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Properties

Label 5550.2.a.bo.1.1
Level $5550$
Weight $2$
Character 5550.1
Self dual yes
Analytic conductor $44.317$
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] = [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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1110)
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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +5.00000 q^{11} +1.00000 q^{12} +1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} +1.00000 q^{21} +5.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} +1.00000 q^{27} +1.00000 q^{28} -3.00000 q^{29} +1.00000 q^{31} +1.00000 q^{32} +5.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} -1.00000 q^{37} -1.00000 q^{41} +1.00000 q^{42} +7.00000 q^{43} +5.00000 q^{44} +4.00000 q^{46} -4.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} +1.00000 q^{51} +3.00000 q^{53} +1.00000 q^{54} +1.00000 q^{56} -3.00000 q^{58} -8.00000 q^{59} +5.00000 q^{61} +1.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +5.00000 q^{66} -4.00000 q^{67} +1.00000 q^{68} +4.00000 q^{69} -6.00000 q^{71} +1.00000 q^{72} +10.0000 q^{73} -1.00000 q^{74} +5.00000 q^{77} +1.00000 q^{81} -1.00000 q^{82} -8.00000 q^{83} +1.00000 q^{84} +7.00000 q^{86} -3.00000 q^{87} +5.00000 q^{88} +8.00000 q^{89} +4.00000 q^{92} +1.00000 q^{93} -4.00000 q^{94} +1.00000 q^{96} -1.00000 q^{97} -6.00000 q^{98} +5.00000 q^{99} +O(q^{100})\)

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
\(7\) 1.00000 0.377964 0.188982 − 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 − 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536 0.121268 − 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 5.00000 1.06600
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) −3.00000 −0.557086 −0.278543 − 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605 0.0898027 − 0.995960i \(-0.471376\pi\)
0.0898027 + 0.995960i \(0.471376\pi\)
\(32\) 1.00000 0.176777
\(33\) 5.00000 0.870388
\(34\) 1.00000 0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.00000 −0.156174 −0.0780869 − 0.996947i \(-0.524881\pi\)
−0.0780869 + 0.996947i \(0.524881\pi\)
\(42\) 1.00000 0.154303
\(43\) 7.00000 1.06749 0.533745 − 0.845645i \(-0.320784\pi\)
0.533745 + 0.845645i \(0.320784\pi\)
\(44\) 5.00000 0.753778
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) −4.00000 −0.583460 −0.291730 − 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 1.00000 0.140028
\(52\) 0 0
\(53\) 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
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −3.00000 −0.393919
\(59\) −8.00000 −1.04151 −0.520756 − 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 5.00000 0.640184 0.320092 − 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) 1.00000 0.127000
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 5.00000 0.615457
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 1.00000 0.121268
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 1.00000 0.117851
\(73\) 10.0000 1.17041 0.585206 − 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) 0 0
\(77\) 5.00000 0.569803
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −1.00000 −0.110432
\(83\) −8.00000 −0.878114 −0.439057 − 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) 7.00000 0.754829
\(87\) −3.00000 −0.321634
\(88\) 5.00000 0.533002
\(89\) 8.00000 0.847998 0.423999 − 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.00000 0.417029
\(93\) 1.00000 0.103695
\(94\) −4.00000 −0.412568
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −1.00000 −0.101535 −0.0507673 − 0.998711i \(-0.516167\pi\)
−0.0507673 + 0.998711i \(0.516167\pi\)
\(98\) −6.00000 −0.606092
\(99\) 5.00000 0.502519
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 − 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 1.00000 0.0990148
\(103\) 8.00000 0.788263 0.394132 − 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 3.00000 0.291386
\(107\) 6.00000 0.580042 0.290021 − 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 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
\(111\) −1.00000 −0.0949158
