Properties

Label 2550.2.a.h.1.1
Level $2550$
Weight $2$
Character 2550.1
Self dual yes
Analytic conductor $20.362$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2550.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -5.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +3.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} +1.00000 q^{19} -3.00000 q^{21} +5.00000 q^{22} -6.00000 q^{23} -1.00000 q^{24} -2.00000 q^{26} +1.00000 q^{27} -3.00000 q^{28} +10.0000 q^{29} +5.00000 q^{31} -1.00000 q^{32} -5.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} +3.00000 q^{37} -1.00000 q^{38} +2.00000 q^{39} +6.00000 q^{41} +3.00000 q^{42} +1.00000 q^{43} -5.00000 q^{44} +6.00000 q^{46} -3.00000 q^{47} +1.00000 q^{48} +2.00000 q^{49} -1.00000 q^{51} +2.00000 q^{52} +1.00000 q^{53} -1.00000 q^{54} +3.00000 q^{56} +1.00000 q^{57} -10.0000 q^{58} +8.00000 q^{59} -2.00000 q^{61} -5.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} +5.00000 q^{66} +11.0000 q^{67} -1.00000 q^{68} -6.00000 q^{69} +6.00000 q^{71} -1.00000 q^{72} +12.0000 q^{73} -3.00000 q^{74} +1.00000 q^{76} +15.0000 q^{77} -2.00000 q^{78} +5.00000 q^{79} +1.00000 q^{81} -6.00000 q^{82} -18.0000 q^{83} -3.00000 q^{84} -1.00000 q^{86} +10.0000 q^{87} +5.00000 q^{88} +12.0000 q^{89} -6.00000 q^{91} -6.00000 q^{92} +5.00000 q^{93} +3.00000 q^{94} -1.00000 q^{96} +14.0000 q^{97} -2.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)

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
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) −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
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 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 \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 5.00000 1.06600
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) −3.00000 −0.566947
\(29\) 10.0000 1.85695 0.928477 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) −1.00000 −0.176777
\(33\) −5.00000 −0.870388
\(34\) 1.00000 0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 3.00000 0.493197 0.246598 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) −1.00000 −0.162221
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 3.00000 0.462910
\(43\) 1.00000 0.152499 0.0762493 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) −5.00000 −0.753778
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 1.00000 0.144338
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) −1.00000 −0.140028
\(52\) 2.00000 0.277350
\(53\) 1.00000 0.137361 0.0686803 0.997639i \(-0.478121\pi\)
0.0686803 + 0.997639i \(0.478121\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 3.00000 0.400892
\(57\) 1.00000 0.132453
\(58\) −10.0000 −1.31306
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −5.00000 −0.635001
\(63\) −3.00000 −0.377964
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 5.00000 0.615457
\(67\) 11.0000 1.34386 0.671932 0.740613i \(-0.265465\pi\)
0.671932 + 0.740613i \(0.265465\pi\)
\(68\) −1.00000 −0.121268
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) −1.00000 −0.117851
\(73\) 12.0000 1.40449 0.702247 0.711934i \(-0.252180\pi\)
0.702247 + 0.711934i \(0.252180\pi\)
\(74\) −3.00000 −0.348743
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 15.0000 1.70941
\(78\) −2.00000 −0.226455
\(79\) 5.00000 0.562544 0.281272 0.959628i \(-0.409244\pi\)
0.281272 + 0.959628i \(0.409244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) −18.0000 −1.97576 −0.987878 0.155230i \(-0.950388\pi\)
−0.987878 + 0.155230i \(0.950388\pi\)
\(84\) −3.00000 −0.327327
\(85\) 0 0
\(86\) −1.00000 −0.107833
\(87\) 10.0000 1.07211
\(88\) 5.00000 0.533002
\(89\) 12.0000 1.27200 0.635999 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) −6.00000 −0.625543
\(93\) 5.00000 0.518476
\(94\) 3.00000 0.309426
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) −2.00000 −0.202031
\(99\) −5.00000 −0.502519
\(100\) 0 0
\(101\) 11.0000 1.09454 0.547270 0.836956i \(-0.315667\pi\)
0.547270 + 0.836956i \(0.315667\pi\)
\(102\) 1.00000 0.0990148
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −1.00000 −0.0971286
