Properties

Label 1014.2.a.f.1.1
Level $1014$
Weight $2$
Character 1014.1
Self dual yes
Analytic conductor $8.097$
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] = [1014,2,Mod(1,1014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1014, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1014.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1014.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: \(8.09683076496\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1014.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} -6.00000 q^{11} +1.00000 q^{12} -2.00000 q^{14} -3.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} +1.00000 q^{18} -2.00000 q^{19} -3.00000 q^{20} -2.00000 q^{21} -6.00000 q^{22} -6.00000 q^{23} +1.00000 q^{24} +4.00000 q^{25} +1.00000 q^{27} -2.00000 q^{28} +3.00000 q^{29} -3.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} -6.00000 q^{33} -3.00000 q^{34} +6.00000 q^{35} +1.00000 q^{36} +7.00000 q^{37} -2.00000 q^{38} -3.00000 q^{40} +3.00000 q^{41} -2.00000 q^{42} -10.0000 q^{43} -6.00000 q^{44} -3.00000 q^{45} -6.00000 q^{46} -6.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +4.00000 q^{50} -3.00000 q^{51} +3.00000 q^{53} +1.00000 q^{54} +18.0000 q^{55} -2.00000 q^{56} -2.00000 q^{57} +3.00000 q^{58} -3.00000 q^{60} -7.00000 q^{61} +4.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -6.00000 q^{66} +10.0000 q^{67} -3.00000 q^{68} -6.00000 q^{69} +6.00000 q^{70} -6.00000 q^{71} +1.00000 q^{72} +13.0000 q^{73} +7.00000 q^{74} +4.00000 q^{75} -2.00000 q^{76} +12.0000 q^{77} -4.00000 q^{79} -3.00000 q^{80} +1.00000 q^{81} +3.00000 q^{82} +6.00000 q^{83} -2.00000 q^{84} +9.00000 q^{85} -10.0000 q^{86} +3.00000 q^{87} -6.00000 q^{88} -18.0000 q^{89} -3.00000 q^{90} -6.00000 q^{92} +4.00000 q^{93} -6.00000 q^{94} +6.00000 q^{95} +1.00000 q^{96} -14.0000 q^{97} -3.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 1.00000 0.408248
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −3.00000 −0.948683
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −2.00000 −0.534522
\(15\) −3.00000 −0.774597
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 1.00000 0.235702
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) −3.00000 −0.670820
\(21\) −2.00000 −0.436436
\(22\) −6.00000 −1.27920
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 1.00000 0.204124
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −2.00000 −0.377964
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) −3.00000 −0.547723
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) −6.00000 −1.04447
\(34\) −3.00000 −0.514496
\(35\) 6.00000 1.01419
\(36\) 1.00000 0.166667
\(37\) 7.00000 1.15079 0.575396 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) −2.00000 −0.308607
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) −6.00000 −0.904534
\(45\) −3.00000 −0.447214
\(46\) −6.00000 −0.884652
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 4.00000 0.565685
\(51\) −3.00000 −0.420084
\(52\) 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
\(55\) 18.0000 2.42712
\(56\) −2.00000 −0.267261
\(57\) −2.00000 −0.264906
\(58\) 3.00000 0.393919
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −3.00000 −0.387298
\(61\) −7.00000 −0.896258 −0.448129 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 4.00000 0.508001
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −6.00000 −0.738549
\(67\) 10.0000 1.22169 0.610847 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) −3.00000 −0.363803
\(69\) −6.00000 −0.722315
\(70\) 6.00000 0.717137
\(71\) −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 \(-0.224823\pi\)
0.760767 + 0.649025i \(0.224823\pi\)
\(74\) 7.00000 0.813733
\(75\) 4.00000 0.461880
\(76\) −2.00000 −0.229416
\(77\) 12.0000 1.36753
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −3.00000 −0.335410
\(81\) 1.00000 0.111111
\(82\) 3.00000 0.331295
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) −2.00000 −0.218218
\(85\) 9.00000 0.976187
\(86\) −10.0000 −1.07833
\(87\) 3.00000 0.321634
\(88\) −6.00000 −0.639602
\(89\) −18.0000 −1.90800 −0.953998 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) −3.00000 −0.316228
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) 4.00000 0.414781
\(94\) −6.00000 −0.618853
\(95\) 6.00000 0.615587
\(96\) 1.00000 0.102062
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) −3.00000 −0.303046
\(99\) −6.00000 −0.603023
\(100\) 4.00000 0.400000
