Properties

Label 1014.2.a.h.1.1
Level $1014$
Weight $2$
Character 1014.1
Self dual yes
Analytic conductor $8.097$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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
Root \(-1.73205\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.73205 q^{5} +1.00000 q^{6} +4.73205 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.73205 q^{5} +1.00000 q^{6} +4.73205 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.73205 q^{10} +4.73205 q^{11} -1.00000 q^{12} -4.73205 q^{14} +1.73205 q^{15} +1.00000 q^{16} -5.19615 q^{17} -1.00000 q^{18} +1.26795 q^{19} -1.73205 q^{20} -4.73205 q^{21} -4.73205 q^{22} +2.19615 q^{23} +1.00000 q^{24} -2.00000 q^{25} -1.00000 q^{27} +4.73205 q^{28} -3.00000 q^{29} -1.73205 q^{30} +2.53590 q^{31} -1.00000 q^{32} -4.73205 q^{33} +5.19615 q^{34} -8.19615 q^{35} +1.00000 q^{36} +3.00000 q^{37} -1.26795 q^{38} +1.73205 q^{40} -0.464102 q^{41} +4.73205 q^{42} +6.19615 q^{43} +4.73205 q^{44} -1.73205 q^{45} -2.19615 q^{46} -1.26795 q^{47} -1.00000 q^{48} +15.3923 q^{49} +2.00000 q^{50} +5.19615 q^{51} +3.00000 q^{53} +1.00000 q^{54} -8.19615 q^{55} -4.73205 q^{56} -1.26795 q^{57} +3.00000 q^{58} +13.8564 q^{59} +1.73205 q^{60} +4.80385 q^{61} -2.53590 q^{62} +4.73205 q^{63} +1.00000 q^{64} +4.73205 q^{66} +10.7321 q^{67} -5.19615 q^{68} -2.19615 q^{69} +8.19615 q^{70} +8.19615 q^{71} -1.00000 q^{72} -12.1244 q^{73} -3.00000 q^{74} +2.00000 q^{75} +1.26795 q^{76} +22.3923 q^{77} -12.3923 q^{79} -1.73205 q^{80} +1.00000 q^{81} +0.464102 q^{82} -11.6603 q^{83} -4.73205 q^{84} +9.00000 q^{85} -6.19615 q^{86} +3.00000 q^{87} -4.73205 q^{88} +2.53590 q^{89} +1.73205 q^{90} +2.19615 q^{92} -2.53590 q^{93} +1.26795 q^{94} -2.19615 q^{95} +1.00000 q^{96} +6.00000 q^{97} -15.3923 q^{98} +4.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9} + 6 q^{11} - 2 q^{12} - 6 q^{14} + 2 q^{16} - 2 q^{18} + 6 q^{19} - 6 q^{21} - 6 q^{22} - 6 q^{23} + 2 q^{24} - 4 q^{25} - 2 q^{27} + 6 q^{28} - 6 q^{29} + 12 q^{31} - 2 q^{32} - 6 q^{33} - 6 q^{35} + 2 q^{36} + 6 q^{37} - 6 q^{38} + 6 q^{41} + 6 q^{42} + 2 q^{43} + 6 q^{44} + 6 q^{46} - 6 q^{47} - 2 q^{48} + 10 q^{49} + 4 q^{50} + 6 q^{53} + 2 q^{54} - 6 q^{55} - 6 q^{56} - 6 q^{57} + 6 q^{58} + 20 q^{61} - 12 q^{62} + 6 q^{63} + 2 q^{64} + 6 q^{66} + 18 q^{67} + 6 q^{69} + 6 q^{70} + 6 q^{71} - 2 q^{72} - 6 q^{74} + 4 q^{75} + 6 q^{76} + 24 q^{77} - 4 q^{79} + 2 q^{81} - 6 q^{82} - 6 q^{83} - 6 q^{84} + 18 q^{85} - 2 q^{86} + 6 q^{87} - 6 q^{88} + 12 q^{89} - 6 q^{92} - 12 q^{93} + 6 q^{94} + 6 q^{95} + 2 q^{96} + 12 q^{97} - 10 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) −1.73205 −0.774597 −0.387298 0.921954i \(-0.626592\pi\)
−0.387298 + 0.921954i \(0.626592\pi\)
\(6\) 1.00000 0.408248
\(7\) 4.73205 1.78855 0.894274 0.447521i \(-0.147693\pi\)
0.894274 + 0.447521i \(0.147693\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.73205 0.547723
\(11\) 4.73205 1.42677 0.713384 0.700774i \(-0.247162\pi\)
0.713384 + 0.700774i \(0.247162\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −4.73205 −1.26469
\(15\) 1.73205 0.447214
\(16\) 1.00000 0.250000
\(17\) −5.19615 −1.26025 −0.630126 0.776493i \(-0.716997\pi\)
−0.630126 + 0.776493i \(0.716997\pi\)
\(18\) −1.00000 −0.235702
\(19\) 1.26795 0.290887 0.145444 0.989367i \(-0.453539\pi\)
0.145444 + 0.989367i \(0.453539\pi\)
\(20\) −1.73205 −0.387298
\(21\) −4.73205 −1.03262
\(22\) −4.73205 −1.00888
\(23\) 2.19615 0.457929 0.228965 0.973435i \(-0.426466\pi\)
0.228965 + 0.973435i \(0.426466\pi\)
\(24\) 1.00000 0.204124
\(25\) −2.00000 −0.400000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 4.73205 0.894274
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) −1.73205 −0.316228
\(31\) 2.53590 0.455461 0.227730 0.973724i \(-0.426870\pi\)
0.227730 + 0.973724i \(0.426870\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.73205 −0.823744
\(34\) 5.19615 0.891133
\(35\) −8.19615 −1.38540
\(36\) 1.00000 0.166667
\(37\) 3.00000 0.493197 0.246598 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) −1.26795 −0.205689
\(39\) 0 0
\(40\) 1.73205 0.273861
\(41\) −0.464102 −0.0724805 −0.0362402 0.999343i \(-0.511538\pi\)
−0.0362402 + 0.999343i \(0.511538\pi\)
\(42\) 4.73205 0.730171
\(43\) 6.19615 0.944904 0.472452 0.881356i \(-0.343369\pi\)
0.472452 + 0.881356i \(0.343369\pi\)
\(44\) 4.73205 0.713384
\(45\) −1.73205 −0.258199
\(46\) −2.19615 −0.323805
\(47\) −1.26795 −0.184949 −0.0924747 0.995715i \(-0.529478\pi\)
−0.0924747 + 0.995715i \(0.529478\pi\)
\(48\) −1.00000 −0.144338
\(49\) 15.3923 2.19890
\(50\) 2.00000 0.282843
\(51\) 5.19615 0.727607
\(52\) 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
\(55\) −8.19615 −1.10517
\(56\) −4.73205 −0.632347
\(57\) −1.26795 −0.167944
\(58\) 3.00000 0.393919
\(59\) 13.8564 1.80395 0.901975 0.431788i \(-0.142117\pi\)
0.901975 + 0.431788i \(0.142117\pi\)
\(60\) 1.73205 0.223607
\(61\) 4.80385 0.615070 0.307535 0.951537i \(-0.400496\pi\)
0.307535 + 0.951537i \(0.400496\pi\)
\(62\) −2.53590 −0.322059
\(63\) 4.73205 0.596182
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 4.73205 0.582475