\(112\) 1.00000 0.0944911
\(113\) −21.0000 −1.97551 −0.987757 − 0.156001i \(-0.950140\pi\)
−0.987757 + 0.156001i \(0.950140\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −3.00000 −0.278543
\(117\) 0 0
\(118\) −8.00000 −0.736460
\(119\) 1.00000 0.0916698
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 5.00000 0.452679
\(123\) −1.00000 −0.0901670
\(124\) 1.00000 0.0898027
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) −8.00000 −0.709885 −0.354943 − 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 7.00000 0.616316
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 − 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 5.00000 0.435194
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 1.00000 0.0857493
\(137\) 18.0000 1.53784 0.768922 − 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 4.00000 0.340503
\(139\) −11.0000 −0.933008 −0.466504 − 0.884519i \(-0.654487\pi\)
−0.466504 + 0.884519i \(0.654487\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) −6.00000 −0.503509
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 10.0000 0.827606
\(147\) −6.00000 −0.494872
\(148\) −1.00000 −0.0821995
\(149\) 4.00000 0.327693 0.163846 − 0.986486i \(-0.447610\pi\)
0.163846 + 0.986486i \(0.447610\pi\)
\(150\) 0 0
\(151\) −2.00000 −0.162758 −0.0813788 − 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) 0 0
\(153\) 1.00000 0.0808452
\(154\) 5.00000 0.402911
\(155\) 0 0
\(156\) 0 0
\(157\) 23.0000 1.83560 0.917800 − 0.397043i \(-0.129964\pi\)
0.917800 + 0.397043i \(0.129964\pi\)
\(158\) 0 0
\(159\) 3.00000 0.237915
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 1.00000 0.0785674
\(163\) −1.00000 −0.0783260 −0.0391630 − 0.999233i \(-0.512469\pi\)
−0.0391630 + 0.999233i \(0.512469\pi\)
\(164\) −1.00000 −0.0780869
\(165\) 0 0
\(166\) −8.00000 −0.620920
\(167\) −6.00000 −0.464294 −0.232147 − 0.972681i \(-0.574575\pi\)
−0.232147 + 0.972681i \(0.574575\pi\)
\(168\) 1.00000 0.0771517
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 7.00000 0.533745
\(173\) −13.0000 −0.988372 −0.494186 − 0.869356i \(-0.664534\pi\)
−0.494186 + 0.869356i \(0.664534\pi\)
\(174\) −3.00000 −0.227429
\(175\) 0 0
\(176\) 5.00000 0.376889
\(177\) −8.00000 −0.601317
\(178\) 8.00000 0.599625
\(179\) 10.0000 0.747435 0.373718 − 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) −16.0000 −1.18927 −0.594635 − 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 0 0
\(183\) 5.00000 0.369611
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) 1.00000 0.0733236
\(187\) 5.00000 0.365636
\(188\) −4.00000 −0.291730
\(189\) 1.00000 0.0727393
\(190\) 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
\(193\) 10.0000 0.719816 0.359908 − 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) −1.00000 −0.0717958
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 26.0000 1.85242 0.926212 − 0.377004i \(-0.123046\pi\)
0.926212 + 0.377004i \(0.123046\pi\)
\(198\) 5.00000 0.355335
\(199\) 24.0000 1.70131 0.850657 − 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 10.0000 0.703598
\(203\) −3.00000 −0.210559
\(204\) 1.00000 0.0700140
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −21.0000 −1.44570 −0.722850 − 0.691005i \(-0.757168\pi\)
−0.722850 + 0.691005i \(0.757168\pi\)
\(212\) 3.00000 0.206041
\(213\) −6.00000 −0.411113
\(214\) 6.00000 0.410152
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 1.00000 0.0678844
\(218\) 1.00000 0.0677285
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) 0 0
\(222\) −1.00000 −0.0671156
\(223\) −5.00000 −0.334825 −0.167412 − 0.985887i \(-0.553541\pi\)
−0.167412 + 0.985887i \(0.553541\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −21.0000 −1.39690
\(227\) 1.00000 0.0663723 0.0331862 − 0.999449i \(-0.489435\pi\)
0.0331862 + 0.999449i \(0.489435\pi\)
\(228\) 0 0
\(229\) −16.0000 −1.05731 −0.528655 − 0.848837i \(-0.677303\pi\)
−0.528655 + 0.848837i \(0.677303\pi\)
\(230\) 0 0
\(231\) 5.00000 0.328976
\(232\) −3.00000 −0.196960
\(233\) −10.0000 −0.655122 −0.327561 − 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −8.00000 −0.520756
\(237\) 0 0
\(238\) 1.00000 0.0648204
\(239\) 11.0000 0.711531 0.355765 − 0.934575i \(-0.384220\pi\)
