↑

Properties

Label 2850.2.a.w.1.1
Level $2850$
Weight $2$
Character 2850.1
Self dual yes
Analytic conductor $22.757$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2850,2,Mod(1,2850)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2850, 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("2850.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2850.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: \(22.7573645761\)
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 \(\chi\) \(=\) 2850.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{12} -4.00000 q^{14} +1.00000 q^{16} -8.00000 q^{17} +1.00000 q^{18} +1.00000 q^{19} -4.00000 q^{21} -1.00000 q^{22} -3.00000 q^{23} +1.00000 q^{24} +1.00000 q^{27} -4.00000 q^{28} -1.00000 q^{29} +1.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} -8.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} +1.00000 q^{38} -10.0000 q^{41} -4.00000 q^{42} -8.00000 q^{43} -1.00000 q^{44} -3.00000 q^{46} +1.00000 q^{48} +9.00000 q^{49} -8.00000 q^{51} +3.00000 q^{53} +1.00000 q^{54} -4.00000 q^{56} +1.00000 q^{57} -1.00000 q^{58} +4.00000 q^{59} +5.00000 q^{61} +1.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} -1.00000 q^{66} -5.00000 q^{67} -8.00000 q^{68} -3.00000 q^{69} -6.00000 q^{71} +1.00000 q^{72} -13.0000 q^{73} -2.00000 q^{74} +1.00000 q^{76} +4.00000 q^{77} -5.00000 q^{79} +1.00000 q^{81} -10.0000 q^{82} +11.0000 q^{83} -4.00000 q^{84} -8.00000 q^{86} -1.00000 q^{87} -1.00000 q^{88} +3.00000 q^{89} -3.00000 q^{92} +1.00000 q^{93} +1.00000 q^{96} +10.0000 q^{97} +9.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 − 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −8.00000 −1.94029 −0.970143 − 0.242536i \(-0.922021\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) −1.00000 −0.213201
\(23\) −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
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −4.00000 −0.755929
\(29\) −1.00000 −0.185695 −0.0928477 − 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 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
\(33\) −1.00000 −0.174078
\(34\) −8.00000 −1.37199
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) −4.00000 −0.617213
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −3.00000 −0.442326
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) −8.00000 −1.12022
\(52\) 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
\(56\) −4.00000 −0.534522
\(57\) 1.00000 0.132453
\(58\) −1.00000 −0.131306
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 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
\(63\) −4.00000 −0.503953
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.00000 −0.123091
\(67\) −5.00000 −0.610847 −0.305424 − 0.952217i \(-0.598798\pi\)
−0.305424 + 0.952217i \(0.598798\pi\)
\(68\) −8.00000 −0.970143
\(69\) −3.00000 −0.361158
\(70\) 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
\(73\) −13.0000 −1.52153 −0.760767 − 0.649025i \(-0.775177\pi\)
−0.760767 + 0.649025i \(0.775177\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) −5.00000 −0.562544 −0.281272 − 0.959628i \(-0.590756\pi\)
−0.281272 + 0.959628i \(0.590756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) 11.0000 1.20741 0.603703 − 0.797209i \(-0.293691\pi\)
0.603703 + 0.797209i \(0.293691\pi\)
\(84\) −4.00000 −0.436436
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) −1.00000 −0.107211
\(88\) −1.00000 −0.106600
\(89\) 3.00000 0.317999 0.159000 − 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −3.00000 −0.312772
\(93\) 1.00000 0.103695
\(94\) 0 0
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 9.00000 0.909137
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −12.0000 −1.19404 −0.597022 − 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) −8.00000 −0.792118
\(103\) 1.00000 0.0985329 0.0492665 − 0.998786i \(-0.484312\pi\)
0.0492665 + 0.998786i \(0.484312\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 3.00000 0.291386
\(107\) 14.0000 1.35343 0.676716 − 0.736245i \(-0.263403\pi\)
0.676716 + 0.736245i \(0.263403\pi\)
\(108\) 1.00000 0.0962250
\(109\) 6.00000 0.574696 0.287348 − 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) −4.00000 −0.377964