\(67\) 10.7321 1.31113 0.655564 0.755139i \(-0.272431\pi\)
0.655564 + 0.755139i \(0.272431\pi\)
\(68\) −5.19615 −0.630126
\(69\) −2.19615 −0.264386
\(70\) 8.19615 0.979628
\(71\) 8.19615 0.972704 0.486352 0.873763i \(-0.338327\pi\)
0.486352 + 0.873763i \(0.338327\pi\)
\(72\) −1.00000 −0.117851
\(73\) −12.1244 −1.41905 −0.709524 0.704681i \(-0.751090\pi\)
−0.709524 + 0.704681i \(0.751090\pi\)
\(74\) −3.00000 −0.348743
\(75\) 2.00000 0.230940
\(76\) 1.26795 0.145444
\(77\) 22.3923 2.55184
\(78\) 0 0
\(79\) −12.3923 −1.39424 −0.697122 0.716953i \(-0.745536\pi\)
−0.697122 + 0.716953i \(0.745536\pi\)
\(80\) −1.73205 −0.193649
\(81\) 1.00000 0.111111
\(82\) 0.464102 0.0512514
\(83\) −11.6603 −1.27988 −0.639940 0.768425i \(-0.721041\pi\)
−0.639940 + 0.768425i \(0.721041\pi\)
\(84\) −4.73205 −0.516309
\(85\) 9.00000 0.976187
\(86\) −6.19615 −0.668148
\(87\) 3.00000 0.321634
\(88\) −4.73205 −0.504438
\(89\) 2.53590 0.268805 0.134402 0.990927i \(-0.457089\pi\)
0.134402 + 0.990927i \(0.457089\pi\)
\(90\) 1.73205 0.182574
\(91\) 0 0
\(92\) 2.19615 0.228965
\(93\) −2.53590 −0.262960
\(94\) 1.26795 0.130779
\(95\) −2.19615 −0.225320
\(96\) 1.00000 0.102062
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) −15.3923 −1.55486
\(99\) 4.73205 0.475589
\(100\) −2.00000 −0.200000
\(101\) −1.39230 −0.138540 −0.0692698 0.997598i \(-0.522067\pi\)
−0.0692698 + 0.997598i \(0.522067\pi\)
\(102\) −5.19615 −0.514496
\(103\) −4.19615 −0.413459 −0.206730 0.978398i \(-0.566282\pi\)
−0.206730 + 0.978398i \(0.566282\pi\)
\(104\) 0 0
\(105\) 8.19615 0.799863
\(106\) −3.00000 −0.291386
\(107\) 8.19615 0.792352 0.396176 0.918175i \(-0.370337\pi\)
0.396176 + 0.918175i \(0.370337\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 16.3923 1.57010 0.785049 0.619434i \(-0.212638\pi\)
0.785049 + 0.619434i \(0.212638\pi\)
\(110\) 8.19615 0.781472
\(111\) −3.00000 −0.284747
\(112\) 4.73205 0.447137
\(113\) 11.1962 1.05325 0.526623 0.850099i \(-0.323458\pi\)
0.526623 + 0.850099i \(0.323458\pi\)
\(114\) 1.26795 0.118754
\(115\) −3.80385 −0.354711
\(116\) −3.00000 −0.278543
\(117\) 0 0
\(118\) −13.8564 −1.27559
\(119\) −24.5885 −2.25402
\(120\) −1.73205 −0.158114
\(121\) 11.3923 1.03566
\(122\) −4.80385 −0.434920
\(123\) 0.464102 0.0418466
\(124\) 2.53590 0.227730
\(125\) 12.1244 1.08444
\(126\) −4.73205 −0.421565
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −6.19615 −0.545541
\(130\) 0 0
\(131\) 16.3923 1.43220 0.716101 0.697997i \(-0.245925\pi\)
0.716101 + 0.697997i \(0.245925\pi\)
\(132\) −4.73205 −0.411872
\(133\) 6.00000 0.520266
\(134\) −10.7321 −0.927108
\(135\) 1.73205 0.149071
\(136\) 5.19615 0.445566
\(137\) −9.00000 −0.768922 −0.384461 0.923141i \(-0.625613\pi\)
−0.384461 + 0.923141i \(0.625613\pi\)
\(138\) 2.19615 0.186949
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −8.19615 −0.692701
\(141\) 1.26795 0.106781
\(142\) −8.19615 −0.687806
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 5.19615 0.431517
\(146\) 12.1244 1.00342
\(147\) −15.3923 −1.26954
\(148\) 3.00000 0.246598
\(149\) −18.1244 −1.48481 −0.742403 0.669954i \(-0.766314\pi\)
−0.742403 + 0.669954i \(0.766314\pi\)
\(150\) −2.00000 −0.163299
\(151\) 7.26795 0.591457 0.295729 0.955272i \(-0.404438\pi\)
0.295729 + 0.955272i \(0.404438\pi\)
\(152\) −1.26795 −0.102844
\(153\) −5.19615 −0.420084
\(154\) −22.3923 −1.80442
\(155\) −4.39230 −0.352798
\(156\) 0 0
\(157\) −3.19615 −0.255081 −0.127540 0.991833i \(-0.540708\pi\)
−0.127540 + 0.991833i \(0.540708\pi\)
\(158\) 12.3923 0.985879
\(159\) −3.00000 −0.237915
\(160\) 1.73205 0.136931
\(161\) 10.3923 0.819028
\(162\) −1.00000 −0.0785674
\(163\) 9.46410 0.741286 0.370643 0.928775i \(-0.379137\pi\)
0.370643 + 0.928775i \(0.379137\pi\)
\(164\) −0.464102 −0.0362402
\(165\) 8.19615 0.638070
\(166\) 11.6603 0.905011
\(167\) −2.53590 −0.196234 −0.0981169 0.995175i \(-0.531282\pi\)
−0.0981169 + 0.995175i \(0.531282\pi\)
\(168\) 4.73205 0.365086
\(169\) 0 0
\(170\) −9.00000 −0.690268
\(171\) 1.26795 0.0969625
\(172\) 6.19615 0.472452
\(173\) 16.3923 1.24628 0.623142 0.782109i \(-0.285856\pi\)
0.623142 + 0.782109i \(0.285856\pi\)
\(174\) −3.00000 −0.227429
\(175\) −9.46410 −0.715419
\(176\) 4.73205 0.356692
\(177\) −13.8564 −1.04151
\(178\) −2.53590 −0.190074
\(179\) −8.19615 −0.612609 −0.306305 0.951934i \(-0.599093\pi\)
−0.306305 + 0.951934i \(0.599093\pi\)
\(180\) −1.73205 −0.129099
\(181\) −11.5885 −0.861363 −0.430682 0.902504i \(-0.641727\pi\)
−0.430682 + 0.902504i \(0.641727\pi\)
\(182\) 0 0
\(183\) −4.80385 −0.355111
\(184\) −2.19615 −0.161903
\(185\) −5.19615 −0.382029
\(186\) 2.53590 0.185941
\(187\) −24.5885 −1.79809
\(188\) −1.26795 −0.0924747
\(189\) −4.73205 −0.344206
\(190\) 2.19615 0.159326
\(191\) 20.7846 1.50392 0.751961 0.659208i \(-0.229108\pi\)
0.751961 + 0.659208i \(0.229108\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −12.8038 −0.921641 −0.460821 0.887493i \(-0.652445\pi\)
−0.460821 + 0.887493i \(0.652445\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) 15.3923 1.09945
\(197\) 6.92820 0.493614 0.246807 0.969065i \(-0.420619\pi\)
0.246807 + 0.969065i \(0.420619\pi\)
\(198\) −4.73205 −0.336292
