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Properties

Label 5550.2.a.u.1.1
Level $5550$
Weight $2$
Character 5550.1
Self dual yes
Analytic conductor $44.317$
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] = [5550,2,Mod(1,5550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5550, 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("5550.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5550.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: \(44.3169731218\)
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 \(\chi\) \(=\) 5550.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} +1.00000 q^{12} -4.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -3.00000 q^{19} +4.00000 q^{21} +2.00000 q^{22} -3.00000 q^{23} -1.00000 q^{24} +4.00000 q^{26} +1.00000 q^{27} +4.00000 q^{28} -1.00000 q^{32} -2.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} -1.00000 q^{37} +3.00000 q^{38} -4.00000 q^{39} -3.00000 q^{41} -4.00000 q^{42} +1.00000 q^{43} -2.00000 q^{44} +3.00000 q^{46} -6.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} -2.00000 q^{51} -4.00000 q^{52} +3.00000 q^{53} -1.00000 q^{54} -4.00000 q^{56} -3.00000 q^{57} -7.00000 q^{59} -8.00000 q^{61} +4.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} +4.00000 q^{67} -2.00000 q^{68} -3.00000 q^{69} -12.0000 q^{71} -1.00000 q^{72} -9.00000 q^{73} +1.00000 q^{74} -3.00000 q^{76} -8.00000 q^{77} +4.00000 q^{78} +3.00000 q^{79} +1.00000 q^{81} +3.00000 q^{82} +8.00000 q^{83} +4.00000 q^{84} -1.00000 q^{86} +2.00000 q^{88} -8.00000 q^{89} -16.0000 q^{91} -3.00000 q^{92} +6.00000 q^{94} -1.00000 q^{96} +2.00000 q^{97} -9.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 1.00000 0.288675
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −1.00000 −0.235702
\(19\) −3.00000 −0.688247 −0.344124 − 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 2.00000 0.426401
\(23\) −3.00000 −0.625543 −0.312772 − 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) 1.00000 0.192450
\(28\) 4.00000 0.755929
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.00000 −0.348155
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −1.00000 −0.164399
\(38\) 3.00000 0.486664
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 − 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) −4.00000 −0.617213
\(43\) 1.00000 0.152499 0.0762493 − 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 3.00000 0.442326
\(47\) −6.00000 −0.875190 −0.437595 − 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 1.00000 0.144338
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) −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
\(56\) −4.00000 −0.534522
\(57\) −3.00000 −0.397360
\(58\) 0 0
\(59\) −7.00000 −0.911322 −0.455661 − 0.890153i \(-0.650597\pi\)
−0.455661 + 0.890153i \(0.650597\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −2.00000 −0.242536
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) −1.00000 −0.117851
\(73\) −9.00000 −1.05337 −0.526685 − 0.850060i \(-0.676565\pi\)
−0.526685 + 0.850060i \(0.676565\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −3.00000 −0.344124
\(77\) −8.00000 −0.911685
\(78\) 4.00000 0.452911
\(79\) 3.00000 0.337526 0.168763 − 0.985657i \(-0.446023\pi\)
0.168763 + 0.985657i \(0.446023\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 3.00000 0.331295
\(83\) 8.00000 0.878114 0.439057 − 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) −1.00000 −0.107833
\(87\) 0 0
\(88\) 2.00000 0.213201
\(89\) −8.00000 −0.847998 −0.423999 − 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) −16.0000 −1.67726
\(92\) −3.00000 −0.312772
\(93\) 0 0
\(94\) 6.00000 0.618853
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −9.00000 −0.909137
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) −15.0000 −1.49256 −0.746278 − 0.665635i \(-0.768161\pi\)
−0.746278 + 0.665635i \(0.768161\pi\)
\(102\) 2.00000 0.198030
\(103\) −1.00000 −0.0985329 −0.0492665 − 0.998786i \(-0.515688\pi\)
−0.0492665 + 0.998786i \(0.515688\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) −3.00000 −0.291386
\(107\) −6.00000 −0.580042 −0.290021 − 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 1.00000 0.0962250
\(109\) −2.00000 −0.191565 −0.0957826 − 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) −1.00000 −0.0949158