\(113\) −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
\(116\) −1.00000 −0.0928477
\(117\) 0 0
\(118\) 4.00000 0.368230
\(119\) 32.0000 2.93344
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 5.00000 0.452679
\(123\) −10.0000 −0.901670
\(124\) 1.00000 0.0898027
\(125\) 0 0
\(126\) −4.00000 −0.356348
\(127\) −17.0000 −1.50851 −0.754253 − 0.656584i \(-0.772001\pi\)
−0.754253 + 0.656584i \(0.772001\pi\)
\(128\) 1.00000 0.0883883
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) −15.0000 −1.31056 −0.655278 − 0.755388i \(-0.727449\pi\)
−0.655278 + 0.755388i \(0.727449\pi\)
\(132\) −1.00000 −0.0870388
\(133\) −4.00000 −0.346844
\(134\) −5.00000 −0.431934
\(135\) 0 0
\(136\) −8.00000 −0.685994
\(137\) 4.00000 0.341743 0.170872 − 0.985293i \(-0.445342\pi\)
0.170872 + 0.985293i \(0.445342\pi\)
\(138\) −3.00000 −0.255377
\(139\) 6.00000 0.508913 0.254457 − 0.967084i \(-0.418103\pi\)
0.254457 + 0.967084i \(0.418103\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −6.00000 −0.503509
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −13.0000 −1.07589
\(147\) 9.00000 0.742307
\(148\) −2.00000 −0.164399
\(149\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 1.00000 0.0811107
\(153\) −8.00000 −0.646762
\(154\) 4.00000 0.322329
\(155\) 0 0
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 − 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) −5.00000 −0.397779
\(159\) 3.00000 0.237915
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) 1.00000 0.0785674
\(163\) 10.0000 0.783260 0.391630 − 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) 11.0000 0.853766
\(167\) −22.0000 −1.70241 −0.851206 − 0.524832i \(-0.824128\pi\)
−0.851206 + 0.524832i \(0.824128\pi\)
\(168\) −4.00000 −0.308607
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) −8.00000 −0.609994
\(173\) −11.0000 −0.836315 −0.418157 − 0.908375i \(-0.637324\pi\)
−0.418157 + 0.908375i \(0.637324\pi\)
\(174\) −1.00000 −0.0758098
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 4.00000 0.300658
\(178\) 3.00000 0.224860
\(179\) 20.0000 1.49487 0.747435 − 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) 12.0000 0.891953 0.445976 − 0.895045i \(-0.352856\pi\)
0.445976 + 0.895045i \(0.352856\pi\)
\(182\) 0 0
\(183\) 5.00000 0.369611
\(184\) −3.00000 −0.221163
\(185\) 0 0
\(186\) 1.00000 0.0733236
\(187\) 8.00000 0.585018
\(188\) 0 0
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) 13.0000 0.940647 0.470323 − 0.882494i \(-0.344137\pi\)
0.470323 + 0.882494i \(0.344137\pi\)
\(192\) 1.00000 0.0721688
\(193\) −14.0000 −1.00774 −0.503871 − 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 26.0000 1.84309 0.921546 − 0.388270i \(-0.126927\pi\)
0.921546 + 0.388270i \(0.126927\pi\)
\(200\) 0 0
\(201\) −5.00000 −0.352673
\(202\) −12.0000 −0.844317
\(203\) 4.00000 0.280745
\(204\) −8.00000 −0.560112
\(205\) 0 0
\(206\) 1.00000 0.0696733
\(207\) −3.00000 −0.208514
\(208\) 0 0
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 25.0000 1.72107 0.860535 − 0.509390i \(-0.170129\pi\)
0.860535 + 0.509390i \(0.170129\pi\)
\(212\) 3.00000 0.206041
\(213\) −6.00000 −0.411113
\(214\) 14.0000 0.957020
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −4.00000 −0.271538
\(218\) 6.00000 0.406371
\(219\) −13.0000 −0.878459
\(220\) 0 0
\(221\) 0 0
\(222\) −2.00000 −0.134231
\(223\) 29.0000 1.94198 0.970992 − 0.239113i \(-0.0768565\pi\)
0.970992 + 0.239113i \(0.0768565\pi\)
\(224\) −4.00000 −0.267261
\(225\) 0 0
\(226\) −1.00000 −0.0665190
\(227\) −30.0000 −1.99117 −0.995585 − 0.0938647i \(-0.970078\pi\)
−0.995585 + 0.0938647i \(0.970078\pi\)
\(228\) 1.00000 0.0662266
\(229\) 13.0000 0.859064 0.429532 − 0.903052i \(-0.358679\pi\)
0.429532 + 0.903052i \(0.358679\pi\)
\(230\) 0 0
\(231\) 4.00000 0.263181
\(232\) −1.00000 −0.0656532
\(233\) −4.00000 −0.262049 −0.131024 − 0.991379i \(-0.541827\pi\)
−0.131024 + 0.991379i \(0.541827\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) −5.00000 −0.324785
\(238\) 32.0000 2.07425
\(239\) 24.0000 1.55243 0.776215 − 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 − 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) −10.0000 −0.642824
\(243\) 1.00000 0.0641500
