Properties

Label 5225.2.a.g.1.1
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
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] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.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: \(41.7218350561\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1045)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 5225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.414214 q^{2} +2.00000 q^{3} -1.82843 q^{4} -0.828427 q^{6} -0.828427 q^{7} +1.58579 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.414214 q^{2} +2.00000 q^{3} -1.82843 q^{4} -0.828427 q^{6} -0.828427 q^{7} +1.58579 q^{8} +1.00000 q^{9} -1.00000 q^{11} -3.65685 q^{12} +6.82843 q^{13} +0.343146 q^{14} +3.00000 q^{16} +6.82843 q^{17} -0.414214 q^{18} -1.00000 q^{19} -1.65685 q^{21} +0.414214 q^{22} +7.65685 q^{23} +3.17157 q^{24} -2.82843 q^{26} -4.00000 q^{27} +1.51472 q^{28} -4.82843 q^{29} -6.82843 q^{31} -4.41421 q^{32} -2.00000 q^{33} -2.82843 q^{34} -1.82843 q^{36} -8.48528 q^{37} +0.414214 q^{38} +13.6569 q^{39} +6.48528 q^{41} +0.686292 q^{42} -0.828427 q^{43} +1.82843 q^{44} -3.17157 q^{46} +11.6569 q^{47} +6.00000 q^{48} -6.31371 q^{49} +13.6569 q^{51} -12.4853 q^{52} +10.8284 q^{53} +1.65685 q^{54} -1.31371 q^{56} -2.00000 q^{57} +2.00000 q^{58} -2.82843 q^{59} -2.00000 q^{61} +2.82843 q^{62} -0.828427 q^{63} -4.17157 q^{64} +0.828427 q^{66} +6.00000 q^{67} -12.4853 q^{68} +15.3137 q^{69} -14.8284 q^{71} +1.58579 q^{72} +1.17157 q^{73} +3.51472 q^{74} +1.82843 q^{76} +0.828427 q^{77} -5.65685 q^{78} +5.65685 q^{79} -11.0000 q^{81} -2.68629 q^{82} -6.48528 q^{83} +3.02944 q^{84} +0.343146 q^{86} -9.65685 q^{87} -1.58579 q^{88} -4.34315 q^{89} -5.65685 q^{91} -14.0000 q^{92} -13.6569 q^{93} -4.82843 q^{94} -8.82843 q^{96} +1.17157 q^{97} +2.61522 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 4 q^{6} + 4 q^{7} + 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 4 q^{6} + 4 q^{7} + 6 q^{8} + 2 q^{9} - 2 q^{11} + 4 q^{12} + 8 q^{13} + 12 q^{14} + 6 q^{16} + 8 q^{17} + 2 q^{18} - 2 q^{19} + 8 q^{21} - 2 q^{22} + 4 q^{23} + 12 q^{24} - 8 q^{27} + 20 q^{28} - 4 q^{29} - 8 q^{31} - 6 q^{32} - 4 q^{33} + 2 q^{36} - 2 q^{38} + 16 q^{39} - 4 q^{41} + 24 q^{42} + 4 q^{43} - 2 q^{44} - 12 q^{46} + 12 q^{47} + 12 q^{48} + 10 q^{49} + 16 q^{51} - 8 q^{52} + 16 q^{53} - 8 q^{54} + 20 q^{56} - 4 q^{57} + 4 q^{58} - 4 q^{61} + 4 q^{63} - 14 q^{64} - 4 q^{66} + 12 q^{67} - 8 q^{68} + 8 q^{69} - 24 q^{71} + 6 q^{72} + 8 q^{73} + 24 q^{74} - 2 q^{76} - 4 q^{77} - 22 q^{81} - 28 q^{82} + 4 q^{83} + 40 q^{84} + 12 q^{86} - 8 q^{87} - 6 q^{88} - 20 q^{89} - 28 q^{92} - 16 q^{93} - 4 q^{94} - 12 q^{96} + 8 q^{97} + 42 q^{98} - 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.414214 −0.292893 −0.146447 0.989219i \(-0.546784\pi\)