\(112\) 4.00000 0.377964
\(113\) −10.0000 −0.940721 −0.470360 − 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 3.00000 0.280976
\(115\) 0 0
\(116\) 0 0
\(117\) −4.00000 −0.369800
\(118\) 7.00000 0.644402
\(119\) −8.00000 −0.733359
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 8.00000 0.724286
\(123\) −3.00000 −0.270501
\(124\) 0 0
\(125\) 0 0
\(126\) −4.00000 −0.356348
\(127\) 18.0000 1.59724 0.798621 − 0.601834i \(-0.205563\pi\)
0.798621 + 0.601834i \(0.205563\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 1.00000 0.0880451
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 − 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) −2.00000 −0.174078
\(133\) −12.0000 −1.04053
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) −2.00000 −0.170872 −0.0854358 − 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 3.00000 0.255377
\(139\) 8.00000 0.678551 0.339276 − 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 12.0000 1.00702
\(143\) 8.00000 0.668994
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 9.00000 0.744845
\(147\) 9.00000 0.742307
\(148\) −1.00000 −0.0821995
\(149\) 17.0000 1.39269 0.696347 − 0.717705i \(-0.254807\pi\)
0.696347 + 0.717705i \(0.254807\pi\)
\(150\) 0 0
\(151\) −18.0000 −1.46482 −0.732410 − 0.680864i \(-0.761604\pi\)
−0.732410 + 0.680864i \(0.761604\pi\)
\(152\) 3.00000 0.243332
\(153\) −2.00000 −0.161690
\(154\) 8.00000 0.644658
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) 19.0000 1.51637 0.758183 − 0.652042i \(-0.226088\pi\)
0.758183 + 0.652042i \(0.226088\pi\)
\(158\) −3.00000 −0.238667
\(159\) 3.00000 0.237915
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) −1.00000 −0.0785674
\(163\) 13.0000 1.01824 0.509119 − 0.860696i \(-0.329971\pi\)
0.509119 + 0.860696i \(0.329971\pi\)
\(164\) −3.00000 −0.234261
\(165\) 0 0
\(166\) −8.00000 −0.620920
\(167\) −13.0000 −1.00597 −0.502985 − 0.864295i \(-0.667765\pi\)
−0.502985 + 0.864295i \(0.667765\pi\)
\(168\) −4.00000 −0.308607
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −3.00000 −0.229416
\(172\) 1.00000 0.0762493
\(173\) 14.0000 1.06440 0.532200 − 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) −7.00000 −0.526152
\(178\) 8.00000 0.599625
\(179\) −4.00000 −0.298974 −0.149487 − 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) 5.00000 0.371647 0.185824 − 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) 16.0000 1.18600
\(183\) −8.00000 −0.591377
\(184\) 3.00000 0.221163
\(185\) 0 0
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) −6.00000 −0.437595
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) −19.0000 −1.37479 −0.687396 − 0.726283i \(-0.741246\pi\)
−0.687396 + 0.726283i \(0.741246\pi\)
\(192\) 1.00000 0.0721688
\(193\) 16.0000 1.15171 0.575853 − 0.817554i \(-0.304670\pi\)
0.575853 + 0.817554i \(0.304670\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 17.0000 1.21120 0.605600 − 0.795769i \(-0.292933\pi\)
0.605600 + 0.795769i \(0.292933\pi\)
\(198\) 2.00000 0.142134
\(199\) 3.00000 0.212664 0.106332 − 0.994331i \(-0.466089\pi\)
0.106332 + 0.994331i \(0.466089\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 15.0000 1.05540
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) 1.00000 0.0696733
\(207\) −3.00000 −0.208514
\(208\) −4.00000 −0.277350
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) −26.0000 −1.78991 −0.894957 − 0.446153i \(-0.852794\pi\)
−0.894957 + 0.446153i \(0.852794\pi\)
\(212\) 3.00000 0.206041
\(213\) −12.0000 −0.822226
\(214\) 6.00000 0.410152
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 2.00000 0.135457
\(219\) −9.00000 −0.608164
\(220\) 0 0
\(221\) 8.00000 0.538138
\(222\) 1.00000 0.0671156
\(223\) −2.00000 −0.133930 −0.0669650 − 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) −4.00000 −0.267261
\(225\) 0 0
\(226\) 10.0000 0.665190
\(227\) −7.00000 −0.464606 −0.232303 − 0.972643i \(-0.574626\pi\)
−0.232303 + 0.972643i \(0.574626\pi\)
\(228\) −3.00000 −0.198680
\(229\) 15.0000 0.991228 0.495614 − 0.868543i \(-0.334943\pi\)
0.495614 + 0.868543i \(0.334943\pi\)
\(230\) 0 0
\(231\) −8.00000 −0.526361
\(232\) 0 0
\(233\) −11.0000 −0.720634 −0.360317 − 0.932830i \(-0.617331\pi\)
−0.360317 + 0.932830i \(0.617331\pi\)