\(244\) 5.00000 0.320092
\(245\) 0 0
\(246\) −10.0000 −0.637577
\(247\) 0 0
\(248\) 1.00000 0.0635001
\(249\) 11.0000 0.697097
\(250\) 0 0
\(251\) 16.0000 1.00991 0.504956 − 0.863145i \(-0.331509\pi\)
0.504956 + 0.863145i \(0.331509\pi\)
\(252\) −4.00000 −0.251976
\(253\) 3.00000 0.188608
\(254\) −17.0000 −1.06667
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −27.0000 −1.68421 −0.842107 − 0.539311i \(-0.818685\pi\)
−0.842107 + 0.539311i \(0.818685\pi\)
\(258\) −8.00000 −0.498058
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) −15.0000 −0.926703
\(263\) −19.0000 −1.17159 −0.585795 − 0.810459i \(-0.699218\pi\)
−0.585795 + 0.810459i \(0.699218\pi\)
\(264\) −1.00000 −0.0615457
\(265\) 0 0
\(266\) −4.00000 −0.245256
\(267\) 3.00000 0.183597
\(268\) −5.00000 −0.305424
\(269\) −26.0000 −1.58525 −0.792624 − 0.609711i \(-0.791286\pi\)
−0.792624 + 0.609711i \(0.791286\pi\)
\(270\) 0 0
\(271\) −14.0000 −0.850439 −0.425220 − 0.905090i \(-0.639803\pi\)
−0.425220 + 0.905090i \(0.639803\pi\)
\(272\) −8.00000 −0.485071
\(273\) 0 0
\(274\) 4.00000 0.241649
\(275\) 0 0
\(276\) −3.00000 −0.180579
\(277\) 17.0000 1.02143 0.510716 − 0.859750i \(-0.329381\pi\)
0.510716 + 0.859750i \(0.329381\pi\)
\(278\) 6.00000 0.359856
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) −15.0000 −0.894825 −0.447412 − 0.894328i \(-0.647654\pi\)
−0.447412 + 0.894328i \(0.647654\pi\)
\(282\) 0 0
\(283\) 32.0000 1.90220 0.951101 − 0.308879i \(-0.0999539\pi\)
0.951101 + 0.308879i \(0.0999539\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 0 0
\(287\) 40.0000 2.36113
\(288\) 1.00000 0.0589256
\(289\) 47.0000 2.76471
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) −13.0000 −0.760767
\(293\) 1.00000 0.0584206 0.0292103 − 0.999573i \(-0.490701\pi\)
0.0292103 + 0.999573i \(0.490701\pi\)
\(294\) 9.00000 0.524891
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 32.0000 1.84445
\(302\) 0 0
\(303\) −12.0000 −0.689382
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) −8.00000 −0.457330
\(307\) −5.00000 −0.285365 −0.142683 − 0.989769i \(-0.545573\pi\)
−0.142683 + 0.989769i \(0.545573\pi\)
\(308\) 4.00000 0.227921
\(309\) 1.00000 0.0568880
\(310\) 0 0
\(311\) −16.0000 −0.907277 −0.453638 − 0.891186i \(-0.649874\pi\)
−0.453638 + 0.891186i \(0.649874\pi\)
\(312\) 0 0
\(313\) −21.0000 −1.18699 −0.593495 − 0.804838i \(-0.702252\pi\)
−0.593495 + 0.804838i \(0.702252\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) −5.00000 −0.281272
\(317\) 9.00000 0.505490 0.252745 − 0.967533i \(-0.418667\pi\)
0.252745 + 0.967533i \(0.418667\pi\)
\(318\) 3.00000 0.168232
\(319\) 1.00000 0.0559893
\(320\) 0 0
\(321\) 14.0000 0.781404
\(322\) 12.0000 0.668734
\(323\) −8.00000 −0.445132
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 10.0000 0.553849
\(327\) 6.00000 0.331801
\(328\) −10.0000 −0.552158
\(329\) 0 0
\(330\) 0 0
\(331\) −1.00000 −0.0549650 −0.0274825 − 0.999622i \(-0.508749\pi\)
−0.0274825 + 0.999622i \(0.508749\pi\)
\(332\) 11.0000 0.603703
\(333\) −2.00000 −0.109599
\(334\) −22.0000 −1.20379
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) −16.0000 −0.871576 −0.435788 − 0.900049i \(-0.643530\pi\)
−0.435788 + 0.900049i \(0.643530\pi\)
\(338\) −13.0000 −0.707107
\(339\) −1.00000 −0.0543125
\(340\) 0 0
\(341\) −1.00000 −0.0541530
\(342\) 1.00000 0.0540738
\(343\) −8.00000 −0.431959
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) −11.0000 −0.591364
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −1.00000 −0.0536056
\(349\) −25.0000 −1.33822 −0.669110 − 0.743164i \(-0.733324\pi\)
−0.669110 + 0.743164i \(0.733324\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 4.00000 0.212598
\(355\) 0 0
\(356\) 3.00000 0.159000
\(357\) 32.0000 1.69362
\(358\) 20.0000 1.05703
\(359\) 16.0000 0.844448 0.422224 − 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 12.0000 0.630706
\(363\) −10.0000 −0.524864
\(364\) 0 0
\(365\) 0 0
\(366\) 5.00000 0.261354
\(367\) −22.0000 −1.14839 −0.574195 − 0.818718i \(-0.694685\pi\)
−0.574195 + 0.818718i \(0.694685\pi\)
\(368\) −3.00000 −0.156386
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 1.00000 0.0518476
\(373\) 32.0000 1.65690 0.828449 − 0.560065i \(-0.189224\pi\)
