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Properties

Label 9150.2.a.n.1.1
Level $9150$
Weight $2$
Character 9150.1
Self dual yes
Analytic conductor $73.063$
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] = [9150,2,Mod(1,9150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9150, 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("9150.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9150 = 2 \cdot 3 \cdot 5^{2} \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9150.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: \(73.0631178495\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 366)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 9150.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{12} +5.00000 q^{13} -3.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -8.00000 q^{19} +3.00000 q^{21} +1.00000 q^{22} -5.00000 q^{23} -1.00000 q^{24} -5.00000 q^{26} +1.00000 q^{27} +3.00000 q^{28} -4.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} -4.00000 q^{37} +8.00000 q^{38} +5.00000 q^{39} +3.00000 q^{41} -3.00000 q^{42} -4.00000 q^{43} -1.00000 q^{44} +5.00000 q^{46} -2.00000 q^{47} +1.00000 q^{48} +2.00000 q^{49} -2.00000 q^{51} +5.00000 q^{52} -1.00000 q^{54} -3.00000 q^{56} -8.00000 q^{57} -7.00000 q^{59} +1.00000 q^{61} +4.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} +1.00000 q^{66} +13.0000 q^{67} -2.00000 q^{68} -5.00000 q^{69} -16.0000 q^{71} -1.00000 q^{72} -9.00000 q^{73} +4.00000 q^{74} -8.00000 q^{76} -3.00000 q^{77} -5.00000 q^{78} -1.00000 q^{79} +1.00000 q^{81} -3.00000 q^{82} -14.0000 q^{83} +3.00000 q^{84} +4.00000 q^{86} +1.00000 q^{88} -4.00000 q^{89} +15.0000 q^{91} -5.00000 q^{92} -4.00000 q^{93} +2.00000 q^{94} -1.00000 q^{96} -14.0000 q^{97} -2.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
\(7\) 3.00000 1.13389 0.566947 − 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 − 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 1.00000 0.288675
\(13\) 5.00000 1.38675 0.693375 − 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) −3.00000 −0.801784
\(15\) 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
\(19\) −8.00000 −1.83533 −0.917663 − 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 1.00000 0.213201
\(23\) −5.00000 −1.04257 −0.521286 − 0.853382i \(-0.674548\pi\)
−0.521286 + 0.853382i \(0.674548\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −5.00000 −0.980581
\(27\) 1.00000 0.192450
\(28\) 3.00000 0.566947
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −4.00000 −0.657596 −0.328798 − 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 8.00000 1.29777
\(39\) 5.00000 0.800641
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 − 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) −3.00000 −0.462910
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 5.00000 0.737210
\(47\) −2.00000 −0.291730 −0.145865 − 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 1.00000 0.144338
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 5.00000 0.693375
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −3.00000 −0.400892
\(57\) −8.00000 −1.05963
\(58\) 0 0
\(59\) −7.00000 −0.911322 −0.455661 − 0.890153i \(-0.650597\pi\)
−0.455661 + 0.890153i \(0.650597\pi\)
\(60\) 0 0
\(61\) 1.00000 0.128037
\(62\) 4.00000 0.508001
\(63\) 3.00000 0.377964
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.00000 0.123091
\(67\) 13.0000 1.58820 0.794101 − 0.607785i \(-0.207942\pi\)
0.794101 + 0.607785i \(0.207942\pi\)
\(68\) −2.00000 −0.242536
\(69\) −5.00000 −0.601929
\(70\) 0 0
\(71\) −16.0000 −1.89885 −0.949425 − 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) −1.00000 −0.117851
\(73\) −9.00000 −1.05337 −0.526685 − 0.850060i \(-0.676565\pi\)
−0.526685 + 0.850060i \(0.676565\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) −8.00000 −0.917663
\(77\) −3.00000 −0.341882
\(78\) −5.00000 −0.566139
\(79\) −1.00000 −0.112509 −0.0562544 − 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −3.00000 −0.331295
\(83\) −14.0000 −1.53670 −0.768350 − 0.640030i \(-0.778922\pi\)
−0.768350 + 0.640030i \(0.778922\pi\)
\(84\) 3.00000 0.327327
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) −4.00000 −0.423999 −0.212000 − 0.977270i \(-0.567998\pi\)
−0.212000 + 0.977270i \(0.567998\pi\)
\(90\) 0 0
\(91\) 15.0000 1.57243
\(92\) −5.00000 −0.521286
\(93\) −4.00000 −0.414781