\(234\) 4.00000 0.261488
\(235\) 0 0
\(236\) −7.00000 −0.455661
\(237\) 3.00000 0.194871
\(238\) 8.00000 0.518563
\(239\) 3.00000 0.194054 0.0970269 − 0.995282i \(-0.469067\pi\)
0.0970269 + 0.995282i \(0.469067\pi\)
\(240\) 0 0
\(241\) 24.0000 1.54598 0.772988 − 0.634421i \(-0.218761\pi\)
0.772988 + 0.634421i \(0.218761\pi\)
\(242\) 7.00000 0.449977
\(243\) 1.00000 0.0641500
\(244\) −8.00000 −0.512148
\(245\) 0 0
\(246\) 3.00000 0.191273
\(247\) 12.0000 0.763542
\(248\) 0 0
\(249\) 8.00000 0.506979
\(250\) 0 0
\(251\) −9.00000 −0.568075 −0.284037 − 0.958813i \(-0.591674\pi\)
−0.284037 + 0.958813i \(0.591674\pi\)
\(252\) 4.00000 0.251976
\(253\) 6.00000 0.377217
\(254\) −18.0000 −1.12942
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 − 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) −1.00000 −0.0622573
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 0 0
\(262\) 8.00000 0.494242
\(263\) 6.00000 0.369976 0.184988 − 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 2.00000 0.123091
\(265\) 0 0
\(266\) 12.0000 0.735767
\(267\) −8.00000 −0.489592
\(268\) 4.00000 0.244339
\(269\) −3.00000 −0.182913 −0.0914566 − 0.995809i \(-0.529152\pi\)
−0.0914566 + 0.995809i \(0.529152\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 − 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) −2.00000 −0.121268
\(273\) −16.0000 −0.968364
\(274\) 2.00000 0.120824
\(275\) 0 0
\(276\) −3.00000 −0.180579
\(277\) −32.0000 −1.92269 −0.961347 − 0.275340i \(-0.911209\pi\)
−0.961347 + 0.275340i \(0.911209\pi\)
\(278\) −8.00000 −0.479808
\(279\) 0 0
\(280\) 0 0
\(281\) 4.00000 0.238620 0.119310 − 0.992857i \(-0.461932\pi\)
0.119310 + 0.992857i \(0.461932\pi\)
\(282\) 6.00000 0.357295
\(283\) 17.0000 1.01055 0.505273 − 0.862960i \(-0.331392\pi\)
0.505273 + 0.862960i \(0.331392\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) −8.00000 −0.473050
\(287\) −12.0000 −0.708338
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) −9.00000 −0.526685
\(293\) 7.00000 0.408944 0.204472 − 0.978872i \(-0.434452\pi\)
0.204472 + 0.978872i \(0.434452\pi\)
\(294\) −9.00000 −0.524891
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) −2.00000 −0.116052
\(298\) −17.0000 −0.984784
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 18.0000 1.03578
\(303\) −15.0000 −0.861727
\(304\) −3.00000 −0.172062
\(305\) 0 0
\(306\) 2.00000 0.114332
\(307\) 6.00000 0.342438 0.171219 − 0.985233i \(-0.445229\pi\)
0.171219 + 0.985233i \(0.445229\pi\)
\(308\) −8.00000 −0.455842
\(309\) −1.00000 −0.0568880
\(310\) 0 0
\(311\) −7.00000 −0.396934 −0.198467 − 0.980108i \(-0.563596\pi\)
−0.198467 + 0.980108i \(0.563596\pi\)
\(312\) 4.00000 0.226455
\(313\) −30.0000 −1.69570 −0.847850 − 0.530236i \(-0.822103\pi\)
−0.847850 + 0.530236i \(0.822103\pi\)
\(314\) −19.0000 −1.07223
\(315\) 0 0
\(316\) 3.00000 0.168763
\(317\) −13.0000 −0.730153 −0.365076 − 0.930978i \(-0.618957\pi\)
−0.365076 + 0.930978i \(0.618957\pi\)
\(318\) −3.00000 −0.168232
\(319\) 0 0
\(320\) 0 0
\(321\) −6.00000 −0.334887
\(322\) 12.0000 0.668734
\(323\) 6.00000 0.333849
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −13.0000 −0.720003
\(327\) −2.00000 −0.110600
\(328\) 3.00000 0.165647
\(329\) −24.0000 −1.32316
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 − 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 8.00000 0.439057
\(333\) −1.00000 −0.0547997
\(334\) 13.0000 0.711328
\(335\) 0 0
\(336\) 4.00000 0.218218
\(337\) −9.00000 −0.490261 −0.245131 − 0.969490i \(-0.578831\pi\)
−0.245131 + 0.969490i \(0.578831\pi\)
\(338\) −3.00000 −0.163178
\(339\) −10.0000 −0.543125
\(340\) 0 0
\(341\) 0 0
\(342\) 3.00000 0.162221
\(343\) 8.00000 0.431959
\(344\) −1.00000 −0.0539164
\(345\) 0 0
\(346\) −14.0000 −0.752645
\(347\) 9.00000 0.483145 0.241573 − 0.970383i \(-0.422337\pi\)
0.241573 + 0.970383i \(0.422337\pi\)
\(348\) 0 0
\(349\) 9.00000 0.481759 0.240879 − 0.970555i \(-0.422564\pi\)
0.240879 + 0.970555i \(0.422564\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 2.00000 0.106600
\(353\) 16.0000 0.851594 0.425797 − 0.904819i \(-0.359994\pi\)
0.425797 + 0.904819i \(0.359994\pi\)
\(354\) 7.00000 0.372046
\(355\) 0 0
\(356\) −8.00000 −0.423999
\(357\) −8.00000 −0.423405
\(358\) 4.00000 0.211407