\(101\) 15.0000 1.49256 0.746278 0.665635i \(-0.231839\pi\)
0.746278 + 0.665635i \(0.231839\pi\)
\(102\) −3.00000 −0.297044
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 0 0
\(105\) 6.00000 0.585540
\(106\) 3.00000 0.291386
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 1.00000 0.0962250
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 18.0000 1.71623
\(111\) 7.00000 0.664411
\(112\) −2.00000 −0.188982
\(113\) −3.00000 −0.282216 −0.141108 0.989994i \(-0.545067\pi\)
−0.141108 + 0.989994i \(0.545067\pi\)
\(114\) −2.00000 −0.187317
\(115\) 18.0000 1.67851
\(116\) 3.00000 0.278543
\(117\) 0 0
\(118\) 0 0
\(119\) 6.00000 0.550019
\(120\) −3.00000 −0.273861
\(121\) 25.0000 2.27273
\(122\) −7.00000 −0.633750
\(123\) 3.00000 0.270501
\(124\) 4.00000 0.359211
\(125\) 3.00000 0.268328
\(126\) −2.00000 −0.178174
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 1.00000 0.0883883
\(129\) −10.0000 −0.880451
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) −6.00000 −0.522233
\(133\) 4.00000 0.346844
\(134\) 10.0000 0.863868
\(135\) −3.00000 −0.258199
\(136\) −3.00000 −0.257248
\(137\) −9.00000 −0.768922 −0.384461 0.923141i \(-0.625613\pi\)
−0.384461 + 0.923141i \(0.625613\pi\)
\(138\) −6.00000 −0.510754
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 6.00000 0.507093
\(141\) −6.00000 −0.505291
\(142\) −6.00000 −0.503509
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −9.00000 −0.747409
\(146\) 13.0000 1.07589
\(147\) −3.00000 −0.247436
\(148\) 7.00000 0.575396
\(149\) 9.00000 0.737309 0.368654 0.929567i \(-0.379819\pi\)
0.368654 + 0.929567i \(0.379819\pi\)
\(150\) 4.00000 0.326599
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) −2.00000 −0.162221
\(153\) −3.00000 −0.242536
\(154\) 12.0000 0.966988
\(155\) −12.0000 −0.963863
\(156\) 0 0
\(157\) 5.00000 0.399043 0.199522 0.979893i \(-0.436061\pi\)
0.199522 + 0.979893i \(0.436061\pi\)
\(158\) −4.00000 −0.318223
\(159\) 3.00000 0.237915
\(160\) −3.00000 −0.237171
\(161\) 12.0000 0.945732
\(162\) 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
\(165\) 18.0000 1.40130
\(166\) 6.00000 0.465690
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −2.00000 −0.154303
\(169\) 0 0
\(170\) 9.00000 0.690268
\(171\) −2.00000 −0.152944
\(172\) −10.0000 −0.762493
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 3.00000 0.227429
\(175\) −8.00000 −0.604743
\(176\) −6.00000 −0.452267
\(177\) 0 0
\(178\) −18.0000 −1.34916
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) −3.00000 −0.223607
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 0 0
\(183\) −7.00000 −0.517455
\(184\) −6.00000 −0.442326
\(185\) −21.0000 −1.54395
\(186\) 4.00000 0.293294
\(187\) 18.0000 1.31629
\(188\) −6.00000 −0.437595
\(189\) −2.00000 −0.145479
\(190\) 6.00000 0.435286
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 1.00000 0.0721688
\(193\) −23.0000 −1.65558 −0.827788 0.561041i \(-0.810401\pi\)
−0.827788 + 0.561041i \(0.810401\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) −6.00000 −0.426401
\(199\) −10.0000 −0.708881 −0.354441 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) 4.00000 0.282843
\(201\) 10.0000 0.705346
\(202\) 15.0000 1.05540
\(203\) −6.00000 −0.421117
\(204\) −3.00000 −0.210042
\(205\) −9.00000 −0.628587
\(206\) 14.0000 0.975426
\(207\) −6.00000 −0.417029
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 6.00000 0.414039
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 3.00000 0.206041
\(213\) −6.00000 −0.411113
\(214\) −6.00000 −0.410152
\(215\) 30.0000 2.04598
\(216\) 1.00000 0.0680414
\(217\) −8.00000 −0.543075
\(218\) −14.0000 −0.948200
\(219\) 13.0000 0.878459
\(220\) 18.0000 1.21356
\(221\) 0 0
\(222\) 7.00000 0.469809
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) −2.00000 −0.133631
\(225\) 4.00000 0.266667
\(226\) −3.00000 −0.199557
\(227\) −18.0000 −1.19470 −0.597351 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) −2.00000 −0.132453
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 18.0000 1.18688
\(231\) 12.0000 0.789542
\(232\) 3.00000 0.196960
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 18.0000 1.17419
\(236\) 0 0
\(237\) −4.00000 −0.259828
\(238\) 6.00000 0.388922
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) −3.00000 −0.193649
\(241\) 1.00000 0.0644157 0.0322078 0.999481i \(-0.489746\pi\)
0.0322078 + 0.999481i \(0.489746\pi\)
\(242\) 25.0000 1.60706
\(243\) 1.00000 0.0641500
\(244\) −7.00000 −0.448129
\(245\) 9.00000 0.574989
\(246\) 3.00000 0.191273