\(107\) 5.00000 0.483368 0.241684 0.970355i \(-0.422300\pi\)
0.241684 + 0.970355i \(0.422300\pi\)
\(108\) 1.00000 0.0962250
\(109\) 5.00000 0.478913 0.239457 0.970907i \(-0.423031\pi\)
0.239457 + 0.970907i \(0.423031\pi\)
\(110\) 0 0
\(111\) 3.00000 0.284747
\(112\) −3.00000 −0.283473
\(113\) −1.00000 −0.0940721 −0.0470360 0.998893i \(-0.514978\pi\)
−0.0470360 + 0.998893i \(0.514978\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) 10.0000 0.928477
\(117\) 2.00000 0.184900
\(118\) −8.00000 −0.736460
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 2.00000 0.181071
\(123\) 6.00000 0.541002
\(124\) 5.00000 0.449013
\(125\) 0 0
\(126\) 3.00000 0.267261
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 1.00000 0.0880451
\(130\) 0 0
\(131\) −20.0000 −1.74741 −0.873704 0.486458i \(-0.838289\pi\)
−0.873704 + 0.486458i \(0.838289\pi\)
\(132\) −5.00000 −0.435194
\(133\) −3.00000 −0.260133
\(134\) −11.0000 −0.950255
\(135\) 0 0
\(136\) 1.00000 0.0857493
\(137\) −16.0000 −1.36697 −0.683486 0.729964i \(-0.739537\pi\)
−0.683486 + 0.729964i \(0.739537\pi\)
\(138\) 6.00000 0.510754
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) −3.00000 −0.252646
\(142\) −6.00000 −0.503509
\(143\) −10.0000 −0.836242
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −12.0000 −0.993127
\(147\) 2.00000 0.164957
\(148\) 3.00000 0.246598
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −1.00000 −0.0808452
\(154\) −15.0000 −1.20873
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) −5.00000 −0.397779
\(159\) 1.00000 0.0793052
\(160\) 0 0
\(161\) 18.0000 1.41860
\(162\) −1.00000 −0.0785674
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 18.0000 1.39707
\(167\) 18.0000 1.39288 0.696441 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) 3.00000 0.231455
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 1.00000 0.0762493
\(173\) −20.0000 −1.52057 −0.760286 0.649589i \(-0.774941\pi\)
−0.760286 + 0.649589i \(0.774941\pi\)
\(174\) −10.0000 −0.758098
\(175\) 0 0
\(176\) −5.00000 −0.376889
\(177\) 8.00000 0.601317
\(178\) −12.0000 −0.899438
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −17.0000 −1.26360 −0.631800 0.775131i \(-0.717684\pi\)
−0.631800 + 0.775131i \(0.717684\pi\)
\(182\) 6.00000 0.444750
\(183\) −2.00000 −0.147844
\(184\) 6.00000 0.442326
\(185\) 0 0
\(186\) −5.00000 −0.366618
\(187\) 5.00000 0.365636
\(188\) −3.00000 −0.218797
\(189\) −3.00000 −0.218218
\(190\) 0 0
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) 1.00000 0.0721688
\(193\) 24.0000 1.72756 0.863779 0.503871i \(-0.168091\pi\)
0.863779 + 0.503871i \(0.168091\pi\)
\(194\) −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
\(199\) 17.0000 1.20510 0.602549 0.798082i \(-0.294152\pi\)
0.602549 + 0.798082i \(0.294152\pi\)
\(200\) 0 0
\(201\) 11.0000 0.775880
\(202\) −11.0000 −0.773957
\(203\) −30.0000 −2.10559
\(204\) −1.00000 −0.0700140
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) −6.00000 −0.417029
\(208\) 2.00000 0.138675
\(209\) −5.00000 −0.345857
\(210\) 0 0
\(211\) 18.0000 1.23917 0.619586 0.784929i \(-0.287301\pi\)
0.619586 + 0.784929i \(0.287301\pi\)
\(212\) 1.00000 0.0686803
\(213\) 6.00000 0.411113
\(214\) −5.00000 −0.341793
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) −15.0000 −1.01827
\(218\) −5.00000 −0.338643
\(219\) 12.0000 0.810885
\(220\) 0 0
\(221\) −2.00000 −0.134535
\(222\) −3.00000 −0.201347
\(223\) −18.0000 −1.20537 −0.602685 0.797980i \(-0.705902\pi\)
−0.602685 + 0.797980i \(0.705902\pi\)
\(224\) 3.00000 0.200446
\(225\) 0 0
\(226\) 1.00000 0.0665190
\(227\) 27.0000 1.79205 0.896026 0.444001i \(-0.146441\pi\)
0.896026 + 0.444001i \(0.146441\pi\)
\(228\) 1.00000 0.0662266
\(229\) 24.0000 1.58596 0.792982 0.609245i \(-0.208527\pi\)
0.792982 + 0.609245i \(0.208527\pi\)
\(230\) 0 0
\(231\) 15.0000 0.986928
\(232\) −10.0000 −0.656532
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 8.00000 0.520756
\(237\) 5.00000 0.324785
\(238\) −3.00000 −0.194461
\(239\) 13.0000 0.840900 0.420450 0.907316i \(-0.361872\pi\)
0.420450 + 0.907316i \(0.361872\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) −14.0000 −0.899954