0.355765 + 0.934575i \(0.384220\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 − 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 14.0000 0.899954
\(243\) 1.00000 0.0641500
\(244\) 5.00000 0.320092
\(245\) 0 0
\(246\) −1.00000 −0.0637577
\(247\) 0 0
\(248\) 1.00000 0.0635001
\(249\) −8.00000 −0.506979
\(250\) 0 0
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 1.00000 0.0629941
\(253\) 20.0000 1.25739
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 − 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 7.00000 0.435801
\(259\) −1.00000 −0.0621370
\(260\) 0 0
\(261\) −3.00000 −0.185695
\(262\) −6.00000 −0.370681
\(263\) 19.0000 1.17159 0.585795 − 0.810459i \(-0.300782\pi\)
0.585795 + 0.810459i \(0.300782\pi\)
\(264\) 5.00000 0.307729
\(265\) 0 0
\(266\) 0 0
\(267\) 8.00000 0.489592
\(268\) −4.00000 −0.244339
\(269\) −12.0000 −0.731653 −0.365826 − 0.930683i \(-0.619214\pi\)
−0.365826 + 0.930683i \(0.619214\pi\)
\(270\) 0 0
\(271\) −14.0000 −0.850439 −0.425220 − 0.905090i \(-0.639803\pi\)
−0.425220 + 0.905090i \(0.639803\pi\)
\(272\) 1.00000 0.0606339
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) −8.00000 −0.480673 −0.240337 − 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) −11.0000 −0.659736
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 − 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) −4.00000 −0.238197
\(283\) −16.0000 −0.951101 −0.475551 − 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 0 0
\(287\) −1.00000 −0.0590281
\(288\) 1.00000 0.0589256
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) −1.00000 −0.0586210
\(292\) 10.0000 0.585206
\(293\) 9.00000 0.525786 0.262893 − 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) −6.00000 −0.349927
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) 5.00000 0.290129
\(298\) 4.00000 0.231714
\(299\) 0 0
\(300\) 0 0
\(301\) 7.00000 0.403473
\(302\) −2.00000 −0.115087
\(303\) 10.0000 0.574485
\(304\) 0 0
\(305\) 0 0
\(306\) 1.00000 0.0571662
\(307\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 5.00000 0.284901
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) 15.0000 0.850572 0.425286 − 0.905059i \(-0.360174\pi\)
0.425286 + 0.905059i \(0.360174\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 − 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 23.0000 1.29797
\(315\) 0 0
\(316\) 0 0
\(317\) −5.00000 −0.280828 −0.140414 − 0.990093i \(-0.544843\pi\)
−0.140414 + 0.990093i \(0.544843\pi\)
\(318\) 3.00000 0.168232
\(319\) −15.0000 −0.839839
\(320\) 0 0
\(321\) 6.00000 0.334887
\(322\) 4.00000 0.222911
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −1.00000 −0.0553849
\(327\) 1.00000 0.0553001
\(328\) −1.00000 −0.0552158
\(329\) −4.00000 −0.220527
\(330\) 0 0
\(331\) −26.0000 −1.42909 −0.714545 − 0.699590i \(-0.753366\pi\)
−0.714545 + 0.699590i \(0.753366\pi\)
\(332\) −8.00000 −0.439057
\(333\) −1.00000 −0.0547997
\(334\) −6.00000 −0.328305
\(335\) 0 0
\(336\) 1.00000 0.0545545
\(337\) 16.0000 0.871576 0.435788 − 0.900049i \(-0.356470\pi\)
0.435788 + 0.900049i \(0.356470\pi\)
\(338\) −13.0000 −0.707107
\(339\) −21.0000 −1.14056
\(340\) 0 0
\(341\) 5.00000 0.270765
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) 7.00000 0.377415
\(345\) 0 0
\(346\) −13.0000 −0.698884
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −3.00000 −0.160817
\(349\) −2.00000 −0.107058 −0.0535288 − 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5.00000 0.266501
\(353\) −31.0000 −1.64996 −0.824982 − 0.565159i \(-0.808815\pi\)
−0.824982 + 0.565159i \(0.808815\pi\)
\(354\) −8.00000 −0.425195
\(355\) 0 0
\(356\) 8.00000 0.423999
\(357\) 1.00000 0.0529256
\(358\) 10.0000 0.528516
\(359\) 6.00000 0.316668 0.158334 − 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) −16.0000 −0.840941
\(363\) 14.0000 0.734809
\(364\) 0 0
\(365\) 0 0
\(366\) 5.00000 0.261354
\(367\) 13.0000 0.678594 0.339297 − 0.940679i \(-0.389811\pi\)
0.339297 + 0.940679i \(0.389811\pi\)
\(368\) 4.00000 0.208514
\(369\) −1.00000 −0.0520579
\(370\) 0 0
\(371\) 3.00000 0.155752
\(372\) 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
\(376\) −4.00000 −0.206284
\(377\) 0 0