\(247\) 0 0
\(248\) 4.00000 0.254000
\(249\) 6.00000 0.380235
\(250\) 3.00000 0.189737
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) −2.00000 −0.125988
\(253\) 36.0000 2.26330
\(254\) −4.00000 −0.250982
\(255\) 9.00000 0.563602
\(256\) 1.00000 0.0625000
\(257\) −3.00000 −0.187135 −0.0935674 0.995613i \(-0.529827\pi\)
−0.0935674 + 0.995613i \(0.529827\pi\)
\(258\) −10.0000 −0.622573
\(259\) −14.0000 −0.869918
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) 0 0
\(263\) −6.00000 −0.369976 −0.184988 0.982741i \(-0.559225\pi\)
−0.184988 + 0.982741i \(0.559225\pi\)
\(264\) −6.00000 −0.369274
\(265\) −9.00000 −0.552866
\(266\) 4.00000 0.245256
\(267\) −18.0000 −1.10158
\(268\) 10.0000 0.610847
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) −3.00000 −0.182574
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) −3.00000 −0.181902
\(273\) 0 0
\(274\) −9.00000 −0.543710
\(275\) −24.0000 −1.44725
\(276\) −6.00000 −0.361158
\(277\) 17.0000 1.02143 0.510716 0.859750i \(-0.329381\pi\)
0.510716 + 0.859750i \(0.329381\pi\)
\(278\) −4.00000 −0.239904
\(279\) 4.00000 0.239474
\(280\) 6.00000 0.358569
\(281\) −9.00000 −0.536895 −0.268447 0.963294i \(-0.586511\pi\)
−0.268447 + 0.963294i \(0.586511\pi\)
\(282\) −6.00000 −0.357295
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) −6.00000 −0.356034
\(285\) 6.00000 0.355409
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 1.00000 0.0589256
\(289\) −8.00000 −0.470588
\(290\) −9.00000 −0.528498
\(291\) −14.0000 −0.820695
\(292\) 13.0000 0.760767
\(293\) 21.0000 1.22683 0.613417 0.789760i \(-0.289795\pi\)
0.613417 + 0.789760i \(0.289795\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) 7.00000 0.406867
\(297\) −6.00000 −0.348155
\(298\) 9.00000 0.521356
\(299\) 0 0
\(300\) 4.00000 0.230940
\(301\) 20.0000 1.15278
\(302\) 10.0000 0.575435
\(303\) 15.0000 0.861727
\(304\) −2.00000 −0.114708
\(305\) 21.0000 1.20246
\(306\) −3.00000 −0.171499
\(307\) 10.0000 0.570730 0.285365 0.958419i \(-0.407885\pi\)
0.285365 + 0.958419i \(0.407885\pi\)
\(308\) 12.0000 0.683763
\(309\) 14.0000 0.796432
\(310\) −12.0000 −0.681554
\(311\) −30.0000 −1.70114 −0.850572 0.525859i \(-0.823744\pi\)
−0.850572 + 0.525859i \(0.823744\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 5.00000 0.282166
\(315\) 6.00000 0.338062
\(316\) −4.00000 −0.225018
\(317\) −3.00000 −0.168497 −0.0842484 0.996445i \(-0.526849\pi\)
−0.0842484 + 0.996445i \(0.526849\pi\)
\(318\) 3.00000 0.168232
\(319\) −18.0000 −1.00781
\(320\) −3.00000 −0.167705
\(321\) −6.00000 −0.334887
\(322\) 12.0000 0.668734
\(323\) 6.00000 0.333849
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 4.00000 0.221540
\(327\) −14.0000 −0.774202
\(328\) 3.00000 0.165647
\(329\) 12.0000 0.661581
\(330\) 18.0000 0.990867
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 6.00000 0.329293
\(333\) 7.00000 0.383598
\(334\) 0 0
\(335\) −30.0000 −1.63908
\(336\) −2.00000 −0.109109
\(337\) 23.0000 1.25289 0.626445 0.779466i \(-0.284509\pi\)
0.626445 + 0.779466i \(0.284509\pi\)
\(338\) 0 0
\(339\) −3.00000 −0.162938
\(340\) 9.00000 0.488094
\(341\) −24.0000 −1.29967
\(342\) −2.00000 −0.108148
\(343\) 20.0000 1.07990
\(344\) −10.0000 −0.539164
\(345\) 18.0000 0.969087
\(346\) −6.00000 −0.322562
\(347\) −30.0000 −1.61048 −0.805242 0.592946i \(-0.797965\pi\)
−0.805242 + 0.592946i \(0.797965\pi\)
\(348\) 3.00000 0.160817
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) −8.00000 −0.427618
\(351\) 0 0
\(352\) −6.00000 −0.319801
\(353\) 15.0000 0.798369 0.399185 0.916871i \(-0.369293\pi\)
0.399185 + 0.916871i \(0.369293\pi\)
\(354\) 0 0
\(355\) 18.0000 0.955341
\(356\) −18.0000 −0.953998
\(357\) 6.00000 0.317554
\(358\) 6.00000 0.317110
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) −3.00000 −0.158114
\(361\) −15.0000 −0.789474
\(362\) −7.00000 −0.367912
\(363\) 25.0000 1.31216
\(364\) 0 0
\(365\) −39.0000 −2.04135
\(366\) −7.00000 −0.365896
\(367\) 2.00000 0.104399 0.0521996 0.998637i \(-0.483377\pi\)
0.0521996 + 0.998637i \(0.483377\pi\)
\(368\) −6.00000 −0.312772
\(369\) 3.00000 0.156174
\(370\) −21.0000 −1.09174
\(371\) −6.00000 −0.311504
\(372\) 4.00000 0.207390
\(373\) 29.0000 1.50156 0.750782 0.660551i \(-0.229677\pi\)
0.750782 + 0.660551i \(0.229677\pi\)
\(374\) 18.0000 0.930758
\(375\) 3.00000 0.154919
\(376\) −6.00000 −0.309426
\(377\) 0 0
\(378\) −2.00000 −0.102869
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 6.00000 0.307794
\(381\) −4.00000 −0.204926