\(94\) 2.00000 0.206284
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −14.0000 −1.42148 −0.710742 − 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) −2.00000 −0.202031
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −8.00000 −0.796030 −0.398015 − 0.917379i \(-0.630301\pi\)
−0.398015 + 0.917379i \(0.630301\pi\)
\(102\) 2.00000 0.198030
\(103\) 14.0000 1.37946 0.689730 − 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) −5.00000 −0.490290
\(105\) 0 0
\(106\) 0 0
\(107\) 2.00000 0.193347 0.0966736 − 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) 1.00000 0.0962250
\(109\) −15.0000 −1.43674 −0.718370 − 0.695662i \(-0.755111\pi\)
−0.718370 + 0.695662i \(0.755111\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 3.00000 0.283473
\(113\) −11.0000 −1.03479 −0.517396 − 0.855746i \(-0.673099\pi\)
−0.517396 + 0.855746i \(0.673099\pi\)
\(114\) 8.00000 0.749269
\(115\) 0 0
\(116\) 0 0
\(117\) 5.00000 0.462250
\(118\) 7.00000 0.644402
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) −1.00000 −0.0905357
\(123\) 3.00000 0.270501
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) −3.00000 −0.267261
\(127\) 22.0000 1.95218 0.976092 − 0.217357i \(-0.0697436\pi\)
0.976092 + 0.217357i \(0.0697436\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 10.0000 0.873704 0.436852 − 0.899533i \(-0.356093\pi\)
0.436852 + 0.899533i \(0.356093\pi\)
\(132\) −1.00000 −0.0870388
\(133\) −24.0000 −2.08106
\(134\) −13.0000 −1.12303
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) −15.0000 −1.28154 −0.640768 − 0.767734i \(-0.721384\pi\)
−0.640768 + 0.767734i \(0.721384\pi\)
\(138\) 5.00000 0.425628
\(139\) −5.00000 −0.424094 −0.212047 − 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) 0 0
\(141\) −2.00000 −0.168430
\(142\) 16.0000 1.34269
\(143\) −5.00000 −0.418121
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 9.00000 0.744845
\(147\) 2.00000 0.164957
\(148\) −4.00000 −0.328798
\(149\) −21.0000 −1.72039 −0.860194 − 0.509968i \(-0.829657\pi\)
−0.860194 + 0.509968i \(0.829657\pi\)
\(150\) 0 0
\(151\) −9.00000 −0.732410 −0.366205 − 0.930534i \(-0.619343\pi\)
−0.366205 + 0.930534i \(0.619343\pi\)
\(152\) 8.00000 0.648886
\(153\) −2.00000 −0.161690
\(154\) 3.00000 0.241747
\(155\) 0 0
\(156\) 5.00000 0.400320
\(157\) −6.00000 −0.478852 −0.239426 − 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) 1.00000 0.0795557
\(159\) 0 0
\(160\) 0 0
\(161\) −15.0000 −1.18217
\(162\) −1.00000 −0.0785674
\(163\) 4.00000 0.313304 0.156652 − 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 3.00000 0.234261
\(165\) 0 0
\(166\) 14.0000 1.08661
\(167\) 18.0000 1.39288 0.696441 − 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) −3.00000 −0.231455
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) −8.00000 −0.611775
\(172\) −4.00000 −0.304997
\(173\) 22.0000 1.67263 0.836315 − 0.548250i \(-0.184706\pi\)
0.836315 + 0.548250i \(0.184706\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) −7.00000 −0.526152
\(178\) 4.00000 0.299813
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 26.0000 1.93256 0.966282 − 0.257485i \(-0.0828937\pi\)
0.966282 + 0.257485i \(0.0828937\pi\)
\(182\) −15.0000 −1.11187
\(183\) 1.00000 0.0739221
\(184\) 5.00000 0.368605
\(185\) 0 0
\(186\) 4.00000 0.293294
\(187\) 2.00000 0.146254
\(188\) −2.00000 −0.145865
\(189\) 3.00000 0.218218
\(190\) 0 0
\(191\) −15.0000 −1.08536 −0.542681 − 0.839939i \(-0.682591\pi\)
−0.542681 + 0.839939i \(0.682591\pi\)
\(192\) 1.00000 0.0721688
\(193\) 18.0000 1.29567 0.647834 − 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) −21.0000 −1.49619 −0.748094 − 0.663593i \(-0.769031\pi\)
−0.748094 + 0.663593i \(0.769031\pi\)
\(198\) 1.00000 0.0710669
\(199\) 18.0000 1.27599 0.637993 − 0.770042i \(-0.279765\pi\)
0.637993 + 0.770042i \(0.279765\pi\)
\(200\) 0 0
\(201\) 13.0000 0.916949
\(202\) 8.00000 0.562878
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) −14.0000 −0.975426
\(207\) −5.00000 −0.347524
\(208\) 5.00000 0.346688
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 − 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 0 0
\(213\) −16.0000 −1.09630
\(214\) −2.00000 −0.136717
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) −12.0000 −0.814613
\(218\) 15.0000 1.01593
\(219\) −9.00000 −0.608164
\(220\) 0 0
\(221\) −10.0000 −0.672673
\(222\) 4.00000 0.268462