0.828449 + 0.560065i \(0.189224\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) −4.00000 −0.205738
\(379\) −32.0000 −1.64373 −0.821865 − 0.569683i \(-0.807066\pi\)
−0.821865 + 0.569683i \(0.807066\pi\)
\(380\) 0 0
\(381\) −17.0000 −0.870936
\(382\) 13.0000 0.665138
\(383\) −18.0000 −0.919757 −0.459879 − 0.887982i \(-0.652107\pi\)
−0.459879 + 0.887982i \(0.652107\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) −8.00000 −0.406663
\(388\) 10.0000 0.507673
\(389\) −22.0000 −1.11544 −0.557722 − 0.830028i \(-0.688325\pi\)
−0.557722 + 0.830028i \(0.688325\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) 9.00000 0.454569
\(393\) −15.0000 −0.756650
\(394\) 0 0
\(395\) 0 0
\(396\) −1.00000 −0.0502519
\(397\) 35.0000 1.75660 0.878300 − 0.478110i \(-0.158678\pi\)
0.878300 + 0.478110i \(0.158678\pi\)
\(398\) 26.0000 1.30326
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) −5.00000 −0.249688 −0.124844 − 0.992176i \(-0.539843\pi\)
−0.124844 + 0.992176i \(0.539843\pi\)
\(402\) −5.00000 −0.249377
\(403\) 0 0
\(404\) −12.0000 −0.597022
\(405\) 0 0
\(406\) 4.00000 0.198517
\(407\) 2.00000 0.0991363
\(408\) −8.00000 −0.396059
\(409\) −18.0000 −0.890043 −0.445021 − 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 0 0
\(411\) 4.00000 0.197305
\(412\) 1.00000 0.0492665
\(413\) −16.0000 −0.787309
\(414\) −3.00000 −0.147442
\(415\) 0 0
\(416\) 0 0
\(417\) 6.00000 0.293821
\(418\) −1.00000 −0.0489116
\(419\) −12.0000 −0.586238 −0.293119 − 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 − 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 25.0000 1.21698
\(423\) 0 0
\(424\) 3.00000 0.145693
\(425\) 0 0
\(426\) −6.00000 −0.290701
\(427\) −20.0000 −0.967868
\(428\) 14.0000 0.676716
\(429\) 0 0
\(430\) 0 0
\(431\) −14.0000 −0.674356 −0.337178 − 0.941441i \(-0.609472\pi\)
−0.337178 + 0.941441i \(0.609472\pi\)
\(432\) 1.00000 0.0481125
\(433\) −22.0000 −1.05725 −0.528626 − 0.848855i \(-0.677293\pi\)
−0.528626 + 0.848855i \(0.677293\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) 6.00000 0.287348
\(437\) −3.00000 −0.143509
\(438\) −13.0000 −0.621164
\(439\) 7.00000 0.334092 0.167046 − 0.985949i \(-0.446577\pi\)
0.167046 + 0.985949i \(0.446577\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) 3.00000 0.142534 0.0712672 − 0.997457i \(-0.477296\pi\)
0.0712672 + 0.997457i \(0.477296\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) 29.0000 1.37319
\(447\) 0 0
\(448\) −4.00000 −0.188982
\(449\) −23.0000 −1.08544 −0.542719 − 0.839915i \(-0.682605\pi\)
−0.542719 + 0.839915i \(0.682605\pi\)
\(450\) 0 0
\(451\) 10.0000 0.470882
\(452\) −1.00000 −0.0470360
\(453\) 0 0
\(454\) −30.0000 −1.40797
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) −22.0000 −1.02912 −0.514558 − 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 13.0000 0.607450
\(459\) −8.00000 −0.373408
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 − 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 4.00000 0.186097
\(463\) 4.00000 0.185896 0.0929479 − 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 0 0
\(466\) −4.00000 −0.185296
\(467\) −3.00000 −0.138823 −0.0694117 − 0.997588i \(-0.522112\pi\)
−0.0694117 + 0.997588i \(0.522112\pi\)
\(468\) 0 0
\(469\) 20.0000 0.923514
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 4.00000 0.184115
\(473\) 8.00000 0.367840
\(474\) −5.00000 −0.229658
\(475\) 0 0
\(476\) 32.0000 1.46672
\(477\) 3.00000 0.137361
\(478\) 24.0000 1.09773
\(479\) 21.0000 0.959514 0.479757 − 0.877401i \(-0.340725\pi\)
0.479757 + 0.877401i \(0.340725\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 8.00000 0.364390
\(483\) 12.0000 0.546019
\(484\) −10.0000 −0.454545
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −32.0000 −1.45006 −0.725029 − 0.688718i \(-0.758174\pi\)
−0.725029 + 0.688718i \(0.758174\pi\)
\(488\) 5.00000 0.226339
\(489\) 10.0000 0.452216
\(490\) 0 0
\(491\) 4.00000 0.180517 0.0902587 − 0.995918i \(-0.471231\pi\)
0.0902587 + 0.995918i \(0.471231\pi\)
\(492\) −10.0000 −0.450835
\(493\) 8.00000 0.360302
\(494\) 0 0
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) 24.0000 1.07655
\(498\) 11.0000 0.492922