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 2.00000 0.127257
\(248\) −5.00000 −0.317500
\(249\) −18.0000 −1.14070
\(250\) 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
\(253\) 30.0000 1.88608
\(254\) 12.0000 0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) −1.00000 −0.0622573
\(259\) −9.00000 −0.559233
\(260\) 0 0
\(261\) 10.0000 0.618984
\(262\) 20.0000 1.23560
\(263\) −3.00000 −0.184988 −0.0924940 0.995713i \(-0.529484\pi\)
−0.0924940 + 0.995713i \(0.529484\pi\)
\(264\) 5.00000 0.307729
\(265\) 0 0
\(266\) 3.00000 0.183942
\(267\) 12.0000 0.734388
\(268\) 11.0000 0.671932
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 14.0000 0.850439 0.425220 0.905090i \(-0.360197\pi\)
0.425220 + 0.905090i \(0.360197\pi\)
\(272\) −1.00000 −0.0606339
\(273\) −6.00000 −0.363137
\(274\) 16.0000 0.966595
\(275\) 0 0
\(276\) −6.00000 −0.361158
\(277\) 31.0000 1.86261 0.931305 0.364241i \(-0.118672\pi\)
0.931305 + 0.364241i \(0.118672\pi\)
\(278\) −8.00000 −0.479808
\(279\) 5.00000 0.299342
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 3.00000 0.178647
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 10.0000 0.591312
\(287\) −18.0000 −1.06251
\(288\) −1.00000 −0.0589256
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) 12.0000 0.702247
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) −2.00000 −0.116642
\(295\) 0 0
\(296\) −3.00000 −0.174371
\(297\) −5.00000 −0.290129
\(298\) 6.00000 0.347571
\(299\) −12.0000 −0.693978
\(300\) 0 0
\(301\) −3.00000 −0.172917
\(302\) −8.00000 −0.460348
\(303\) 11.0000 0.631933
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 1.00000 0.0571662
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 15.0000 0.854704
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −2.00000 −0.113410 −0.0567048 0.998391i \(-0.518059\pi\)
−0.0567048 + 0.998391i \(0.518059\pi\)
\(312\) −2.00000 −0.113228
\(313\) 34.0000 1.92179 0.960897 0.276907i \(-0.0893093\pi\)
0.960897 + 0.276907i \(0.0893093\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 5.00000 0.281272
\(317\) 12.0000 0.673987 0.336994 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) −1.00000 −0.0560772
\(319\) −50.0000 −2.79946
\(320\) 0 0
\(321\) 5.00000 0.279073
\(322\) −18.0000 −1.00310
\(323\) −1.00000 −0.0556415
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 0 0
\(327\) 5.00000 0.276501
\(328\) −6.00000 −0.331295
\(329\) 9.00000 0.496186
\(330\) 0 0
\(331\) −25.0000 −1.37412 −0.687062 0.726599i \(-0.741100\pi\)
−0.687062 + 0.726599i \(0.741100\pi\)
\(332\) −18.0000 −0.987878
\(333\) 3.00000 0.164399
\(334\) −18.0000 −0.984916
\(335\) 0 0
\(336\) −3.00000 −0.163663
\(337\) −12.0000 −0.653682 −0.326841 0.945079i \(-0.605984\pi\)
−0.326841 + 0.945079i \(0.605984\pi\)
\(338\) 9.00000 0.489535
\(339\) −1.00000 −0.0543125
\(340\) 0 0
\(341\) −25.0000 −1.35383
\(342\) −1.00000 −0.0540738
\(343\) 15.0000 0.809924
\(344\) −1.00000 −0.0539164
\(345\) 0 0
\(346\) 20.0000 1.07521
\(347\) 23.0000 1.23470 0.617352 0.786687i \(-0.288205\pi\)
0.617352 + 0.786687i \(0.288205\pi\)
\(348\) 10.0000 0.536056
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 5.00000 0.266501
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) −8.00000 −0.425195
\(355\) 0 0
\(356\) 12.0000 0.635999
\(357\) 3.00000 0.158777
\(358\) 12.0000 0.634220
\(359\) 15.0000 0.791670 0.395835 0.918322i \(-0.370455\pi\)
0.395835 + 0.918322i \(0.370455\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 17.0000 0.893500
\(363\) 14.0000 0.734809
\(364\) −6.00000 −0.314485
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) −5.00000 −0.260998 −0.130499 0.991448i \(-0.541658\pi\)
−0.130499 + 0.991448i \(0.541658\pi\)
\(368\) −6.00000 −0.312772
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) −3.00000 −0.155752
\(372\) 5.00000 0.259238
\(373\) 18.0000 0.932005 0.466002 0.884783i \(-0.345694\pi\)
0.466002 + 0.884783i \(0.345694\pi\)
\(374\) −5.00000 −0.258544
\(375\) 0 0
\(376\) 3.00000 0.154713
\(377\) 20.0000 1.03005
\(378\) 3.00000 0.154303
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) −3.00000 −0.153493
