Properties

Label 414.2.a.g.1.2
Level $414$
Weight $2$
Character 414.1
Self dual yes
Analytic conductor $3.306$
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] = [414,2,Mod(1,414)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(414, 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("414.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 414.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: \(3.30580664368\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.64575\) of defining polynomial
Character \(\chi\) \(=\) 414.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +3.64575 q^{5} +2.00000 q^{7} +1.00000 q^{8} +3.64575 q^{10} -3.64575 q^{11} -5.29150 q^{13} +2.00000 q^{14} +1.00000 q^{16} -7.29150 q^{17} +5.64575 q^{19} +3.64575 q^{20} -3.64575 q^{22} -1.00000 q^{23} +8.29150 q^{25} -5.29150 q^{26} +2.00000 q^{28} +1.29150 q^{29} +9.29150 q^{31} +1.00000 q^{32} -7.29150 q^{34} +7.29150 q^{35} -8.93725 q^{37} +5.64575 q^{38} +3.64575 q^{40} -6.00000 q^{41} +5.64575 q^{43} -3.64575 q^{44} -1.00000 q^{46} -6.00000 q^{47} -3.00000 q^{49} +8.29150 q^{50} -5.29150 q^{52} -3.64575 q^{53} -13.2915 q^{55} +2.00000 q^{56} +1.29150 q^{58} +5.64575 q^{61} +9.29150 q^{62} +1.00000 q^{64} -19.2915 q^{65} +0.937254 q^{67} -7.29150 q^{68} +7.29150 q^{70} +6.00000 q^{71} +3.29150 q^{73} -8.93725 q^{74} +5.64575 q^{76} -7.29150 q^{77} -12.5830 q^{79} +3.64575 q^{80} -6.00000 q^{82} +8.35425 q^{83} -26.5830 q^{85} +5.64575 q^{86} -3.64575 q^{88} -10.5830 q^{91} -1.00000 q^{92} -6.00000 q^{94} +20.5830 q^{95} +9.29150 q^{97} -3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{7} + 2 q^{8} + 2 q^{10} - 2 q^{11} + 4 q^{14} + 2 q^{16} - 4 q^{17} + 6 q^{19} + 2 q^{20} - 2 q^{22} - 2 q^{23} + 6 q^{25} + 4 q^{28} - 8 q^{29} + 8 q^{31} + 2 q^{32}+ \cdots - 6 q^{98}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 3.64575 1.63043 0.815215 0.579159i \(-0.196619\pi\)
0.815215 + 0.579159i \(0.196619\pi\)
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 3.64575 1.15289
\(11\) −3.64575 −1.09924 −0.549618 0.835416i \(-0.685227\pi\)
−0.549618 + 0.835416i \(0.685227\pi\)
\(12\) 0 0
\(13\) −5.29150 −1.46760 −0.733799 0.679366i \(-0.762255\pi\)
−0.733799 + 0.679366i \(0.762255\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −7.29150 −1.76845 −0.884225 0.467062i \(-0.845312\pi\)
−0.884225 + 0.467062i \(0.845312\pi\)
\(18\) 0 0
\(19\) 5.64575 1.29522 0.647612 0.761970i \(-0.275768\pi\)
0.647612 + 0.761970i \(0.275768\pi\)
\(20\) 3.64575 0.815215
\(21\) 0 0
\(22\) −3.64575 −0.777277
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 8.29150 1.65830
\(26\) −5.29150 −1.03775
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) 1.29150 0.239826 0.119913 0.992784i \(-0.461738\pi\)
0.119913 + 0.992784i \(0.461738\pi\)
\(30\) 0 0
\(31\) 9.29150 1.66880 0.834402 0.551157i \(-0.185813\pi\)
0.834402 + 0.551157i \(0.185813\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −7.29150 −1.25048
\(35\) 7.29150 1.23249
\(36\) 0 0
\(37\) −8.93725 −1.46928 −0.734638 0.678460i \(-0.762648\pi\)
−0.734638 + 0.678460i \(0.762648\pi\)
\(38\) 5.64575 0.915862
\(39\) 0 0
\(40\) 3.64575 0.576444
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 5.64575 0.860969 0.430485 0.902598i \(-0.358343\pi\)
0.430485 + 0.902598i \(0.358343\pi\)
\(44\) −3.64575 −0.549618
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 8.29150 1.17260
\(51\) 0 0
\(52\) −5.29150 −0.733799
\(53\) −3.64575 −0.500782 −0.250391 0.968145i \(-0.580559\pi\)
−0.250391 + 0.968145i \(0.580559\pi\)
\(54\) 0 0
\(55\) −13.2915 −1.79223
\(56\) 2.00000 0.267261
\(57\) 0 0
\(58\) 1.29150 0.169583
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 5.64575 0.722864 0.361432 0.932398i \(-0.382288\pi\)
0.361432 + 0.932398i \(0.382288\pi\)
\(62\) 9.29150 1.18002
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −19.2915 −2.39282
\(66\) 0 0
\(67\) 0.937254 0.114504 0.0572519 0.998360i \(-0.481766\pi\)
0.0572519 + 0.998360i \(0.481766\pi\)
\(68\) −7.29150 −0.884225
\(69\) 0 0
\(70\) 7.29150 0.871501
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) 3.29150 0.385241 0.192621 0.981273i \(-0.438301\pi\)
0.192621 + 0.981273i \(0.438301\pi\)
\(74\) −8.93725 −1.03893
\(75\) 0 0
\(76\) 5.64575 0.647612
\(77\) −7.29150 −0.830944
\(78\) 0 0