\(223\) 21.0000 1.40626 0.703132 − 0.711059i \(-0.251784\pi\)
0.703132 + 0.711059i \(0.251784\pi\)
\(224\) −3.00000 −0.200446
\(225\) 0 0
\(226\) 11.0000 0.731709
\(227\) 15.0000 0.995585 0.497792 − 0.867296i \(-0.334144\pi\)
0.497792 + 0.867296i \(0.334144\pi\)
\(228\) −8.00000 −0.529813
\(229\) −17.0000 −1.12339 −0.561696 − 0.827344i \(-0.689851\pi\)
−0.561696 + 0.827344i \(0.689851\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) 0 0
\(233\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) −5.00000 −0.326860
\(235\) 0 0
\(236\) −7.00000 −0.455661
\(237\) −1.00000 −0.0649570
\(238\) 6.00000 0.388922
\(239\) −12.0000 −0.776215 −0.388108 − 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 25.0000 1.61039 0.805196 − 0.593009i \(-0.202060\pi\)
0.805196 + 0.593009i \(0.202060\pi\)
\(242\) 10.0000 0.642824
\(243\) 1.00000 0.0641500
\(244\) 1.00000 0.0640184
\(245\) 0 0
\(246\) −3.00000 −0.191273
\(247\) −40.0000 −2.54514
\(248\) 4.00000 0.254000
\(249\) −14.0000 −0.887214
\(250\) 0 0
\(251\) 8.00000 0.504956 0.252478 − 0.967603i \(-0.418755\pi\)
0.252478 + 0.967603i \(0.418755\pi\)
\(252\) 3.00000 0.188982
\(253\) 5.00000 0.314347
\(254\) −22.0000 −1.38040
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 − 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 4.00000 0.249029
\(259\) −12.0000 −0.745644
\(260\) 0 0
\(261\) 0 0
\(262\) −10.0000 −0.617802
\(263\) −24.0000 −1.47990 −0.739952 − 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 1.00000 0.0615457
\(265\) 0 0
\(266\) 24.0000 1.47153
\(267\) −4.00000 −0.244796
\(268\) 13.0000 0.794101
\(269\) −18.0000 −1.09748 −0.548740 − 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 − 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) −2.00000 −0.121268
\(273\) 15.0000 0.907841
\(274\) 15.0000 0.906183
\(275\) 0 0
\(276\) −5.00000 −0.300965
\(277\) −22.0000 −1.32185 −0.660926 − 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 5.00000 0.299880
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 24.0000 1.43172 0.715860 − 0.698244i \(-0.246035\pi\)
0.715860 + 0.698244i \(0.246035\pi\)
\(282\) 2.00000 0.119098
\(283\) 14.0000 0.832214 0.416107 − 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) −16.0000 −0.949425
\(285\) 0 0
\(286\) 5.00000 0.295656
\(287\) 9.00000 0.531253
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −14.0000 −0.820695
\(292\) −9.00000 −0.526685
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) −2.00000 −0.116642
\(295\) 0 0
\(296\) 4.00000 0.232495
\(297\) −1.00000 −0.0580259
\(298\) 21.0000 1.21650
\(299\) −25.0000 −1.44579
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) 9.00000 0.517892
\(303\) −8.00000 −0.459588
\(304\) −8.00000 −0.458831
\(305\) 0 0
\(306\) 2.00000 0.114332
\(307\) −19.0000 −1.08439 −0.542194 − 0.840254i \(-0.682406\pi\)
−0.542194 + 0.840254i \(0.682406\pi\)
\(308\) −3.00000 −0.170941
\(309\) 14.0000 0.796432
\(310\) 0 0
\(311\) 35.0000 1.98467 0.992334 − 0.123585i \(-0.0394392\pi\)
0.992334 + 0.123585i \(0.0394392\pi\)
\(312\) −5.00000 −0.283069
\(313\) 8.00000 0.452187 0.226093 − 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) 6.00000 0.338600
\(315\) 0 0
\(316\) −1.00000 −0.0562544
\(317\) −10.0000 −0.561656 −0.280828 − 0.959758i \(-0.590609\pi\)
−0.280828 + 0.959758i \(0.590609\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 2.00000 0.111629
\(322\) 15.0000 0.835917
\(323\) 16.0000 0.890264
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) −15.0000 −0.829502
\(328\) −3.00000 −0.165647
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 9.00000 0.494685 0.247342 − 0.968928i \(-0.420443\pi\)
0.247342 + 0.968928i \(0.420443\pi\)
\(332\) −14.0000 −0.768350
\(333\) −4.00000 −0.219199
\(334\) −18.0000 −0.984916
\(335\) 0 0
\(336\) 3.00000 0.163663
\(337\) 16.0000 0.871576 0.435788 − 0.900049i \(-0.356470\pi\)
0.435788 + 0.900049i \(0.356470\pi\)
\(338\) −12.0000 −0.652714
\(339\) −11.0000 −0.597438
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) 8.00000 0.432590
\(343\) −15.0000 −0.809924
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) −22.0000 −1.18273
\(347\) 4.00000 0.214731 0.107366 − 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) 0 0
\(349\) −22.0000 −1.17763 −0.588817 − 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) 0 0
\(351\) 5.00000 0.266880
\(352\) 1.00000 0.0533002
