0.894768i \(-0.647341\pi\)
−0.446531 + 0.894768i \(0.647341\pi\)
\(984\) −10.0000 −0.318788
\(985\) 0 0
\(986\) −4.00000 −0.127386
\(987\) −27.0000 −0.859419
\(988\) −4.00000 −0.127257
\(989\) −44.0000 −1.39912
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 − 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 1.00000 0.0317500
\(993\) −5.00000 −0.158670
\(994\) −42.0000 −1.33216
\(995\) 0 0
\(996\) −8.00000 −0.253490
\(997\) −25.0000 −0.791758 −0.395879 − 0.918303i \(-0.629560\pi\)
−0.395879 + 0.918303i \(0.629560\pi\)
\(998\) 38.0000 1.20287
\(999\) 9.00000 0.284747
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 2550.2.a.f.1.1 ✓ 1
3.2 odd 2 7650.2.a.ch.1.1 1
5.2 odd 4 2550.2.d.l.2449.1 2
5.3 odd 4 2550.2.d.l.2449.2 2
5.4 even 2 2550.2.a.bb.1.1 yes 1
15.14 odd 2 7650.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2550.2.a.f.1.1 ✓ 1 1.1 even 1 trivial
2550.2.a.bb.1.1 yes 1 5.4 even 2
2550.2.d.l.2449.1 2 5.2 odd 4
2550.2.d.l.2449.2 2 5.3 odd 4
7650.2.a.f.1.1 1 15.14 odd 2
7650.2.a.ch.1.1 1 3.2 odd 2