−0.146447 + 0.989219i \(0.546784\pi\)
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) −1.82843 −0.914214
\(5\) 0 0
\(6\) −0.828427 −0.338204
\(7\) −0.828427 −0.313116 −0.156558 0.987669i \(-0.550040\pi\)
−0.156558 + 0.987669i \(0.550040\pi\)
\(8\) 1.58579 0.560660
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −3.65685 −1.05564
\(13\) 6.82843 1.89386 0.946932 0.321433i \(-0.104164\pi\)
0.946932 + 0.321433i \(0.104164\pi\)
\(14\) 0.343146 0.0917096
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 6.82843 1.65614 0.828068 0.560627i \(-0.189440\pi\)
0.828068 + 0.560627i \(0.189440\pi\)
\(18\) −0.414214 −0.0976311
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −1.65685 −0.361555
\(22\) 0.414214 0.0883106
\(23\) 7.65685 1.59656 0.798282 0.602284i \(-0.205742\pi\)
0.798282 + 0.602284i \(0.205742\pi\)
\(24\) 3.17157 0.647395
\(25\) 0 0
\(26\) −2.82843 −0.554700
\(27\) −4.00000 −0.769800
\(28\) 1.51472 0.286255
\(29\) −4.82843 −0.896616 −0.448308 0.893879i \(-0.647973\pi\)
−0.448308 + 0.893879i \(0.647973\pi\)
\(30\) 0 0
\(31\) −6.82843 −1.22642 −0.613211 0.789919i \(-0.710122\pi\)
−0.613211 + 0.789919i \(0.710122\pi\)
\(32\) −4.41421 −0.780330
\(33\) −2.00000 −0.348155
\(34\) −2.82843 −0.485071
\(35\) 0 0
\(36\) −1.82843 −0.304738
\(37\) −8.48528 −1.39497 −0.697486 0.716599i \(-0.745698\pi\)
−0.697486 + 0.716599i \(0.745698\pi\)
\(38\) 0.414214 0.0671943
\(39\) 13.6569 2.18685
\(40\) 0 0
\(41\) 6.48528 1.01283 0.506415 0.862290i \(-0.330970\pi\)
0.506415 + 0.862290i \(0.330970\pi\)
\(42\) 0.686292 0.105897
\(43\) −0.828427 −0.126334 −0.0631670 0.998003i \(-0.520120\pi\)
−0.0631670 + 0.998003i \(0.520120\pi\)
\(44\) 1.82843 0.275646
\(45\) 0 0
\(46\) −3.17157 −0.467623
\(47\) 11.6569 1.70033 0.850163 0.526519i \(-0.176503\pi\)
0.850163 + 0.526519i \(0.176503\pi\)
\(48\) 6.00000 0.866025
\(49\) −6.31371 −0.901958
\(50\) 0 0
\(51\) 13.6569 1.91234
\(52\) −12.4853 −1.73140
\(53\) 10.8284 1.48740 0.743699 0.668514i \(-0.233069\pi\)
0.743699 + 0.668514i \(0.233069\pi\)
\(54\) 1.65685 0.225469
\(55\) 0 0
\(56\) −1.31371 −0.175552
\(57\) −2.00000 −0.264906
\(58\) 2.00000 0.262613
\(59\) −2.82843 −0.368230 −0.184115 0.982905i \(-0.558942\pi\)
−0.184115 + 0.982905i \(0.558942\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 2.82843 0.359211
\(63\) −0.828427 −0.104372
\(64\) −4.17157 −0.521447
\(65\) 0 0
\(66\) 0.828427 0.101972
\(67\) 6.00000 0.733017 0.366508 0.930415i \(-0.380553\pi\)