\(383\) 8.00000 0.408781 0.204390 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −24.0000 −1.22157
\(387\) 1.00000 0.0508329
\(388\) 14.0000 0.710742
\(389\) −5.00000 −0.253510 −0.126755 0.991934i \(-0.540456\pi\)
−0.126755 + 0.991934i \(0.540456\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) −2.00000 −0.101015
\(393\) −20.0000 −1.00887
\(394\) 12.0000 0.604551
\(395\) 0 0
\(396\) −5.00000 −0.251259
\(397\) −21.0000 −1.05396 −0.526980 0.849878i \(-0.676676\pi\)
−0.526980 + 0.849878i \(0.676676\pi\)
\(398\) −17.0000 −0.852133
\(399\) −3.00000 −0.150188
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) −11.0000 −0.548630
\(403\) 10.0000 0.498135
\(404\) 11.0000 0.547270
\(405\) 0 0
\(406\) 30.0000 1.48888
\(407\) −15.0000 −0.743522
\(408\) 1.00000 0.0495074
\(409\) −2.00000 −0.0988936 −0.0494468 0.998777i \(-0.515746\pi\)
−0.0494468 + 0.998777i \(0.515746\pi\)
\(410\) 0 0
\(411\) −16.0000 −0.789222
\(412\) −4.00000 −0.197066
\(413\) −24.0000 −1.18096
\(414\) 6.00000 0.294884
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 8.00000 0.391762
\(418\) 5.00000 0.244558
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 0 0
\(421\) 18.0000 0.877266 0.438633 0.898666i \(-0.355463\pi\)
0.438633 + 0.898666i \(0.355463\pi\)
\(422\) −18.0000 −0.876226
\(423\) −3.00000 −0.145865
\(424\) −1.00000 −0.0485643
\(425\) 0 0
\(426\) −6.00000 −0.290701
\(427\) 6.00000 0.290360
\(428\) 5.00000 0.241684
\(429\) −10.0000 −0.482805
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 1.00000 0.0481125
\(433\) −21.0000 −1.00920 −0.504598 0.863355i \(-0.668359\pi\)
−0.504598 + 0.863355i \(0.668359\pi\)
\(434\) 15.0000 0.720023
\(435\) 0 0
\(436\) 5.00000 0.239457
\(437\) −6.00000 −0.287019
\(438\) −12.0000 −0.573382
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 2.00000 0.0951303
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 3.00000 0.142374
\(445\) 0 0
\(446\) 18.0000 0.852325
\(447\) −6.00000 −0.283790
\(448\) −3.00000 −0.141737
\(449\) 21.0000 0.991051 0.495526 0.868593i \(-0.334975\pi\)
0.495526 + 0.868593i \(0.334975\pi\)
\(450\) 0 0
\(451\) −30.0000 −1.41264
\(452\) −1.00000 −0.0470360
\(453\) 8.00000 0.375873
\(454\) −27.0000 −1.26717
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) 25.0000 1.16945 0.584725 0.811231i \(-0.301202\pi\)
0.584725 + 0.811231i \(0.301202\pi\)
\(458\) −24.0000 −1.12145
\(459\) −1.00000 −0.0466760
\(460\) 0 0
\(461\) −15.0000 −0.698620 −0.349310 0.937007i \(-0.613584\pi\)
−0.349310 + 0.937007i \(0.613584\pi\)
\(462\) −15.0000 −0.697863
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 10.0000 0.464238
\(465\) 0 0
\(466\) −10.0000 −0.463241
\(467\) −18.0000 −0.832941 −0.416470 0.909149i \(-0.636733\pi\)
−0.416470 + 0.909149i \(0.636733\pi\)
\(468\) 2.00000 0.0924500
\(469\) −33.0000 −1.52380
\(470\) 0 0
\(471\) 0 0
\(472\) −8.00000 −0.368230
\(473\) −5.00000 −0.229900
\(474\) −5.00000 −0.229658
\(475\) 0 0
\(476\) 3.00000 0.137505
\(477\) 1.00000 0.0457869
\(478\) −13.0000 −0.594606
\(479\) 26.0000 1.18797 0.593985 0.804476i \(-0.297554\pi\)
0.593985 + 0.804476i \(0.297554\pi\)
\(480\) 0 0
\(481\) 6.00000 0.273576
\(482\) 0 0
\(483\) 18.0000 0.819028
\(484\) 14.0000 0.636364
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 40.0000 1.81257 0.906287 0.422664i \(-0.138905\pi\)
0.906287 + 0.422664i \(0.138905\pi\)
\(488\) 2.00000 0.0905357
\(489\) 0 0
\(490\) 0 0
\(491\) 4.00000 0.180517 0.0902587 0.995918i \(-0.471231\pi\)
0.0902587 + 0.995918i \(0.471231\pi\)
\(492\) 6.00000 0.270501
\(493\) −10.0000 −0.450377
\(494\) −2.00000 −0.0899843
\(495\) 0 0
\(496\) 5.00000 0.224507
\(497\) −18.0000 −0.807410
\(498\) 18.0000 0.806599
\(499\) 22.0000 0.984855 0.492428 0.870353i \(-0.336110\pi\)
0.492428 + 0.870353i \(0.336110\pi\)
\(500\) 0 0
\(501\) 18.0000 0.804181
\(502\) 18.0000 0.803379
\(503\) −6.00000 −0.267527 −0.133763 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 3.00000 0.133631
\(505\) 0 0
\(506\) −30.0000 −1.33366
\(507\) −9.00000 −0.399704
\(508\) −12.0000 −0.532414
\(509\) 9.00000 0.398918 0.199459 0.979906i \(-0.436082\pi\)