\(378\) 1.00000 0.0514344
\(379\) 20.0000 1.02733 0.513665 − 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 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
\(386\) 10.0000 0.508987
\(387\) 7.00000 0.355830
\(388\) −1.00000 −0.0507673
\(389\) 23.0000 1.16615 0.583073 − 0.812420i \(-0.301850\pi\)
0.583073 + 0.812420i \(0.301850\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) −6.00000 −0.303046
\(393\) −6.00000 −0.302660
\(394\) 26.0000 1.30986
\(395\) 0 0
\(396\) 5.00000 0.251259
\(397\) 22.0000 1.10415 0.552074 − 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 24.0000 1.20301
\(399\) 0 0
\(400\) 0 0
\(401\) 10.0000 0.499376 0.249688 − 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) −4.00000 −0.199502
\(403\) 0 0
\(404\) 10.0000 0.497519
\(405\) 0 0
\(406\) −3.00000 −0.148888
\(407\) −5.00000 −0.247841
\(408\) 1.00000 0.0495074
\(409\) −40.0000 −1.97787 −0.988936 − 0.148340i \(-0.952607\pi\)
−0.988936 + 0.148340i \(0.952607\pi\)
\(410\) 0 0
\(411\) 18.0000 0.887875
\(412\) 8.00000 0.394132
\(413\) −8.00000 −0.393654
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) 0 0
\(417\) −11.0000 −0.538672
\(418\) 0 0
\(419\) 20.0000 0.977064 0.488532 − 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) 0 0
\(421\) −38.0000 −1.85201 −0.926003 − 0.377515i \(-0.876779\pi\)
−0.926003 + 0.377515i \(0.876779\pi\)
\(422\) −21.0000 −1.02226
\(423\) −4.00000 −0.194487
\(424\) 3.00000 0.145693
\(425\) 0 0
\(426\) −6.00000 −0.290701
\(427\) 5.00000 0.241967
\(428\) 6.00000 0.290021
\(429\) 0 0
\(430\) 0 0
\(431\) −13.0000 −0.626188 −0.313094 − 0.949722i \(-0.601365\pi\)
−0.313094 + 0.949722i \(0.601365\pi\)
\(432\) 1.00000 0.0481125
\(433\) −20.0000 −0.961139 −0.480569 − 0.876957i \(-0.659570\pi\)
−0.480569 + 0.876957i \(0.659570\pi\)
\(434\) 1.00000 0.0480015
\(435\) 0 0
\(436\) 1.00000 0.0478913
\(437\) 0 0
\(438\) 10.0000 0.477818
\(439\) −33.0000 −1.57500 −0.787502 − 0.616312i \(-0.788626\pi\)
−0.787502 + 0.616312i \(0.788626\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) −6.00000 −0.285069 −0.142534 − 0.989790i \(-0.545525\pi\)
−0.142534 + 0.989790i \(0.545525\pi\)
\(444\) −1.00000 −0.0474579
\(445\) 0 0
\(446\) −5.00000 −0.236757
\(447\) 4.00000 0.189194
\(448\) 1.00000 0.0472456
\(449\) −24.0000 −1.13263 −0.566315 − 0.824189i \(-0.691631\pi\)
−0.566315 + 0.824189i \(0.691631\pi\)
\(450\) 0 0
\(451\) −5.00000 −0.235441
\(452\) −21.0000 −0.987757
\(453\) −2.00000 −0.0939682
\(454\) 1.00000 0.0469323
\(455\) 0 0
\(456\) 0 0
\(457\) 19.0000 0.888783 0.444391 − 0.895833i \(-0.353420\pi\)
0.444391 + 0.895833i \(0.353420\pi\)
\(458\) −16.0000 −0.747631
\(459\) 1.00000 0.0466760
\(460\) 0 0
\(461\) 3.00000 0.139724 0.0698620 − 0.997557i \(-0.477744\pi\)
0.0698620 + 0.997557i \(0.477744\pi\)
\(462\) 5.00000 0.232621
\(463\) −24.0000 −1.11537 −0.557687 − 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) −3.00000 −0.139272
\(465\) 0 0
\(466\) −10.0000 −0.463241
\(467\) 27.0000 1.24941 0.624705 − 0.780860i \(-0.285219\pi\)
0.624705 + 0.780860i \(0.285219\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) 23.0000 1.05978
\(472\) −8.00000 −0.368230
\(473\) 35.0000 1.60930
\(474\) 0 0
\(475\) 0 0
\(476\) 1.00000 0.0458349
\(477\) 3.00000 0.137361
\(478\) 11.0000 0.503128
\(479\) −28.0000 −1.27935 −0.639676 − 0.768644i \(-0.720932\pi\)
−0.639676 + 0.768644i \(0.720932\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −2.00000 −0.0910975
\(483\) 4.00000 0.182006
\(484\) 14.0000 0.636364
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −12.0000 −0.543772 −0.271886 − 0.962329i \(-0.587647\pi\)
−0.271886 + 0.962329i \(0.587647\pi\)
\(488\) 5.00000 0.226339
\(489\) −1.00000 −0.0452216
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 − 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) −1.00000 −0.0450835
\(493\) −3.00000 −0.135113
\(494\) 0 0
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) −6.00000 −0.269137
\(498\) −8.00000 −0.358489
\(499\) −30.0000 −1.34298 −0.671492 − 0.741012i \(-0.734346\pi\)
−0.671492 + 0.741012i \(0.734346\pi\)
\(500\) 0 0
\(501\) −6.00000 −0.268060
\(502\) 0 0