\(499\) −4.00000 −0.179065 −0.0895323 − 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) −22.0000 −0.982888
\(502\) 16.0000 0.714115
\(503\) −4.00000 −0.178351 −0.0891756 − 0.996016i \(-0.528423\pi\)
−0.0891756 + 0.996016i \(0.528423\pi\)
\(504\) −4.00000 −0.178174
\(505\) 0 0
\(506\) 3.00000 0.133366
\(507\) −13.0000 −0.577350
\(508\) −17.0000 −0.754253
\(509\) 25.0000 1.10811 0.554053 − 0.832482i \(-0.313081\pi\)
0.554053 + 0.832482i \(0.313081\pi\)
\(510\) 0 0
\(511\) 52.0000 2.30034
\(512\) 1.00000 0.0441942
\(513\) 1.00000 0.0441511
\(514\) −27.0000 −1.19092
\(515\) 0 0
\(516\) −8.00000 −0.352180
\(517\) 0 0
\(518\) 8.00000 0.351500
\(519\) −11.0000 −0.482846
\(520\) 0 0
\(521\) −1.00000 −0.0438108 −0.0219054 − 0.999760i \(-0.506973\pi\)
−0.0219054 + 0.999760i \(0.506973\pi\)
\(522\) −1.00000 −0.0437688
\(523\) 4.00000 0.174908 0.0874539 − 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) −15.0000 −0.655278
\(525\) 0 0
\(526\) −19.0000 −0.828439
\(527\) −8.00000 −0.348485
\(528\) −1.00000 −0.0435194
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) −4.00000 −0.173422
\(533\) 0 0
\(534\) 3.00000 0.129823
\(535\) 0 0
\(536\) −5.00000 −0.215967
\(537\) 20.0000 0.863064
\(538\) −26.0000 −1.12094
\(539\) −9.00000 −0.387657
\(540\) 0 0
\(541\) 27.0000 1.16082 0.580410 − 0.814324i \(-0.302892\pi\)
0.580410 + 0.814324i \(0.302892\pi\)
\(542\) −14.0000 −0.601351
\(543\) 12.0000 0.514969
\(544\) −8.00000 −0.342997
\(545\) 0 0
\(546\) 0 0
\(547\) 21.0000 0.897895 0.448948 − 0.893558i \(-0.351799\pi\)
0.448948 + 0.893558i \(0.351799\pi\)
\(548\) 4.00000 0.170872
\(549\) 5.00000 0.213395
\(550\) 0 0
\(551\) −1.00000 −0.0426014
\(552\) −3.00000 −0.127688
\(553\) 20.0000 0.850487
\(554\) 17.0000 0.722261
\(555\) 0 0
\(556\) 6.00000 0.254457
\(557\) 36.0000 1.52537 0.762684 − 0.646771i \(-0.223881\pi\)
0.762684 + 0.646771i \(0.223881\pi\)
\(558\) 1.00000 0.0423334
\(559\) 0 0
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) −15.0000 −0.632737
\(563\) 22.0000 0.927189 0.463595 − 0.886047i \(-0.346559\pi\)
0.463595 + 0.886047i \(0.346559\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 32.0000 1.34506
\(567\) −4.00000 −0.167984
\(568\) −6.00000 −0.251754
\(569\) 6.00000 0.251533 0.125767 − 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 6.00000 0.251092 0.125546 − 0.992088i \(-0.459932\pi\)
0.125546 + 0.992088i \(0.459932\pi\)
\(572\) 0 0
\(573\) 13.0000 0.543083
\(574\) 40.0000 1.66957
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −21.0000 −0.874241 −0.437121 − 0.899403i \(-0.644002\pi\)
−0.437121 + 0.899403i \(0.644002\pi\)
\(578\) 47.0000 1.95494
\(579\) −14.0000 −0.581820
\(580\) 0 0
\(581\) −44.0000 −1.82543
\(582\) 10.0000 0.414513
\(583\) −3.00000 −0.124247
\(584\) −13.0000 −0.537944
\(585\) 0 0
\(586\) 1.00000 0.0413096
\(587\) −33.0000 −1.36206 −0.681028 − 0.732257i \(-0.738467\pi\)
−0.681028 + 0.732257i \(0.738467\pi\)
\(588\) 9.00000 0.371154
\(589\) 1.00000 0.0412043
\(590\) 0 0
\(591\) 0 0
\(592\) −2.00000 −0.0821995
\(593\) −20.0000 −0.821302 −0.410651 − 0.911793i \(-0.634698\pi\)
−0.410651 + 0.911793i \(0.634698\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) 0 0
\(597\) 26.0000 1.06411
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 − 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −38.0000 −1.55005 −0.775026 − 0.631929i \(-0.782263\pi\)
−0.775026 + 0.631929i \(0.782263\pi\)
\(602\) 32.0000 1.30422
\(603\) −5.00000 −0.203616
\(604\) 0 0
\(605\) 0 0
\(606\) −12.0000 −0.487467
\(607\) 27.0000 1.09590 0.547948 − 0.836512i \(-0.315409\pi\)
0.547948 + 0.836512i \(0.315409\pi\)
\(608\) 1.00000 0.0405554
\(609\) 4.00000 0.162088
\(610\) 0 0
\(611\) 0 0
\(612\) −8.00000 −0.323381
\(613\) 38.0000 1.53481 0.767403 − 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) −5.00000 −0.201784
\(615\) 0 0
\(616\) 4.00000 0.161165
\(617\) 6.00000 0.241551 0.120775 − 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 1.00000 0.0402259
\(619\) −20.0000 −0.803868 −0.401934 − 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) −3.00000 −0.120386
\(622\) −16.0000 −0.641542
\(623\) −12.0000 −0.480770
\(624\) 0 0
\(625\) 0 0
\(626\) −21.0000 −0.839329
\(627\) −1.00000 −0.0399362
\(628\) 2.00000 0.0798087