\(199\) 8.58846 0.608820 0.304410 0.952541i \(-0.401541\pi\)
0.304410 + 0.952541i \(0.401541\pi\)
\(200\) 2.00000 0.141421
\(201\) −10.7321 −0.756980
\(202\) 1.39230 0.0979622
\(203\) −14.1962 −0.996375
\(204\) 5.19615 0.363803
\(205\) 0.803848 0.0561432
\(206\) 4.19615 0.292360
\(207\) 2.19615 0.152643
\(208\) 0 0
\(209\) 6.00000 0.415029
\(210\) −8.19615 −0.565588
\(211\) −3.60770 −0.248364 −0.124182 0.992259i \(-0.539631\pi\)
−0.124182 + 0.992259i \(0.539631\pi\)
\(212\) 3.00000 0.206041
\(213\) −8.19615 −0.561591
\(214\) −8.19615 −0.560277
\(215\) −10.7321 −0.731920
\(216\) 1.00000 0.0680414
\(217\) 12.0000 0.814613
\(218\) −16.3923 −1.11023
\(219\) 12.1244 0.819288
\(220\) −8.19615 −0.552584
\(221\) 0 0
\(222\) 3.00000 0.201347
\(223\) −18.9282 −1.26753 −0.633763 0.773527i \(-0.718491\pi\)
−0.633763 + 0.773527i \(0.718491\pi\)
\(224\) −4.73205 −0.316173
\(225\) −2.00000 −0.133333
\(226\) −11.1962 −0.744757
\(227\) −9.80385 −0.650704 −0.325352 0.945593i \(-0.605483\pi\)
−0.325352 + 0.945593i \(0.605483\pi\)
\(228\) −1.26795 −0.0839720
\(229\) 19.8564 1.31215 0.656074 0.754696i \(-0.272216\pi\)
0.656074 + 0.754696i \(0.272216\pi\)
\(230\) 3.80385 0.250818
\(231\) −22.3923 −1.47331
\(232\) 3.00000 0.196960
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 2.19615 0.143261
\(236\) 13.8564 0.901975
\(237\) 12.3923 0.804967
\(238\) 24.5885 1.59383
\(239\) −24.5885 −1.59050 −0.795248 0.606285i \(-0.792659\pi\)
−0.795248 + 0.606285i \(0.792659\pi\)
\(240\) 1.73205 0.111803
\(241\) 0.803848 0.0517804 0.0258902 0.999665i \(-0.491758\pi\)
0.0258902 + 0.999665i \(0.491758\pi\)
\(242\) −11.3923 −0.732325
\(243\) −1.00000 −0.0641500
\(244\) 4.80385 0.307535
\(245\) −26.6603 −1.70326
\(246\) −0.464102 −0.0295900
\(247\) 0 0
\(248\) −2.53590 −0.161030
\(249\) 11.6603 0.738939
\(250\) −12.1244 −0.766812
\(251\) −4.39230 −0.277240 −0.138620 0.990346i \(-0.544267\pi\)
−0.138620 + 0.990346i \(0.544267\pi\)
\(252\) 4.73205 0.298091
\(253\) 10.3923 0.653359
\(254\) 4.00000 0.250982
\(255\) −9.00000 −0.563602
\(256\) 1.00000 0.0625000
\(257\) 12.8038 0.798682 0.399341 0.916802i \(-0.369239\pi\)
0.399341 + 0.916802i \(0.369239\pi\)
\(258\) 6.19615 0.385756
\(259\) 14.1962 0.882106
\(260\) 0 0
\(261\) −3.00000 −0.185695
\(262\) −16.3923 −1.01272
\(263\) −2.19615 −0.135421 −0.0677103 0.997705i \(-0.521569\pi\)
−0.0677103 + 0.997705i \(0.521569\pi\)
\(264\) 4.73205 0.291238
\(265\) −5.19615 −0.319197
\(266\) −6.00000 −0.367884
\(267\) −2.53590 −0.155194
\(268\) 10.7321 0.655564
\(269\) −28.3923 −1.73111 −0.865555 0.500814i \(-0.833034\pi\)
−0.865555 + 0.500814i \(0.833034\pi\)
\(270\) −1.73205 −0.105409
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) −5.19615 −0.315063
\(273\) 0 0
\(274\) 9.00000 0.543710
\(275\) −9.46410 −0.570707
\(276\) −2.19615 −0.132193
\(277\) 15.1962 0.913048 0.456524 0.889711i \(-0.349094\pi\)
0.456524 + 0.889711i \(0.349094\pi\)
\(278\) 4.00000 0.239904
\(279\) 2.53590 0.151820
\(280\) 8.19615 0.489814
\(281\) 24.4641 1.45941 0.729703 0.683764i \(-0.239658\pi\)
0.729703 + 0.683764i \(0.239658\pi\)
\(282\) −1.26795 −0.0755053
\(283\) −30.1962 −1.79497 −0.897487 0.441040i \(-0.854610\pi\)
−0.897487 + 0.441040i \(0.854610\pi\)
\(284\) 8.19615 0.486352
\(285\) 2.19615 0.130089
\(286\) 0 0
\(287\) −2.19615 −0.129635
\(288\) −1.00000 −0.0589256
\(289\) 10.0000 0.588235
\(290\) −5.19615 −0.305129
\(291\) −6.00000 −0.351726
\(292\) −12.1244 −0.709524
\(293\) −14.6603 −0.856461 −0.428231 0.903669i \(-0.640863\pi\)
−0.428231 + 0.903669i \(0.640863\pi\)
\(294\) 15.3923 0.897697
\(295\) −24.0000 −1.39733
\(296\) −3.00000 −0.174371
\(297\) −4.73205 −0.274581
\(298\) 18.1244 1.04992
\(299\) 0 0
\(300\) 2.00000 0.115470
\(301\) 29.3205 1.69001
\(302\) −7.26795 −0.418223
\(303\) 1.39230 0.0799858
\(304\) 1.26795 0.0727219
\(305\) −8.32051 −0.476431
\(306\) 5.19615 0.297044
\(307\) 10.7321 0.612510 0.306255 0.951949i \(-0.400924\pi\)
0.306255 + 0.951949i \(0.400924\pi\)
\(308\) 22.3923 1.27592
\(309\) 4.19615 0.238711
\(310\) 4.39230 0.249466
\(311\) 2.19615 0.124532 0.0622662 0.998060i \(-0.480167\pi\)
0.0622662 + 0.998060i \(0.480167\pi\)
\(312\) 0 0
\(313\) −24.3923 −1.37873 −0.689367 0.724412i \(-0.742111\pi\)
−0.689367 + 0.724412i \(0.742111\pi\)
\(314\) 3.19615 0.180369
\(315\) −8.19615 −0.461801
\(316\) −12.3923 −0.697122
\(317\) 6.12436 0.343978 0.171989 0.985099i \(-0.444981\pi\)
0.171989 + 0.985099i \(0.444981\pi\)
\(318\) 3.00000 0.168232
\(319\) −14.1962 −0.794832
\(320\) −1.73205 −0.0968246
\(321\) −8.19615 −0.457465
\(322\) −10.3923 −0.579141
\(323\) −6.58846 −0.366592
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −9.46410 −0.524168
\(327\) −16.3923 −0.906497
\(328\) 0.464102 0.0256257
\(329\) −6.00000 −0.330791
\(330\) −8.19615 −0.451183
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) −11.6603 −0.639940
\(333\) 3.00000 0.164399
\(334\) 2.53590 0.138758
\(335\) −18.5885 −1.01560
\(336\) −4.73205 −0.258155
\(337\) −31.0000 −1.68868 −0.844339 0.535810i \(-0.820006\pi\)
−0.844339 + 0.535810i \(0.820006\pi\)
\(338\) 0 0
\(339\) −11.1962 −0.608092
\(340\) 9.00000 0.488094
\(341\) 12.0000 0.649836
\(342\) −1.26795 −0.0685628