\(503\) 6.00000 0.267527 0.133763 − 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) 20.0000 0.889108
\(507\) −13.0000 −0.577350
\(508\) −8.00000 −0.354943
\(509\) −42.0000 −1.86162 −0.930809 − 0.365507i \(-0.880896\pi\)
−0.930809 + 0.365507i \(0.880896\pi\)
\(510\) 0 0
\(511\) 10.0000 0.442374
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 6.00000 0.264649
\(515\) 0 0
\(516\) 7.00000 0.308158
\(517\) −20.0000 −0.879599
\(518\) −1.00000 −0.0439375
\(519\) −13.0000 −0.570637
\(520\) 0 0
\(521\) −35.0000 −1.53338 −0.766689 − 0.642019i \(-0.778097\pi\)
−0.766689 + 0.642019i \(0.778097\pi\)
\(522\) −3.00000 −0.131306
\(523\) 20.0000 0.874539 0.437269 − 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) −6.00000 −0.262111
\(525\) 0 0
\(526\) 19.0000 0.828439
\(527\) 1.00000 0.0435607
\(528\) 5.00000 0.217597
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −8.00000 −0.347170
\(532\) 0 0
\(533\) 0 0
\(534\) 8.00000 0.346194
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 10.0000 0.431532
\(538\) −12.0000 −0.517357
\(539\) −30.0000 −1.29219
\(540\) 0 0
\(541\) 34.0000 1.46177 0.730887 − 0.682498i \(-0.239107\pi\)
0.730887 + 0.682498i \(0.239107\pi\)
\(542\) −14.0000 −0.601351
\(543\) −16.0000 −0.686626
\(544\) 1.00000 0.0428746
\(545\) 0 0
\(546\) 0 0
\(547\) −41.0000 −1.75303 −0.876517 − 0.481371i \(-0.840139\pi\)
−0.876517 + 0.481371i \(0.840139\pi\)
\(548\) 18.0000 0.768922
\(549\) 5.00000 0.213395
\(550\) 0 0
\(551\) 0 0
\(552\) 4.00000 0.170251
\(553\) 0 0
\(554\) −8.00000 −0.339887
\(555\) 0 0
\(556\) −11.0000 −0.466504
\(557\) 6.00000 0.254228 0.127114 − 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) 1.00000 0.0423334
\(559\) 0 0
\(560\) 0 0
\(561\) 5.00000 0.211100
\(562\) 18.0000 0.759284
\(563\) −19.0000 −0.800755 −0.400377 − 0.916350i \(-0.631121\pi\)
−0.400377 + 0.916350i \(0.631121\pi\)
\(564\) −4.00000 −0.168430
\(565\) 0 0
\(566\) −16.0000 −0.672530
\(567\) 1.00000 0.0419961
\(568\) −6.00000 −0.251754
\(569\) −10.0000 −0.419222 −0.209611 − 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) −29.0000 −1.21361 −0.606806 − 0.794850i \(-0.707550\pi\)
−0.606806 + 0.794850i \(0.707550\pi\)
\(572\) 0 0
\(573\) 3.00000 0.125327
\(574\) −1.00000 −0.0417392
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 22.0000 0.915872 0.457936 − 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) −16.0000 −0.665512
\(579\) 10.0000 0.415586
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) −1.00000 −0.0414513
\(583\) 15.0000 0.621237
\(584\) 10.0000 0.413803
\(585\) 0 0
\(586\) 9.00000 0.371787
\(587\) −33.0000 −1.36206 −0.681028 − 0.732257i \(-0.738467\pi\)
−0.681028 + 0.732257i \(0.738467\pi\)
\(588\) −6.00000 −0.247436
\(589\) 0 0
\(590\) 0 0
\(591\) 26.0000 1.06950
\(592\) −1.00000 −0.0410997
\(593\) −24.0000 −0.985562 −0.492781 − 0.870153i \(-0.664020\pi\)
−0.492781 + 0.870153i \(0.664020\pi\)
\(594\) 5.00000 0.205152
\(595\) 0 0
\(596\) 4.00000 0.163846
\(597\) 24.0000 0.982255
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 − 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −7.00000 −0.285536 −0.142768 − 0.989756i \(-0.545600\pi\)
−0.142768 + 0.989756i \(0.545600\pi\)
\(602\) 7.00000 0.285299
\(603\) −4.00000 −0.162893
\(604\) −2.00000 −0.0813788
\(605\) 0 0
\(606\) 10.0000 0.406222
\(607\) −10.0000 −0.405887 −0.202944 − 0.979190i \(-0.565051\pi\)
−0.202944 + 0.979190i \(0.565051\pi\)
\(608\) 0 0
\(609\) −3.00000 −0.121566
\(610\) 0 0
\(611\) 0 0
\(612\) 1.00000 0.0404226
\(613\) 37.0000 1.49442 0.747208 − 0.664590i \(-0.231394\pi\)
0.747208 + 0.664590i \(0.231394\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 5.00000 0.201456
\(617\) 34.0000 1.36879 0.684394 − 0.729112i \(-0.260067\pi\)
0.684394 + 0.729112i \(0.260067\pi\)
\(618\) 8.00000 0.321807
\(619\) −31.0000 −1.24600 −0.622998 − 0.782224i \(-0.714085\pi\)
−0.622998 + 0.782224i \(0.714085\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 15.0000 0.601445
\(623\) 8.00000 0.320513
\(624\) 0 0
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) 23.0000 0.917800
\(629\) −1.00000 −0.0398726
\(630\) 0 0
\(631\) 5.00000 0.199047 0.0995234 − 0.995035i \(-0.468268\pi\)
0.0995234 + 0.995035i \(0.468268\pi\)