\(79\) −12.5830 −1.41570 −0.707849 0.706363i \(-0.750334\pi\)
−0.707849 + 0.706363i \(0.750334\pi\)
\(80\) 3.64575 0.407607
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) 8.35425 0.916998 0.458499 0.888695i \(-0.348387\pi\)
0.458499 + 0.888695i \(0.348387\pi\)
\(84\) 0 0
\(85\) −26.5830 −2.88333
\(86\) 5.64575 0.608797
\(87\) 0 0
\(88\) −3.64575 −0.388638
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −10.5830 −1.10940
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) 20.5830 2.11177
\(96\) 0 0
\(97\) 9.29150 0.943409 0.471705 0.881757i \(-0.343639\pi\)
0.471705 + 0.881757i \(0.343639\pi\)
\(98\) −3.00000 −0.303046
\(99\) 0 0
\(100\) 8.29150 0.829150
\(101\) 1.29150 0.128509 0.0642547 0.997934i \(-0.479533\pi\)
0.0642547 + 0.997934i \(0.479533\pi\)
\(102\) 0 0
\(103\) −17.2915 −1.70378 −0.851891 0.523719i \(-0.824544\pi\)
−0.851891 + 0.523719i \(0.824544\pi\)
\(104\) −5.29150 −0.518875
\(105\) 0 0
\(106\) −3.64575 −0.354107
\(107\) 8.35425 0.807636 0.403818 0.914839i \(-0.367683\pi\)
0.403818 + 0.914839i \(0.367683\pi\)
\(108\) 0 0
\(109\) 8.22876 0.788172 0.394086 0.919074i \(-0.371061\pi\)
0.394086 + 0.919074i \(0.371061\pi\)
\(110\) −13.2915 −1.26730
\(111\) 0 0
\(112\) 2.00000 0.188982
\(113\) 14.5830 1.37185 0.685927 0.727670i \(-0.259397\pi\)
0.685927 + 0.727670i \(0.259397\pi\)
\(114\) 0 0
\(115\) −3.64575 −0.339968
\(116\) 1.29150 0.119913
\(117\) 0 0
\(118\) 0 0
\(119\) −14.5830 −1.33682
\(120\) 0 0
\(121\) 2.29150 0.208318
\(122\) 5.64575 0.511142
\(123\) 0 0
\(124\) 9.29150 0.834402
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 10.5830 0.939090 0.469545 0.882909i \(-0.344418\pi\)
0.469545 + 0.882909i \(0.344418\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −19.2915 −1.69198
\(131\) −9.87451 −0.862740 −0.431370 0.902175i \(-0.641970\pi\)
−0.431370 + 0.902175i \(0.641970\pi\)
\(132\) 0 0
\(133\) 11.2915 0.979097
\(134\) 0.937254 0.0809664
\(135\) 0 0
\(136\) −7.29150 −0.625241
\(137\) 9.87451 0.843636 0.421818 0.906680i \(-0.361392\pi\)
0.421818 + 0.906680i \(0.361392\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 7.29150 0.616244
\(141\) 0 0
\(142\) 6.00000 0.503509
\(143\) 19.2915 1.61324
\(144\) 0 0
\(145\) 4.70850 0.391019
\(146\) 3.29150 0.272407
\(147\) 0 0
\(148\) −8.93725 −0.734638
\(149\) −15.6458 −1.28175 −0.640875 0.767645i \(-0.721428\pi\)
−0.640875 + 0.767645i \(0.721428\pi\)
\(150\) 0 0
\(151\) 6.70850 0.545930 0.272965 0.962024i \(-0.411996\pi\)
0.272965 + 0.962024i \(0.411996\pi\)
\(152\) 5.64575 0.457931
\(153\) 0 0
\(154\) −7.29150 −0.587566
\(155\) 33.8745 2.72087
\(156\) 0 0
\(157\) −1.64575 −0.131345 −0.0656726 0.997841i \(-0.520919\pi\)
−0.0656726 + 0.997841i \(0.520919\pi\)
\(158\) −12.5830 −1.00105
\(159\) 0 0
\(160\) 3.64575 0.288222
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) −11.2915 −0.884419 −0.442209 0.896912i \(-0.645805\pi\)
−0.442209 + 0.896912i \(0.645805\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 8.35425 0.648415
\(167\) −23.1660 −1.79264 −0.896320 0.443408i \(-0.853769\pi\)
−0.896320 + 0.443408i \(0.853769\pi\)
\(168\) 0 0
\(169\) 15.0000 1.15385
\(170\) −26.5830 −2.03882
\(171\) 0 0
\(172\) 5.64575 0.430485
\(173\) −15.8745 −1.20692 −0.603458 0.797395i \(-0.706211\pi\)
−0.603458 + 0.797395i \(0.706211\pi\)
\(174\) 0 0
\(175\) 16.5830 1.25356
\(176\) −3.64575 −0.274809
\(177\) 0 0
\(178\) 0 0
\(179\) −4.70850 −0.351930 −0.175965 0.984396i \(-0.556304\pi\)
−0.175965 + 0.984396i \(0.556304\pi\)
\(180\) 0 0
\(181\) 10.3542 0.769625 0.384813 0.922995i \(-0.374266\pi\)
0.384813 + 0.922995i \(0.374266\pi\)
\(182\) −10.5830 −0.784465
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) −32.5830 −2.39555
\(186\) 0 0
\(187\) 26.5830 1.94394
\(188\) −6.00000 −0.437595
\(189\) 0 0
\(190\) 20.5830 1.49325
\(191\) 7.29150 0.527595 0.263797 0.964578i \(-0.415025\pi\)
0.263797 + 0.964578i \(0.415025\pi\)
\(192\) 0 0
\(193\) −22.0000 −1.58359 −0.791797 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) 9.29150 0.667091
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 14.0000 0.992434 0.496217 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) 8.29150 0.586298
\(201\) 0 0
\(202\) 1.29150 0.0908698
\(203\) 2.58301 0.181291
\(204\) 0 0
\(205\) −21.8745 −1.52778
\(206\) −17.2915 −1.20476
\(207\) 0 0