\(382\) −12.0000 −0.613973
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 1.00000 0.0510310
\(385\) −36.0000 −1.83473
\(386\) −23.0000 −1.17067
\(387\) −10.0000 −0.508329
\(388\) −14.0000 −0.710742
\(389\) 39.0000 1.97738 0.988689 0.149979i \(-0.0479205\pi\)
0.988689 + 0.149979i \(0.0479205\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) −3.00000 −0.151523
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) 12.0000 0.603786
\(396\) −6.00000 −0.301511
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) −10.0000 −0.501255
\(399\) 4.00000 0.200250
\(400\) 4.00000 0.200000
\(401\) 3.00000 0.149813 0.0749064 0.997191i \(-0.476134\pi\)
0.0749064 + 0.997191i \(0.476134\pi\)
\(402\) 10.0000 0.498755
\(403\) 0 0
\(404\) 15.0000 0.746278
\(405\) −3.00000 −0.149071
\(406\) −6.00000 −0.297775
\(407\) −42.0000 −2.08186
\(408\) −3.00000 −0.148522
\(409\) 1.00000 0.0494468 0.0247234 0.999694i \(-0.492129\pi\)
0.0247234 + 0.999694i \(0.492129\pi\)
\(410\) −9.00000 −0.444478
\(411\) −9.00000 −0.443937
\(412\) 14.0000 0.689730
\(413\) 0 0
\(414\) −6.00000 −0.294884
\(415\) −18.0000 −0.883585
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 12.0000 0.586939
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 6.00000 0.292770
\(421\) −29.0000 −1.41337 −0.706687 0.707527i \(-0.749811\pi\)
−0.706687 + 0.707527i \(0.749811\pi\)
\(422\) −16.0000 −0.778868
\(423\) −6.00000 −0.291730
\(424\) 3.00000 0.145693
\(425\) −12.0000 −0.582086
\(426\) −6.00000 −0.290701
\(427\) 14.0000 0.677507
\(428\) −6.00000 −0.290021
\(429\) 0 0
\(430\) 30.0000 1.44673
\(431\) 6.00000 0.289010 0.144505 0.989504i \(-0.453841\pi\)
0.144505 + 0.989504i \(0.453841\pi\)
\(432\) 1.00000 0.0481125
\(433\) −13.0000 −0.624740 −0.312370 0.949960i \(-0.601123\pi\)
−0.312370 + 0.949960i \(0.601123\pi\)
\(434\) −8.00000 −0.384012
\(435\) −9.00000 −0.431517
\(436\) −14.0000 −0.670478
\(437\) 12.0000 0.574038
\(438\) 13.0000 0.621164
\(439\) 14.0000 0.668184 0.334092 0.942541i \(-0.391570\pi\)
0.334092 + 0.942541i \(0.391570\pi\)
\(440\) 18.0000 0.858116
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 7.00000 0.332205
\(445\) 54.0000 2.55985
\(446\) −8.00000 −0.378811
\(447\) 9.00000 0.425685
\(448\) −2.00000 −0.0944911
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 4.00000 0.188562
\(451\) −18.0000 −0.847587
\(452\) −3.00000 −0.141108
\(453\) 10.0000 0.469841
\(454\) −18.0000 −0.844782
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) −11.0000 −0.514558 −0.257279 0.966337i \(-0.582826\pi\)
−0.257279 + 0.966337i \(0.582826\pi\)
\(458\) 22.0000 1.02799
\(459\) −3.00000 −0.140028
\(460\) 18.0000 0.839254
\(461\) −15.0000 −0.698620 −0.349310 0.937007i \(-0.613584\pi\)
−0.349310 + 0.937007i \(0.613584\pi\)
\(462\) 12.0000 0.558291
\(463\) −38.0000 −1.76601 −0.883005 0.469364i \(-0.844483\pi\)
−0.883005 + 0.469364i \(0.844483\pi\)
\(464\) 3.00000 0.139272
\(465\) −12.0000 −0.556487
\(466\) −6.00000 −0.277945
\(467\) −18.0000 −0.832941 −0.416470 0.909149i \(-0.636733\pi\)
−0.416470 + 0.909149i \(0.636733\pi\)
\(468\) 0 0
\(469\) −20.0000 −0.923514
\(470\) 18.0000 0.830278
\(471\) 5.00000 0.230388
\(472\) 0 0
\(473\) 60.0000 2.75880
\(474\) −4.00000 −0.183726
\(475\) −8.00000 −0.367065
\(476\) 6.00000 0.275010
\(477\) 3.00000 0.137361
\(478\) −6.00000 −0.274434
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) −3.00000 −0.136931
\(481\) 0 0
\(482\) 1.00000 0.0455488
\(483\) 12.0000 0.546019
\(484\) 25.0000 1.13636
\(485\) 42.0000 1.90712
\(486\) 1.00000 0.0453609
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) −7.00000 −0.316875
\(489\) 4.00000 0.180886
\(490\) 9.00000 0.406579
\(491\) −18.0000 −0.812329 −0.406164 0.913800i \(-0.633134\pi\)
−0.406164 + 0.913800i \(0.633134\pi\)
\(492\) 3.00000 0.135250
\(493\) −9.00000 −0.405340
\(494\) 0 0
\(495\) 18.0000 0.809040
\(496\) 4.00000 0.179605
\(497\) 12.0000 0.538274
\(498\) 6.00000 0.268866
\(499\) −32.0000 −1.43252 −0.716258 0.697835i \(-0.754147\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) 3.00000 0.134164
\(501\) 0 0
\(502\) −12.0000 −0.535586
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) −2.00000 −0.0890871
\(505\) −45.0000 −2.00247
\(506\) 36.0000 1.60040
\(507\) 0 0
\(508\) −4.00000 −0.177471
\(509\) −3.00000 −0.132973 −0.0664863 0.997787i \(-0.521179\pi\)
−0.0664863 + 0.997787i \(0.521179\pi\)
\(510\) 9.00000 0.398527
\(511\) −26.0000 −1.15017
\(512\) 1.00000 0.0441942
\(513\) −2.00000 −0.0883022
\(514\) −3.00000 −0.132324