\(343\) 39.7128 2.14429
\(344\) −6.19615 −0.334074
\(345\) 3.80385 0.204792
\(346\) −16.3923 −0.881256
\(347\) −12.5885 −0.675784 −0.337892 0.941185i \(-0.609714\pi\)
−0.337892 + 0.941185i \(0.609714\pi\)
\(348\) 3.00000 0.160817
\(349\) 2.53590 0.135744 0.0678718 0.997694i \(-0.478379\pi\)
0.0678718 + 0.997694i \(0.478379\pi\)
\(350\) 9.46410 0.505878
\(351\) 0 0
\(352\) −4.73205 −0.252219
\(353\) 5.78461 0.307884 0.153942 0.988080i \(-0.450803\pi\)
0.153942 + 0.988080i \(0.450803\pi\)
\(354\) 13.8564 0.736460
\(355\) −14.1962 −0.753454
\(356\) 2.53590 0.134402
\(357\) 24.5885 1.30136
\(358\) 8.19615 0.433180
\(359\) 22.0526 1.16389 0.581945 0.813228i \(-0.302292\pi\)
0.581945 + 0.813228i \(0.302292\pi\)
\(360\) 1.73205 0.0912871
\(361\) −17.3923 −0.915384
\(362\) 11.5885 0.609076
\(363\) −11.3923 −0.597941
\(364\) 0 0
\(365\) 21.0000 1.09919
\(366\) 4.80385 0.251101
\(367\) −24.1962 −1.26303 −0.631514 0.775364i \(-0.717566\pi\)
−0.631514 + 0.775364i \(0.717566\pi\)
\(368\) 2.19615 0.114482
\(369\) −0.464102 −0.0241602
\(370\) 5.19615 0.270135
\(371\) 14.1962 0.737028
\(372\) −2.53590 −0.131480
\(373\) 23.9808 1.24168 0.620838 0.783939i \(-0.286792\pi\)
0.620838 + 0.783939i \(0.286792\pi\)
\(374\) 24.5885 1.27144
\(375\) −12.1244 −0.626099
\(376\) 1.26795 0.0653895
\(377\) 0 0
\(378\) 4.73205 0.243390
\(379\) −18.2487 −0.937373 −0.468687 0.883364i \(-0.655273\pi\)
−0.468687 + 0.883364i \(0.655273\pi\)
\(380\) −2.19615 −0.112660
\(381\) 4.00000 0.204926
\(382\) −20.7846 −1.06343
\(383\) 11.3205 0.578451 0.289225 0.957261i \(-0.406602\pi\)
0.289225 + 0.957261i \(0.406602\pi\)
\(384\) 1.00000 0.0510310
\(385\) −38.7846 −1.97665
\(386\) 12.8038 0.651699
\(387\) 6.19615 0.314968
\(388\) 6.00000 0.304604
\(389\) −13.3923 −0.679017 −0.339508 0.940603i \(-0.610261\pi\)
−0.339508 + 0.940603i \(0.610261\pi\)
\(390\) 0 0
\(391\) −11.4115 −0.577107
\(392\) −15.3923 −0.777429
\(393\) −16.3923 −0.826882
\(394\) −6.92820 −0.349038
\(395\) 21.4641 1.07998
\(396\) 4.73205 0.237795
\(397\) −16.3923 −0.822706 −0.411353 0.911476i \(-0.634944\pi\)
−0.411353 + 0.911476i \(0.634944\pi\)
\(398\) −8.58846 −0.430500
\(399\) −6.00000 −0.300376
\(400\) −2.00000 −0.100000
\(401\) 21.0000 1.04869 0.524345 0.851506i \(-0.324310\pi\)
0.524345 + 0.851506i \(0.324310\pi\)
\(402\) 10.7321 0.535266
\(403\) 0 0
\(404\) −1.39230 −0.0692698
\(405\) −1.73205 −0.0860663
\(406\) 14.1962 0.704543
\(407\) 14.1962 0.703677
\(408\) −5.19615 −0.257248
\(409\) −3.33975 −0.165140 −0.0825699 0.996585i \(-0.526313\pi\)
−0.0825699 + 0.996585i \(0.526313\pi\)
\(410\) −0.803848 −0.0396992
\(411\) 9.00000 0.443937
\(412\) −4.19615 −0.206730
\(413\) 65.5692 3.22645
\(414\) −2.19615 −0.107935
\(415\) 20.1962 0.991390
\(416\) 0 0
\(417\) 4.00000 0.195881
\(418\) −6.00000 −0.293470
\(419\) 16.3923 0.800816 0.400408 0.916337i \(-0.368868\pi\)
0.400408 + 0.916337i \(0.368868\pi\)
\(420\) 8.19615 0.399931
\(421\) 0.464102 0.0226189 0.0113095 0.999936i \(-0.496400\pi\)
0.0113095 + 0.999936i \(0.496400\pi\)
\(422\) 3.60770 0.175620
\(423\) −1.26795 −0.0616498
\(424\) −3.00000 −0.145693
\(425\) 10.3923 0.504101
\(426\) 8.19615 0.397105
\(427\) 22.7321 1.10008
\(428\) 8.19615 0.396176
\(429\) 0 0
\(430\) 10.7321 0.517545
\(431\) 27.8038 1.33926 0.669632 0.742693i \(-0.266452\pi\)
0.669632 + 0.742693i \(0.266452\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −33.7846 −1.62358 −0.811792 0.583946i \(-0.801508\pi\)
−0.811792 + 0.583946i \(0.801508\pi\)
\(434\) −12.0000 −0.576018
\(435\) −5.19615 −0.249136
\(436\) 16.3923 0.785049
\(437\) 2.78461 0.133206
\(438\) −12.1244 −0.579324
\(439\) 16.5885 0.791724 0.395862 0.918310i \(-0.370446\pi\)
0.395862 + 0.918310i \(0.370446\pi\)
\(440\) 8.19615 0.390736
\(441\) 15.3923 0.732967
\(442\) 0 0
\(443\) −4.39230 −0.208685 −0.104342 0.994541i \(-0.533274\pi\)
−0.104342 + 0.994541i \(0.533274\pi\)
\(444\) −3.00000 −0.142374
\(445\) −4.39230 −0.208215
\(446\) 18.9282 0.896276
\(447\) 18.1244 0.857253
\(448\) 4.73205 0.223568
\(449\) 33.4641 1.57927 0.789634 0.613578i \(-0.210270\pi\)
0.789634 + 0.613578i \(0.210270\pi\)
\(450\) 2.00000 0.0942809
\(451\) −2.19615 −0.103413
\(452\) 11.1962 0.526623
\(453\) −7.26795 −0.341478
\(454\) 9.80385 0.460117
\(455\) 0 0
\(456\) 1.26795 0.0593772
\(457\) 19.9808 0.934661 0.467330 0.884083i \(-0.345216\pi\)
0.467330 + 0.884083i \(0.345216\pi\)
\(458\) −19.8564 −0.927829
\(459\) 5.19615 0.242536
\(460\) −3.80385 −0.177355
\(461\) −19.9808 −0.930597 −0.465298 0.885154i \(-0.654053\pi\)
−0.465298 + 0.885154i \(0.654053\pi\)
\(462\) 22.3923 1.04178
\(463\) −26.1962 −1.21744 −0.608719 0.793386i \(-0.708316\pi\)
−0.608719 + 0.793386i \(0.708316\pi\)
\(464\) −3.00000 −0.139272
\(465\) 4.39230 0.203688
\(466\) 18.0000 0.833834
\(467\) 36.5885 1.69311 0.846556 0.532300i \(-0.178672\pi\)
0.846556 + 0.532300i \(0.178672\pi\)
\(468\) 0 0
\(469\) 50.7846 2.34502
\(470\) −2.19615 −0.101301
\(471\) 3.19615 0.147271
\(472\) −13.8564 −0.637793
\(473\) 29.3205 1.34816
\(474\) −12.3923 −0.569197
\(475\) −2.53590 −0.116355
\(476\) −24.5885 −1.12701
\(477\) 3.00000 0.137361
\(478\) 24.5885 1.12465