0.199459 + 0.979906i \(0.436082\pi\)
\(510\) 0 0
\(511\) −36.0000 −1.59255
\(512\) −1.00000 −0.0441942
\(513\) 1.00000 0.0441511
\(514\) 2.00000 0.0882162
\(515\) 0 0
\(516\) 1.00000 0.0440225
\(517\) 15.0000 0.659699
\(518\) 9.00000 0.395437
\(519\) −20.0000 −0.877903
\(520\) 0 0
\(521\) −39.0000 −1.70862 −0.854311 0.519763i \(-0.826020\pi\)
−0.854311 + 0.519763i \(0.826020\pi\)
\(522\) −10.0000 −0.437688
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) −20.0000 −0.873704
\(525\) 0 0
\(526\) 3.00000 0.130806
\(527\) −5.00000 −0.217803
\(528\) −5.00000 −0.217597
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 8.00000 0.347170
\(532\) −3.00000 −0.130066
\(533\) 12.0000 0.519778
\(534\) −12.0000 −0.519291
\(535\) 0 0
\(536\) −11.0000 −0.475128
\(537\) −12.0000 −0.517838
\(538\) 6.00000 0.258678
\(539\) −10.0000 −0.430730
\(540\) 0 0
\(541\) −41.0000 −1.76273 −0.881364 0.472438i \(-0.843374\pi\)
−0.881364 + 0.472438i \(0.843374\pi\)
\(542\) −14.0000 −0.601351
\(543\) −17.0000 −0.729540
\(544\) 1.00000 0.0428746
\(545\) 0 0
\(546\) 6.00000 0.256776
\(547\) −22.0000 −0.940652 −0.470326 0.882493i \(-0.655864\pi\)
−0.470326 + 0.882493i \(0.655864\pi\)
\(548\) −16.0000 −0.683486
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 10.0000 0.426014
\(552\) 6.00000 0.255377
\(553\) −15.0000 −0.637865
\(554\) −31.0000 −1.31706
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) −17.0000 −0.720313 −0.360157 0.932892i \(-0.617277\pi\)
−0.360157 + 0.932892i \(0.617277\pi\)
\(558\) −5.00000 −0.211667
\(559\) 2.00000 0.0845910
\(560\) 0 0
\(561\) 5.00000 0.211100
\(562\) 30.0000 1.26547
\(563\) 8.00000 0.337160 0.168580 0.985688i \(-0.446082\pi\)
0.168580 + 0.985688i \(0.446082\pi\)
\(564\) −3.00000 −0.126323
\(565\) 0 0
\(566\) −14.0000 −0.588464
\(567\) −3.00000 −0.125988
\(568\) −6.00000 −0.251754
\(569\) −36.0000 −1.50920 −0.754599 0.656186i \(-0.772169\pi\)
−0.754599 + 0.656186i \(0.772169\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) −10.0000 −0.418121
\(573\) 3.00000 0.125327
\(574\) 18.0000 0.751305
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −33.0000 −1.37381 −0.686904 0.726748i \(-0.741031\pi\)
−0.686904 + 0.726748i \(0.741031\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 24.0000 0.997406
\(580\) 0 0
\(581\) 54.0000 2.24030
\(582\) −14.0000 −0.580319
\(583\) −5.00000 −0.207079
\(584\) −12.0000 −0.496564
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) 18.0000 0.742940 0.371470 0.928445i \(-0.378854\pi\)
0.371470 + 0.928445i \(0.378854\pi\)
\(588\) 2.00000 0.0824786
\(589\) 5.00000 0.206021
\(590\) 0 0
\(591\) −12.0000 −0.493614
\(592\) 3.00000 0.123299
\(593\) −12.0000 −0.492781 −0.246390 0.969171i \(-0.579245\pi\)
−0.246390 + 0.969171i \(0.579245\pi\)
\(594\) 5.00000 0.205152
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 17.0000 0.695764
\(598\) 12.0000 0.490716
\(599\) 35.0000 1.43006 0.715031 0.699093i \(-0.246413\pi\)
0.715031 + 0.699093i \(0.246413\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 3.00000 0.122271
\(603\) 11.0000 0.447955
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) −11.0000 −0.446844
\(607\) −20.0000 −0.811775 −0.405887 0.913923i \(-0.633038\pi\)
−0.405887 + 0.913923i \(0.633038\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −30.0000 −1.21566
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) −1.00000 −0.0404226
\(613\) 26.0000 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) 20.0000 0.807134
\(615\) 0 0
\(616\) −15.0000 −0.604367
\(617\) 13.0000 0.523360 0.261680 0.965155i \(-0.415723\pi\)
0.261680 + 0.965155i \(0.415723\pi\)
\(618\) 4.00000 0.160904
\(619\) −8.00000 −0.321547 −0.160774 0.986991i \(-0.551399\pi\)
−0.160774 + 0.986991i \(0.551399\pi\)
\(620\) 0 0
\(621\) −6.00000 −0.240772
\(622\) 2.00000 0.0801927
\(623\) −36.0000 −1.44231
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) −34.0000 −1.35891
\(627\) −5.00000 −0.199681
\(628\) 0 0
\(629\) −3.00000 −0.119618
\(630\) 0 0
\(631\) −14.0000 −0.557331 −0.278666 0.960388i \(-0.589892\pi\)
−0.278666 + 0.960388i \(0.589892\pi\)