\(353\) −25.0000 −1.33062 −0.665308 − 0.746569i \(-0.731700\pi\)
−0.665308 + 0.746569i \(0.731700\pi\)
\(354\) 7.00000 0.372046
\(355\) 0 0
\(356\) −4.00000 −0.212000
\(357\) −6.00000 −0.317554
\(358\) 0 0
\(359\) 36.0000 1.90001 0.950004 − 0.312239i \(-0.101079\pi\)
0.950004 + 0.312239i \(0.101079\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) −26.0000 −1.36653
\(363\) −10.0000 −0.524864
\(364\) 15.0000 0.786214
\(365\) 0 0
\(366\) −1.00000 −0.0522708
\(367\) −36.0000 −1.87918 −0.939592 − 0.342296i \(-0.888796\pi\)
−0.939592 + 0.342296i \(0.888796\pi\)
\(368\) −5.00000 −0.260643
\(369\) 3.00000 0.156174
\(370\) 0 0
\(371\) 0 0
\(372\) −4.00000 −0.207390
\(373\) −4.00000 −0.207112 −0.103556 − 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) −2.00000 −0.103418
\(375\) 0 0
\(376\) 2.00000 0.103142
\(377\) 0 0
\(378\) −3.00000 −0.154303
\(379\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 22.0000 1.12709
\(382\) 15.0000 0.767467
\(383\) −7.00000 −0.357683 −0.178842 − 0.983878i \(-0.557235\pi\)
−0.178842 + 0.983878i \(0.557235\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −18.0000 −0.916176
\(387\) −4.00000 −0.203331
\(388\) −14.0000 −0.710742
\(389\) 18.0000 0.912636 0.456318 − 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) 10.0000 0.505722
\(392\) −2.00000 −0.101015
\(393\) 10.0000 0.504433
\(394\) 21.0000 1.05796
\(395\) 0 0
\(396\) −1.00000 −0.0502519
\(397\) 2.00000 0.100377 0.0501886 − 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) −18.0000 −0.902258
\(399\) −24.0000 −1.20150
\(400\) 0 0
\(401\) 6.00000 0.299626 0.149813 − 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) −13.0000 −0.648381
\(403\) −20.0000 −0.996271
\(404\) −8.00000 −0.398015
\(405\) 0 0
\(406\) 0 0
\(407\) 4.00000 0.198273
\(408\) 2.00000 0.0990148
\(409\) −20.0000 −0.988936 −0.494468 − 0.869196i \(-0.664637\pi\)
−0.494468 + 0.869196i \(0.664637\pi\)
\(410\) 0 0
\(411\) −15.0000 −0.739895
\(412\) 14.0000 0.689730
\(413\) −21.0000 −1.03334
\(414\) 5.00000 0.245737
\(415\) 0 0
\(416\) −5.00000 −0.245145
\(417\) −5.00000 −0.244851
\(418\) −8.00000 −0.391293
\(419\) 4.00000 0.195413 0.0977064 − 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 − 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) −20.0000 −0.973585
\(423\) −2.00000 −0.0972433
\(424\) 0 0
\(425\) 0 0
\(426\) 16.0000 0.775203
\(427\) 3.00000 0.145180
\(428\) 2.00000 0.0966736
\(429\) −5.00000 −0.241402
\(430\) 0 0
\(431\) −2.00000 −0.0963366 −0.0481683 − 0.998839i \(-0.515338\pi\)
−0.0481683 + 0.998839i \(0.515338\pi\)
\(432\) 1.00000 0.0481125
\(433\) 2.00000 0.0961139 0.0480569 − 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 12.0000 0.576018
\(435\) 0 0
\(436\) −15.0000 −0.718370
\(437\) 40.0000 1.91346
\(438\) 9.00000 0.430037
\(439\) −12.0000 −0.572729 −0.286364 − 0.958121i \(-0.592447\pi\)
−0.286364 + 0.958121i \(0.592447\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 10.0000 0.475651
\(443\) 16.0000 0.760183 0.380091 − 0.924949i \(-0.375893\pi\)
0.380091 + 0.924949i \(0.375893\pi\)
\(444\) −4.00000 −0.189832
\(445\) 0 0
\(446\) −21.0000 −0.994379
\(447\) −21.0000 −0.993266
\(448\) 3.00000 0.141737
\(449\) −17.0000 −0.802280 −0.401140 − 0.916017i \(-0.631386\pi\)
−0.401140 + 0.916017i \(0.631386\pi\)
\(450\) 0 0
\(451\) −3.00000 −0.141264
\(452\) −11.0000 −0.517396
\(453\) −9.00000 −0.422857
\(454\) −15.0000 −0.703985
\(455\) 0 0
\(456\) 8.00000 0.374634
\(457\) −8.00000 −0.374224 −0.187112 − 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) 17.0000 0.794358
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) −6.00000 −0.279448 −0.139724 − 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 3.00000 0.139573
\(463\) −8.00000 −0.371792 −0.185896 − 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 25.0000 1.15686 0.578431 − 0.815731i \(-0.303665\pi\)
0.578431 + 0.815731i \(0.303665\pi\)
\(468\) 5.00000 0.231125
\(469\) 39.0000 1.80085
\(470\) 0 0
\(471\) −6.00000 −0.276465
\(472\) 7.00000 0.322201
\(473\) 4.00000 0.183920
\(474\) 1.00000 0.0459315
\(475\) 0 0
\(476\) −6.00000 −0.275010
\(477\) 0 0
\(478\) 12.0000 0.548867
\(479\) −36.0000 −1.64488 −0.822441 − 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) 0 0
\(481\) −20.0000 −0.911922
\(482\) −25.0000 −1.13872
\(483\) −15.0000 −0.682524
\(484\) −10.0000 −0.454545