\(479\) −35.3205 −1.61384 −0.806918 0.590664i \(-0.798866\pi\)
−0.806918 + 0.590664i \(0.798866\pi\)
\(480\) −1.73205 −0.0790569
\(481\) 0 0
\(482\) −0.803848 −0.0366143
\(483\) −10.3923 −0.472866
\(484\) 11.3923 0.517832
\(485\) −10.3923 −0.471890
\(486\) 1.00000 0.0453609
\(487\) 9.12436 0.413464 0.206732 0.978398i \(-0.433717\pi\)
0.206732 + 0.978398i \(0.433717\pi\)
\(488\) −4.80385 −0.217460
\(489\) −9.46410 −0.427981
\(490\) 26.6603 1.20439
\(491\) 0.588457 0.0265567 0.0132784 0.999912i \(-0.495773\pi\)
0.0132784 + 0.999912i \(0.495773\pi\)
\(492\) 0.464102 0.0209233
\(493\) 15.5885 0.702069
\(494\) 0 0
\(495\) −8.19615 −0.368390
\(496\) 2.53590 0.113865
\(497\) 38.7846 1.73973
\(498\) −11.6603 −0.522508
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 12.1244 0.542218
\(501\) 2.53590 0.113296
\(502\) 4.39230 0.196038
\(503\) −18.5885 −0.828818 −0.414409 0.910091i \(-0.636012\pi\)
−0.414409 + 0.910091i \(0.636012\pi\)
\(504\) −4.73205 −0.210782
\(505\) 2.41154 0.107312
\(506\) −10.3923 −0.461994
\(507\) 0 0
\(508\) −4.00000 −0.177471
\(509\) 9.33975 0.413977 0.206988 0.978343i \(-0.433634\pi\)
0.206988 + 0.978343i \(0.433634\pi\)
\(510\) 9.00000 0.398527
\(511\) −57.3731 −2.53804
\(512\) −1.00000 −0.0441942
\(513\) −1.26795 −0.0559813
\(514\) −12.8038 −0.564754
\(515\) 7.26795 0.320264
\(516\) −6.19615 −0.272770
\(517\) −6.00000 −0.263880
\(518\) −14.1962 −0.623743
\(519\) −16.3923 −0.719542
\(520\) 0 0
\(521\) −18.8038 −0.823812 −0.411906 0.911226i \(-0.635137\pi\)
−0.411906 + 0.911226i \(0.635137\pi\)
\(522\) 3.00000 0.131306
\(523\) 1.41154 0.0617225 0.0308612 0.999524i \(-0.490175\pi\)
0.0308612 + 0.999524i \(0.490175\pi\)
\(524\) 16.3923 0.716101
\(525\) 9.46410 0.413047
\(526\) 2.19615 0.0957568
\(527\) −13.1769 −0.573995
\(528\) −4.73205 −0.205936
\(529\) −18.1769 −0.790301
\(530\) 5.19615 0.225706
\(531\) 13.8564 0.601317
\(532\) 6.00000 0.260133
\(533\) 0 0
\(534\) 2.53590 0.109739
\(535\) −14.1962 −0.613753
\(536\) −10.7321 −0.463554
\(537\) 8.19615 0.353690
\(538\) 28.3923 1.22408
\(539\) 72.8372 3.13732
\(540\) 1.73205 0.0745356
\(541\) 16.8564 0.724714 0.362357 0.932039i \(-0.381972\pi\)
0.362357 + 0.932039i \(0.381972\pi\)
\(542\) 0 0
\(543\) 11.5885 0.497308
\(544\) 5.19615 0.222783
\(545\) −28.3923 −1.21619
\(546\) 0 0
\(547\) −6.19615 −0.264928 −0.132464 0.991188i \(-0.542289\pi\)
−0.132464 + 0.991188i \(0.542289\pi\)
\(548\) −9.00000 −0.384461
\(549\) 4.80385 0.205023
\(550\) 9.46410 0.403551
\(551\) −3.80385 −0.162049
\(552\) 2.19615 0.0934745
\(553\) −58.6410 −2.49367
\(554\) −15.1962 −0.645623
\(555\) 5.19615 0.220564
\(556\) −4.00000 −0.169638
\(557\) 22.2679 0.943523 0.471762 0.881726i \(-0.343618\pi\)
0.471762 + 0.881726i \(0.343618\pi\)
\(558\) −2.53590 −0.107353
\(559\) 0 0
\(560\) −8.19615 −0.346351
\(561\) 24.5885 1.03813
\(562\) −24.4641 −1.03196
\(563\) 8.78461 0.370227 0.185114 0.982717i \(-0.440735\pi\)
0.185114 + 0.982717i \(0.440735\pi\)
\(564\) 1.26795 0.0533903
\(565\) −19.3923 −0.815840
\(566\) 30.1962 1.26924
\(567\) 4.73205 0.198727
\(568\) −8.19615 −0.343903
\(569\) −32.7846 −1.37440 −0.687201 0.726467i \(-0.741161\pi\)
−0.687201 + 0.726467i \(0.741161\pi\)
\(570\) −2.19615 −0.0919867
\(571\) −13.8038 −0.577673 −0.288837 0.957378i \(-0.593268\pi\)
−0.288837 + 0.957378i \(0.593268\pi\)
\(572\) 0 0
\(573\) −20.7846 −0.868290
\(574\) 2.19615 0.0916656
\(575\) −4.39230 −0.183172
\(576\) 1.00000 0.0416667
\(577\) −16.2679 −0.677244 −0.338622 0.940923i \(-0.609961\pi\)
−0.338622 + 0.940923i \(0.609961\pi\)
\(578\) −10.0000 −0.415945
\(579\) 12.8038 0.532110
\(580\) 5.19615 0.215758
\(581\) −55.1769 −2.28912
\(582\) 6.00000 0.248708
\(583\) 14.1962 0.587945
\(584\) 12.1244 0.501709
\(585\) 0 0
\(586\) 14.6603 0.605610
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) −15.3923 −0.634768
\(589\) 3.21539 0.132488
\(590\) 24.0000 0.988064
\(591\) −6.92820 −0.284988
\(592\) 3.00000 0.123299
\(593\) −46.8564 −1.92416 −0.962081 0.272764i \(-0.912062\pi\)
−0.962081 + 0.272764i \(0.912062\pi\)
\(594\) 4.73205 0.194158
\(595\) 42.5885 1.74596
\(596\) −18.1244 −0.742403
\(597\) −8.58846 −0.351502
\(598\) 0 0
\(599\) −4.39230 −0.179465 −0.0897324 0.995966i \(-0.528601\pi\)
−0.0897324 + 0.995966i \(0.528601\pi\)
\(600\) −2.00000 −0.0816497
\(601\) 21.7846 0.888613 0.444306 0.895875i \(-0.353450\pi\)
0.444306 + 0.895875i \(0.353450\pi\)
\(602\) −29.3205 −1.19501
\(603\) 10.7321 0.437043
\(604\) 7.26795 0.295729
\(605\) −19.7321 −0.802222
\(606\) −1.39230 −0.0565585
\(607\) −48.7846 −1.98011 −0.990053 0.140694i \(-0.955067\pi\)
−0.990053 + 0.140694i \(0.955067\pi\)
\(608\) −1.26795 −0.0514221
\(609\) 14.1962 0.575257
\(610\) 8.32051 0.336888
\(611\) 0 0
\(612\) −5.19615 −0.210042
\(613\) −40.8564 −1.65017 −0.825087 0.565005i \(-0.808874\pi\)
−0.825087 + 0.565005i \(0.808874\pi\)
\(614\) −10.7321 −0.433110
\(615\) −0.803848 −0.0324143
\(616\) −22.3923 −0.902212
\(617\) 10.6077 0.427050 0.213525 0.976938i \(-0.431506\pi\)
0.213525 + 0.976938i \(0.431506\pi\)
\(618\) −4.19615 −0.168794
\(619\) −7.60770 −0.305779 −0.152890 0.988243i \(-0.548858\pi\)
−0.152890 + 0.988243i \(0.548858\pi\)
\(620\) −4.39230 −0.176399