\(632\) −5.00000 −0.198889
\(633\) 18.0000 0.715436
\(634\) −12.0000 −0.476581
\(635\) 0 0
\(636\) 1.00000 0.0396526
\(637\) 4.00000 0.158486
\(638\) 50.0000 1.97952
\(639\) 6.00000 0.237356
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) −5.00000 −0.197334
\(643\) −34.0000 −1.34083 −0.670415 0.741987i \(-0.733884\pi\)
−0.670415 + 0.741987i \(0.733884\pi\)
\(644\) 18.0000 0.709299
\(645\) 0 0
\(646\) 1.00000 0.0393445
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −40.0000 −1.57014
\(650\) 0 0
\(651\) −15.0000 −0.587896
\(652\) 0 0
\(653\) 16.0000 0.626128 0.313064 0.949732i \(-0.398644\pi\)
0.313064 + 0.949732i \(0.398644\pi\)
\(654\) −5.00000 −0.195515
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) 12.0000 0.468165
\(658\) −9.00000 −0.350857
\(659\) 16.0000 0.623272 0.311636 0.950202i \(-0.399123\pi\)
0.311636 + 0.950202i \(0.399123\pi\)
\(660\) 0 0
\(661\) 8.00000 0.311164 0.155582 0.987823i \(-0.450275\pi\)
0.155582 + 0.987823i \(0.450275\pi\)
\(662\) 25.0000 0.971653
\(663\) −2.00000 −0.0776736
\(664\) 18.0000 0.698535
\(665\) 0 0
\(666\) −3.00000 −0.116248
\(667\) −60.0000 −2.32321
\(668\) 18.0000 0.696441
\(669\) −18.0000 −0.695920
\(670\) 0 0
\(671\) 10.0000 0.386046
\(672\) 3.00000 0.115728
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 12.0000 0.462223
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −12.0000 −0.461197 −0.230599 0.973049i \(-0.574068\pi\)
−0.230599 + 0.973049i \(0.574068\pi\)
\(678\) 1.00000 0.0384048
\(679\) −42.0000 −1.61181
\(680\) 0 0
\(681\) 27.0000 1.03464
\(682\) 25.0000 0.957299
\(683\) 8.00000 0.306111 0.153056 0.988218i \(-0.451089\pi\)
0.153056 + 0.988218i \(0.451089\pi\)
\(684\) 1.00000 0.0382360
\(685\) 0 0
\(686\) −15.0000 −0.572703
\(687\) 24.0000 0.915657
\(688\) 1.00000 0.0381246
\(689\) 2.00000 0.0761939
\(690\) 0 0
\(691\) −48.0000 −1.82601 −0.913003 0.407953i \(-0.866243\pi\)
−0.913003 + 0.407953i \(0.866243\pi\)
\(692\) −20.0000 −0.760286
\(693\) 15.0000 0.569803
\(694\) −23.0000 −0.873068
\(695\) 0 0
\(696\) −10.0000 −0.379049
\(697\) −6.00000 −0.227266
\(698\) 14.0000 0.529908
\(699\) 10.0000 0.378235
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 3.00000 0.113147
\(704\) −5.00000 −0.188445
\(705\) 0 0
\(706\) −2.00000 −0.0752710
\(707\) −33.0000 −1.24109
\(708\) 8.00000 0.300658
\(709\) 25.0000 0.938895 0.469447 0.882960i \(-0.344453\pi\)
0.469447 + 0.882960i \(0.344453\pi\)
\(710\) 0 0
\(711\) 5.00000 0.187515
\(712\) −12.0000 −0.449719
\(713\) −30.0000 −1.12351
\(714\) −3.00000 −0.112272
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 13.0000 0.485494
\(718\) −15.0000 −0.559795
\(719\) 6.00000 0.223762 0.111881 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) 0 0
\(721\) 12.0000 0.446903
\(722\) 18.0000 0.669891
\(723\) 0 0
\(724\) −17.0000 −0.631800
\(725\) 0 0
\(726\) −14.0000 −0.519589
\(727\) −2.00000 −0.0741759 −0.0370879 0.999312i \(-0.511808\pi\)
−0.0370879 + 0.999312i \(0.511808\pi\)
\(728\) 6.00000 0.222375
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −1.00000 −0.0369863
\(732\) −2.00000 −0.0739221
\(733\) 16.0000 0.590973 0.295487 0.955347i \(-0.404518\pi\)
0.295487 + 0.955347i \(0.404518\pi\)
\(734\) 5.00000 0.184553
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) −55.0000 −2.02595
\(738\) −6.00000 −0.220863
\(739\) −19.0000 −0.698926 −0.349463 0.936950i \(-0.613636\pi\)
−0.349463 + 0.936950i \(0.613636\pi\)
\(740\) 0 0
\(741\) 2.00000 0.0734718
\(742\) 3.00000 0.110133
\(743\) −18.0000 −0.660356 −0.330178 0.943919i \(-0.607109\pi\)
−0.330178 + 0.943919i \(0.607109\pi\)
\(744\) −5.00000 −0.183309
\(745\) 0 0
\(746\) −18.0000 −0.659027
\(747\) −18.0000 −0.658586
\(748\) 5.00000 0.182818
\(749\) −15.0000 −0.548088
\(750\) 0 0
\(751\) −48.0000 −1.75154 −0.875772 0.482724i \(-0.839647\pi\)
−0.875772 + 0.482724i \(0.839647\pi\)
\(752\) −3.00000 −0.109399
\(753\) −18.0000 −0.655956
\(754\) −20.0000 −0.728357
\(755\) 0 0
\(756\) −3.00000 −0.109109
\(757\) −14.0000 −0.508839 −0.254419 0.967094i \(-0.581884\pi\)
−0.254419 + 0.967094i \(0.581884\pi\)
\(758\) 8.00000 0.290573
\(759\) 30.0000 1.08893
\(760\) 0 0