\(208\) −5.29150 −0.366900
\(209\) −20.5830 −1.42376
\(210\) 0 0
\(211\) 0.708497 0.0487750 0.0243875 0.999703i \(-0.492236\pi\)
0.0243875 + 0.999703i \(0.492236\pi\)
\(212\) −3.64575 −0.250391
\(213\) 0 0
\(214\) 8.35425 0.571085
\(215\) 20.5830 1.40375
\(216\) 0 0
\(217\) 18.5830 1.26150
\(218\) 8.22876 0.557322
\(219\) 0 0
\(220\) −13.2915 −0.896113
\(221\) 38.5830 2.59537
\(222\) 0 0
\(223\) 11.8745 0.795176 0.397588 0.917564i \(-0.369847\pi\)
0.397588 + 0.917564i \(0.369847\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) 14.5830 0.970047
\(227\) 22.9373 1.52240 0.761200 0.648518i \(-0.224611\pi\)
0.761200 + 0.648518i \(0.224611\pi\)
\(228\) 0 0
\(229\) 24.9373 1.64790 0.823950 0.566662i \(-0.191766\pi\)
0.823950 + 0.566662i \(0.191766\pi\)
\(230\) −3.64575 −0.240394
\(231\) 0 0
\(232\) 1.29150 0.0847913
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) −21.8745 −1.42694
\(236\) 0 0
\(237\) 0 0
\(238\) −14.5830 −0.945276
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) −5.29150 −0.340856 −0.170428 0.985370i \(-0.554515\pi\)
−0.170428 + 0.985370i \(0.554515\pi\)
\(242\) 2.29150 0.147303
\(243\) 0 0
\(244\) 5.64575 0.361432
\(245\) −10.9373 −0.698756
\(246\) 0 0
\(247\) −29.8745 −1.90087
\(248\) 9.29150 0.590011
\(249\) 0 0
\(250\) 12.0000 0.758947
\(251\) 20.3542 1.28475 0.642374 0.766391i \(-0.277949\pi\)
0.642374 + 0.766391i \(0.277949\pi\)
\(252\) 0 0
\(253\) 3.64575 0.229206
\(254\) 10.5830 0.664037
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −3.41699 −0.213146 −0.106573 0.994305i \(-0.533988\pi\)
−0.106573 + 0.994305i \(0.533988\pi\)
\(258\) 0 0
\(259\) −17.8745 −1.11067
\(260\) −19.2915 −1.19641
\(261\) 0 0
\(262\) −9.87451 −0.610049
\(263\) 9.41699 0.580677 0.290338 0.956924i \(-0.406232\pi\)
0.290338 + 0.956924i \(0.406232\pi\)
\(264\) 0 0
\(265\) −13.2915 −0.816491
\(266\) 11.2915 0.692326
\(267\) 0 0
\(268\) 0.937254 0.0572519
\(269\) 32.5830 1.98662 0.993310 0.115474i \(-0.0368389\pi\)
0.993310 + 0.115474i \(0.0368389\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) −7.29150 −0.442112
\(273\) 0 0
\(274\) 9.87451 0.596541
\(275\) −30.2288 −1.82286
\(276\) 0 0
\(277\) −5.29150 −0.317936 −0.158968 0.987284i \(-0.550817\pi\)
−0.158968 + 0.987284i \(0.550817\pi\)
\(278\) −4.00000 −0.239904
\(279\) 0 0
\(280\) 7.29150 0.435751
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 15.5203 0.922584 0.461292 0.887248i \(-0.347386\pi\)
0.461292 + 0.887248i \(0.347386\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 19.2915 1.14073
\(287\) −12.0000 −0.708338
\(288\) 0 0
\(289\) 36.1660 2.12741
\(290\) 4.70850 0.276492
\(291\) 0 0
\(292\) 3.29150 0.192621
\(293\) 5.77124 0.337160 0.168580 0.985688i \(-0.446082\pi\)
0.168580 + 0.985688i \(0.446082\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −8.93725 −0.519467
\(297\) 0 0
\(298\) −15.6458 −0.906334
\(299\) 5.29150 0.306015
\(300\) 0 0
\(301\) 11.2915 0.650831
\(302\) 6.70850 0.386031
\(303\) 0 0
\(304\) 5.64575 0.323806
\(305\) 20.5830 1.17858
\(306\) 0 0
\(307\) −1.87451 −0.106984 −0.0534919 0.998568i \(-0.517035\pi\)
−0.0534919 + 0.998568i \(0.517035\pi\)
\(308\) −7.29150 −0.415472
\(309\) 0 0
\(310\) 33.8745 1.92394
\(311\) 29.1660 1.65385 0.826926 0.562310i \(-0.190087\pi\)
0.826926 + 0.562310i \(0.190087\pi\)
\(312\) 0 0
\(313\) −5.29150 −0.299093 −0.149547 0.988755i \(-0.547781\pi\)
−0.149547 + 0.988755i \(0.547781\pi\)
\(314\) −1.64575 −0.0928751
\(315\) 0 0
\(316\) −12.5830 −0.707849
\(317\) −10.7085 −0.601449 −0.300725 0.953711i \(-0.597228\pi\)
−0.300725 + 0.953711i \(0.597228\pi\)
\(318\) 0 0
\(319\) −4.70850 −0.263625
\(320\) 3.64575 0.203804
\(321\) 0 0
\(322\) −2.00000 −0.111456
\(323\) −41.1660 −2.29054
\(324\) 0 0
\(325\) −43.8745 −2.43372
\(326\) −11.2915 −0.625378
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) 5.87451 0.322892 0.161446 0.986882i \(-0.448384\pi\)
0.161446 + 0.986882i \(0.448384\pi\)
\(332\) 8.35425 0.458499
\(333\) 0 0
\(334\) −23.1660 −1.26759
\(335\) 3.41699 0.186690
\(336\) 0 0
\(337\) −27.1660 −1.47983 −0.739913 0.672702i \(-0.765134\pi\)
−0.739913 + 0.672702i \(0.765134\pi\)
\(338\) 15.0000 0.815892
\(339\) 0 0
\(340\) −26.5830 −1.44167
\(341\) −33.8745 −1.83441
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 5.64575 0.304399
\(345\) 0 0