\(359\) −6.00000 −0.316668 −0.158334 − 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) −5.00000 −0.262794
\(363\) −7.00000 −0.367405
\(364\) −16.0000 −0.838628
\(365\) 0 0
\(366\) 8.00000 0.418167
\(367\) −2.00000 −0.104399 −0.0521996 − 0.998637i \(-0.516623\pi\)
−0.0521996 + 0.998637i \(0.516623\pi\)
\(368\) −3.00000 −0.156386
\(369\) −3.00000 −0.156174
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) −22.0000 −1.13912 −0.569558 − 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) −4.00000 −0.206835
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) 0 0
\(378\) −4.00000 −0.205738
\(379\) −14.0000 −0.719132 −0.359566 − 0.933120i \(-0.617075\pi\)
−0.359566 + 0.933120i \(0.617075\pi\)
\(380\) 0 0
\(381\) 18.0000 0.922168
\(382\) 19.0000 0.972125
\(383\) −21.0000 −1.07305 −0.536525 − 0.843884i \(-0.680263\pi\)
−0.536525 + 0.843884i \(0.680263\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −16.0000 −0.814379
\(387\) 1.00000 0.0508329
\(388\) 2.00000 0.101535
\(389\) −2.00000 −0.101404 −0.0507020 − 0.998714i \(-0.516146\pi\)
−0.0507020 + 0.998714i \(0.516146\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) −9.00000 −0.454569
\(393\) −8.00000 −0.403547
\(394\) −17.0000 −0.856448
\(395\) 0 0
\(396\) −2.00000 −0.100504
\(397\) 15.0000 0.752828 0.376414 − 0.926451i \(-0.377157\pi\)
0.376414 + 0.926451i \(0.377157\pi\)
\(398\) −3.00000 −0.150376
\(399\) −12.0000 −0.600751
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 − 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) −4.00000 −0.199502
\(403\) 0 0
\(404\) −15.0000 −0.746278
\(405\) 0 0
\(406\) 0 0
\(407\) 2.00000 0.0991363
\(408\) 2.00000 0.0990148
\(409\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) −1.00000 −0.0492665
\(413\) −28.0000 −1.37779
\(414\) 3.00000 0.147442
\(415\) 0 0
\(416\) 4.00000 0.196116
\(417\) 8.00000 0.391762
\(418\) −6.00000 −0.293470
\(419\) 2.00000 0.0977064 0.0488532 − 0.998806i \(-0.484443\pi\)
0.0488532 + 0.998806i \(0.484443\pi\)
\(420\) 0 0
\(421\) −12.0000 −0.584844 −0.292422 − 0.956289i \(-0.594461\pi\)
−0.292422 + 0.956289i \(0.594461\pi\)
\(422\) 26.0000 1.26566
\(423\) −6.00000 −0.291730
\(424\) −3.00000 −0.145693
\(425\) 0 0
\(426\) 12.0000 0.581402
\(427\) −32.0000 −1.54859
\(428\) −6.00000 −0.290021
\(429\) 8.00000 0.386244
\(430\) 0 0
\(431\) −3.00000 −0.144505 −0.0722525 − 0.997386i \(-0.523019\pi\)
−0.0722525 + 0.997386i \(0.523019\pi\)
\(432\) 1.00000 0.0481125
\(433\) 30.0000 1.44171 0.720854 − 0.693087i \(-0.243750\pi\)
0.720854 + 0.693087i \(0.243750\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) 9.00000 0.430528
\(438\) 9.00000 0.430037
\(439\) 21.0000 1.00228 0.501138 − 0.865368i \(-0.332915\pi\)
0.501138 + 0.865368i \(0.332915\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) −8.00000 −0.380521
\(443\) −10.0000 −0.475114 −0.237557 − 0.971374i \(-0.576347\pi\)
−0.237557 + 0.971374i \(0.576347\pi\)
\(444\) −1.00000 −0.0474579
\(445\) 0 0
\(446\) 2.00000 0.0947027
\(447\) 17.0000 0.804072
\(448\) 4.00000 0.188982
\(449\) 18.0000 0.849473 0.424736 − 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) −10.0000 −0.470360
\(453\) −18.0000 −0.845714
\(454\) 7.00000 0.328526
\(455\) 0 0
\(456\) 3.00000 0.140488
\(457\) −8.00000 −0.374224 −0.187112 − 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) −15.0000 −0.700904
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) −10.0000 −0.465746 −0.232873 − 0.972507i \(-0.574813\pi\)
−0.232873 + 0.972507i \(0.574813\pi\)
\(462\) 8.00000 0.372194
\(463\) −4.00000 −0.185896 −0.0929479 − 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 11.0000 0.509565
\(467\) 12.0000 0.555294 0.277647 − 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) −4.00000 −0.184900
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) 19.0000 0.875474
\(472\) 7.00000 0.322201
\(473\) −2.00000 −0.0919601
\(474\) −3.00000 −0.137795
\(475\) 0 0
\(476\) −8.00000 −0.366679
\(477\) 3.00000 0.137361
\(478\) −3.00000 −0.137217
\(479\) −24.0000 −1.09659 −0.548294 − 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) −24.0000 −1.09317
\(483\) −12.0000 −0.546019
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 8.00000 0.362515 0.181257 − 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 8.00000 0.362143
\(489\) 13.0000 0.587880
\(490\) 0 0