\(761\) 36.0000 1.30500 0.652499 0.757789i \(-0.273720\pi\)
0.652499 + 0.757789i \(0.273720\pi\)
\(762\) 12.0000 0.434714
\(763\) −15.0000 −0.543036
\(764\) 3.00000 0.108536
\(765\) 0 0
\(766\) −8.00000 −0.289052
\(767\) 16.0000 0.577727
\(768\) 1.00000 0.0360844
\(769\) −37.0000 −1.33425 −0.667127 0.744944i \(-0.732476\pi\)
−0.667127 + 0.744944i \(0.732476\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) 24.0000 0.863779
\(773\) 34.0000 1.22290 0.611448 0.791285i \(-0.290588\pi\)
0.611448 + 0.791285i \(0.290588\pi\)
\(774\) −1.00000 −0.0359443
\(775\) 0 0
\(776\) −14.0000 −0.502571
\(777\) −9.00000 −0.322873
\(778\) 5.00000 0.179259
\(779\) 6.00000 0.214972
\(780\) 0 0
\(781\) −30.0000 −1.07348
\(782\) −6.00000 −0.214560
\(783\) 10.0000 0.357371
\(784\) 2.00000 0.0714286
\(785\) 0 0
\(786\) 20.0000 0.713376
\(787\) −18.0000 −0.641631 −0.320815 0.947142i \(-0.603957\pi\)
−0.320815 + 0.947142i \(0.603957\pi\)
\(788\) −12.0000 −0.427482
\(789\) −3.00000 −0.106803
\(790\) 0 0
\(791\) 3.00000 0.106668
\(792\) 5.00000 0.177667
\(793\) −4.00000 −0.142044
\(794\) 21.0000 0.745262
\(795\) 0 0
\(796\) 17.0000 0.602549
\(797\) 11.0000 0.389640 0.194820 0.980839i \(-0.437588\pi\)
0.194820 + 0.980839i \(0.437588\pi\)
\(798\) 3.00000 0.106199
\(799\) 3.00000 0.106132
\(800\) 0 0
\(801\) 12.0000 0.423999
\(802\) 18.0000 0.635602
\(803\) −60.0000 −2.11735
\(804\) 11.0000 0.387940
\(805\) 0 0
\(806\) −10.0000 −0.352235
\(807\) −6.00000 −0.211210
\(808\) −11.0000 −0.386979
\(809\) −39.0000 −1.37117 −0.685583 0.727994i \(-0.740453\pi\)
−0.685583 + 0.727994i \(0.740453\pi\)
\(810\) 0 0
\(811\) 56.0000 1.96643 0.983213 0.182462i \(-0.0584065\pi\)
0.983213 + 0.182462i \(0.0584065\pi\)
\(812\) −30.0000 −1.05279
\(813\) 14.0000 0.491001
\(814\) 15.0000 0.525750
\(815\) 0 0
\(816\) −1.00000 −0.0350070
\(817\) 1.00000 0.0349856
\(818\) 2.00000 0.0699284
\(819\) −6.00000 −0.209657
\(820\) 0 0
\(821\) 54.0000 1.88461 0.942306 0.334751i \(-0.108652\pi\)
0.942306 + 0.334751i \(0.108652\pi\)
\(822\) 16.0000 0.558064
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) 53.0000 1.84299 0.921495 0.388390i \(-0.126968\pi\)
0.921495 + 0.388390i \(0.126968\pi\)
\(828\) −6.00000 −0.208514
\(829\) −30.0000 −1.04194 −0.520972 0.853574i \(-0.674430\pi\)
−0.520972 + 0.853574i \(0.674430\pi\)
\(830\) 0 0
\(831\) 31.0000 1.07538
\(832\) 2.00000 0.0693375
\(833\) −2.00000 −0.0692959
\(834\) −8.00000 −0.277017
\(835\) 0 0
\(836\) −5.00000 −0.172929
\(837\) 5.00000 0.172825
\(838\) −24.0000 −0.829066
\(839\) −8.00000 −0.276191 −0.138095 0.990419i \(-0.544098\pi\)
−0.138095 + 0.990419i \(0.544098\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) −18.0000 −0.620321
\(843\) −30.0000 −1.03325
\(844\) 18.0000 0.619586
\(845\) 0 0
\(846\) 3.00000 0.103142
\(847\) −42.0000 −1.44314
\(848\) 1.00000 0.0343401
\(849\) 14.0000 0.480479
\(850\) 0 0
\(851\) −18.0000 −0.617032
\(852\) 6.00000 0.205557
\(853\) 37.0000 1.26686 0.633428 0.773802i \(-0.281647\pi\)
0.633428 + 0.773802i \(0.281647\pi\)
\(854\) −6.00000 −0.205316
\(855\) 0 0
\(856\) −5.00000 −0.170896
\(857\) 21.0000 0.717346 0.358673 0.933463i \(-0.383229\pi\)
0.358673 + 0.933463i \(0.383229\pi\)
\(858\) 10.0000 0.341394
\(859\) −23.0000 −0.784750 −0.392375 0.919805i \(-0.628346\pi\)
−0.392375 + 0.919805i \(0.628346\pi\)
\(860\) 0 0
\(861\) −18.0000 −0.613438
\(862\) 0 0
\(863\) 33.0000 1.12333 0.561667 0.827364i \(-0.310160\pi\)
0.561667 + 0.827364i \(0.310160\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 21.0000 0.713609
\(867\) 1.00000 0.0339618
\(868\) −15.0000 −0.509133
\(869\) −25.0000 −0.848067
\(870\) 0 0
\(871\) 22.0000 0.745442
\(872\) −5.00000 −0.169321
\(873\) 14.0000 0.473828
\(874\) 6.00000 0.202953
\(875\) 0 0
\(876\) 12.0000 0.405442
\(877\) −14.0000 −0.472746 −0.236373 0.971662i \(-0.575959\pi\)
−0.236373 + 0.971662i \(0.575959\pi\)
\(878\) −16.0000 −0.539974
\(879\) 18.0000 0.607125
\(880\) 0 0
\(881\) −9.00000 −0.303218 −0.151609 0.988441i \(-0.548445\pi\)
−0.151609 + 0.988441i \(0.548445\pi\)
\(882\) −2.00000 −0.0673435
\(883\) 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