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 16.0000 0.725029 0.362515 − 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) −1.00000 −0.0452679
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 − 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 3.00000 0.135250
\(493\) 0 0
\(494\) 40.0000 1.79969
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) −48.0000 −2.15309
\(498\) 14.0000 0.627355
\(499\) −5.00000 −0.223831 −0.111915 − 0.993718i \(-0.535699\pi\)
−0.111915 + 0.993718i \(0.535699\pi\)
\(500\) 0 0
\(501\) 18.0000 0.804181
\(502\) −8.00000 −0.357057
\(503\) 10.0000 0.445878 0.222939 − 0.974832i \(-0.428435\pi\)
0.222939 + 0.974832i \(0.428435\pi\)
\(504\) −3.00000 −0.133631
\(505\) 0 0
\(506\) −5.00000 −0.222277
\(507\) 12.0000 0.532939
\(508\) 22.0000 0.976092
\(509\) −10.0000 −0.443242 −0.221621 − 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 0 0
\(511\) −27.0000 −1.19441
\(512\) −1.00000 −0.0441942
\(513\) −8.00000 −0.353209
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 2.00000 0.0879599
\(518\) 12.0000 0.527250
\(519\) 22.0000 0.965693
\(520\) 0 0
\(521\) −22.0000 −0.963837 −0.481919 − 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) 0 0
\(523\) −5.00000 −0.218635 −0.109317 − 0.994007i \(-0.534866\pi\)
−0.109317 + 0.994007i \(0.534866\pi\)
\(524\) 10.0000 0.436852
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) 8.00000 0.348485
\(528\) −1.00000 −0.0435194
\(529\) 2.00000 0.0869565
\(530\) 0 0
\(531\) −7.00000 −0.303774
\(532\) −24.0000 −1.04053
\(533\) 15.0000 0.649722
\(534\) 4.00000 0.173097
\(535\) 0 0
\(536\) −13.0000 −0.561514
\(537\) 0 0
\(538\) 18.0000 0.776035
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) −40.0000 −1.71973 −0.859867 − 0.510518i \(-0.829454\pi\)
−0.859867 + 0.510518i \(0.829454\pi\)
\(542\) −8.00000 −0.343629
\(543\) 26.0000 1.11577
\(544\) 2.00000 0.0857493
\(545\) 0 0
\(546\) −15.0000 −0.641941
\(547\) 35.0000 1.49649 0.748246 − 0.663421i \(-0.230896\pi\)
0.748246 + 0.663421i \(0.230896\pi\)
\(548\) −15.0000 −0.640768
\(549\) 1.00000 0.0426790
\(550\) 0 0
\(551\) 0 0
\(552\) 5.00000 0.212814
\(553\) −3.00000 −0.127573
\(554\) 22.0000 0.934690
\(555\) 0 0
\(556\) −5.00000 −0.212047
\(557\) 12.0000 0.508456 0.254228 − 0.967144i \(-0.418179\pi\)
0.254228 + 0.967144i \(0.418179\pi\)
\(558\) 4.00000 0.169334
\(559\) −20.0000 −0.845910
\(560\) 0 0
\(561\) 2.00000 0.0844401
\(562\) −24.0000 −1.01238
\(563\) −6.00000 −0.252870 −0.126435 − 0.991975i \(-0.540353\pi\)
−0.126435 + 0.991975i \(0.540353\pi\)
\(564\) −2.00000 −0.0842152
\(565\) 0 0
\(566\) −14.0000 −0.588464
\(567\) 3.00000 0.125988
\(568\) 16.0000 0.671345
\(569\) −17.0000 −0.712677 −0.356339 − 0.934357i \(-0.615975\pi\)
−0.356339 + 0.934357i \(0.615975\pi\)
\(570\) 0 0
\(571\) −2.00000 −0.0836974 −0.0418487 − 0.999124i \(-0.513325\pi\)
−0.0418487 + 0.999124i \(0.513325\pi\)
\(572\) −5.00000 −0.209061
\(573\) −15.0000 −0.626634
\(574\) −9.00000 −0.375653
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −2.00000 −0.0832611 −0.0416305 − 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 13.0000 0.540729
\(579\) 18.0000 0.748054
\(580\) 0 0
\(581\) −42.0000 −1.74245
\(582\) 14.0000 0.580319
\(583\) 0 0
\(584\) 9.00000 0.372423
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) −12.0000 −0.495293 −0.247647 − 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 2.00000 0.0824786
\(589\) 32.0000 1.31854
\(590\) 0 0
\(591\) −21.0000 −0.863825
\(592\) −4.00000 −0.164399
\(593\) 6.00000 0.246390 0.123195 − 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 1.00000 0.0410305
\(595\) 0 0
\(596\) −21.0000 −0.860194
\(597\) 18.0000 0.736691
\(598\) 25.0000 1.02233
\(599\) 39.0000 1.59350 0.796748 − 0.604311i \(-0.206552\pi\)
0.796748 + 0.604311i \(0.206552\pi\)
\(600\) 0 0
\(601\) −3.00000 −0.122373 −0.0611863 − 0.998126i \(-0.519488\pi\)
−0.0611863 + 0.998126i \(0.519488\pi\)
\(602\) 12.0000 0.489083
\(603\) 13.0000 0.529401
\(604\) −9.00000 −0.366205
\(605\) 0 0
\(606\) 8.00000 0.324978
\(607\) 4.00000 0.162355 0.0811775 − 0.996700i \(-0.474132\pi\)
0.0811775 + 0.996700i \(0.474132\pi\)
\(608\) 8.00000 0.324443
\(609\) 0 0
\(610\) 0 0
\(611\) −10.0000 −0.404557
\(612\) −2.00000 −0.0808452
\(613\) 38.0000 1.53481 0.767403 − 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) 19.0000 0.766778
\(615\) 0 0
\(616\) 3.00000 0.120873