\(491\) 30.0000 1.35388 0.676941 − 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) −3.00000 −0.135250
\(493\) 0 0
\(494\) −12.0000 −0.539906
\(495\) 0 0
\(496\) 0 0
\(497\) −48.0000 −2.15309
\(498\) −8.00000 −0.358489
\(499\) −11.0000 −0.492428 −0.246214 − 0.969216i \(-0.579187\pi\)
−0.246214 + 0.969216i \(0.579187\pi\)
\(500\) 0 0
\(501\) −13.0000 −0.580797
\(502\) 9.00000 0.401690
\(503\) −32.0000 −1.42681 −0.713405 − 0.700752i \(-0.752848\pi\)
−0.713405 + 0.700752i \(0.752848\pi\)
\(504\) −4.00000 −0.178174
\(505\) 0 0
\(506\) −6.00000 −0.266733
\(507\) 3.00000 0.133235
\(508\) 18.0000 0.798621
\(509\) 15.0000 0.664863 0.332432 − 0.943127i \(-0.392131\pi\)
0.332432 + 0.943127i \(0.392131\pi\)
\(510\) 0 0
\(511\) −36.0000 −1.59255
\(512\) −1.00000 −0.0441942
\(513\) −3.00000 −0.132453
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) 1.00000 0.0440225
\(517\) 12.0000 0.527759
\(518\) 4.00000 0.175750
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) 9.00000 0.394297 0.197149 − 0.980374i \(-0.436832\pi\)
0.197149 + 0.980374i \(0.436832\pi\)
\(522\) 0 0
\(523\) 12.0000 0.524723 0.262362 − 0.964970i \(-0.415499\pi\)
0.262362 + 0.964970i \(0.415499\pi\)
\(524\) −8.00000 −0.349482
\(525\) 0 0
\(526\) −6.00000 −0.261612
\(527\) 0 0
\(528\) −2.00000 −0.0870388
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) −7.00000 −0.303774
\(532\) −12.0000 −0.520266
\(533\) 12.0000 0.519778
\(534\) 8.00000 0.346194
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) −4.00000 −0.172613
\(538\) 3.00000 0.129339
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 − 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) −2.00000 −0.0859074
\(543\) 5.00000 0.214571
\(544\) 2.00000 0.0857493
\(545\) 0 0
\(546\) 16.0000 0.684737
\(547\) 20.0000 0.855138 0.427569 − 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) −2.00000 −0.0854358
\(549\) −8.00000 −0.341432
\(550\) 0 0
\(551\) 0 0
\(552\) 3.00000 0.127688
\(553\) 12.0000 0.510292
\(554\) 32.0000 1.35955
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) 12.0000 0.508456 0.254228 − 0.967144i \(-0.418179\pi\)
0.254228 + 0.967144i \(0.418179\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) 4.00000 0.168880
\(562\) −4.00000 −0.168730
\(563\) 19.0000 0.800755 0.400377 − 0.916350i \(-0.368879\pi\)
0.400377 + 0.916350i \(0.368879\pi\)
\(564\) −6.00000 −0.252646
\(565\) 0 0
\(566\) −17.0000 −0.714563
\(567\) 4.00000 0.167984
\(568\) 12.0000 0.503509
\(569\) −46.0000 −1.92842 −0.964210 − 0.265139i \(-0.914582\pi\)
−0.964210 + 0.265139i \(0.914582\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 − 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 8.00000 0.334497
\(573\) −19.0000 −0.793736
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 20.0000 0.832611 0.416305 − 0.909225i \(-0.363325\pi\)
0.416305 + 0.909225i \(0.363325\pi\)
\(578\) 13.0000 0.540729
\(579\) 16.0000 0.664937
\(580\) 0 0
\(581\) 32.0000 1.32758
\(582\) −2.00000 −0.0829027
\(583\) −6.00000 −0.248495
\(584\) 9.00000 0.372423
\(585\) 0 0
\(586\) −7.00000 −0.289167
\(587\) −33.0000 −1.36206 −0.681028 − 0.732257i \(-0.738467\pi\)
−0.681028 + 0.732257i \(0.738467\pi\)
\(588\) 9.00000 0.371154
\(589\) 0 0
\(590\) 0 0
\(591\) 17.0000 0.699287
\(592\) −1.00000 −0.0410997
\(593\) 21.0000 0.862367 0.431183 − 0.902264i \(-0.358096\pi\)
0.431183 + 0.902264i \(0.358096\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) 17.0000 0.696347
\(597\) 3.00000 0.122782
\(598\) −12.0000 −0.490716
\(599\) 14.0000 0.572024 0.286012 − 0.958226i \(-0.407670\pi\)
0.286012 + 0.958226i \(0.407670\pi\)
\(600\) 0 0
\(601\) −13.0000 −0.530281 −0.265141 − 0.964210i \(-0.585418\pi\)
−0.265141 + 0.964210i \(0.585418\pi\)
\(602\) −4.00000 −0.163028
\(603\) 4.00000 0.162893
\(604\) −18.0000 −0.732410
\(605\) 0 0
\(606\) 15.0000 0.609333
\(607\) 24.0000 0.974130 0.487065 − 0.873366i \(-0.338067\pi\)
0.487065 + 0.873366i \(0.338067\pi\)
\(608\) 3.00000 0.121666
\(609\) 0 0
\(610\) 0 0
\(611\) 24.0000 0.970936
\(612\) −2.00000 −0.0808452
\(613\) −41.0000 −1.65597 −0.827987 − 0.560747i \(-0.810514\pi\)
−0.827987 + 0.560747i \(0.810514\pi\)
\(614\) −6.00000 −0.242140
\(615\) 0 0
\(616\) 8.00000 0.322329
\(617\) −22.0000 −0.885687 −0.442843 − 0.896599i \(-0.646030\pi\)