\(629\) 16.0000 0.637962
\(630\) 0 0
\(631\) 44.0000 1.75161 0.875806 − 0.482663i \(-0.160330\pi\)
0.875806 + 0.482663i \(0.160330\pi\)
\(632\) −5.00000 −0.198889
\(633\) 25.0000 0.993661
\(634\) 9.00000 0.357436
\(635\) 0 0
\(636\) 3.00000 0.118958
\(637\) 0 0
\(638\) 1.00000 0.0395904
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) 6.00000 0.236986 0.118493 − 0.992955i \(-0.462194\pi\)
0.118493 + 0.992955i \(0.462194\pi\)
\(642\) 14.0000 0.552536
\(643\) 2.00000 0.0788723 0.0394362 − 0.999222i \(-0.487444\pi\)
0.0394362 + 0.999222i \(0.487444\pi\)
\(644\) 12.0000 0.472866
\(645\) 0 0
\(646\) −8.00000 −0.314756
\(647\) 43.0000 1.69050 0.845252 − 0.534368i \(-0.179450\pi\)
0.845252 + 0.534368i \(0.179450\pi\)
\(648\) 1.00000 0.0392837
\(649\) −4.00000 −0.157014
\(650\) 0 0
\(651\) −4.00000 −0.156772
\(652\) 10.0000 0.391630
\(653\) −24.0000 −0.939193 −0.469596 − 0.882881i \(-0.655601\pi\)
−0.469596 + 0.882881i \(0.655601\pi\)
\(654\) 6.00000 0.234619
\(655\) 0 0
\(656\) −10.0000 −0.390434
\(657\) −13.0000 −0.507178
\(658\) 0 0
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −8.00000 −0.311164 −0.155582 − 0.987823i \(-0.549725\pi\)
−0.155582 + 0.987823i \(0.549725\pi\)
\(662\) −1.00000 −0.0388661
\(663\) 0 0
\(664\) 11.0000 0.426883
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 3.00000 0.116160
\(668\) −22.0000 −0.851206
\(669\) 29.0000 1.12120
\(670\) 0 0
\(671\) −5.00000 −0.193023
\(672\) −4.00000 −0.154303
\(673\) 16.0000 0.616755 0.308377 − 0.951264i \(-0.400214\pi\)
0.308377 + 0.951264i \(0.400214\pi\)
\(674\) −16.0000 −0.616297
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) 3.00000 0.115299 0.0576497 − 0.998337i \(-0.481639\pi\)
0.0576497 + 0.998337i \(0.481639\pi\)
\(678\) −1.00000 −0.0384048
\(679\) −40.0000 −1.53506
\(680\) 0 0
\(681\) −30.0000 −1.14960
\(682\) −1.00000 −0.0382920
\(683\) −6.00000 −0.229584 −0.114792 − 0.993390i \(-0.536620\pi\)
−0.114792 + 0.993390i \(0.536620\pi\)
\(684\) 1.00000 0.0382360
\(685\) 0 0
\(686\) −8.00000 −0.305441
\(687\) 13.0000 0.495981
\(688\) −8.00000 −0.304997
\(689\) 0 0
\(690\) 0 0
\(691\) 10.0000 0.380418 0.190209 − 0.981744i \(-0.439083\pi\)
0.190209 + 0.981744i \(0.439083\pi\)
\(692\) −11.0000 −0.418157
\(693\) 4.00000 0.151947
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) −1.00000 −0.0379049
\(697\) 80.0000 3.03022
\(698\) −25.0000 −0.946264
\(699\) −4.00000 −0.151294
\(700\) 0 0
\(701\) −8.00000 −0.302156 −0.151078 − 0.988522i \(-0.548274\pi\)
−0.151078 + 0.988522i \(0.548274\pi\)
\(702\) 0 0
\(703\) −2.00000 −0.0754314
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 0 0
\(707\) 48.0000 1.80523
\(708\) 4.00000 0.150329
\(709\) −35.0000 −1.31445 −0.657226 − 0.753693i \(-0.728270\pi\)
−0.657226 + 0.753693i \(0.728270\pi\)
\(710\) 0 0
\(711\) −5.00000 −0.187515
\(712\) 3.00000 0.112430
\(713\) −3.00000 −0.112351
\(714\) 32.0000 1.19757
\(715\) 0 0
\(716\) 20.0000 0.747435
\(717\) 24.0000 0.896296
\(718\) 16.0000 0.597115
\(719\) 29.0000 1.08152 0.540759 − 0.841178i \(-0.318137\pi\)
0.540759 + 0.841178i \(0.318137\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) 1.00000 0.0372161
\(723\) 8.00000 0.297523
\(724\) 12.0000 0.445976
\(725\) 0 0
\(726\) −10.0000 −0.371135
\(727\) −18.0000 −0.667583 −0.333792 − 0.942647i \(-0.608328\pi\)
−0.333792 + 0.942647i \(0.608328\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 64.0000 2.36713
\(732\) 5.00000 0.184805
\(733\) 23.0000 0.849524 0.424762 − 0.905305i \(-0.360358\pi\)
0.424762 + 0.905305i \(0.360358\pi\)
\(734\) −22.0000 −0.812035
\(735\) 0 0
\(736\) −3.00000 −0.110581
\(737\) 5.00000 0.184177
\(738\) −10.0000 −0.368105
\(739\) 4.00000 0.147142 0.0735712 − 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −12.0000 −0.440534
\(743\) −24.0000 −0.880475 −0.440237 − 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 1.00000 0.0366618
\(745\) 0 0
\(746\) 32.0000 1.17160
\(747\) 11.0000 0.402469
\(748\) 8.00000 0.292509
\(749\) −56.0000 −2.04620
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 − 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 0 0
\(753\) 16.0000 0.583072
\(754\) 0 0
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) 23.0000 0.835949 0.417975 − 0.908459i \(-0.362740\pi\)