\(617\) −10.0000 −0.402585 −0.201292 − 0.979531i \(-0.564514\pi\)
−0.201292 + 0.979531i \(0.564514\pi\)
\(618\) −14.0000 −0.563163
\(619\) −14.0000 −0.562708 −0.281354 − 0.959604i \(-0.590783\pi\)
−0.281354 + 0.959604i \(0.590783\pi\)
\(620\) 0 0
\(621\) −5.00000 −0.200643
\(622\) −35.0000 −1.40337
\(623\) −12.0000 −0.480770
\(624\) 5.00000 0.200160
\(625\) 0 0
\(626\) −8.00000 −0.319744
\(627\) 8.00000 0.319489
\(628\) −6.00000 −0.239426
\(629\) 8.00000 0.318981
\(630\) 0 0
\(631\) 25.0000 0.995234 0.497617 − 0.867397i \(-0.334208\pi\)
0.497617 + 0.867397i \(0.334208\pi\)
\(632\) 1.00000 0.0397779
\(633\) 20.0000 0.794929
\(634\) 10.0000 0.397151
\(635\) 0 0
\(636\) 0 0
\(637\) 10.0000 0.396214
\(638\) 0 0
\(639\) −16.0000 −0.632950
\(640\) 0 0
\(641\) −44.0000 −1.73790 −0.868948 − 0.494904i \(-0.835203\pi\)
−0.868948 + 0.494904i \(0.835203\pi\)
\(642\) −2.00000 −0.0789337
\(643\) 8.00000 0.315489 0.157745 − 0.987480i \(-0.449578\pi\)
0.157745 + 0.987480i \(0.449578\pi\)
\(644\) −15.0000 −0.591083
\(645\) 0 0
\(646\) −16.0000 −0.629512
\(647\) 3.00000 0.117942 0.0589711 − 0.998260i \(-0.481218\pi\)
0.0589711 + 0.998260i \(0.481218\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 7.00000 0.274774
\(650\) 0 0
\(651\) −12.0000 −0.470317
\(652\) 4.00000 0.156652
\(653\) 14.0000 0.547862 0.273931 − 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) 15.0000 0.586546
\(655\) 0 0
\(656\) 3.00000 0.117130
\(657\) −9.00000 −0.351123
\(658\) 6.00000 0.233904
\(659\) 6.00000 0.233727 0.116863 − 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 0 0
\(661\) 32.0000 1.24466 0.622328 − 0.782757i \(-0.286187\pi\)
0.622328 + 0.782757i \(0.286187\pi\)
\(662\) −9.00000 −0.349795
\(663\) −10.0000 −0.388368
\(664\) 14.0000 0.543305
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) 0 0
\(668\) 18.0000 0.696441
\(669\) 21.0000 0.811907
\(670\) 0 0
\(671\) −1.00000 −0.0386046
\(672\) −3.00000 −0.115728
\(673\) 14.0000 0.539660 0.269830 − 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) −16.0000 −0.616297
\(675\) 0 0
\(676\) 12.0000 0.461538
\(677\) −30.0000 −1.15299 −0.576497 − 0.817099i \(-0.695581\pi\)
−0.576497 + 0.817099i \(0.695581\pi\)
\(678\) 11.0000 0.422452
\(679\) −42.0000 −1.61181
\(680\) 0 0
\(681\) 15.0000 0.574801
\(682\) −4.00000 −0.153168
\(683\) −48.0000 −1.83667 −0.918334 − 0.395805i \(-0.870466\pi\)
−0.918334 + 0.395805i \(0.870466\pi\)
\(684\) −8.00000 −0.305888
\(685\) 0 0
\(686\) 15.0000 0.572703
\(687\) −17.0000 −0.648590
\(688\) −4.00000 −0.152499
\(689\) 0 0
\(690\) 0 0
\(691\) −24.0000 −0.913003 −0.456502 − 0.889723i \(-0.650898\pi\)
−0.456502 + 0.889723i \(0.650898\pi\)
\(692\) 22.0000 0.836315
\(693\) −3.00000 −0.113961
\(694\) −4.00000 −0.151838
\(695\) 0 0
\(696\) 0 0
\(697\) −6.00000 −0.227266
\(698\) 22.0000 0.832712
\(699\) 0 0
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 − 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) −5.00000 −0.188713
\(703\) 32.0000 1.20690
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 25.0000 0.940887
\(707\) −24.0000 −0.902613
\(708\) −7.00000 −0.263076
\(709\) −14.0000 −0.525781 −0.262891 − 0.964826i \(-0.584676\pi\)
−0.262891 + 0.964826i \(0.584676\pi\)
\(710\) 0 0
\(711\) −1.00000 −0.0375029
\(712\) 4.00000 0.149906
\(713\) 20.0000 0.749006
\(714\) 6.00000 0.224544
\(715\) 0 0
\(716\) 0 0
\(717\) −12.0000 −0.448148
\(718\) −36.0000 −1.34351
\(719\) −20.0000 −0.745874 −0.372937 − 0.927857i \(-0.621649\pi\)
−0.372937 + 0.927857i \(0.621649\pi\)
\(720\) 0 0
\(721\) 42.0000 1.56416
\(722\) −45.0000 −1.67473
\(723\) 25.0000 0.929760
\(724\) 26.0000 0.966282
\(725\) 0 0
\(726\) 10.0000 0.371135
\(727\) −34.0000 −1.26099 −0.630495 − 0.776193i \(-0.717148\pi\)
−0.630495 + 0.776193i \(0.717148\pi\)
\(728\) −15.0000 −0.555937
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) 1.00000 0.0369611
\(733\) 1.00000 0.0369358 0.0184679 − 0.999829i \(-0.494121\pi\)
0.0184679 + 0.999829i \(0.494121\pi\)
\(734\) 36.0000 1.32878
\(735\) 0 0
\(736\) 5.00000 0.184302
\(737\) −13.0000 −0.478861
\(738\) −3.00000 −0.110432
\(739\) 19.0000 0.698926 0.349463 − 0.936950i \(-0.386364\pi\)
0.349463 + 0.936950i \(0.386364\pi\)
\(740\) 0 0
\(741\) −40.0000 −1.46944
\(742\) 0 0
\(743\) 31.0000 1.13728 0.568640 − 0.822587i \(-0.307470\pi\)
0.568640 + 0.822587i \(0.307470\pi\)
\(744\) 4.00000 0.146647
\(745\) 0 0
\(746\) 4.00000 0.146450