−0.442843 + 0.896599i \(0.646030\pi\)
\(618\) 1.00000 0.0402259
\(619\) 4.00000 0.160774 0.0803868 − 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) −3.00000 −0.120386
\(622\) 7.00000 0.280674
\(623\) −32.0000 −1.28205
\(624\) −4.00000 −0.160128
\(625\) 0 0
\(626\) 30.0000 1.19904
\(627\) 6.00000 0.239617
\(628\) 19.0000 0.758183
\(629\) 2.00000 0.0797452
\(630\) 0 0
\(631\) −28.0000 −1.11466 −0.557331 − 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) −3.00000 −0.119334
\(633\) −26.0000 −1.03341
\(634\) 13.0000 0.516296
\(635\) 0 0
\(636\) 3.00000 0.118958
\(637\) −36.0000 −1.42637
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) 9.00000 0.355479 0.177739 − 0.984078i \(-0.443122\pi\)
0.177739 + 0.984078i \(0.443122\pi\)
\(642\) 6.00000 0.236801
\(643\) 20.0000 0.788723 0.394362 − 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) −12.0000 −0.472866
\(645\) 0 0
\(646\) −6.00000 −0.236067
\(647\) −33.0000 −1.29736 −0.648682 − 0.761060i \(-0.724679\pi\)
−0.648682 + 0.761060i \(0.724679\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 14.0000 0.549548
\(650\) 0 0
\(651\) 0 0
\(652\) 13.0000 0.509119
\(653\) −24.0000 −0.939193 −0.469596 − 0.882881i \(-0.655601\pi\)
−0.469596 + 0.882881i \(0.655601\pi\)
\(654\) 2.00000 0.0782062
\(655\) 0 0
\(656\) −3.00000 −0.117130
\(657\) −9.00000 −0.351123
\(658\) 24.0000 0.935617
\(659\) 34.0000 1.32445 0.662226 − 0.749304i \(-0.269612\pi\)
0.662226 + 0.749304i \(0.269612\pi\)
\(660\) 0 0
\(661\) −26.0000 −1.01128 −0.505641 − 0.862744i \(-0.668744\pi\)
−0.505641 + 0.862744i \(0.668744\pi\)
\(662\) 8.00000 0.310929
\(663\) 8.00000 0.310694
\(664\) −8.00000 −0.310460
\(665\) 0 0
\(666\) 1.00000 0.0387492
\(667\) 0 0
\(668\) −13.0000 −0.502985
\(669\) −2.00000 −0.0773245
\(670\) 0 0
\(671\) 16.0000 0.617673
\(672\) −4.00000 −0.154303
\(673\) 31.0000 1.19496 0.597481 − 0.801883i \(-0.296168\pi\)
0.597481 + 0.801883i \(0.296168\pi\)
\(674\) 9.00000 0.346667
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 27.0000 1.03769 0.518847 − 0.854867i \(-0.326361\pi\)
0.518847 + 0.854867i \(0.326361\pi\)
\(678\) 10.0000 0.384048
\(679\) 8.00000 0.307012
\(680\) 0 0
\(681\) −7.00000 −0.268241
\(682\) 0 0
\(683\) −48.0000 −1.83667 −0.918334 − 0.395805i \(-0.870466\pi\)
−0.918334 + 0.395805i \(0.870466\pi\)
\(684\) −3.00000 −0.114708
\(685\) 0 0
\(686\) −8.00000 −0.305441
\(687\) 15.0000 0.572286
\(688\) 1.00000 0.0381246
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 14.0000 0.532585 0.266293 − 0.963892i \(-0.414201\pi\)
0.266293 + 0.963892i \(0.414201\pi\)
\(692\) 14.0000 0.532200
\(693\) −8.00000 −0.303895
\(694\) −9.00000 −0.341635
\(695\) 0 0
\(696\) 0 0
\(697\) 6.00000 0.227266
\(698\) −9.00000 −0.340655
\(699\) −11.0000 −0.416058
\(700\) 0 0
\(701\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 4.00000 0.150970
\(703\) 3.00000 0.113147
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) −16.0000 −0.602168
\(707\) −60.0000 −2.25653
\(708\) −7.00000 −0.263076
\(709\) 22.0000 0.826227 0.413114 − 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 0 0
\(711\) 3.00000 0.112509
\(712\) 8.00000 0.299813
\(713\) 0 0
\(714\) 8.00000 0.299392
\(715\) 0 0
\(716\) −4.00000 −0.149487
\(717\) 3.00000 0.112037
\(718\) 6.00000 0.223918
\(719\) 4.00000 0.149175 0.0745874 − 0.997214i \(-0.476236\pi\)
0.0745874 + 0.997214i \(0.476236\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) 10.0000 0.372161
\(723\) 24.0000 0.892570
\(724\) 5.00000 0.185824
\(725\) 0 0
\(726\) 7.00000 0.259794
\(727\) −24.0000 −0.890111 −0.445055 − 0.895503i \(-0.646816\pi\)
−0.445055 + 0.895503i \(0.646816\pi\)
\(728\) 16.0000 0.592999
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −2.00000 −0.0739727
\(732\) −8.00000 −0.295689
\(733\) 10.0000 0.369358 0.184679 − 0.982799i \(-0.440875\pi\)
0.184679 + 0.982799i \(0.440875\pi\)
\(734\) 2.00000 0.0738213
\(735\) 0 0
\(736\) 3.00000 0.110581
\(737\) −8.00000 −0.294684
\(738\) 3.00000 0.110432
\(739\) 38.0000 1.39785 0.698926 − 0.715194i \(-0.253662\pi\)
0.698926 + 0.715194i \(0.253662\pi\)
\(740\) 0 0
\(741\) 12.0000 0.440831
\(742\) −12.0000 −0.440534
\(743\) 16.0000 0.586983 0.293492 − 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 22.0000 0.805477
\(747\) 8.00000 0.292705