\(621\) −2.19615 −0.0881286
\(622\) −2.19615 −0.0880577
\(623\) 12.0000 0.480770
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 24.3923 0.974913
\(627\) −6.00000 −0.239617
\(628\) −3.19615 −0.127540
\(629\) −15.5885 −0.621552
\(630\) 8.19615 0.326543
\(631\) 25.8564 1.02933 0.514664 0.857392i \(-0.327917\pi\)
0.514664 + 0.857392i \(0.327917\pi\)
\(632\) 12.3923 0.492939
\(633\) 3.60770 0.143393
\(634\) −6.12436 −0.243229
\(635\) 6.92820 0.274937
\(636\) −3.00000 −0.118958
\(637\) 0 0
\(638\) 14.1962 0.562031
\(639\) 8.19615 0.324235
\(640\) 1.73205 0.0684653
\(641\) −30.8038 −1.21668 −0.608339 0.793677i \(-0.708164\pi\)
−0.608339 + 0.793677i \(0.708164\pi\)
\(642\) 8.19615 0.323476
\(643\) −27.7128 −1.09289 −0.546443 0.837496i \(-0.684019\pi\)
−0.546443 + 0.837496i \(0.684019\pi\)
\(644\) 10.3923 0.409514
\(645\) 10.7321 0.422574
\(646\) 6.58846 0.259219
\(647\) 13.1769 0.518038 0.259019 0.965872i \(-0.416601\pi\)
0.259019 + 0.965872i \(0.416601\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 65.5692 2.57382
\(650\) 0 0
\(651\) −12.0000 −0.470317
\(652\) 9.46410 0.370643
\(653\) −49.1769 −1.92444 −0.962221 0.272271i \(-0.912225\pi\)
−0.962221 + 0.272271i \(0.912225\pi\)
\(654\) 16.3923 0.640990
\(655\) −28.3923 −1.10938
\(656\) −0.464102 −0.0181201
\(657\) −12.1244 −0.473016
\(658\) 6.00000 0.233904
\(659\) 25.1769 0.980753 0.490377 0.871511i \(-0.336859\pi\)
0.490377 + 0.871511i \(0.336859\pi\)
\(660\) 8.19615 0.319035
\(661\) 9.00000 0.350059 0.175030 0.984563i \(-0.443998\pi\)
0.175030 + 0.984563i \(0.443998\pi\)
\(662\) −12.0000 −0.466393
\(663\) 0 0
\(664\) 11.6603 0.452506
\(665\) −10.3923 −0.402996
\(666\) −3.00000 −0.116248
\(667\) −6.58846 −0.255106
\(668\) −2.53590 −0.0981169
\(669\) 18.9282 0.731807
\(670\) 18.5885 0.718135
\(671\) 22.7321 0.877561
\(672\) 4.73205 0.182543
\(673\) 1.00000 0.0385472 0.0192736 0.999814i \(-0.493865\pi\)
0.0192736 + 0.999814i \(0.493865\pi\)
\(674\) 31.0000 1.19408
\(675\) 2.00000 0.0769800
\(676\) 0 0
\(677\) 4.39230 0.168810 0.0844050 0.996432i \(-0.473101\pi\)
0.0844050 + 0.996432i \(0.473101\pi\)
\(678\) 11.1962 0.429986
\(679\) 28.3923 1.08960
\(680\) −9.00000 −0.345134
\(681\) 9.80385 0.375684
\(682\) −12.0000 −0.459504
\(683\) 27.7128 1.06040 0.530201 0.847872i \(-0.322117\pi\)
0.530201 + 0.847872i \(0.322117\pi\)
\(684\) 1.26795 0.0484812
\(685\) 15.5885 0.595604
\(686\) −39.7128 −1.51624
\(687\) −19.8564 −0.757569
\(688\) 6.19615 0.236226
\(689\) 0 0
\(690\) −3.80385 −0.144810
\(691\) −19.5167 −0.742449 −0.371224 0.928543i \(-0.621062\pi\)
−0.371224 + 0.928543i \(0.621062\pi\)
\(692\) 16.3923 0.623142
\(693\) 22.3923 0.850613
\(694\) 12.5885 0.477851
\(695\) 6.92820 0.262802
\(696\) −3.00000 −0.113715
\(697\) 2.41154 0.0913437
\(698\) −2.53590 −0.0959852
\(699\) 18.0000 0.680823
\(700\) −9.46410 −0.357709
\(701\) 4.39230 0.165895 0.0829475 0.996554i \(-0.473567\pi\)
0.0829475 + 0.996554i \(0.473567\pi\)
\(702\) 0 0
\(703\) 3.80385 0.143465
\(704\) 4.73205 0.178346
\(705\) −2.19615 −0.0827119
\(706\) −5.78461 −0.217707
\(707\) −6.58846 −0.247784
\(708\) −13.8564 −0.520756
\(709\) 3.24871 0.122008 0.0610040 0.998138i \(-0.480570\pi\)
0.0610040 + 0.998138i \(0.480570\pi\)
\(710\) 14.1962 0.532772
\(711\) −12.3923 −0.464748
\(712\) −2.53590 −0.0950368
\(713\) 5.56922 0.208569
\(714\) −24.5885 −0.920200
\(715\) 0 0
\(716\) −8.19615 −0.306305
\(717\) 24.5885 0.918273
\(718\) −22.0526 −0.822994
\(719\) 52.3923 1.95390 0.976952 0.213461i \(-0.0684736\pi\)
0.976952 + 0.213461i \(0.0684736\pi\)
\(720\) −1.73205 −0.0645497
\(721\) −19.8564 −0.739491
\(722\) 17.3923 0.647275
\(723\) −0.803848 −0.0298954
\(724\) −11.5885 −0.430682
\(725\) 6.00000 0.222834
\(726\) 11.3923 0.422808
\(727\) −24.1962 −0.897386 −0.448693 0.893686i \(-0.648110\pi\)
−0.448693 + 0.893686i \(0.648110\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −21.0000 −0.777245
\(731\) −32.1962 −1.19082
\(732\) −4.80385 −0.177555
\(733\) −14.3205 −0.528940 −0.264470 0.964394i \(-0.585197\pi\)
−0.264470 + 0.964394i \(0.585197\pi\)
\(734\) 24.1962 0.893096
\(735\) 26.6603 0.983378
\(736\) −2.19615 −0.0809513
\(737\) 50.7846 1.87068
\(738\) 0.464102 0.0170838
\(739\) −18.9282 −0.696285 −0.348143 0.937442i \(-0.613188\pi\)
−0.348143 + 0.937442i \(0.613188\pi\)
\(740\) −5.19615 −0.191014
\(741\) 0 0
\(742\) −14.1962 −0.521157
\(743\) 4.39230 0.161138 0.0805690 0.996749i \(-0.474326\pi\)
0.0805690 + 0.996749i \(0.474326\pi\)
\(744\) 2.53590 0.0929705
\(745\) 31.3923 1.15013
\(746\) −23.9808 −0.877998
\(747\) −11.6603 −0.426626
\(748\) −24.5885 −0.899043
\(749\) 38.7846 1.41716
\(750\) 12.1244 0.442719
\(751\) −24.9808 −0.911561 −0.455780 0.890092i \(-0.650640\pi\)
−0.455780 + 0.890092i \(0.650640\pi\)
\(752\) −1.26795 −0.0462373
\(753\) 4.39230 0.160064
\(754\) 0 0
\(755\) −12.5885 −0.458141
\(756\) −4.73205 −0.172103
\(757\) 18.7846 0.682738 0.341369 0.939929i \(-0.389109\pi\)
0.341369 + 0.939929i \(0.389109\pi\)
\(758\) 18.2487 0.662823
\(759\) −10.3923 −0.377217
\(760\) 2.19615 0.0796628
\(761\) −4.39230 −0.159221 −0.0796105 0.996826i \(-0.525368\pi\)
−0.0796105 + 0.996826i \(0.525368\pi\)
\(762\) −4.00000 −0.144905