0.417975 + 0.908459i \(0.362740\pi\)
\(758\) −32.0000 −1.16229
\(759\) 3.00000 0.108893
\(760\) 0 0
\(761\) −26.0000 −0.942499 −0.471250 − 0.882000i \(-0.656197\pi\)
−0.471250 + 0.882000i \(0.656197\pi\)
\(762\) −17.0000 −0.615845
\(763\) −24.0000 −0.868858
\(764\) 13.0000 0.470323
\(765\) 0 0
\(766\) −18.0000 −0.650366
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −45.0000 −1.62274 −0.811371 − 0.584532i \(-0.801278\pi\)
−0.811371 + 0.584532i \(0.801278\pi\)
\(770\) 0 0
\(771\) −27.0000 −0.972381
\(772\) −14.0000 −0.503871
\(773\) 54.0000 1.94225 0.971123 − 0.238581i \(-0.0766824\pi\)
0.971123 + 0.238581i \(0.0766824\pi\)
\(774\) −8.00000 −0.287554
\(775\) 0 0
\(776\) 10.0000 0.358979
\(777\) 8.00000 0.286998
\(778\) −22.0000 −0.788738
\(779\) −10.0000 −0.358287
\(780\) 0 0
\(781\) 6.00000 0.214697
\(782\) 24.0000 0.858238
\(783\) −1.00000 −0.0357371
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) −15.0000 −0.535032
\(787\) 1.00000 0.0356462 0.0178231 − 0.999841i \(-0.494326\pi\)
0.0178231 + 0.999841i \(0.494326\pi\)
\(788\) 0 0
\(789\) −19.0000 −0.676418
\(790\) 0 0
\(791\) 4.00000 0.142224
\(792\) −1.00000 −0.0355335
\(793\) 0 0
\(794\) 35.0000 1.24210
\(795\) 0 0
\(796\) 26.0000 0.921546
\(797\) −46.0000 −1.62940 −0.814702 − 0.579880i \(-0.803099\pi\)
−0.814702 + 0.579880i \(0.803099\pi\)
\(798\) −4.00000 −0.141598
\(799\) 0 0
\(800\) 0 0
\(801\) 3.00000 0.106000
\(802\) −5.00000 −0.176556
\(803\) 13.0000 0.458760
\(804\) −5.00000 −0.176336
\(805\) 0 0
\(806\) 0 0
\(807\) −26.0000 −0.915243
\(808\) −12.0000 −0.422159
\(809\) −16.0000 −0.562530 −0.281265 − 0.959630i \(-0.590754\pi\)
−0.281265 + 0.959630i \(0.590754\pi\)
\(810\) 0 0
\(811\) −49.0000 −1.72062 −0.860311 − 0.509769i \(-0.829731\pi\)
−0.860311 + 0.509769i \(0.829731\pi\)
\(812\) 4.00000 0.140372
\(813\) −14.0000 −0.491001
\(814\) 2.00000 0.0701000
\(815\) 0 0
\(816\) −8.00000 −0.280056
\(817\) −8.00000 −0.279885
\(818\) −18.0000 −0.629355
\(819\) 0 0
\(820\) 0 0
\(821\) −10.0000 −0.349002 −0.174501 − 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) 4.00000 0.139516
\(823\) −26.0000 −0.906303 −0.453152 − 0.891434i \(-0.649700\pi\)
−0.453152 + 0.891434i \(0.649700\pi\)
\(824\) 1.00000 0.0348367
\(825\) 0 0
\(826\) −16.0000 −0.556711
\(827\) −28.0000 −0.973655 −0.486828 − 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) −3.00000 −0.104257
\(829\) 26.0000 0.903017 0.451509 − 0.892267i \(-0.350886\pi\)
0.451509 + 0.892267i \(0.350886\pi\)
\(830\) 0 0
\(831\) 17.0000 0.589723
\(832\) 0 0
\(833\) −72.0000 −2.49465
\(834\) 6.00000 0.207763
\(835\) 0 0
\(836\) −1.00000 −0.0345857
\(837\) 1.00000 0.0345651
\(838\) −12.0000 −0.414533
\(839\) −48.0000 −1.65714 −0.828572 − 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 22.0000 0.758170
\(843\) −15.0000 −0.516627
\(844\) 25.0000 0.860535
\(845\) 0 0
\(846\) 0 0
\(847\) 40.0000 1.37442
\(848\) 3.00000 0.103020
\(849\) 32.0000 1.09824
\(850\) 0 0
\(851\) 6.00000 0.205677
\(852\) −6.00000 −0.205557
\(853\) 46.0000 1.57501 0.787505 − 0.616308i \(-0.211372\pi\)
0.787505 + 0.616308i \(0.211372\pi\)
\(854\) −20.0000 −0.684386
\(855\) 0 0
\(856\) 14.0000 0.478510
\(857\) −18.0000 −0.614868 −0.307434 − 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) 34.0000 1.16007 0.580033 − 0.814593i \(-0.303040\pi\)
0.580033 + 0.814593i \(0.303040\pi\)
\(860\) 0 0
\(861\) 40.0000 1.36320
\(862\) −14.0000 −0.476842
\(863\) 28.0000 0.953131 0.476566 − 0.879139i \(-0.341881\pi\)
0.476566 + 0.879139i \(0.341881\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −22.0000 −0.747590
\(867\) 47.0000 1.59620
\(868\) −4.00000 −0.135769
\(869\) 5.00000 0.169613
\(870\) 0 0
\(871\) 0 0
\(872\) 6.00000 0.203186
\(873\) 10.0000 0.338449
\(874\) −3.00000 −0.101477
\(875\) 0 0
\(876\) −13.0000 −0.439229
\(877\) 42.0000 1.41824 0.709120 − 0.705088i \(-0.249093\pi\)
0.709120 + 0.705088i \(0.249093\pi\)
\(878\) 7.00000 0.236239
\(879\) 1.00000 0.0337292
\(880\) 0 0
\(881\) −8.00000 −0.269527 −0.134763 − 0.990878i \(-0.543027\pi\)
−0.134763 + 0.990878i \(0.543027\pi\)
\(882\) 9.00000 0.303046
\(883\) −26.0000 −0.874970 −0.437485 − 0.899226i \(-0.644131\pi\)