\(632\) 0 0
\(633\) −21.0000 −0.834675
\(634\) −5.00000 −0.198575
\(635\) 0 0
\(636\) 3.00000 0.118958
\(637\) 0 0
\(638\) −15.0000 −0.593856
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) −37.0000 −1.46141 −0.730706 − 0.682692i \(-0.760809\pi\)
−0.730706 + 0.682692i \(0.760809\pi\)
\(642\) 6.00000 0.236801
\(643\) −21.0000 −0.828159 −0.414080 − 0.910241i \(-0.635896\pi\)
−0.414080 + 0.910241i \(0.635896\pi\)
\(644\) 4.00000 0.157622
\(645\) 0 0
\(646\) 0 0
\(647\) 42.0000 1.65119 0.825595 − 0.564263i \(-0.190840\pi\)
0.825595 + 0.564263i \(0.190840\pi\)
\(648\) 1.00000 0.0392837
\(649\) −40.0000 −1.57014
\(650\) 0 0
\(651\) 1.00000 0.0391931
\(652\) −1.00000 −0.0391630
\(653\) 10.0000 0.391330 0.195665 − 0.980671i \(-0.437313\pi\)
0.195665 + 0.980671i \(0.437313\pi\)
\(654\) 1.00000 0.0391031
\(655\) 0 0
\(656\) −1.00000 −0.0390434
\(657\) 10.0000 0.390137
\(658\) −4.00000 −0.155936
\(659\) −32.0000 −1.24654 −0.623272 − 0.782006i \(-0.714197\pi\)
−0.623272 + 0.782006i \(0.714197\pi\)
\(660\) 0 0
\(661\) −13.0000 −0.505641 −0.252821 − 0.967513i \(-0.581358\pi\)
−0.252821 + 0.967513i \(0.581358\pi\)
\(662\) −26.0000 −1.01052
\(663\) 0 0
\(664\) −8.00000 −0.310460
\(665\) 0 0
\(666\) −1.00000 −0.0387492
\(667\) −12.0000 −0.464642
\(668\) −6.00000 −0.232147
\(669\) −5.00000 −0.193311
\(670\) 0 0
\(671\) 25.0000 0.965114
\(672\) 1.00000 0.0385758
\(673\) 36.0000 1.38770 0.693849 − 0.720121i \(-0.255914\pi\)
0.693849 + 0.720121i \(0.255914\pi\)
\(674\) 16.0000 0.616297
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) −2.00000 −0.0768662 −0.0384331 − 0.999261i \(-0.512237\pi\)
−0.0384331 + 0.999261i \(0.512237\pi\)
\(678\) −21.0000 −0.806500
\(679\) −1.00000 −0.0383765
\(680\) 0 0
\(681\) 1.00000 0.0383201
\(682\) 5.00000 0.191460
\(683\) 37.0000 1.41577 0.707883 − 0.706330i \(-0.249650\pi\)
0.707883 + 0.706330i \(0.249650\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −13.0000 −0.496342
\(687\) −16.0000 −0.610438
\(688\) 7.00000 0.266872
\(689\) 0 0
\(690\) 0 0
\(691\) −5.00000 −0.190209 −0.0951045 − 0.995467i \(-0.530319\pi\)
−0.0951045 + 0.995467i \(0.530319\pi\)
\(692\) −13.0000 −0.494186
\(693\) 5.00000 0.189934
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) −3.00000 −0.113715
\(697\) −1.00000 −0.0378777
\(698\) −2.00000 −0.0757011
\(699\) −10.0000 −0.378235
\(700\) 0 0
\(701\) 26.0000 0.982006 0.491003 − 0.871158i \(-0.336630\pi\)
0.491003 + 0.871158i \(0.336630\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 5.00000 0.188445
\(705\) 0 0
\(706\) −31.0000 −1.16670
\(707\) 10.0000 0.376089
\(708\) −8.00000 −0.300658
\(709\) 9.00000 0.338002 0.169001 − 0.985616i \(-0.445946\pi\)
0.169001 + 0.985616i \(0.445946\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 8.00000 0.299813
\(713\) 4.00000 0.149801
\(714\) 1.00000 0.0374241
\(715\) 0 0
\(716\) 10.0000 0.373718
\(717\) 11.0000 0.410803
\(718\) 6.00000 0.223918
\(719\) −28.0000 −1.04422 −0.522112 − 0.852877i \(-0.674856\pi\)
−0.522112 + 0.852877i \(0.674856\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) −19.0000 −0.707107
\(723\) −2.00000 −0.0743808
\(724\) −16.0000 −0.594635
\(725\) 0 0
\(726\) 14.0000 0.519589
\(727\) −42.0000 −1.55769 −0.778847 − 0.627214i \(-0.784195\pi\)
−0.778847 + 0.627214i \(0.784195\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 7.00000 0.258904
\(732\) 5.00000 0.184805
\(733\) −19.0000 −0.701781 −0.350891 − 0.936416i \(-0.614121\pi\)
−0.350891 + 0.936416i \(0.614121\pi\)
\(734\) 13.0000 0.479839
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) −20.0000 −0.736709
\(738\) −1.00000 −0.0368105
\(739\) 37.0000 1.36107 0.680534 − 0.732717i \(-0.261748\pi\)
0.680534 + 0.732717i \(0.261748\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 3.00000 0.110133
\(743\) 15.0000 0.550297 0.275148 − 0.961402i \(-0.411273\pi\)
0.275148 + 0.961402i \(0.411273\pi\)
\(744\) 1.00000 0.0366618
\(745\) 0 0
\(746\) 6.00000 0.219676
\(747\) −8.00000 −0.292705
\(748\) 5.00000 0.182818
\(749\) 6.00000 0.219235
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 − 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) −4.00000 −0.145865