\(748\) 4.00000 0.146254
\(749\) −24.0000 −0.876941
\(750\) 0 0
\(751\) −50.0000 −1.82453 −0.912263 − 0.409605i \(-0.865667\pi\)
−0.912263 + 0.409605i \(0.865667\pi\)
\(752\) −6.00000 −0.218797
\(753\) −9.00000 −0.327978
\(754\) 0 0
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) 16.0000 0.581530 0.290765 − 0.956795i \(-0.406090\pi\)
0.290765 + 0.956795i \(0.406090\pi\)
\(758\) 14.0000 0.508503
\(759\) 6.00000 0.217786
\(760\) 0 0
\(761\) 15.0000 0.543750 0.271875 − 0.962333i \(-0.412356\pi\)
0.271875 + 0.962333i \(0.412356\pi\)
\(762\) −18.0000 −0.652071
\(763\) −8.00000 −0.289619
\(764\) −19.0000 −0.687396
\(765\) 0 0
\(766\) 21.0000 0.758761
\(767\) 28.0000 1.01102
\(768\) 1.00000 0.0360844
\(769\) −32.0000 −1.15395 −0.576975 − 0.816762i \(-0.695767\pi\)
−0.576975 + 0.816762i \(0.695767\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 16.0000 0.575853
\(773\) 6.00000 0.215805 0.107903 − 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) −1.00000 −0.0359443
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) −4.00000 −0.143499
\(778\) 2.00000 0.0717035
\(779\) 9.00000 0.322458
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) −6.00000 −0.214560
\(783\) 0 0
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) 8.00000 0.285351
\(787\) −18.0000 −0.641631 −0.320815 − 0.947142i \(-0.603957\pi\)
−0.320815 + 0.947142i \(0.603957\pi\)
\(788\) 17.0000 0.605600
\(789\) 6.00000 0.213606
\(790\) 0 0
\(791\) −40.0000 −1.42224
\(792\) 2.00000 0.0710669
\(793\) 32.0000 1.13635
\(794\) −15.0000 −0.532330
\(795\) 0 0
\(796\) 3.00000 0.106332
\(797\) −14.0000 −0.495905 −0.247953 − 0.968772i \(-0.579758\pi\)
−0.247953 + 0.968772i \(0.579758\pi\)
\(798\) 12.0000 0.424795
\(799\) 12.0000 0.424529
\(800\) 0 0
\(801\) −8.00000 −0.282666
\(802\) −30.0000 −1.05934
\(803\) 18.0000 0.635206
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 0 0
\(807\) −3.00000 −0.105605
\(808\) 15.0000 0.527698
\(809\) −48.0000 −1.68759 −0.843795 − 0.536666i \(-0.819684\pi\)
−0.843795 + 0.536666i \(0.819684\pi\)
\(810\) 0 0
\(811\) 10.0000 0.351147 0.175574 − 0.984466i \(-0.443822\pi\)
0.175574 + 0.984466i \(0.443822\pi\)
\(812\) 0 0
\(813\) 2.00000 0.0701431
\(814\) −2.00000 −0.0701000
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) −3.00000 −0.104957
\(818\) 0 0
\(819\) −16.0000 −0.559085
\(820\) 0 0
\(821\) 51.0000 1.77991 0.889956 − 0.456046i \(-0.150735\pi\)
0.889956 + 0.456046i \(0.150735\pi\)
\(822\) 2.00000 0.0697580
\(823\) −8.00000 −0.278862 −0.139431 − 0.990232i \(-0.544527\pi\)
−0.139431 + 0.990232i \(0.544527\pi\)
\(824\) 1.00000 0.0348367
\(825\) 0 0
\(826\) 28.0000 0.974245
\(827\) 12.0000 0.417281 0.208640 − 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) −3.00000 −0.104257
\(829\) −44.0000 −1.52818 −0.764092 − 0.645108i \(-0.776812\pi\)
−0.764092 + 0.645108i \(0.776812\pi\)
\(830\) 0 0
\(831\) −32.0000 −1.11007
\(832\) −4.00000 −0.138675
\(833\) −18.0000 −0.623663
\(834\) −8.00000 −0.277017
\(835\) 0 0
\(836\) 6.00000 0.207514
\(837\) 0 0
\(838\) −2.00000 −0.0690889
\(839\) 48.0000 1.65714 0.828572 − 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 12.0000 0.413547
\(843\) 4.00000 0.137767
\(844\) −26.0000 −0.894957
\(845\) 0 0
\(846\) 6.00000 0.206284
\(847\) −28.0000 −0.962091
\(848\) 3.00000 0.103020
\(849\) 17.0000 0.583438
\(850\) 0 0
\(851\) 3.00000 0.102839
\(852\) −12.0000 −0.411113
\(853\) 10.0000 0.342393 0.171197 − 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 32.0000 1.09502
\(855\) 0 0
\(856\) 6.00000 0.205076
\(857\) −12.0000 −0.409912 −0.204956 − 0.978771i \(-0.565705\pi\)
−0.204956 + 0.978771i \(0.565705\pi\)
\(858\) −8.00000 −0.273115
\(859\) −11.0000 −0.375315 −0.187658 − 0.982235i \(-0.560090\pi\)
−0.187658 + 0.982235i \(0.560090\pi\)
\(860\) 0 0
\(861\) −12.0000 −0.408959
\(862\) 3.00000 0.102180
\(863\) 6.00000 0.204242 0.102121 − 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −30.0000 −1.01944
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) −6.00000 −0.203536
\(870\) 0 0
\(871\) −16.0000 −0.542139
\(872\) 2.00000 0.0677285
\(873\) 2.00000 0.0676897
\(874\) −9.00000 −0.304430
\(875\) 0 0
\(876\) −9.00000 −0.304082
\(877\) −49.0000 −1.65461 −0.827306 − 0.561751i \(-0.810128\pi\)
−0.827306 + 0.561751i \(0.810128\pi\)