−0.437485 + 0.899226i \(0.644131\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 3.00000 0.100787
\(887\) −48.0000 −1.61168 −0.805841 − 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 68.0000 2.28065
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 29.0000 0.970992
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −4.00000 −0.133631
\(897\) 0 0
\(898\) −23.0000 −0.767520
\(899\) −1.00000 −0.0333519
\(900\) 0 0
\(901\) −24.0000 −0.799556
\(902\) 10.0000 0.332964
\(903\) 32.0000 1.06489
\(904\) −1.00000 −0.0332595
\(905\) 0 0
\(906\) 0 0
\(907\) −48.0000 −1.59381 −0.796907 − 0.604102i \(-0.793532\pi\)
−0.796907 + 0.604102i \(0.793532\pi\)
\(908\) −30.0000 −0.995585
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) 18.0000 0.596367 0.298183 − 0.954509i \(-0.403619\pi\)
0.298183 + 0.954509i \(0.403619\pi\)
\(912\) 1.00000 0.0331133
\(913\) −11.0000 −0.364047
\(914\) −22.0000 −0.727695
\(915\) 0 0
\(916\) 13.0000 0.429532
\(917\) 60.0000 1.98137
\(918\) −8.00000 −0.264039
\(919\) −2.00000 −0.0659739 −0.0329870 − 0.999456i \(-0.510502\pi\)
−0.0329870 + 0.999456i \(0.510502\pi\)
\(920\) 0 0
\(921\) −5.00000 −0.164756
\(922\) −12.0000 −0.395199
\(923\) 0 0
\(924\) 4.00000 0.131590
\(925\) 0 0
\(926\) 4.00000 0.131448
\(927\) 1.00000 0.0328443
\(928\) −1.00000 −0.0328266
\(929\) −12.0000 −0.393707 −0.196854 − 0.980433i \(-0.563072\pi\)
−0.196854 + 0.980433i \(0.563072\pi\)
\(930\) 0 0
\(931\) 9.00000 0.294963
\(932\) −4.00000 −0.131024
\(933\) −16.0000 −0.523816
\(934\) −3.00000 −0.0981630
\(935\) 0 0
\(936\) 0 0
\(937\) 42.0000 1.37208 0.686040 − 0.727564i \(-0.259347\pi\)
0.686040 + 0.727564i \(0.259347\pi\)
\(938\) 20.0000 0.653023
\(939\) −21.0000 −0.685309
\(940\) 0 0
\(941\) −19.0000 −0.619382 −0.309691 − 0.950837i \(-0.600226\pi\)
−0.309691 + 0.950837i \(0.600226\pi\)
\(942\) 2.00000 0.0651635
\(943\) 30.0000 0.976934
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 8.00000 0.260102
\(947\) −52.0000 −1.68977 −0.844886 − 0.534946i \(-0.820332\pi\)
−0.844886 + 0.534946i \(0.820332\pi\)
\(948\) −5.00000 −0.162392
\(949\) 0 0
\(950\) 0 0
\(951\) 9.00000 0.291845
\(952\) 32.0000 1.03713
\(953\) −15.0000 −0.485898 −0.242949 − 0.970039i \(-0.578115\pi\)
−0.242949 + 0.970039i \(0.578115\pi\)
\(954\) 3.00000 0.0971286
\(955\) 0 0
\(956\) 24.0000 0.776215
\(957\) 1.00000 0.0323254
\(958\) 21.0000 0.678479
\(959\) −16.0000 −0.516667
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) 14.0000 0.451144
\(964\) 8.00000 0.257663
\(965\) 0 0
\(966\) 12.0000 0.386094
\(967\) 38.0000 1.22200 0.610999 − 0.791632i \(-0.290768\pi\)
0.610999 + 0.791632i \(0.290768\pi\)
\(968\) −10.0000 −0.321412
\(969\) −8.00000 −0.256997
\(970\) 0 0
\(971\) −8.00000 −0.256732 −0.128366 − 0.991727i \(-0.540973\pi\)
−0.128366 + 0.991727i \(0.540973\pi\)
\(972\) 1.00000 0.0320750
\(973\) −24.0000 −0.769405
\(974\) −32.0000 −1.02535
\(975\) 0 0
\(976\) 5.00000 0.160046
\(977\) 2.00000 0.0639857 0.0319928 − 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) 10.0000 0.319765
\(979\) −3.00000 −0.0958804
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) 4.00000 0.127645
\(983\) 30.0000 0.956851 0.478426 − 0.878128i \(-0.341208\pi\)
0.478426 + 0.878128i \(0.341208\pi\)
\(984\) −10.0000 −0.318788
\(985\) 0 0
\(986\) 8.00000 0.254772
\(987\) 0 0
\(988\) 0 0
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) −25.0000 −0.794151 −0.397076 − 0.917786i \(-0.629975\pi\)
−0.397076 + 0.917786i \(0.629975\pi\)
\(992\) 1.00000 0.0317500
\(993\) −1.00000 −0.0317340
\(994\) 24.0000 0.761234
\(995\) 0 0
\(996\) 11.0000 0.348548
\(997\) −13.0000 −0.411714 −0.205857 − 0.978582i \(-0.565998\pi\)
−0.205857 + 0.978582i \(0.565998\pi\)
\(998\) −4.00000 −0.126618
\(999\) −2.00000 −0.0632772
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2850.2.a.w.1.1 yes 1
3.2 odd 2 8550.2.a.c.1.1 1
5.2 odd 4 2850.2.d.o.799.2 2
5.3 odd 4 2850.2.d.o.799.1 2
5.4 even 2 2850.2.a.f.1.1 ✓ 1
15.14 odd 2 8550.2.a.bk.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2850.2.a.f.1.1 ✓ 1 5.4 even 2
2850.2.a.w.1.1 yes 1 1.1 even 1 trivial
2850.2.d.o.799.1 2 5.3 odd 4
2850.2.d.o.799.2 2 5.2 odd 4
8550.2.a.c.1.1 1 3.2 odd 2
8550.2.a.bk.1.1 1 15.14 odd 2