\(747\) −14.0000 −0.512233
\(748\) 2.00000 0.0731272
\(749\) 6.00000 0.219235
\(750\) 0 0
\(751\) −12.0000 −0.437886 −0.218943 − 0.975738i \(-0.570261\pi\)
−0.218943 + 0.975738i \(0.570261\pi\)
\(752\) −2.00000 −0.0729325
\(753\) 8.00000 0.291536
\(754\) 0 0
\(755\) 0 0
\(756\) 3.00000 0.109109
\(757\) 23.0000 0.835949 0.417975 − 0.908459i \(-0.362740\pi\)
0.417975 + 0.908459i \(0.362740\pi\)
\(758\) 0 0
\(759\) 5.00000 0.181489
\(760\) 0 0
\(761\) −10.0000 −0.362500 −0.181250 − 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) −22.0000 −0.796976
\(763\) −45.0000 −1.62911
\(764\) −15.0000 −0.542681
\(765\) 0 0
\(766\) 7.00000 0.252920
\(767\) −35.0000 −1.26378
\(768\) 1.00000 0.0360844
\(769\) −10.0000 −0.360609 −0.180305 − 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 18.0000 0.647834
\(773\) −18.0000 −0.647415 −0.323708 − 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) 14.0000 0.502571
\(777\) −12.0000 −0.430498
\(778\) −18.0000 −0.645331
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 16.0000 0.572525
\(782\) −10.0000 −0.357599
\(783\) 0 0
\(784\) 2.00000 0.0714286
\(785\) 0 0
\(786\) −10.0000 −0.356688
\(787\) 32.0000 1.14068 0.570338 − 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) −21.0000 −0.748094
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) −33.0000 −1.17334
\(792\) 1.00000 0.0355335
\(793\) 5.00000 0.177555
\(794\) −2.00000 −0.0709773
\(795\) 0 0
\(796\) 18.0000 0.637993
\(797\) −29.0000 −1.02723 −0.513616 − 0.858020i \(-0.671695\pi\)
−0.513616 + 0.858020i \(0.671695\pi\)
\(798\) 24.0000 0.849591
\(799\) 4.00000 0.141510
\(800\) 0 0
\(801\) −4.00000 −0.141333
\(802\) −6.00000 −0.211867
\(803\) 9.00000 0.317603
\(804\) 13.0000 0.458475
\(805\) 0 0
\(806\) 20.0000 0.704470
\(807\) −18.0000 −0.633630
\(808\) 8.00000 0.281439
\(809\) 23.0000 0.808637 0.404318 − 0.914618i \(-0.367509\pi\)
0.404318 + 0.914618i \(0.367509\pi\)
\(810\) 0 0
\(811\) −7.00000 −0.245803 −0.122902 − 0.992419i \(-0.539220\pi\)
−0.122902 + 0.992419i \(0.539220\pi\)
\(812\) 0 0
\(813\) 8.00000 0.280572
\(814\) −4.00000 −0.140200
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) 32.0000 1.11954
\(818\) 20.0000 0.699284
\(819\) 15.0000 0.524142
\(820\) 0 0
\(821\) −30.0000 −1.04701 −0.523504 − 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) 15.0000 0.523185
\(823\) −40.0000 −1.39431 −0.697156 − 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) −14.0000 −0.487713
\(825\) 0 0
\(826\) 21.0000 0.730683
\(827\) 48.0000 1.66912 0.834562 − 0.550914i \(-0.185721\pi\)
0.834562 + 0.550914i \(0.185721\pi\)
\(828\) −5.00000 −0.173762
\(829\) −7.00000 −0.243120 −0.121560 − 0.992584i \(-0.538790\pi\)
−0.121560 + 0.992584i \(0.538790\pi\)
\(830\) 0 0
\(831\) −22.0000 −0.763172
\(832\) 5.00000 0.173344
\(833\) −4.00000 −0.138592
\(834\) 5.00000 0.173136
\(835\) 0 0
\(836\) 8.00000 0.276686
\(837\) −4.00000 −0.138260
\(838\) −4.00000 −0.138178
\(839\) 20.0000 0.690477 0.345238 − 0.938515i \(-0.387798\pi\)
0.345238 + 0.938515i \(0.387798\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) −2.00000 −0.0689246
\(843\) 24.0000 0.826604
\(844\) 20.0000 0.688428
\(845\) 0 0
\(846\) 2.00000 0.0687614
\(847\) −30.0000 −1.03081
\(848\) 0 0
\(849\) 14.0000 0.480479
\(850\) 0 0
\(851\) 20.0000 0.685591
\(852\) −16.0000 −0.548151
\(853\) −39.0000 −1.33533 −0.667667 − 0.744460i \(-0.732707\pi\)
−0.667667 + 0.744460i \(0.732707\pi\)
\(854\) −3.00000 −0.102658
\(855\) 0 0
\(856\) −2.00000 −0.0683586
\(857\) 53.0000 1.81045 0.905223 − 0.424937i \(-0.139704\pi\)
0.905223 + 0.424937i \(0.139704\pi\)
\(858\) 5.00000 0.170697
\(859\) −14.0000 −0.477674 −0.238837 − 0.971060i \(-0.576766\pi\)
−0.238837 + 0.971060i \(0.576766\pi\)
\(860\) 0 0
\(861\) 9.00000 0.306719
\(862\) 2.00000 0.0681203
\(863\) 56.0000 1.90626 0.953131 − 0.302558i \(-0.0978405\pi\)
0.953131 + 0.302558i \(0.0978405\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −2.00000 −0.0679628
\(867\) −13.0000 −0.441503
\(868\) −12.0000 −0.407307
\(869\) 1.00000 0.0339227
\(870\) 0 0
\(871\) 65.0000 2.20244
\(872\) 15.0000 0.507964
\(873\) −14.0000 −0.473828
\(874\) −40.0000 −1.35302
\(875\) 0 0
\(876\) −9.00000 −0.304082
\(877\) 40.0000 1.35070 0.675352 − 0.737496i \(-0.263992\pi\)
0.675352 + 0.737496i \(0.263992\pi\)
\(878\) 12.0000 0.404980
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) −35.0000 −1.17918 −0.589590 − 0.807703i \(-0.700711\pi\)