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 1.00000 0.0363696
\(757\) −20.0000 −0.726912 −0.363456 − 0.931611i \(-0.618403\pi\)
−0.363456 + 0.931611i \(0.618403\pi\)
\(758\) 20.0000 0.726433
\(759\) 20.0000 0.725954
\(760\) 0 0
\(761\) −21.0000 −0.761249 −0.380625 − 0.924730i \(-0.624291\pi\)
−0.380625 + 0.924730i \(0.624291\pi\)
\(762\) −8.00000 −0.289809
\(763\) 1.00000 0.0362024
\(764\) 3.00000 0.108536
\(765\) 0 0
\(766\) 8.00000 0.289052
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 10.0000 0.359908
\(773\) −27.0000 −0.971123 −0.485561 − 0.874203i \(-0.661385\pi\)
−0.485561 + 0.874203i \(0.661385\pi\)
\(774\) 7.00000 0.251610
\(775\) 0 0
\(776\) −1.00000 −0.0358979
\(777\) −1.00000 −0.0358748
\(778\) 23.0000 0.824590
\(779\) 0 0
\(780\) 0 0
\(781\) −30.0000 −1.07348
\(782\) 4.00000 0.143040
\(783\) −3.00000 −0.107211
\(784\) −6.00000 −0.214286
\(785\) 0 0
\(786\) −6.00000 −0.214013
\(787\) 10.0000 0.356462 0.178231 − 0.983989i \(-0.442963\pi\)
0.178231 + 0.983989i \(0.442963\pi\)
\(788\) 26.0000 0.926212
\(789\) 19.0000 0.676418
\(790\) 0 0
\(791\) −21.0000 −0.746674
\(792\) 5.00000 0.177667
\(793\) 0 0
\(794\) 22.0000 0.780751
\(795\) 0 0
\(796\) 24.0000 0.850657
\(797\) −8.00000 −0.283375 −0.141687 − 0.989911i \(-0.545253\pi\)
−0.141687 + 0.989911i \(0.545253\pi\)
\(798\) 0 0
\(799\) −4.00000 −0.141510
\(800\) 0 0
\(801\) 8.00000 0.282666
\(802\) 10.0000 0.353112
\(803\) 50.0000 1.76446
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 0 0
\(807\) −12.0000 −0.422420
\(808\) 10.0000 0.351799
\(809\) 48.0000 1.68759 0.843795 − 0.536666i \(-0.180316\pi\)
0.843795 + 0.536666i \(0.180316\pi\)
\(810\) 0 0
\(811\) −8.00000 −0.280918 −0.140459 − 0.990086i \(-0.544858\pi\)
−0.140459 + 0.990086i \(0.544858\pi\)
\(812\) −3.00000 −0.105279
\(813\) −14.0000 −0.491001
\(814\) −5.00000 −0.175250
\(815\) 0 0
\(816\) 1.00000 0.0350070
\(817\) 0 0
\(818\) −40.0000 −1.39857
\(819\) 0 0
\(820\) 0 0
\(821\) −50.0000 −1.74501 −0.872506 − 0.488603i \(-0.837507\pi\)
−0.872506 + 0.488603i \(0.837507\pi\)
\(822\) 18.0000 0.627822
\(823\) −24.0000 −0.836587 −0.418294 − 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) 8.00000 0.278693
\(825\) 0 0
\(826\) −8.00000 −0.278356
\(827\) −35.0000 −1.21707 −0.608535 − 0.793527i \(-0.708242\pi\)
−0.608535 + 0.793527i \(0.708242\pi\)
\(828\) 4.00000 0.139010
\(829\) 21.0000 0.729360 0.364680 − 0.931133i \(-0.381178\pi\)
0.364680 + 0.931133i \(0.381178\pi\)
\(830\) 0 0
\(831\) −8.00000 −0.277517
\(832\) 0 0
\(833\) −6.00000 −0.207888
\(834\) −11.0000 −0.380899
\(835\) 0 0
\(836\) 0 0
\(837\) 1.00000 0.0345651
\(838\) 20.0000 0.690889
\(839\) 40.0000 1.38095 0.690477 − 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −38.0000 −1.30957
\(843\) 18.0000 0.619953
\(844\) −21.0000 −0.722850
\(845\) 0 0
\(846\) −4.00000 −0.137523
\(847\) 14.0000 0.481046
\(848\) 3.00000 0.103020
\(849\) −16.0000 −0.549119
\(850\) 0 0
\(851\) −4.00000 −0.137118
\(852\) −6.00000 −0.205557
\(853\) 2.00000 0.0684787 0.0342393 − 0.999414i \(-0.489099\pi\)
0.0342393 + 0.999414i \(0.489099\pi\)
\(854\) 5.00000 0.171096
\(855\) 0 0
\(856\) 6.00000 0.205076
\(857\) 27.0000 0.922302 0.461151 − 0.887322i \(-0.347437\pi\)
0.461151 + 0.887322i \(0.347437\pi\)
\(858\) 0 0
\(859\) 8.00000 0.272956 0.136478 − 0.990643i \(-0.456422\pi\)
0.136478 + 0.990643i \(0.456422\pi\)
\(860\) 0 0
\(861\) −1.00000 −0.0340799
\(862\) −13.0000 −0.442782
\(863\) 21.0000 0.714848 0.357424 − 0.933942i \(-0.383655\pi\)
0.357424 + 0.933942i \(0.383655\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −20.0000 −0.679628
\(867\) −16.0000 −0.543388
\(868\) 1.00000 0.0339422
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 1.00000 0.0338643
\(873\) −1.00000 −0.0338449
\(874\) 0 0
\(875\) 0 0
\(876\) 10.0000 0.337869
\(877\) −13.0000 −0.438979 −0.219489 − 0.975615i \(-0.570439\pi\)
−0.219489 + 0.975615i \(0.570439\pi\)
\(878\) −33.0000 −1.11370
\(879\) 9.00000 0.303562
\(880\) 0 0
\(881\) −45.0000 −1.51609 −0.758044 − 0.652203i \(-0.773845\pi\)
−0.758044 + 0.652203i \(0.773845\pi\)
\(882\) −6.00000 −0.202031
\(883\) 33.0000 1.11054 0.555269 − 0.831671i \(-0.312615\pi\)