\(515\) −42.0000 −1.85074
\(516\) −10.0000 −0.440225
\(517\) 36.0000 1.58328
\(518\) −14.0000 −0.615125
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) 33.0000 1.44576 0.722878 0.690976i \(-0.242819\pi\)
0.722878 + 0.690976i \(0.242819\pi\)
\(522\) 3.00000 0.131306
\(523\) −34.0000 −1.48672 −0.743358 0.668894i \(-0.766768\pi\)
−0.743358 + 0.668894i \(0.766768\pi\)
\(524\) 0 0
\(525\) −8.00000 −0.349149
\(526\) −6.00000 −0.261612
\(527\) −12.0000 −0.522728
\(528\) −6.00000 −0.261116
\(529\) 13.0000 0.565217
\(530\) −9.00000 −0.390935
\(531\) 0 0
\(532\) 4.00000 0.173422
\(533\) 0 0
\(534\) −18.0000 −0.778936
\(535\) 18.0000 0.778208
\(536\) 10.0000 0.431934
\(537\) 6.00000 0.258919
\(538\) 18.0000 0.776035
\(539\) 18.0000 0.775315
\(540\) −3.00000 −0.129099
\(541\) −29.0000 −1.24681 −0.623404 0.781900i \(-0.714251\pi\)
−0.623404 + 0.781900i \(0.714251\pi\)
\(542\) 16.0000 0.687259
\(543\) −7.00000 −0.300399
\(544\) −3.00000 −0.128624
\(545\) 42.0000 1.79908
\(546\) 0 0
\(547\) −34.0000 −1.45374 −0.726868 0.686778i \(-0.759025\pi\)
−0.726868 + 0.686778i \(0.759025\pi\)
\(548\) −9.00000 −0.384461
\(549\) −7.00000 −0.298753
\(550\) −24.0000 −1.02336
\(551\) −6.00000 −0.255609
\(552\) −6.00000 −0.255377
\(553\) 8.00000 0.340195
\(554\) 17.0000 0.722261
\(555\) −21.0000 −0.891400
\(556\) −4.00000 −0.169638
\(557\) −3.00000 −0.127114 −0.0635570 0.997978i \(-0.520244\pi\)
−0.0635570 + 0.997978i \(0.520244\pi\)
\(558\) 4.00000 0.169334
\(559\) 0 0
\(560\) 6.00000 0.253546
\(561\) 18.0000 0.759961
\(562\) −9.00000 −0.379642
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) −6.00000 −0.252646
\(565\) 9.00000 0.378633
\(566\) 14.0000 0.588464
\(567\) −2.00000 −0.0839921
\(568\) −6.00000 −0.251754
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 6.00000 0.251312
\(571\) −22.0000 −0.920671 −0.460336 0.887745i \(-0.652271\pi\)
−0.460336 + 0.887745i \(0.652271\pi\)
\(572\) 0 0
\(573\) −12.0000 −0.501307
\(574\) −6.00000 −0.250435
\(575\) −24.0000 −1.00087
\(576\) 1.00000 0.0416667
\(577\) −11.0000 −0.457936 −0.228968 0.973434i \(-0.573535\pi\)
−0.228968 + 0.973434i \(0.573535\pi\)
\(578\) −8.00000 −0.332756
\(579\) −23.0000 −0.955847
\(580\) −9.00000 −0.373705
\(581\) −12.0000 −0.497844
\(582\) −14.0000 −0.580319
\(583\) −18.0000 −0.745484
\(584\) 13.0000 0.537944
\(585\) 0 0
\(586\) 21.0000 0.867502
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) −3.00000 −0.123718
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 7.00000 0.287698
\(593\) −9.00000 −0.369586 −0.184793 0.982777i \(-0.559161\pi\)
−0.184793 + 0.982777i \(0.559161\pi\)
\(594\) −6.00000 −0.246183
\(595\) −18.0000 −0.737928
\(596\) 9.00000 0.368654
\(597\) −10.0000 −0.409273
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 4.00000 0.163299
\(601\) −37.0000 −1.50926 −0.754631 0.656150i \(-0.772184\pi\)
−0.754631 + 0.656150i \(0.772184\pi\)
\(602\) 20.0000 0.815139
\(603\) 10.0000 0.407231
\(604\) 10.0000 0.406894
\(605\) −75.0000 −3.04918
\(606\) 15.0000 0.609333
\(607\) 32.0000 1.29884 0.649420 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) −2.00000 −0.0811107
\(609\) −6.00000 −0.243132
\(610\) 21.0000 0.850265
\(611\) 0 0
\(612\) −3.00000 −0.121268
\(613\) 31.0000 1.25208 0.626039 0.779792i \(-0.284675\pi\)
0.626039 + 0.779792i \(0.284675\pi\)
\(614\) 10.0000 0.403567
\(615\) −9.00000 −0.362915
\(616\) 12.0000 0.483494
\(617\) 15.0000 0.603877 0.301939 0.953327i \(-0.402366\pi\)
0.301939 + 0.953327i \(0.402366\pi\)
\(618\) 14.0000 0.563163
\(619\) −8.00000 −0.321547 −0.160774 0.986991i \(-0.551399\pi\)
−0.160774 + 0.986991i \(0.551399\pi\)
\(620\) −12.0000 −0.481932
\(621\) −6.00000 −0.240772
\(622\) −30.0000 −1.20289
\(623\) 36.0000 1.44231
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) −10.0000 −0.399680
\(627\) 12.0000 0.479234
\(628\) 5.00000 0.199522
\(629\) −21.0000 −0.837325
\(630\) 6.00000 0.239046
\(631\) −20.0000 −0.796187 −0.398094 0.917345i \(-0.630328\pi\)
−0.398094 + 0.917345i \(0.630328\pi\)
\(632\) −4.00000 −0.159111
\(633\) −16.0000 −0.635943
\(634\) −3.00000 −0.119145
\(635\) 12.0000 0.476205
\(636\) 3.00000 0.118958
\(637\) 0 0
\(638\) −18.0000 −0.712627
\(639\) −6.00000 −0.237356
\(640\) −3.00000 −0.118585
\(641\) −3.00000 −0.118493 −0.0592464 0.998243i \(-0.518870\pi\)
−0.0592464 + 0.998243i \(0.518870\pi\)
\(642\) −6.00000 −0.236801
\(643\) 16.0000 0.630978 0.315489 0.948929i \(-0.397831\pi\)
0.315489 + 0.948929i \(0.397831\pi\)
\(644\) 12.0000 0.472866