\(346\) −15.8745 −0.853419
\(347\) −4.70850 −0.252765 −0.126383 0.991982i \(-0.540337\pi\)
−0.126383 + 0.991982i \(0.540337\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 16.5830 0.886399
\(351\) 0 0
\(352\) −3.64575 −0.194319
\(353\) −14.5830 −0.776175 −0.388088 0.921623i \(-0.626864\pi\)
−0.388088 + 0.921623i \(0.626864\pi\)
\(354\) 0 0
\(355\) 21.8745 1.16098
\(356\) 0 0
\(357\) 0 0
\(358\) −4.70850 −0.248852
\(359\) −9.87451 −0.521157 −0.260578 0.965453i \(-0.583913\pi\)
−0.260578 + 0.965453i \(0.583913\pi\)
\(360\) 0 0
\(361\) 12.8745 0.677606
\(362\) 10.3542 0.544207
\(363\) 0 0
\(364\) −10.5830 −0.554700
\(365\) 12.0000 0.628109
\(366\) 0 0
\(367\) 31.1660 1.62685 0.813426 0.581668i \(-0.197600\pi\)
0.813426 + 0.581668i \(0.197600\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) −32.5830 −1.69391
\(371\) −7.29150 −0.378556
\(372\) 0 0
\(373\) −11.0627 −0.572807 −0.286404 0.958109i \(-0.592460\pi\)
−0.286404 + 0.958109i \(0.592460\pi\)
\(374\) 26.5830 1.37457
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) −6.83399 −0.351968
\(378\) 0 0
\(379\) 3.52026 0.180824 0.0904118 0.995904i \(-0.471182\pi\)
0.0904118 + 0.995904i \(0.471182\pi\)
\(380\) 20.5830 1.05589
\(381\) 0 0
\(382\) 7.29150 0.373066
\(383\) 31.7490 1.62230 0.811149 0.584839i \(-0.198842\pi\)
0.811149 + 0.584839i \(0.198842\pi\)
\(384\) 0 0
\(385\) −26.5830 −1.35480
\(386\) −22.0000 −1.11977
\(387\) 0 0
\(388\) 9.29150 0.471705
\(389\) 22.9373 1.16296 0.581482 0.813559i \(-0.302473\pi\)
0.581482 + 0.813559i \(0.302473\pi\)
\(390\) 0 0
\(391\) 7.29150 0.368747
\(392\) −3.00000 −0.151523
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) −45.8745 −2.30820
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 14.0000 0.701757
\(399\) 0 0
\(400\) 8.29150 0.414575
\(401\) 21.8745 1.09236 0.546180 0.837668i \(-0.316081\pi\)
0.546180 + 0.837668i \(0.316081\pi\)
\(402\) 0 0
\(403\) −49.1660 −2.44913
\(404\) 1.29150 0.0642547
\(405\) 0 0
\(406\) 2.58301 0.128192
\(407\) 32.5830 1.61508
\(408\) 0 0
\(409\) −27.1660 −1.34327 −0.671636 0.740881i \(-0.734408\pi\)
−0.671636 + 0.740881i \(0.734408\pi\)
\(410\) −21.8745 −1.08030
\(411\) 0 0
\(412\) −17.2915 −0.851891
\(413\) 0 0
\(414\) 0 0
\(415\) 30.4575 1.49510
\(416\) −5.29150 −0.259437
\(417\) 0 0
\(418\) −20.5830 −1.00675
\(419\) −1.06275 −0.0519185 −0.0259593 0.999663i \(-0.508264\pi\)
−0.0259593 + 0.999663i \(0.508264\pi\)
\(420\) 0 0
\(421\) −4.22876 −0.206097 −0.103048 0.994676i \(-0.532860\pi\)
−0.103048 + 0.994676i \(0.532860\pi\)
\(422\) 0.708497 0.0344891
\(423\) 0 0
\(424\) −3.64575 −0.177053
\(425\) −60.4575 −2.93262
\(426\) 0 0
\(427\) 11.2915 0.546434
\(428\) 8.35425 0.403818
\(429\) 0 0
\(430\) 20.5830 0.992601
\(431\) 2.12549 0.102381 0.0511907 0.998689i \(-0.483698\pi\)
0.0511907 + 0.998689i \(0.483698\pi\)
\(432\) 0 0
\(433\) −29.2915 −1.40766 −0.703830 0.710369i \(-0.748528\pi\)
−0.703830 + 0.710369i \(0.748528\pi\)
\(434\) 18.5830 0.892013
\(435\) 0 0
\(436\) 8.22876 0.394086
\(437\) −5.64575 −0.270073
\(438\) 0 0
\(439\) −35.7490 −1.70621 −0.853104 0.521741i \(-0.825283\pi\)
−0.853104 + 0.521741i \(0.825283\pi\)
\(440\) −13.2915 −0.633648
\(441\) 0 0
\(442\) 38.5830 1.83521
\(443\) 17.1660 0.815582 0.407791 0.913075i \(-0.366299\pi\)
0.407791 + 0.913075i \(0.366299\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 11.8745 0.562274
\(447\) 0 0
\(448\) 2.00000 0.0944911
\(449\) −32.5830 −1.53769 −0.768844 0.639437i \(-0.779168\pi\)
−0.768844 + 0.639437i \(0.779168\pi\)
\(450\) 0 0
\(451\) 21.8745 1.03003
\(452\) 14.5830 0.685927
\(453\) 0 0
\(454\) 22.9373 1.07650
\(455\) −38.5830 −1.80880
\(456\) 0 0
\(457\) −31.8745 −1.49103 −0.745513 0.666491i \(-0.767796\pi\)
−0.745513 + 0.666491i \(0.767796\pi\)
\(458\) 24.9373 1.16524
\(459\) 0 0
\(460\) −3.64575 −0.169984
\(461\) 37.7490 1.75815 0.879073 0.476686i \(-0.158162\pi\)
0.879073 + 0.476686i \(0.158162\pi\)
\(462\) 0 0
\(463\) −21.1660 −0.983668 −0.491834 0.870689i \(-0.663673\pi\)
−0.491834 + 0.870689i \(0.663673\pi\)
\(464\) 1.29150 0.0599565
\(465\) 0 0
\(466\) 0 0
\(467\) −8.35425 −0.386589 −0.193294 0.981141i \(-0.561917\pi\)
−0.193294 + 0.981141i \(0.561917\pi\)
\(468\) 0 0
\(469\) 1.87451 0.0865567
\(470\) −21.8745 −1.00900
\(471\) 0 0
\(472\) 0 0
\(473\) −20.5830 −0.946408
\(474\) 0 0