\(763\) 77.5692 2.80819
\(764\) 20.7846 0.751961
\(765\) 9.00000 0.325396
\(766\) −11.3205 −0.409027
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 33.7128 1.21572 0.607858 0.794046i \(-0.292029\pi\)
0.607858 + 0.794046i \(0.292029\pi\)
\(770\) 38.7846 1.39770
\(771\) −12.8038 −0.461119
\(772\) −12.8038 −0.460821
\(773\) −50.7846 −1.82660 −0.913298 0.407293i \(-0.866473\pi\)
−0.913298 + 0.407293i \(0.866473\pi\)
\(774\) −6.19615 −0.222716
\(775\) −5.07180 −0.182184
\(776\) −6.00000 −0.215387
\(777\) −14.1962 −0.509284
\(778\) 13.3923 0.480137
\(779\) −0.588457 −0.0210837
\(780\) 0 0
\(781\) 38.7846 1.38782
\(782\) 11.4115 0.408076
\(783\) 3.00000 0.107211
\(784\) 15.3923 0.549725
\(785\) 5.53590 0.197585
\(786\) 16.3923 0.584694
\(787\) −14.5359 −0.518149 −0.259074 0.965857i \(-0.583417\pi\)
−0.259074 + 0.965857i \(0.583417\pi\)
\(788\) 6.92820 0.246807
\(789\) 2.19615 0.0781851
\(790\) −21.4641 −0.763658
\(791\) 52.9808 1.88378
\(792\) −4.73205 −0.168146
\(793\) 0 0
\(794\) 16.3923 0.581741
\(795\) 5.19615 0.184289
\(796\) 8.58846 0.304410
\(797\) 6.00000 0.212531 0.106265 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) 6.00000 0.212398
\(799\) 6.58846 0.233083
\(800\) 2.00000 0.0707107
\(801\) 2.53590 0.0896016
\(802\) −21.0000 −0.741536
\(803\) −57.3731 −2.02465
\(804\) −10.7321 −0.378490
\(805\) −18.0000 −0.634417
\(806\) 0 0
\(807\) 28.3923 0.999456
\(808\) 1.39230 0.0489811
\(809\) −47.1962 −1.65933 −0.829664 0.558263i \(-0.811468\pi\)
−0.829664 + 0.558263i \(0.811468\pi\)
\(810\) 1.73205 0.0608581
\(811\) 4.39230 0.154235 0.0771173 0.997022i \(-0.475428\pi\)
0.0771173 + 0.997022i \(0.475428\pi\)
\(812\) −14.1962 −0.498187
\(813\) 0 0
\(814\) −14.1962 −0.497575
\(815\) −16.3923 −0.574197
\(816\) 5.19615 0.181902
\(817\) 7.85641 0.274861
\(818\) 3.33975 0.116771
\(819\) 0 0
\(820\) 0.803848 0.0280716
\(821\) −40.6410 −1.41838 −0.709191 0.705017i \(-0.750939\pi\)
−0.709191 + 0.705017i \(0.750939\pi\)
\(822\) −9.00000 −0.313911
\(823\) −32.0000 −1.11545 −0.557725 0.830026i \(-0.688326\pi\)
−0.557725 + 0.830026i \(0.688326\pi\)
\(824\) 4.19615 0.146180
\(825\) 9.46410 0.329498
\(826\) −65.5692 −2.28144
\(827\) −32.1051 −1.11640 −0.558202 0.829705i \(-0.688509\pi\)
−0.558202 + 0.829705i \(0.688509\pi\)
\(828\) 2.19615 0.0763216
\(829\) −11.9808 −0.416109 −0.208055 0.978117i \(-0.566713\pi\)
−0.208055 + 0.978117i \(0.566713\pi\)
\(830\) −20.1962 −0.701019
\(831\) −15.1962 −0.527149
\(832\) 0 0
\(833\) −79.9808 −2.77117
\(834\) −4.00000 −0.138509
\(835\) 4.39230 0.152002
\(836\) 6.00000 0.207514
\(837\) −2.53590 −0.0876535
\(838\) −16.3923 −0.566263
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) −8.19615 −0.282794
\(841\) −20.0000 −0.689655
\(842\) −0.464102 −0.0159940
\(843\) −24.4641 −0.842588
\(844\) −3.60770 −0.124182
\(845\) 0 0
\(846\) 1.26795 0.0435930
\(847\) 53.9090 1.85233
\(848\) 3.00000 0.103020
\(849\) 30.1962 1.03633
\(850\) −10.3923 −0.356453
\(851\) 6.58846 0.225849
\(852\) −8.19615 −0.280796
\(853\) 9.00000 0.308154 0.154077 0.988059i \(-0.450760\pi\)
0.154077 + 0.988059i \(0.450760\pi\)
\(854\) −22.7321 −0.777875
\(855\) −2.19615 −0.0751068
\(856\) −8.19615 −0.280139
\(857\) −54.3731 −1.85735 −0.928674 0.370896i \(-0.879051\pi\)
−0.928674 + 0.370896i \(0.879051\pi\)
\(858\) 0 0
\(859\) 10.5885 0.361273 0.180637 0.983550i \(-0.442184\pi\)
0.180637 + 0.983550i \(0.442184\pi\)
\(860\) −10.7321 −0.365960
\(861\) 2.19615 0.0748447
\(862\) −27.8038 −0.947003
\(863\) 4.48334 0.152615 0.0763073 0.997084i \(-0.475687\pi\)
0.0763073 + 0.997084i \(0.475687\pi\)
\(864\) 1.00000 0.0340207
\(865\) −28.3923 −0.965367
\(866\) 33.7846 1.14805
\(867\) −10.0000 −0.339618
\(868\) 12.0000 0.407307
\(869\) −58.6410 −1.98926
\(870\) 5.19615 0.176166
\(871\) 0 0
\(872\) −16.3923 −0.555113
\(873\) 6.00000 0.203069
\(874\) −2.78461 −0.0941908
\(875\) 57.3731 1.93956
\(876\) 12.1244 0.409644
\(877\) 43.3923 1.46525 0.732627 0.680630i \(-0.238294\pi\)
0.732627 + 0.680630i \(0.238294\pi\)
\(878\) −16.5885 −0.559833
\(879\) 14.6603 0.494478
\(880\) −8.19615 −0.276292
\(881\) 37.9808 1.27960 0.639802 0.768540i \(-0.279016\pi\)
0.639802 + 0.768540i \(0.279016\pi\)
\(882\) −15.3923 −0.518286
\(883\) −24.7846 −0.834069 −0.417034 0.908891i \(-0.636930\pi\)
−0.417034 + 0.908891i \(0.636930\pi\)
\(884\) 0 0
\(885\) 24.0000 0.806751
\(886\) 4.39230 0.147562
\(887\) 48.0000 1.61168 0.805841 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) 3.00000 0.100673
\(889\) −18.9282 −0.634832
\(890\) 4.39230 0.147230
\(891\) 4.73205 0.158530
\(892\) −18.9282 −0.633763
\(893\) −1.60770 −0.0537995
\(894\) −18.1244 −0.606169
\(895\) 14.1962 0.474525
\(896\) −4.73205 −0.158087
\(897\) 0 0
\(898\) −33.4641 −1.11671
\(899\) −7.60770 −0.253731
\(900\) −2.00000 −0.0666667
\(901\) −15.5885 −0.519327
\(902\) 2.19615 0.0731239
\(903\) −29.3205 −0.975725
\(904\) −11.1962 −0.372378
\(905\) 20.0718 0.667209
\(906\) 7.26795 0.241461
\(907\) −41.1769 −1.36726 −0.683629 0.729830i \(-0.739599\pi\)
−0.683629 + 0.729830i \(0.739599\pi\)
\(908\) −9.80385 −0.325352
\(909\) −1.39230 −0.0461798
\(910\) 0 0
\(911\) 37.1769 1.23173 0.615863 0.787853i \(-0.288807\pi\)