−0.589590 + 0.807703i \(0.700711\pi\)
\(882\) −2.00000 −0.0673435
\(883\) −15.0000 −0.504790 −0.252395 − 0.967624i \(-0.581218\pi\)
−0.252395 + 0.967624i \(0.581218\pi\)
\(884\) −10.0000 −0.336336
\(885\) 0 0
\(886\) −16.0000 −0.537531
\(887\) −32.0000 −1.07445 −0.537227 − 0.843437i \(-0.680528\pi\)
−0.537227 + 0.843437i \(0.680528\pi\)
\(888\) 4.00000 0.134231
\(889\) 66.0000 2.21357
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 21.0000 0.703132
\(893\) 16.0000 0.535420
\(894\) 21.0000 0.702345
\(895\) 0 0
\(896\) −3.00000 −0.100223
\(897\) −25.0000 −0.834726
\(898\) 17.0000 0.567297
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 3.00000 0.0998891
\(903\) −12.0000 −0.399335
\(904\) 11.0000 0.365855
\(905\) 0 0
\(906\) 9.00000 0.299005
\(907\) −28.0000 −0.929725 −0.464862 − 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 15.0000 0.497792
\(909\) −8.00000 −0.265343
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 − 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) −8.00000 −0.264906
\(913\) 14.0000 0.463332
\(914\) 8.00000 0.264616
\(915\) 0 0
\(916\) −17.0000 −0.561696
\(917\) 30.0000 0.990687
\(918\) 2.00000 0.0660098
\(919\) −52.0000 −1.71532 −0.857661 − 0.514216i \(-0.828083\pi\)
−0.857661 + 0.514216i \(0.828083\pi\)
\(920\) 0 0
\(921\) −19.0000 −0.626071
\(922\) 6.00000 0.197599
\(923\) −80.0000 −2.63323
\(924\) −3.00000 −0.0986928
\(925\) 0 0
\(926\) 8.00000 0.262896
\(927\) 14.0000 0.459820
\(928\) 0 0
\(929\) 43.0000 1.41078 0.705392 − 0.708817i \(-0.250771\pi\)
0.705392 + 0.708817i \(0.250771\pi\)
\(930\) 0 0
\(931\) −16.0000 −0.524379
\(932\) 0 0
\(933\) 35.0000 1.14585
\(934\) −25.0000 −0.818025
\(935\) 0 0
\(936\) −5.00000 −0.163430
\(937\) 41.0000 1.33941 0.669706 − 0.742627i \(-0.266420\pi\)
0.669706 + 0.742627i \(0.266420\pi\)
\(938\) −39.0000 −1.27340
\(939\) 8.00000 0.261070
\(940\) 0 0
\(941\) 40.0000 1.30396 0.651981 − 0.758235i \(-0.273938\pi\)
0.651981 + 0.758235i \(0.273938\pi\)
\(942\) 6.00000 0.195491
\(943\) −15.0000 −0.488467
\(944\) −7.00000 −0.227831
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) −35.0000 −1.13735 −0.568674 − 0.822563i \(-0.692543\pi\)
−0.568674 + 0.822563i \(0.692543\pi\)
\(948\) −1.00000 −0.0324785
\(949\) −45.0000 −1.46076
\(950\) 0 0
\(951\) −10.0000 −0.324272
\(952\) 6.00000 0.194461
\(953\) −50.0000 −1.61966 −0.809829 − 0.586665i \(-0.800440\pi\)
−0.809829 + 0.586665i \(0.800440\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −12.0000 −0.388108
\(957\) 0 0
\(958\) 36.0000 1.16311
\(959\) −45.0000 −1.45313
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 20.0000 0.644826
\(963\) 2.00000 0.0644491
\(964\) 25.0000 0.805196
\(965\) 0 0
\(966\) 15.0000 0.482617
\(967\) −18.0000 −0.578841 −0.289420 − 0.957202i \(-0.593463\pi\)
−0.289420 + 0.957202i \(0.593463\pi\)
\(968\) 10.0000 0.321412
\(969\) 16.0000 0.513994
\(970\) 0 0
\(971\) −14.0000 −0.449281 −0.224641 − 0.974442i \(-0.572121\pi\)
−0.224641 + 0.974442i \(0.572121\pi\)
\(972\) 1.00000 0.0320750
\(973\) −15.0000 −0.480878
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) 1.00000 0.0320092
\(977\) 30.0000 0.959785 0.479893 − 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) −4.00000 −0.127906
\(979\) 4.00000 0.127841
\(980\) 0 0
\(981\) −15.0000 −0.478913
\(982\) −28.0000 −0.893516
\(983\) 4.00000 0.127580 0.0637901 − 0.997963i \(-0.479681\pi\)
0.0637901 + 0.997963i \(0.479681\pi\)
\(984\) −3.00000 −0.0956365
\(985\) 0 0
\(986\) 0 0
\(987\) −6.00000 −0.190982
\(988\) −40.0000 −1.27257
\(989\) 20.0000 0.635963
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 − 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 4.00000 0.127000
\(993\) 9.00000 0.285606
\(994\) 48.0000 1.52247
\(995\) 0 0
\(996\) −14.0000 −0.443607
\(997\) −2.00000 −0.0633406 −0.0316703 − 0.999498i \(-0.510083\pi\)
−0.0316703 + 0.999498i \(0.510083\pi\)
\(998\) 5.00000 0.158272
\(999\) −4.00000 −0.126554
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 9150.2.a.n.1.1 1
5.4 even 2 366.2.a.d.1.1 ✓ 1
15.14 odd 2 1098.2.a.e.1.1 1
20.19 odd 2 2928.2.a.h.1.1 1
60.59 even 2 8784.2.a.z.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
366.2.a.d.1.1 ✓ 1 5.4 even 2
1098.2.a.e.1.1 1 15.14 odd 2
2928.2.a.h.1.1 1 20.19 odd 2
8784.2.a.z.1.1 1 60.59 even 2
9150.2.a.n.1.1 1 1.1 even 1 trivial