\(475\) 46.8118 2.14787
\(476\) −14.5830 −0.668411
\(477\) 0 0
\(478\) −24.0000 −1.09773
\(479\) −21.4170 −0.978567 −0.489284 0.872125i \(-0.662742\pi\)
−0.489284 + 0.872125i \(0.662742\pi\)
\(480\) 0 0
\(481\) 47.2915 2.15631
\(482\) −5.29150 −0.241021
\(483\) 0 0
\(484\) 2.29150 0.104159
\(485\) 33.8745 1.53816
\(486\) 0 0
\(487\) −21.1660 −0.959123 −0.479562 0.877508i \(-0.659204\pi\)
−0.479562 + 0.877508i \(0.659204\pi\)
\(488\) 5.64575 0.255571
\(489\) 0 0
\(490\) −10.9373 −0.494095
\(491\) −9.87451 −0.445630 −0.222815 0.974861i \(-0.571525\pi\)
−0.222815 + 0.974861i \(0.571525\pi\)
\(492\) 0 0
\(493\) −9.41699 −0.424120
\(494\) −29.8745 −1.34412
\(495\) 0 0
\(496\) 9.29150 0.417201
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) −43.0405 −1.92676 −0.963379 0.268143i \(-0.913590\pi\)
−0.963379 + 0.268143i \(0.913590\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) 20.3542 0.908455
\(503\) 33.8745 1.51039 0.755195 0.655500i \(-0.227542\pi\)
0.755195 + 0.655500i \(0.227542\pi\)
\(504\) 0 0
\(505\) 4.70850 0.209525
\(506\) 3.64575 0.162073
\(507\) 0 0
\(508\) 10.5830 0.469545
\(509\) −3.87451 −0.171735 −0.0858673 0.996307i \(-0.527366\pi\)
−0.0858673 + 0.996307i \(0.527366\pi\)
\(510\) 0 0
\(511\) 6.58301 0.291215
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −3.41699 −0.150717
\(515\) −63.0405 −2.77790
\(516\) 0 0
\(517\) 21.8745 0.962040
\(518\) −17.8745 −0.785361
\(519\) 0 0
\(520\) −19.2915 −0.845988
\(521\) −24.4575 −1.07150 −0.535752 0.844376i \(-0.679972\pi\)
−0.535752 + 0.844376i \(0.679972\pi\)
\(522\) 0 0
\(523\) −20.9373 −0.915522 −0.457761 0.889075i \(-0.651348\pi\)
−0.457761 + 0.889075i \(0.651348\pi\)
\(524\) −9.87451 −0.431370
\(525\) 0 0
\(526\) 9.41699 0.410600
\(527\) −67.7490 −2.95119
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −13.2915 −0.577346
\(531\) 0 0
\(532\) 11.2915 0.489549
\(533\) 31.7490 1.37520
\(534\) 0 0
\(535\) 30.4575 1.31679
\(536\) 0.937254 0.0404832
\(537\) 0 0
\(538\) 32.5830 1.40475
\(539\) 10.9373 0.471101
\(540\) 0 0
\(541\) −0.583005 −0.0250654 −0.0125327 0.999921i \(-0.503989\pi\)
−0.0125327 + 0.999921i \(0.503989\pi\)
\(542\) 20.0000 0.859074
\(543\) 0 0
\(544\) −7.29150 −0.312621
\(545\) 30.0000 1.28506
\(546\) 0 0
\(547\) 27.2915 1.16690 0.583450 0.812149i \(-0.301702\pi\)
0.583450 + 0.812149i \(0.301702\pi\)
\(548\) 9.87451 0.421818
\(549\) 0 0
\(550\) −30.2288 −1.28896
\(551\) 7.29150 0.310628
\(552\) 0 0
\(553\) −25.1660 −1.07017
\(554\) −5.29150 −0.224814
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) −16.1033 −0.682317 −0.341159 0.940006i \(-0.610819\pi\)
−0.341159 + 0.940006i \(0.610819\pi\)
\(558\) 0 0
\(559\) −29.8745 −1.26356
\(560\) 7.29150 0.308122
\(561\) 0 0
\(562\) 0 0
\(563\) 23.3948 0.985972 0.492986 0.870037i \(-0.335905\pi\)
0.492986 + 0.870037i \(0.335905\pi\)
\(564\) 0 0
\(565\) 53.1660 2.23671
\(566\) 15.5203 0.652365
\(567\) 0 0
\(568\) 6.00000 0.251754
\(569\) 19.2915 0.808742 0.404371 0.914595i \(-0.367490\pi\)
0.404371 + 0.914595i \(0.367490\pi\)
\(570\) 0 0
\(571\) 3.06275 0.128172 0.0640860 0.997944i \(-0.479587\pi\)
0.0640860 + 0.997944i \(0.479587\pi\)
\(572\) 19.2915 0.806618
\(573\) 0 0
\(574\) −12.0000 −0.500870
\(575\) −8.29150 −0.345780
\(576\) 0 0
\(577\) 39.2915 1.63573 0.817863 0.575413i \(-0.195158\pi\)
0.817863 + 0.575413i \(0.195158\pi\)
\(578\) 36.1660 1.50431
\(579\) 0 0
\(580\) 4.70850 0.195510
\(581\) 16.7085 0.693185
\(582\) 0 0
\(583\) 13.2915 0.550478
\(584\) 3.29150 0.136203
\(585\) 0 0
\(586\) 5.77124 0.238408
\(587\) 31.2915 1.29154 0.645769 0.763533i \(-0.276537\pi\)
0.645769 + 0.763533i \(0.276537\pi\)
\(588\) 0 0
\(589\) 52.4575 2.16147
\(590\) 0 0
\(591\) 0 0
\(592\) −8.93725 −0.367319
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) −53.1660 −2.17959
\(596\) −15.6458 −0.640875
\(597\) 0 0
\(598\) 5.29150 0.216386
\(599\) −9.41699 −0.384768 −0.192384 0.981320i \(-0.561622\pi\)
−0.192384 + 0.981320i \(0.561622\pi\)
\(600\) 0 0
\(601\) 20.4575 0.834479 0.417240 0.908796i \(-0.362998\pi\)
0.417240 + 0.908796i \(0.362998\pi\)
\(602\) 11.2915 0.460207
\(603\) 0 0
\(604\) 6.70850 0.272965
\(605\) 8.35425 0.339649
\(606\) 0 0
\(607\) 4.12549 0.167449 0.0837243 0.996489i \(-0.473318\pi\)
0.0837243 + 0.996489i \(0.473318\pi\)
\(608\) 5.64575 0.228965
\(609\) 0 0