0.555269 + 0.831671i \(0.312615\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −6.00000 −0.201574
\(887\) −45.0000 −1.51095 −0.755476 − 0.655176i \(-0.772594\pi\)
−0.755476 + 0.655176i \(0.772594\pi\)
\(888\) −1.00000 −0.0335578
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 5.00000 0.167506
\(892\) −5.00000 −0.167412
\(893\) 0 0
\(894\) 4.00000 0.133780
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −24.0000 −0.800890
\(899\) −3.00000 −0.100056
\(900\) 0 0
\(901\) 3.00000 0.0999445
\(902\) −5.00000 −0.166482
\(903\) 7.00000 0.232945
\(904\) −21.0000 −0.698450
\(905\) 0 0
\(906\) −2.00000 −0.0664455
\(907\) 36.0000 1.19536 0.597680 − 0.801735i \(-0.296089\pi\)
0.597680 + 0.801735i \(0.296089\pi\)
\(908\) 1.00000 0.0331862
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 − 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 0 0
\(913\) −40.0000 −1.32381
\(914\) 19.0000 0.628464
\(915\) 0 0
\(916\) −16.0000 −0.528655
\(917\) −6.00000 −0.198137
\(918\) 1.00000 0.0330049
\(919\) 48.0000 1.58337 0.791687 − 0.610927i \(-0.209203\pi\)
0.791687 + 0.610927i \(0.209203\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 3.00000 0.0987997
\(923\) 0 0
\(924\) 5.00000 0.164488
\(925\) 0 0
\(926\) −24.0000 −0.788689
\(927\) 8.00000 0.262754
\(928\) −3.00000 −0.0984798
\(929\) −3.00000 −0.0984268 −0.0492134 − 0.998788i \(-0.515671\pi\)
−0.0492134 + 0.998788i \(0.515671\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −10.0000 −0.327561
\(933\) 15.0000 0.491078
\(934\) 27.0000 0.883467
\(935\) 0 0
\(936\) 0 0
\(937\) −16.0000 −0.522697 −0.261349 − 0.965244i \(-0.584167\pi\)
−0.261349 + 0.965244i \(0.584167\pi\)
\(938\) −4.00000 −0.130605
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) −22.0000 −0.717180 −0.358590 − 0.933495i \(-0.616742\pi\)
−0.358590 + 0.933495i \(0.616742\pi\)
\(942\) 23.0000 0.749380
\(943\) −4.00000 −0.130258
\(944\) −8.00000 −0.260378
\(945\) 0 0
\(946\) 35.0000 1.13795
\(947\) −27.0000 −0.877382 −0.438691 − 0.898638i \(-0.644558\pi\)
−0.438691 + 0.898638i \(0.644558\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −5.00000 −0.162136
\(952\) 1.00000 0.0324102
\(953\) 14.0000 0.453504 0.226752 − 0.973952i \(-0.427189\pi\)
0.226752 + 0.973952i \(0.427189\pi\)
\(954\) 3.00000 0.0971286
\(955\) 0 0
\(956\) 11.0000 0.355765
\(957\) −15.0000 −0.484881
\(958\) −28.0000 −0.904639
\(959\) 18.0000 0.581250
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) 6.00000 0.193347
\(964\) −2.00000 −0.0644157
\(965\) 0 0
\(966\) 4.00000 0.128698
\(967\) −10.0000 −0.321578 −0.160789 − 0.986989i \(-0.551404\pi\)
−0.160789 + 0.986989i \(0.551404\pi\)
\(968\) 14.0000 0.449977
\(969\) 0 0
\(970\) 0 0
\(971\) 39.0000 1.25157 0.625785 − 0.779996i \(-0.284779\pi\)
0.625785 + 0.779996i \(0.284779\pi\)
\(972\) 1.00000 0.0320750
\(973\) −11.0000 −0.352644
\(974\) −12.0000 −0.384505
\(975\) 0 0
\(976\) 5.00000 0.160046
\(977\) −3.00000 −0.0959785 −0.0479893 − 0.998848i \(-0.515281\pi\)
−0.0479893 + 0.998848i \(0.515281\pi\)
\(978\) −1.00000 −0.0319765
\(979\) 40.0000 1.27841
\(980\) 0 0
\(981\) 1.00000 0.0319275
\(982\) 12.0000 0.382935
\(983\) −7.00000 −0.223265 −0.111633 − 0.993750i \(-0.535608\pi\)
−0.111633 + 0.993750i \(0.535608\pi\)
\(984\) −1.00000 −0.0318788
\(985\) 0 0
\(986\) −3.00000 −0.0955395
\(987\) −4.00000 −0.127321
\(988\) 0 0
\(989\) 28.0000 0.890348
\(990\) 0 0
\(991\) −5.00000 −0.158830 −0.0794151 − 0.996842i \(-0.525305\pi\)
−0.0794151 + 0.996842i \(0.525305\pi\)
\(992\) 1.00000 0.0317500
\(993\) −26.0000 −0.825085
\(994\) −6.00000 −0.190308
\(995\) 0 0
\(996\) −8.00000 −0.253490
\(997\) −24.0000 −0.760088 −0.380044 − 0.924968i \(-0.624091\pi\)
−0.380044 + 0.924968i \(0.624091\pi\)
\(998\) −30.0000 −0.949633
\(999\) −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.bo.1.1 1
5.2 odd 4 1110.2.d.c.889.2 yes 2
5.3 odd 4 1110.2.d.c.889.1 ✓ 2
5.4 even 2 5550.2.a.d.1.1 1
15.2 even 4 3330.2.d.d.1999.1 2
15.8 even 4 3330.2.d.d.1999.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.d.c.889.1 ✓ 2 5.3 odd 4
1110.2.d.c.889.2 yes 2 5.2 odd 4
3330.2.d.d.1999.1 2 15.2 even 4
3330.2.d.d.1999.2 2 15.8 even 4
5550.2.a.d.1.1 1 5.4 even 2
5550.2.a.bo.1.1 1 1.1 even 1 trivial