0.615863 + 0.787853i \(0.288807\pi\)
\(912\) −1.26795 −0.0419860
\(913\) −55.1769 −1.82609
\(914\) −19.9808 −0.660905
\(915\) 8.32051 0.275068
\(916\) 19.8564 0.656074
\(917\) 77.5692 2.56156
\(918\) −5.19615 −0.171499
\(919\) 32.3923 1.06852 0.534262 0.845319i \(-0.320590\pi\)
0.534262 + 0.845319i \(0.320590\pi\)
\(920\) 3.80385 0.125409
\(921\) −10.7321 −0.353633
\(922\) 19.9808 0.658031
\(923\) 0 0
\(924\) −22.3923 −0.736653
\(925\) −6.00000 −0.197279
\(926\) 26.1962 0.860859
\(927\) −4.19615 −0.137820
\(928\) 3.00000 0.0984798
\(929\) −34.6077 −1.13544 −0.567721 0.823221i \(-0.692175\pi\)
−0.567721 + 0.823221i \(0.692175\pi\)
\(930\) −4.39230 −0.144029
\(931\) 19.5167 0.639633
\(932\) −18.0000 −0.589610
\(933\) −2.19615 −0.0718988
\(934\) −36.5885 −1.19721
\(935\) 42.5885 1.39279
\(936\) 0 0
\(937\) 5.39230 0.176159 0.0880795 0.996113i \(-0.471927\pi\)
0.0880795 + 0.996113i \(0.471927\pi\)
\(938\) −50.7846 −1.65818
\(939\) 24.3923 0.796013
\(940\) 2.19615 0.0716306
\(941\) 2.78461 0.0907757 0.0453878 0.998969i \(-0.485548\pi\)
0.0453878 + 0.998969i \(0.485548\pi\)
\(942\) −3.19615 −0.104136
\(943\) −1.01924 −0.0331910
\(944\) 13.8564 0.450988
\(945\) 8.19615 0.266621
\(946\) −29.3205 −0.953292
\(947\) −42.9282 −1.39498 −0.697490 0.716595i \(-0.745700\pi\)
−0.697490 + 0.716595i \(0.745700\pi\)
\(948\) 12.3923 0.402483
\(949\) 0 0
\(950\) 2.53590 0.0822754
\(951\) −6.12436 −0.198596
\(952\) 24.5885 0.796916
\(953\) −24.0000 −0.777436 −0.388718 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(954\) −3.00000 −0.0971286
\(955\) −36.0000 −1.16493
\(956\) −24.5885 −0.795248
\(957\) 14.1962 0.458896
\(958\) 35.3205 1.14115
\(959\) −42.5885 −1.37525
\(960\) 1.73205 0.0559017
\(961\) −24.5692 −0.792555
\(962\) 0 0
\(963\) 8.19615 0.264117
\(964\) 0.803848 0.0258902
\(965\) 22.1769 0.713900
\(966\) 10.3923 0.334367
\(967\) −14.8756 −0.478368 −0.239184 0.970974i \(-0.576880\pi\)
−0.239184 + 0.970974i \(0.576880\pi\)
\(968\) −11.3923 −0.366163
\(969\) 6.58846 0.211652
\(970\) 10.3923 0.333677
\(971\) −13.1769 −0.422867 −0.211434 0.977392i \(-0.567813\pi\)
−0.211434 + 0.977392i \(0.567813\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −18.9282 −0.606810
\(974\) −9.12436 −0.292363
\(975\) 0 0
\(976\) 4.80385 0.153767
\(977\) −16.8564 −0.539284 −0.269642 0.962961i \(-0.586905\pi\)
−0.269642 + 0.962961i \(0.586905\pi\)
\(978\) 9.46410 0.302629
\(979\) 12.0000 0.383522
\(980\) −26.6603 −0.851631
\(981\) 16.3923 0.523366
\(982\) −0.588457 −0.0187784
\(983\) −20.7846 −0.662926 −0.331463 0.943468i \(-0.607542\pi\)
−0.331463 + 0.943468i \(0.607542\pi\)
\(984\) −0.464102 −0.0147950
\(985\) −12.0000 −0.382352
\(986\) −15.5885 −0.496438
\(987\) 6.00000 0.190982
\(988\) 0 0
\(989\) 13.6077 0.432700
\(990\) 8.19615 0.260491
\(991\) 29.3731 0.933066 0.466533 0.884504i \(-0.345503\pi\)
0.466533 + 0.884504i \(0.345503\pi\)
\(992\) −2.53590 −0.0805149
\(993\) −12.0000 −0.380808
\(994\) −38.7846 −1.23017
\(995\) −14.8756 −0.471590
\(996\) 11.6603 0.369469
\(997\) 13.1962 0.417926 0.208963 0.977924i \(-0.432991\pi\)
0.208963 + 0.977924i \(0.432991\pi\)
\(998\) 0 0
\(999\) −3.00000 −0.0949158
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.h.1.1 2
3.2 odd 2 3042.2.a.v.1.2 2
4.3 odd 2 8112.2.a.bq.1.1 2
13.2 odd 12 1014.2.i.f.823.1 4
13.3 even 3 1014.2.e.j.529.1 4
13.4 even 6 1014.2.e.h.991.2 4
13.5 odd 4 1014.2.b.d.337.4 4
13.6 odd 12 78.2.i.b.49.2 yes 4
13.7 odd 12 1014.2.i.f.361.1 4
13.8 odd 4 1014.2.b.d.337.1 4
13.9 even 3 1014.2.e.j.991.1 4
13.10 even 6 1014.2.e.h.529.2 4
13.11 odd 12 78.2.i.b.43.2 4
13.12 even 2 1014.2.a.j.1.2 2
39.5 even 4 3042.2.b.l.1351.1 4
39.8 even 4 3042.2.b.l.1351.4 4
39.11 even 12 234.2.l.a.199.1 4
39.32 even 12 234.2.l.a.127.1 4
39.38 odd 2 3042.2.a.s.1.1 2
52.11 even 12 624.2.bv.d.433.2 4
52.19 even 12 624.2.bv.d.49.2 4
52.51 odd 2 8112.2.a.bx.1.2 2
65.19 odd 12 1950.2.bc.c.751.1 4
65.24 odd 12 1950.2.bc.c.901.1 4
65.32 even 12 1950.2.y.h.49.2 4
65.37 even 12 1950.2.y.a.199.1 4
65.58 even 12 1950.2.y.a.49.1 4
65.63 even 12 1950.2.y.h.199.2 4
156.11 odd 12 1872.2.by.k.433.2 4
156.71 odd 12 1872.2.by.k.1297.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.i.b.43.2 4 13.11 odd 12
78.2.i.b.49.2 yes 4 13.6 odd 12
234.2.l.a.127.1 4 39.32 even 12
234.2.l.a.199.1 4 39.11 even 12
624.2.bv.d.49.2 4 52.19 even 12
624.2.bv.d.433.2 4 52.11 even 12
1014.2.a.h.1.1 2 1.1 even 1 trivial
1014.2.a.j.1.2 2 13.12 even 2
1014.2.b.d.337.1 4 13.8 odd 4
1014.2.b.d.337.4 4 13.5 odd 4
1014.2.e.h.529.2 4 13.10 even 6
1014.2.e.h.991.2 4 13.4 even 6
1014.2.e.j.529.1 4 13.3 even 3
1014.2.e.j.991.1 4 13.9 even 3
1014.2.i.f.361.1 4 13.7 odd 12
1014.2.i.f.823.1 4 13.2 odd 12
1872.2.by.k.433.2 4 156.11 odd 12
1872.2.by.k.1297.2 4 156.71 odd 12
1950.2.y.a.49.1 4 65.58 even 12
1950.2.y.a.199.1 4 65.37 even 12
1950.2.y.h.49.2 4 65.32 even 12
1950.2.y.h.199.2 4 65.63 even 12
1950.2.bc.c.751.1 4 65.19 odd 12
1950.2.bc.c.901.1 4 65.24 odd 12
3042.2.a.s.1.1 2 39.38 odd 2
3042.2.a.v.1.2 2 3.2 odd 2
3042.2.b.l.1351.1 4 39.5 even 4
3042.2.b.l.1351.4 4 39.8 even 4
8112.2.a.bq.1.1 2 4.3 odd 2
8112.2.a.bx.1.2 2 52.51 odd 2