\(610\) 20.5830 0.833382
\(611\) 31.7490 1.28443
\(612\) 0 0
\(613\) 12.9373 0.522531 0.261265 0.965267i \(-0.415860\pi\)
0.261265 + 0.965267i \(0.415860\pi\)
\(614\) −1.87451 −0.0756490
\(615\) 0 0
\(616\) −7.29150 −0.293783
\(617\) −12.0000 −0.483102 −0.241551 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) 0 0
\(619\) 17.6458 0.709243 0.354621 0.935010i \(-0.384610\pi\)
0.354621 + 0.935010i \(0.384610\pi\)
\(620\) 33.8745 1.36043
\(621\) 0 0
\(622\) 29.1660 1.16945
\(623\) 0 0
\(624\) 0 0
\(625\) 2.29150 0.0916601
\(626\) −5.29150 −0.211491
\(627\) 0 0
\(628\) −1.64575 −0.0656726
\(629\) 65.1660 2.59834
\(630\) 0 0
\(631\) −7.87451 −0.313479 −0.156740 0.987640i \(-0.550098\pi\)
−0.156740 + 0.987640i \(0.550098\pi\)
\(632\) −12.5830 −0.500525
\(633\) 0 0
\(634\) −10.7085 −0.425289
\(635\) 38.5830 1.53112
\(636\) 0 0
\(637\) 15.8745 0.628971
\(638\) −4.70850 −0.186411
\(639\) 0 0
\(640\) 3.64575 0.144111
\(641\) −24.4575 −0.966014 −0.483007 0.875617i \(-0.660455\pi\)
−0.483007 + 0.875617i \(0.660455\pi\)
\(642\) 0 0
\(643\) 29.6458 1.16911 0.584557 0.811353i \(-0.301268\pi\)
0.584557 + 0.811353i \(0.301268\pi\)
\(644\) −2.00000 −0.0788110
\(645\) 0 0
\(646\) −41.1660 −1.61966
\(647\) 29.1660 1.14663 0.573317 0.819334i \(-0.305656\pi\)
0.573317 + 0.819334i \(0.305656\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −43.8745 −1.72090
\(651\) 0 0
\(652\) −11.2915 −0.442209
\(653\) −15.8745 −0.621217 −0.310609 0.950538i \(-0.600533\pi\)
−0.310609 + 0.950538i \(0.600533\pi\)
\(654\) 0 0
\(655\) −36.0000 −1.40664
\(656\) −6.00000 −0.234261
\(657\) 0 0
\(658\) −12.0000 −0.467809
\(659\) −22.9373 −0.893509 −0.446754 0.894657i \(-0.647420\pi\)
−0.446754 + 0.894657i \(0.647420\pi\)
\(660\) 0 0
\(661\) 22.3542 0.869480 0.434740 0.900556i \(-0.356840\pi\)
0.434740 + 0.900556i \(0.356840\pi\)
\(662\) 5.87451 0.228319
\(663\) 0 0
\(664\) 8.35425 0.324208
\(665\) 41.1660 1.59635
\(666\) 0 0
\(667\) −1.29150 −0.0500072
\(668\) −23.1660 −0.896320
\(669\) 0 0
\(670\) 3.41699 0.132010
\(671\) −20.5830 −0.794598
\(672\) 0 0
\(673\) −23.2915 −0.897821 −0.448911 0.893577i \(-0.648188\pi\)
−0.448911 + 0.893577i \(0.648188\pi\)
\(674\) −27.1660 −1.04640
\(675\) 0 0
\(676\) 15.0000 0.576923
\(677\) 22.9373 0.881550 0.440775 0.897618i \(-0.354704\pi\)
0.440775 + 0.897618i \(0.354704\pi\)
\(678\) 0 0
\(679\) 18.5830 0.713150
\(680\) −26.5830 −1.01941
\(681\) 0 0
\(682\) −33.8745 −1.29712
\(683\) 2.12549 0.0813297 0.0406648 0.999173i \(-0.487052\pi\)
0.0406648 + 0.999173i \(0.487052\pi\)
\(684\) 0 0
\(685\) 36.0000 1.37549
\(686\) −20.0000 −0.763604
\(687\) 0 0
\(688\) 5.64575 0.215242
\(689\) 19.2915 0.734948
\(690\) 0 0
\(691\) −13.8745 −0.527811 −0.263906 0.964549i \(-0.585011\pi\)
−0.263906 + 0.964549i \(0.585011\pi\)
\(692\) −15.8745 −0.603458
\(693\) 0 0
\(694\) −4.70850 −0.178732
\(695\) −14.5830 −0.553165
\(696\) 0 0
\(697\) 43.7490 1.65711
\(698\) 2.00000 0.0757011
\(699\) 0 0
\(700\) 16.5830 0.626779
\(701\) 34.9373 1.31956 0.659781 0.751458i \(-0.270649\pi\)
0.659781 + 0.751458i \(0.270649\pi\)
\(702\) 0 0
\(703\) −50.4575 −1.90304
\(704\) −3.64575 −0.137404
\(705\) 0 0
\(706\) −14.5830 −0.548839
\(707\) 2.58301 0.0971439
\(708\) 0 0
\(709\) −23.0627 −0.866140 −0.433070 0.901360i \(-0.642570\pi\)
−0.433070 + 0.901360i \(0.642570\pi\)
\(710\) 21.8745 0.820936
\(711\) 0 0
\(712\) 0 0
\(713\) −9.29150 −0.347970
\(714\) 0 0
\(715\) 70.3320 2.63027
\(716\) −4.70850 −0.175965
\(717\) 0 0
\(718\) −9.87451 −0.368513
\(719\) −43.7490 −1.63156 −0.815781 0.578360i \(-0.803693\pi\)
−0.815781 + 0.578360i \(0.803693\pi\)
\(720\) 0 0
\(721\) −34.5830 −1.28794
\(722\) 12.8745 0.479140
\(723\) 0 0
\(724\) 10.3542 0.384813
\(725\) 10.7085 0.397704
\(726\) 0 0
\(727\) −44.3320 −1.64418 −0.822092 0.569355i \(-0.807193\pi\)
−0.822092 + 0.569355i \(0.807193\pi\)
\(728\) −10.5830 −0.392232
\(729\) 0 0
\(730\) 12.0000 0.444140
\(731\) −41.1660 −1.52258
\(732\) 0 0
\(733\) 37.3948 1.38121 0.690604 0.723233i \(-0.257345\pi\)
0.690604 + 0.723233i \(0.257345\pi\)
\(734\) 31.1660 1.15036
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) −3.41699 −0.125867
\(738\) 0 0
\(739\) 10.5830 0.389302 0.194651 0.980873i \(-0.437643\pi\)
0.194651 + 0.980873i \(0.437643\pi\)
\(740\) −32.5830 −1.19778
\(741\) 0 0
\(742\) −7.29150 −0.267679