\(645\) 30.0000 1.18125
\(646\) 6.00000 0.236067
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) −8.00000 −0.313545
\(652\) 4.00000 0.156652
\(653\) 42.0000 1.64359 0.821794 0.569785i \(-0.192974\pi\)
0.821794 + 0.569785i \(0.192974\pi\)
\(654\) −14.0000 −0.547443
\(655\) 0 0
\(656\) 3.00000 0.117130
\(657\) 13.0000 0.507178
\(658\) 12.0000 0.467809
\(659\) 24.0000 0.934907 0.467454 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) 18.0000 0.700649
\(661\) −5.00000 −0.194477 −0.0972387 0.995261i \(-0.531001\pi\)
−0.0972387 + 0.995261i \(0.531001\pi\)
\(662\) 4.00000 0.155464
\(663\) 0 0
\(664\) 6.00000 0.232845
\(665\) −12.0000 −0.465340
\(666\) 7.00000 0.271244
\(667\) −18.0000 −0.696963
\(668\) 0 0
\(669\) −8.00000 −0.309298
\(670\) −30.0000 −1.15900
\(671\) 42.0000 1.62139
\(672\) −2.00000 −0.0771517
\(673\) −13.0000 −0.501113 −0.250557 0.968102i \(-0.580614\pi\)
−0.250557 + 0.968102i \(0.580614\pi\)
\(674\) 23.0000 0.885927
\(675\) 4.00000 0.153960
\(676\) 0 0
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) −3.00000 −0.115214
\(679\) 28.0000 1.07454
\(680\) 9.00000 0.345134
\(681\) −18.0000 −0.689761
\(682\) −24.0000 −0.919007
\(683\) 48.0000 1.83667 0.918334 0.395805i \(-0.129534\pi\)
0.918334 + 0.395805i \(0.129534\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 27.0000 1.03162
\(686\) 20.0000 0.763604
\(687\) 22.0000 0.839352
\(688\) −10.0000 −0.381246
\(689\) 0 0
\(690\) 18.0000 0.685248
\(691\) −26.0000 −0.989087 −0.494543 0.869153i \(-0.664665\pi\)
−0.494543 + 0.869153i \(0.664665\pi\)
\(692\) −6.00000 −0.228086
\(693\) 12.0000 0.455842
\(694\) −30.0000 −1.13878
\(695\) 12.0000 0.455186
\(696\) 3.00000 0.113715
\(697\) −9.00000 −0.340899
\(698\) 10.0000 0.378506
\(699\) −6.00000 −0.226941
\(700\) −8.00000 −0.302372
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) −14.0000 −0.528020
\(704\) −6.00000 −0.226134
\(705\) 18.0000 0.677919
\(706\) 15.0000 0.564532
\(707\) −30.0000 −1.12827
\(708\) 0 0
\(709\) −5.00000 −0.187779 −0.0938895 0.995583i \(-0.529930\pi\)
−0.0938895 + 0.995583i \(0.529930\pi\)
\(710\) 18.0000 0.675528
\(711\) −4.00000 −0.150012
\(712\) −18.0000 −0.674579
\(713\) −24.0000 −0.898807
\(714\) 6.00000 0.224544
\(715\) 0 0
\(716\) 6.00000 0.224231
\(717\) −6.00000 −0.224074
\(718\) −6.00000 −0.223918
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) −3.00000 −0.111803
\(721\) −28.0000 −1.04277
\(722\) −15.0000 −0.558242
\(723\) 1.00000 0.0371904
\(724\) −7.00000 −0.260153
\(725\) 12.0000 0.445669
\(726\) 25.0000 0.927837
\(727\) 14.0000 0.519231 0.259616 0.965712i \(-0.416404\pi\)
0.259616 + 0.965712i \(0.416404\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −39.0000 −1.44345
\(731\) 30.0000 1.10959
\(732\) −7.00000 −0.258727
\(733\) 31.0000 1.14501 0.572506 0.819901i \(-0.305971\pi\)
0.572506 + 0.819901i \(0.305971\pi\)
\(734\) 2.00000 0.0738213
\(735\) 9.00000 0.331970
\(736\) −6.00000 −0.221163
\(737\) −60.0000 −2.21013
\(738\) 3.00000 0.110432
\(739\) 16.0000 0.588570 0.294285 0.955718i \(-0.404919\pi\)
0.294285 + 0.955718i \(0.404919\pi\)
\(740\) −21.0000 −0.771975
\(741\) 0 0
\(742\) −6.00000 −0.220267
\(743\) 36.0000 1.32071 0.660356 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) 4.00000 0.146647
\(745\) −27.0000 −0.989203
\(746\) 29.0000 1.06177
\(747\) 6.00000 0.219529
\(748\) 18.0000 0.658145
\(749\) 12.0000 0.438470
\(750\) 3.00000 0.109545
\(751\) 14.0000 0.510867 0.255434 0.966827i \(-0.417782\pi\)
0.255434 + 0.966827i \(0.417782\pi\)
\(752\) −6.00000 −0.218797
\(753\) −12.0000 −0.437304
\(754\) 0 0
\(755\) −30.0000 −1.09181
\(756\) −2.00000 −0.0727393
\(757\) −34.0000 −1.23575 −0.617876 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) −20.0000 −0.726433
\(759\) 36.0000 1.30672
\(760\) 6.00000 0.217643
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) −4.00000 −0.144905
\(763\) 28.0000 1.01367
\(764\) −12.0000 −0.434145
\(765\) 9.00000 0.325396
\(766\) −24.0000 −0.867155
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) −36.0000 −1.29735
\(771\) −3.00000 −0.108042
\(772\) −23.0000 −0.827788
\(773\) 30.0000 1.07903 0.539513 0.841978i \(-0.318609\pi\)
0.539513 + 0.841978i \(0.318609\pi\)
\(774\) −10.0000 −0.359443
\(775\) 16.0000 0.574737
\(776\) −14.0000 −0.502571
\(777\) −14.0000 −0.502247
\(778\) 39.0000 1.39822
\(779\) −6.00000 −0.214972
\(780\) 0 0
\(781\) 36.0000 1.28818
\(782\) 18.0000 0.643679
\(783\) 3.00000 0.107211
\(784\) −3.00000 −0.107143
\(785\) −15.0000 −0.535373