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) −3.00000 −0.100673
\(889\) 36.0000 1.20740
\(890\) 0 0
\(891\) −5.00000 −0.167506
\(892\) −18.0000 −0.602685
\(893\) −3.00000 −0.100391
\(894\) 6.00000 0.200670
\(895\) 0 0
\(896\) 3.00000 0.100223
\(897\) −12.0000 −0.400668
\(898\) −21.0000 −0.700779
\(899\) 50.0000 1.66759
\(900\) 0 0
\(901\) −1.00000 −0.0333148
\(902\) 30.0000 0.998891
\(903\) −3.00000 −0.0998337
\(904\) 1.00000 0.0332595
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) 30.0000 0.996134 0.498067 0.867139i \(-0.334043\pi\)
0.498067 + 0.867139i \(0.334043\pi\)
\(908\) 27.0000 0.896026
\(909\) 11.0000 0.364847
\(910\) 0 0
\(911\) 56.0000 1.85536 0.927681 0.373373i \(-0.121799\pi\)
0.927681 + 0.373373i \(0.121799\pi\)
\(912\) 1.00000 0.0331133
\(913\) 90.0000 2.97857
\(914\) −25.0000 −0.826927
\(915\) 0 0
\(916\) 24.0000 0.792982
\(917\) 60.0000 1.98137
\(918\) 1.00000 0.0330049
\(919\) 44.0000 1.45143 0.725713 0.687998i \(-0.241510\pi\)
0.725713 + 0.687998i \(0.241510\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) 15.0000 0.493999
\(923\) 12.0000 0.394985
\(924\) 15.0000 0.493464
\(925\) 0 0
\(926\) 0 0
\(927\) −4.00000 −0.131377
\(928\) −10.0000 −0.328266
\(929\) 27.0000 0.885841 0.442921 0.896561i \(-0.353942\pi\)
0.442921 + 0.896561i \(0.353942\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) 10.0000 0.327561
\(933\) −2.00000 −0.0654771
\(934\) 18.0000 0.588978
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) −10.0000 −0.326686 −0.163343 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(938\) 33.0000 1.07749
\(939\) 34.0000 1.10955
\(940\) 0 0
\(941\) 22.0000 0.717180 0.358590 0.933495i \(-0.383258\pi\)
0.358590 + 0.933495i \(0.383258\pi\)
\(942\) 0 0
\(943\) −36.0000 −1.17232
\(944\) 8.00000 0.260378
\(945\) 0 0
\(946\) 5.00000 0.162564
\(947\) −61.0000 −1.98223 −0.991117 0.132994i \(-0.957541\pi\)
−0.991117 + 0.132994i \(0.957541\pi\)
\(948\) 5.00000 0.162392
\(949\) 24.0000 0.779073
\(950\) 0 0
\(951\) 12.0000 0.389127
\(952\) −3.00000 −0.0972306
\(953\) 14.0000 0.453504 0.226752 0.973952i \(-0.427189\pi\)
0.226752 + 0.973952i \(0.427189\pi\)
\(954\) −1.00000 −0.0323762
\(955\) 0 0
\(956\) 13.0000 0.420450
\(957\) −50.0000 −1.61627
\(958\) −26.0000 −0.840022
\(959\) 48.0000 1.55000
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) −6.00000 −0.193448
\(963\) 5.00000 0.161123
\(964\) 0 0
\(965\) 0 0
\(966\) −18.0000 −0.579141
\(967\) 36.0000 1.15768 0.578841 0.815440i \(-0.303505\pi\)
0.578841 + 0.815440i \(0.303505\pi\)
\(968\) −14.0000 −0.449977
\(969\) −1.00000 −0.0321246
\(970\) 0 0
\(971\) −38.0000 −1.21948 −0.609739 0.792602i \(-0.708726\pi\)
−0.609739 + 0.792602i \(0.708726\pi\)
\(972\) 1.00000 0.0320750
\(973\) −24.0000 −0.769405
\(974\) −40.0000 −1.28168
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) 10.0000 0.319928 0.159964 0.987123i \(-0.448862\pi\)
0.159964 + 0.987123i \(0.448862\pi\)
\(978\) 0 0
\(979\) −60.0000 −1.91761
\(980\) 0 0
\(981\) 5.00000 0.159638
\(982\) −4.00000 −0.127645
\(983\) 6.00000 0.191370 0.0956851 0.995412i \(-0.469496\pi\)
0.0956851 + 0.995412i \(0.469496\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) 10.0000 0.318465
\(987\) 9.00000 0.286473
\(988\) 2.00000 0.0636285
\(989\) −6.00000 −0.190789
\(990\) 0 0
\(991\) 52.0000 1.65183 0.825917 0.563791i \(-0.190658\pi\)
0.825917 + 0.563791i \(0.190658\pi\)
\(992\) −5.00000 −0.158750
\(993\) −25.0000 −0.793351
\(994\) 18.0000 0.570925
\(995\) 0 0
\(996\) −18.0000 −0.570352
\(997\) 43.0000 1.36182 0.680912 0.732365i \(-0.261584\pi\)
0.680912 + 0.732365i \(0.261584\pi\)
\(998\) −22.0000 −0.696398
\(999\) 3.00000 0.0949158
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.h.1.1 1
3.2 odd 2 7650.2.a.bl.1.1 1
5.2 odd 4 2550.2.d.a.2449.1 2
5.3 odd 4 2550.2.d.a.2449.2 2
5.4 even 2 2550.2.a.z.1.1 yes 1
15.14 odd 2 7650.2.a.be.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2550.2.a.h.1.1 1 1.1 even 1 trivial
2550.2.a.z.1.1 yes 1 5.4 even 2
2550.2.d.a.2449.1 2 5.2 odd 4
2550.2.d.a.2449.2 2 5.3 odd 4
7650.2.a.be.1.1 1 15.14 odd 2
7650.2.a.bl.1.1 1 3.2 odd 2