\(743\) −43.7490 −1.60500 −0.802498 0.596655i \(-0.796496\pi\)
−0.802498 + 0.596655i \(0.796496\pi\)
\(744\) 0 0
\(745\) −57.0405 −2.08980
\(746\) −11.0627 −0.405036
\(747\) 0 0
\(748\) 26.5830 0.971971
\(749\) 16.7085 0.610515
\(750\) 0 0
\(751\) −10.4575 −0.381600 −0.190800 0.981629i \(-0.561108\pi\)
−0.190800 + 0.981629i \(0.561108\pi\)
\(752\) −6.00000 −0.218797
\(753\) 0 0
\(754\) −6.83399 −0.248879
\(755\) 24.4575 0.890100
\(756\) 0 0
\(757\) 32.2288 1.17137 0.585687 0.810537i \(-0.300825\pi\)
0.585687 + 0.810537i \(0.300825\pi\)
\(758\) 3.52026 0.127862
\(759\) 0 0
\(760\) 20.5830 0.746624
\(761\) 12.0000 0.435000 0.217500 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) 0 0
\(763\) 16.4575 0.595802
\(764\) 7.29150 0.263797
\(765\) 0 0
\(766\) 31.7490 1.14714
\(767\) 0 0
\(768\) 0 0
\(769\) −49.0405 −1.76845 −0.884223 0.467065i \(-0.845312\pi\)
−0.884223 + 0.467065i \(0.845312\pi\)
\(770\) −26.5830 −0.957985
\(771\) 0 0
\(772\) −22.0000 −0.791797
\(773\) −3.64575 −0.131129 −0.0655643 0.997848i \(-0.520885\pi\)
−0.0655643 + 0.997848i \(0.520885\pi\)
\(774\) 0 0
\(775\) 77.0405 2.76738
\(776\) 9.29150 0.333546
\(777\) 0 0
\(778\) 22.9373 0.822340
\(779\) −33.8745 −1.21368
\(780\) 0 0
\(781\) −21.8745 −0.782731
\(782\) 7.29150 0.260744
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) −6.00000 −0.214149
\(786\) 0 0
\(787\) −49.6458 −1.76968 −0.884840 0.465895i \(-0.845732\pi\)
−0.884840 + 0.465895i \(0.845732\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) −45.8745 −1.63214
\(791\) 29.1660 1.03702
\(792\) 0 0
\(793\) −29.8745 −1.06087
\(794\) 2.00000 0.0709773
\(795\) 0 0
\(796\) 14.0000 0.496217
\(797\) −1.52026 −0.0538503 −0.0269252 0.999637i \(-0.508572\pi\)
−0.0269252 + 0.999637i \(0.508572\pi\)
\(798\) 0 0
\(799\) 43.7490 1.54773
\(800\) 8.29150 0.293149
\(801\) 0 0
\(802\) 21.8745 0.772416
\(803\) −12.0000 −0.423471
\(804\) 0 0
\(805\) −7.29150 −0.256992
\(806\) −49.1660 −1.73180
\(807\) 0 0
\(808\) 1.29150 0.0454349
\(809\) 12.0000 0.421898 0.210949 0.977497i \(-0.432345\pi\)
0.210949 + 0.977497i \(0.432345\pi\)
\(810\) 0 0
\(811\) −1.87451 −0.0658229 −0.0329114 0.999458i \(-0.510478\pi\)
−0.0329114 + 0.999458i \(0.510478\pi\)
\(812\) 2.58301 0.0906457
\(813\) 0 0
\(814\) 32.5830 1.14203
\(815\) −41.1660 −1.44198
\(816\) 0 0
\(817\) 31.8745 1.11515
\(818\) −27.1660 −0.949837
\(819\) 0 0
\(820\) −21.8745 −0.763891
\(821\) 39.8745 1.39163 0.695815 0.718221i \(-0.255043\pi\)
0.695815 + 0.718221i \(0.255043\pi\)
\(822\) 0 0
\(823\) −2.70850 −0.0944123 −0.0472061 0.998885i \(-0.515032\pi\)
−0.0472061 + 0.998885i \(0.515032\pi\)
\(824\) −17.2915 −0.602378
\(825\) 0 0
\(826\) 0 0
\(827\) −32.8118 −1.14098 −0.570488 0.821306i \(-0.693246\pi\)
−0.570488 + 0.821306i \(0.693246\pi\)
\(828\) 0 0
\(829\) −36.5830 −1.27058 −0.635290 0.772274i \(-0.719119\pi\)
−0.635290 + 0.772274i \(0.719119\pi\)
\(830\) 30.4575 1.05720
\(831\) 0 0
\(832\) −5.29150 −0.183450
\(833\) 21.8745 0.757907
\(834\) 0 0
\(835\) −84.4575 −2.92277
\(836\) −20.5830 −0.711878
\(837\) 0 0
\(838\) −1.06275 −0.0367120
\(839\) 19.7490 0.681812 0.340906 0.940097i \(-0.389266\pi\)
0.340906 + 0.940097i \(0.389266\pi\)
\(840\) 0 0
\(841\) −27.3320 −0.942483
\(842\) −4.22876 −0.145733
\(843\) 0 0
\(844\) 0.708497 0.0243875
\(845\) 54.6863 1.88126
\(846\) 0 0
\(847\) 4.58301 0.157474
\(848\) −3.64575 −0.125196
\(849\) 0 0
\(850\) −60.4575 −2.07368
\(851\) 8.93725 0.306365
\(852\) 0 0
\(853\) −15.1660 −0.519274 −0.259637 0.965706i \(-0.583603\pi\)
−0.259637 + 0.965706i \(0.583603\pi\)
\(854\) 11.2915 0.386387
\(855\) 0 0
\(856\) 8.35425 0.285542
\(857\) −36.0000 −1.22974 −0.614868 0.788630i \(-0.710791\pi\)
−0.614868 + 0.788630i \(0.710791\pi\)
\(858\) 0 0
\(859\) 34.5830 1.17996 0.589978 0.807419i \(-0.299136\pi\)
0.589978 + 0.807419i \(0.299136\pi\)
\(860\) 20.5830 0.701875
\(861\) 0 0
\(862\) 2.12549 0.0723945
\(863\) −53.1660 −1.80979 −0.904896 0.425633i \(-0.860052\pi\)
−0.904896 + 0.425633i \(0.860052\pi\)
\(864\) 0 0
\(865\) −57.8745 −1.96779
\(866\) −29.2915 −0.995366
\(867\) 0 0
\(868\) 18.5830 0.630748
\(869\) 45.8745 1.55619
\(870\) 0 0
\(871\) −4.95948 −0.168046
\(872\) 8.22876 0.278661
\(873\) 0 0
\(874\) −5.64575 −0.190970
\(875\) 24.0000 0.811348
\(876\) 0 0
\(877\) −2.70850 −0.0914595 −0.0457297 0.998954i \(-0.514561\pi\)