\(786\) 0 0
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) −6.00000 −0.213741
\(789\) −6.00000 −0.213606
\(790\) 12.0000 0.426941
\(791\) 6.00000 0.213335
\(792\) −6.00000 −0.213201
\(793\) 0 0
\(794\) −14.0000 −0.496841
\(795\) −9.00000 −0.319197
\(796\) −10.0000 −0.354441
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 4.00000 0.141598
\(799\) 18.0000 0.636794
\(800\) 4.00000 0.141421
\(801\) −18.0000 −0.635999
\(802\) 3.00000 0.105934
\(803\) −78.0000 −2.75256
\(804\) 10.0000 0.352673
\(805\) −36.0000 −1.26883
\(806\) 0 0
\(807\) 18.0000 0.633630
\(808\) 15.0000 0.527698
\(809\) −51.0000 −1.79306 −0.896532 0.442978i \(-0.853922\pi\)
−0.896532 + 0.442978i \(0.853922\pi\)
\(810\) −3.00000 −0.105409
\(811\) 4.00000 0.140459 0.0702295 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) −6.00000 −0.210559
\(813\) 16.0000 0.561144
\(814\) −42.0000 −1.47210
\(815\) −12.0000 −0.420342
\(816\) −3.00000 −0.105021
\(817\) 20.0000 0.699711
\(818\) 1.00000 0.0349642
\(819\) 0 0
\(820\) −9.00000 −0.314294
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) −9.00000 −0.313911
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) 14.0000 0.487713
\(825\) −24.0000 −0.835573
\(826\) 0 0
\(827\) 48.0000 1.66912 0.834562 0.550914i \(-0.185721\pi\)
0.834562 + 0.550914i \(0.185721\pi\)
\(828\) −6.00000 −0.208514
\(829\) 17.0000 0.590434 0.295217 0.955430i \(-0.404608\pi\)
0.295217 + 0.955430i \(0.404608\pi\)
\(830\) −18.0000 −0.624789
\(831\) 17.0000 0.589723
\(832\) 0 0
\(833\) 9.00000 0.311832
\(834\) −4.00000 −0.138509
\(835\) 0 0
\(836\) 12.0000 0.415029
\(837\) 4.00000 0.138260
\(838\) 24.0000 0.829066
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 6.00000 0.207020
\(841\) −20.0000 −0.689655
\(842\) −29.0000 −0.999406
\(843\) −9.00000 −0.309976
\(844\) −16.0000 −0.550743
\(845\) 0 0
\(846\) −6.00000 −0.206284
\(847\) −50.0000 −1.71802
\(848\) 3.00000 0.103020
\(849\) 14.0000 0.480479
\(850\) −12.0000 −0.411597
\(851\) −42.0000 −1.43974
\(852\) −6.00000 −0.205557
\(853\) 19.0000 0.650548 0.325274 0.945620i \(-0.394544\pi\)
0.325274 + 0.945620i \(0.394544\pi\)
\(854\) 14.0000 0.479070
\(855\) 6.00000 0.205196
\(856\) −6.00000 −0.205076
\(857\) 21.0000 0.717346 0.358673 0.933463i \(-0.383229\pi\)
0.358673 + 0.933463i \(0.383229\pi\)
\(858\) 0 0
\(859\) 26.0000 0.887109 0.443554 0.896248i \(-0.353717\pi\)
0.443554 + 0.896248i \(0.353717\pi\)
\(860\) 30.0000 1.02299
\(861\) −6.00000 −0.204479
\(862\) 6.00000 0.204361
\(863\) −18.0000 −0.612727 −0.306364 0.951915i \(-0.599112\pi\)
−0.306364 + 0.951915i \(0.599112\pi\)
\(864\) 1.00000 0.0340207
\(865\) 18.0000 0.612018
\(866\) −13.0000 −0.441758
\(867\) −8.00000 −0.271694
\(868\) −8.00000 −0.271538
\(869\) 24.0000 0.814144
\(870\) −9.00000 −0.305129
\(871\) 0 0
\(872\) −14.0000 −0.474100
\(873\) −14.0000 −0.473828
\(874\) 12.0000 0.405906
\(875\) −6.00000 −0.202837
\(876\) 13.0000 0.439229
\(877\) −41.0000 −1.38447 −0.692236 0.721671i \(-0.743374\pi\)
−0.692236 + 0.721671i \(0.743374\pi\)
\(878\) 14.0000 0.472477
\(879\) 21.0000 0.708312
\(880\) 18.0000 0.606780
\(881\) 33.0000 1.11180 0.555899 0.831250i \(-0.312374\pi\)
0.555899 + 0.831250i \(0.312374\pi\)
\(882\) −3.00000 −0.101015
\(883\) 8.00000 0.269221 0.134611 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −36.0000 −1.20944
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 7.00000 0.234905
\(889\) 8.00000 0.268311
\(890\) 54.0000 1.81008
\(891\) −6.00000 −0.201008
\(892\) −8.00000 −0.267860
\(893\) 12.0000 0.401565
\(894\) 9.00000 0.301005
\(895\) −18.0000 −0.601674
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) −18.0000 −0.600668
\(899\) 12.0000 0.400222
\(900\) 4.00000 0.133333
\(901\) −9.00000 −0.299833
\(902\) −18.0000 −0.599334
\(903\) 20.0000 0.665558
\(904\) −3.00000 −0.0997785
\(905\) 21.0000 0.698064
\(906\) 10.0000 0.332228
\(907\) 44.0000 1.46100 0.730498 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) −18.0000 −0.597351
\(909\) 15.0000 0.497519
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) −2.00000 −0.0662266
\(913\) −36.0000 −1.19143
\(914\) −11.0000 −0.363848
\(915\) 21.0000 0.694239
\(916\) 22.0000 0.726900
\(917\) 0 0
\(918\) −3.00000 −0.0990148
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 18.0000 0.593442
\(921\) 10.0000 0.329511
\(922\) −15.0000 −0.493999
\(923\) 0 0
\(924\) 12.0000 0.394771
\(925\) 28.0000 0.920634
\(926\) −38.0000 −1.24876
\(927\) 14.0000 0.459820
\(928\) 3.00000 0.0984798