\(878\) −21.0000 −0.708716
\(879\) 7.00000 0.236104
\(880\) 0 0
\(881\) −49.0000 −1.65085 −0.825426 − 0.564510i \(-0.809065\pi\)
−0.825426 + 0.564510i \(0.809065\pi\)
\(882\) −9.00000 −0.303046
\(883\) 48.0000 1.61533 0.807664 − 0.589643i \(-0.200731\pi\)
0.807664 + 0.589643i \(0.200731\pi\)
\(884\) 8.00000 0.269069
\(885\) 0 0
\(886\) 10.0000 0.335957
\(887\) 4.00000 0.134307 0.0671534 − 0.997743i \(-0.478608\pi\)
0.0671534 + 0.997743i \(0.478608\pi\)
\(888\) 1.00000 0.0335578
\(889\) 72.0000 2.41480
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) −2.00000 −0.0669650
\(893\) 18.0000 0.602347
\(894\) −17.0000 −0.568565
\(895\) 0 0
\(896\) −4.00000 −0.133631
\(897\) 12.0000 0.400668
\(898\) −18.0000 −0.600668
\(899\) 0 0
\(900\) 0 0
\(901\) −6.00000 −0.199889
\(902\) −6.00000 −0.199778
\(903\) 4.00000 0.133112
\(904\) 10.0000 0.332595
\(905\) 0 0
\(906\) 18.0000 0.598010
\(907\) 31.0000 1.02934 0.514669 − 0.857389i \(-0.327915\pi\)
0.514669 + 0.857389i \(0.327915\pi\)
\(908\) −7.00000 −0.232303
\(909\) −15.0000 −0.497519
\(910\) 0 0
\(911\) −47.0000 −1.55718 −0.778590 − 0.627533i \(-0.784065\pi\)
−0.778590 + 0.627533i \(0.784065\pi\)
\(912\) −3.00000 −0.0993399
\(913\) −16.0000 −0.529523
\(914\) 8.00000 0.264616
\(915\) 0 0
\(916\) 15.0000 0.495614
\(917\) −32.0000 −1.05673
\(918\) 2.00000 0.0660098
\(919\) 16.0000 0.527791 0.263896 − 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 6.00000 0.197707
\(922\) 10.0000 0.329332
\(923\) 48.0000 1.57994
\(924\) −8.00000 −0.263181
\(925\) 0 0
\(926\) 4.00000 0.131448
\(927\) −1.00000 −0.0328443
\(928\) 0 0
\(929\) 26.0000 0.853032 0.426516 − 0.904480i \(-0.359741\pi\)
0.426516 + 0.904480i \(0.359741\pi\)
\(930\) 0 0
\(931\) −27.0000 −0.884889
\(932\) −11.0000 −0.360317
\(933\) −7.00000 −0.229170
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) 4.00000 0.130744
\(937\) 2.00000 0.0653372 0.0326686 − 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) −16.0000 −0.522419
\(939\) −30.0000 −0.979013
\(940\) 0 0
\(941\) 50.0000 1.62995 0.814977 − 0.579494i \(-0.196750\pi\)
0.814977 + 0.579494i \(0.196750\pi\)
\(942\) −19.0000 −0.619053
\(943\) 9.00000 0.293080
\(944\) −7.00000 −0.227831
\(945\) 0 0
\(946\) 2.00000 0.0650256
\(947\) −29.0000 −0.942373 −0.471187 − 0.882034i \(-0.656174\pi\)
−0.471187 + 0.882034i \(0.656174\pi\)
\(948\) 3.00000 0.0974355
\(949\) 36.0000 1.16861
\(950\) 0 0
\(951\) −13.0000 −0.421554
\(952\) 8.00000 0.259281
\(953\) −37.0000 −1.19855 −0.599274 − 0.800544i \(-0.704544\pi\)
−0.599274 + 0.800544i \(0.704544\pi\)
\(954\) −3.00000 −0.0971286
\(955\) 0 0
\(956\) 3.00000 0.0970269
\(957\) 0 0
\(958\) 24.0000 0.775405
\(959\) −8.00000 −0.258333
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −4.00000 −0.128965
\(963\) −6.00000 −0.193347
\(964\) 24.0000 0.772988
\(965\) 0 0
\(966\) 12.0000 0.386094
\(967\) −3.00000 −0.0964735 −0.0482367 − 0.998836i \(-0.515360\pi\)
−0.0482367 + 0.998836i \(0.515360\pi\)
\(968\) 7.00000 0.224989
\(969\) 6.00000 0.192748
\(970\) 0 0
\(971\) 8.00000 0.256732 0.128366 − 0.991727i \(-0.459027\pi\)
0.128366 + 0.991727i \(0.459027\pi\)
\(972\) 1.00000 0.0320750
\(973\) 32.0000 1.02587
\(974\) −8.00000 −0.256337
\(975\) 0 0
\(976\) −8.00000 −0.256074
\(977\) −12.0000 −0.383914 −0.191957 − 0.981403i \(-0.561483\pi\)
−0.191957 + 0.981403i \(0.561483\pi\)
\(978\) −13.0000 −0.415694
\(979\) 16.0000 0.511362
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) −30.0000 −0.957338
\(983\) −52.0000 −1.65854 −0.829271 − 0.558846i \(-0.811244\pi\)
−0.829271 + 0.558846i \(0.811244\pi\)
\(984\) 3.00000 0.0956365
\(985\) 0 0
\(986\) 0 0
\(987\) −24.0000 −0.763928
\(988\) 12.0000 0.381771
\(989\) −3.00000 −0.0953945
\(990\) 0 0
\(991\) 23.0000 0.730619 0.365310 − 0.930886i \(-0.380963\pi\)
0.365310 + 0.930886i \(0.380963\pi\)
\(992\) 0 0
\(993\) −8.00000 −0.253872
\(994\) 48.0000 1.52247
\(995\) 0 0
\(996\) 8.00000 0.253490
\(997\) 12.0000 0.380044 0.190022 − 0.981780i \(-0.439144\pi\)
0.190022 + 0.981780i \(0.439144\pi\)
\(998\) 11.0000 0.348199
\(999\) −1.00000 −0.0316386
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 5550.2.a.u.1.1 ✓ 1
5.4 even 2 5550.2.a.v.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5550.2.a.u.1.1 ✓ 1 1.1 even 1 trivial
5550.2.a.v.1.1 yes 1 5.4 even 2