−0.0457297 + 0.998954i \(0.514561\pi\)
\(878\) −35.7490 −1.20647
\(879\) 0 0
\(880\) −13.2915 −0.448056
\(881\) 31.7490 1.06965 0.534826 0.844962i \(-0.320377\pi\)
0.534826 + 0.844962i \(0.320377\pi\)
\(882\) 0 0
\(883\) 3.29150 0.110768 0.0553839 0.998465i \(-0.482362\pi\)
0.0553839 + 0.998465i \(0.482362\pi\)
\(884\) 38.5830 1.29769
\(885\) 0 0
\(886\) 17.1660 0.576703
\(887\) 37.7490 1.26749 0.633744 0.773543i \(-0.281517\pi\)
0.633744 + 0.773543i \(0.281517\pi\)
\(888\) 0 0
\(889\) 21.1660 0.709885
\(890\) 0 0
\(891\) 0 0
\(892\) 11.8745 0.397588
\(893\) −33.8745 −1.13357
\(894\) 0 0
\(895\) −17.1660 −0.573796
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) −32.5830 −1.08731
\(899\) 12.0000 0.400222
\(900\) 0 0
\(901\) 26.5830 0.885608
\(902\) 21.8745 0.728341
\(903\) 0 0
\(904\) 14.5830 0.485024
\(905\) 37.7490 1.25482
\(906\) 0 0
\(907\) −8.47974 −0.281565 −0.140783 0.990041i \(-0.544962\pi\)
−0.140783 + 0.990041i \(0.544962\pi\)
\(908\) 22.9373 0.761200
\(909\) 0 0
\(910\) −38.5830 −1.27901
\(911\) 7.29150 0.241578 0.120789 0.992678i \(-0.461458\pi\)
0.120789 + 0.992678i \(0.461458\pi\)
\(912\) 0 0
\(913\) −30.4575 −1.00800
\(914\) −31.8745 −1.05432
\(915\) 0 0
\(916\) 24.9373 0.823950
\(917\) −19.7490 −0.652170
\(918\) 0 0
\(919\) −34.0000 −1.12156 −0.560778 0.827966i \(-0.689498\pi\)
−0.560778 + 0.827966i \(0.689498\pi\)
\(920\) −3.64575 −0.120197
\(921\) 0 0
\(922\) 37.7490 1.24320
\(923\) −31.7490 −1.04503
\(924\) 0 0
\(925\) −74.1033 −2.43650
\(926\) −21.1660 −0.695558
\(927\) 0 0
\(928\) 1.29150 0.0423957
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) −16.9373 −0.555096
\(932\) 0 0
\(933\) 0 0
\(934\) −8.35425 −0.273359
\(935\) 96.9150 3.16946
\(936\) 0 0
\(937\) −7.41699 −0.242303 −0.121151 0.992634i \(-0.538659\pi\)
−0.121151 + 0.992634i \(0.538659\pi\)
\(938\) 1.87451 0.0612049
\(939\) 0 0
\(940\) −21.8745 −0.713468
\(941\) −20.8118 −0.678444 −0.339222 0.940706i \(-0.610164\pi\)
−0.339222 + 0.940706i \(0.610164\pi\)
\(942\) 0 0
\(943\) 6.00000 0.195387
\(944\) 0 0
\(945\) 0 0
\(946\) −20.5830 −0.669211
\(947\) −33.8745 −1.10077 −0.550387 0.834910i \(-0.685520\pi\)
−0.550387 + 0.834910i \(0.685520\pi\)
\(948\) 0 0
\(949\) −17.4170 −0.565380
\(950\) 46.8118 1.51877
\(951\) 0 0
\(952\) −14.5830 −0.472638
\(953\) −51.0405 −1.65336 −0.826682 0.562669i \(-0.809775\pi\)
−0.826682 + 0.562669i \(0.809775\pi\)
\(954\) 0 0
\(955\) 26.5830 0.860206
\(956\) −24.0000 −0.776215
\(957\) 0 0
\(958\) −21.4170 −0.691952
\(959\) 19.7490 0.637729
\(960\) 0 0
\(961\) 55.3320 1.78490
\(962\) 47.2915 1.52474
\(963\) 0 0
\(964\) −5.29150 −0.170428
\(965\) −80.2065 −2.58194
\(966\) 0 0
\(967\) −7.87451 −0.253227 −0.126614 0.991952i \(-0.540411\pi\)
−0.126614 + 0.991952i \(0.540411\pi\)
\(968\) 2.29150 0.0736517
\(969\) 0 0
\(970\) 33.8745 1.08764
\(971\) −6.22876 −0.199890 −0.0999452 0.994993i \(-0.531867\pi\)
−0.0999452 + 0.994993i \(0.531867\pi\)
\(972\) 0 0
\(973\) −8.00000 −0.256468
\(974\) −21.1660 −0.678203
\(975\) 0 0
\(976\) 5.64575 0.180716
\(977\) −9.87451 −0.315913 −0.157957 0.987446i \(-0.550491\pi\)
−0.157957 + 0.987446i \(0.550491\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −10.9373 −0.349378
\(981\) 0 0
\(982\) −9.87451 −0.315108
\(983\) −9.87451 −0.314948 −0.157474 0.987523i \(-0.550335\pi\)
−0.157474 + 0.987523i \(0.550335\pi\)
\(984\) 0 0
\(985\) 21.8745 0.696980
\(986\) −9.41699 −0.299898
\(987\) 0 0
\(988\) −29.8745 −0.950435
\(989\) −5.64575 −0.179524
\(990\) 0 0
\(991\) 38.4575 1.22164 0.610822 0.791768i \(-0.290839\pi\)
0.610822 + 0.791768i \(0.290839\pi\)
\(992\) 9.29150 0.295006
\(993\) 0 0
\(994\) 12.0000 0.380617
\(995\) 51.0405 1.61809
\(996\) 0 0
\(997\) −19.4170 −0.614942 −0.307471 0.951557i \(-0.599483\pi\)
−0.307471 + 0.951557i \(0.599483\pi\)
\(998\) −43.0405 −1.36242
\(999\) 0 0
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 414.2.a.g.1.2 yes 2
3.2 odd 2 414.2.a.e.1.1 2
4.3 odd 2 3312.2.a.z.1.2 2
12.11 even 2 3312.2.a.v.1.1 2
23.22 odd 2 9522.2.a.bc.1.1 2
69.68 even 2 9522.2.a.bb.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
414.2.a.e.1.1 2 3.2 odd 2
414.2.a.g.1.2 yes 2 1.1 even 1 trivial
3312.2.a.v.1.1 2 12.11 even 2
3312.2.a.z.1.2 2 4.3 odd 2
9522.2.a.bb.1.2 2 69.68 even 2
9522.2.a.bc.1.1 2 23.22 odd 2