\(929\) −33.0000 −1.08269 −0.541347 0.840799i \(-0.682086\pi\)
−0.541347 + 0.840799i \(0.682086\pi\)
\(930\) −12.0000 −0.393496
\(931\) 6.00000 0.196642
\(932\) −6.00000 −0.196537
\(933\) −30.0000 −0.982156
\(934\) −18.0000 −0.588978
\(935\) −54.0000 −1.76599
\(936\) 0 0
\(937\) 47.0000 1.53542 0.767712 0.640796i \(-0.221395\pi\)
0.767712 + 0.640796i \(0.221395\pi\)
\(938\) −20.0000 −0.653023
\(939\) −10.0000 −0.326338
\(940\) 18.0000 0.587095
\(941\) −42.0000 −1.36916 −0.684580 0.728937i \(-0.740015\pi\)
−0.684580 + 0.728937i \(0.740015\pi\)
\(942\) 5.00000 0.162909
\(943\) −18.0000 −0.586161
\(944\) 0 0
\(945\) 6.00000 0.195180
\(946\) 60.0000 1.95077
\(947\) −24.0000 −0.779895 −0.389948 0.920837i \(-0.627507\pi\)
−0.389948 + 0.920837i \(0.627507\pi\)
\(948\) −4.00000 −0.129914
\(949\) 0 0
\(950\) −8.00000 −0.259554
\(951\) −3.00000 −0.0972817
\(952\) 6.00000 0.194461
\(953\) 54.0000 1.74923 0.874616 0.484817i \(-0.161114\pi\)
0.874616 + 0.484817i \(0.161114\pi\)
\(954\) 3.00000 0.0971286
\(955\) 36.0000 1.16493
\(956\) −6.00000 −0.194054
\(957\) −18.0000 −0.581857
\(958\) 0 0
\(959\) 18.0000 0.581250
\(960\) −3.00000 −0.0968246
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −6.00000 −0.193347
\(964\) 1.00000 0.0322078
\(965\) 69.0000 2.22119
\(966\) 12.0000 0.386094
\(967\) 22.0000 0.707472 0.353736 0.935345i \(-0.384911\pi\)
0.353736 + 0.935345i \(0.384911\pi\)
\(968\) 25.0000 0.803530
\(969\) 6.00000 0.192748
\(970\) 42.0000 1.34854
\(971\) 60.0000 1.92549 0.962746 0.270408i \(-0.0871586\pi\)
0.962746 + 0.270408i \(0.0871586\pi\)
\(972\) 1.00000 0.0320750
\(973\) 8.00000 0.256468
\(974\) −2.00000 −0.0640841
\(975\) 0 0
\(976\) −7.00000 −0.224065
\(977\) 3.00000 0.0959785 0.0479893 0.998848i \(-0.484719\pi\)
0.0479893 + 0.998848i \(0.484719\pi\)
\(978\) 4.00000 0.127906
\(979\) 108.000 3.45169
\(980\) 9.00000 0.287494
\(981\) −14.0000 −0.446986
\(982\) −18.0000 −0.574403
\(983\) 36.0000 1.14822 0.574111 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) 3.00000 0.0956365
\(985\) 18.0000 0.573528
\(986\) −9.00000 −0.286618
\(987\) 12.0000 0.381964
\(988\) 0 0
\(989\) 60.0000 1.90789
\(990\) 18.0000 0.572078
\(991\) 38.0000 1.20711 0.603555 0.797321i \(-0.293750\pi\)
0.603555 + 0.797321i \(0.293750\pi\)
\(992\) 4.00000 0.127000
\(993\) 4.00000 0.126936
\(994\) 12.0000 0.380617
\(995\) 30.0000 0.951064
\(996\) 6.00000 0.190117
\(997\) 5.00000 0.158352 0.0791758 0.996861i \(-0.474771\pi\)
0.0791758 + 0.996861i \(0.474771\pi\)
\(998\) −32.0000 −1.01294
\(999\) 7.00000 0.221470
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 1014.2.a.f.1.1 1
3.2 odd 2 3042.2.a.h.1.1 1
4.3 odd 2 8112.2.a.c.1.1 1
13.2 odd 12 1014.2.i.b.823.2 4
13.3 even 3 1014.2.e.a.529.1 2
13.4 even 6 78.2.e.a.55.1 2
13.5 odd 4 1014.2.b.c.337.1 2
13.6 odd 12 1014.2.i.b.361.1 4
13.7 odd 12 1014.2.i.b.361.2 4
13.8 odd 4 1014.2.b.c.337.2 2
13.9 even 3 1014.2.e.a.991.1 2
13.10 even 6 78.2.e.a.61.1 yes 2
13.11 odd 12 1014.2.i.b.823.1 4
13.12 even 2 1014.2.a.c.1.1 1
39.5 even 4 3042.2.b.h.1351.2 2
39.8 even 4 3042.2.b.h.1351.1 2
39.17 odd 6 234.2.h.a.55.1 2
39.23 odd 6 234.2.h.a.217.1 2
39.38 odd 2 3042.2.a.i.1.1 1
52.23 odd 6 624.2.q.g.529.1 2
52.43 odd 6 624.2.q.g.289.1 2
52.51 odd 2 8112.2.a.m.1.1 1
65.4 even 6 1950.2.i.m.601.1 2
65.17 odd 12 1950.2.z.g.1849.1 4
65.23 odd 12 1950.2.z.g.1699.1 4
65.43 odd 12 1950.2.z.g.1849.2 4
65.49 even 6 1950.2.i.m.451.1 2
65.62 odd 12 1950.2.z.g.1699.2 4
156.23 even 6 1872.2.t.c.1153.1 2
156.95 even 6 1872.2.t.c.289.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.e.a.55.1 2 13.4 even 6
78.2.e.a.61.1 yes 2 13.10 even 6
234.2.h.a.55.1 2 39.17 odd 6
234.2.h.a.217.1 2 39.23 odd 6
624.2.q.g.289.1 2 52.43 odd 6
624.2.q.g.529.1 2 52.23 odd 6
1014.2.a.c.1.1 1 13.12 even 2
1014.2.a.f.1.1 1 1.1 even 1 trivial
1014.2.b.c.337.1 2 13.5 odd 4
1014.2.b.c.337.2 2 13.8 odd 4
1014.2.e.a.529.1 2 13.3 even 3
1014.2.e.a.991.1 2 13.9 even 3
1014.2.i.b.361.1 4 13.6 odd 12
1014.2.i.b.361.2 4 13.7 odd 12
1014.2.i.b.823.1 4 13.11 odd 12
1014.2.i.b.823.2 4 13.2 odd 12
1872.2.t.c.289.1 2 156.95 even 6
1872.2.t.c.1153.1 2 156.23 even 6
1950.2.i.m.451.1 2 65.49 even 6
1950.2.i.m.601.1 2 65.4 even 6
1950.2.z.g.1699.1 4 65.23 odd 12
1950.2.z.g.1699.2 4 65.62 odd 12
1950.2.z.g.1849.1 4 65.17 odd 12
1950.2.z.g.1849.2 4 65.43 odd 12
3042.2.a.h.1.1 1 3.2 odd 2
3042.2.a.i.1.1 1 39.38 odd 2
3042.2.b.h.1351.1 2 39.8 even 4
3042.2.b.h.1351.2 2 39.5 even 4
8112.2.a.c.1.1 1 4.3 odd 2
8112.2.a.m.1.1 1 52.51 odd 2