0.366508 + 0.930415i \(0.380553\pi\)
\(68\) −12.4853 −1.51406
\(69\) 15.3137 1.84355
\(70\) 0 0
\(71\) −14.8284 −1.75981 −0.879905 0.475149i \(-0.842394\pi\)
−0.879905 + 0.475149i \(0.842394\pi\)
\(72\) 1.58579 0.186887
\(73\) 1.17157 0.137122 0.0685611 0.997647i \(-0.478159\pi\)
0.0685611 + 0.997647i \(0.478159\pi\)
\(74\) 3.51472 0.408578
\(75\) 0 0
\(76\) 1.82843 0.209735
\(77\) 0.828427 0.0944080
\(78\) −5.65685 −0.640513
\(79\) 5.65685 0.636446 0.318223 0.948016i \(-0.396914\pi\)
0.318223 + 0.948016i \(0.396914\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) −2.68629 −0.296651
\(83\) −6.48528 −0.711852 −0.355926 0.934514i \(-0.615835\pi\)
−0.355926 + 0.934514i \(0.615835\pi\)
\(84\) 3.02944 0.330539
\(85\) 0 0
\(86\) 0.343146 0.0370024
\(87\) −9.65685 −1.03532
\(88\) −1.58579 −0.169045
\(89\) −4.34315 −0.460373 −0.230186 0.973147i \(-0.573934\pi\)
−0.230186 + 0.973147i \(0.573934\pi\)
\(90\) 0 0
\(91\) −5.65685 −0.592999
\(92\) −14.0000 −1.45960
\(93\) −13.6569 −1.41615
\(94\) −4.82843 −0.498014
\(95\) 0 0
\(96\) −8.82843 −0.901048
\(97\) 1.17157 0.118955 0.0594776 0.998230i \(-0.481057\pi\)
0.0594776 + 0.998230i \(0.481057\pi\)
\(98\) 2.61522 0.264177
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 9.31371 0.926749 0.463374 0.886163i \(-0.346639\pi\)
0.463374 + 0.886163i \(0.346639\pi\)
\(102\) −5.65685 −0.560112
\(103\) −4.34315 −0.427943 −0.213971 0.976840i \(-0.568640\pi\)
−0.213971 + 0.976840i \(0.568640\pi\)
\(104\) 10.8284 1.06181
\(105\) 0 0
\(106\) −4.48528 −0.435649
\(107\) 18.9706 1.83395 0.916977 0.398941i \(-0.130622\pi\)
0.916977 + 0.398941i \(0.130622\pi\)
\(108\) 7.31371 0.703762
\(109\) 0.828427 0.0793489 0.0396745 0.999213i \(-0.487368\pi\)
0.0396745 + 0.999213i \(0.487368\pi\)
\(110\) 0 0
\(111\) −16.9706 −1.61077
\(112\) −2.48528 −0.234837
\(113\) 10.1421 0.954092 0.477046 0.878878i \(-0.341708\pi\)
0.477046 + 0.878878i \(0.341708\pi\)
\(114\) 0.828427 0.0775893
\(115\) 0 0
\(116\) 8.82843 0.819699
\(117\) 6.82843 0.631288
\(118\) 1.17157 0.107852
\(119\) −5.65685 −0.518563
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0.828427 0.0750023
\(123\) 12.9706 1.16952
\(124\) 12.4853 1.12121
\(125\) 0 0
\(126\) 0.343146 0.0305699
\(127\) 14.0000 1.24230 0.621150 0.783692i \(-0.286666\pi\)
0.621150 + 0.783692i \(0.286666\pi\)
\(128\) 10.5563 0.933058
\(129\) −1.65685 −0.145878
\(130\) 0 0
\(131\) 7.31371 0.639002 0.319501 0.947586i \(-0.396485\pi\)
0.319501 + 0.947586i \(0.396485\pi\)
\(132\) 3.65685 0.318288
\(133\) 0.828427 0.0718337
\(134\) −2.48528 −0.214696
\(135\) 0 0
\(136\) 10.8284 0.928530
\(137\) 8.00000 0.683486 0.341743 0.939793i \(-0.388983\pi\)
0.341743 + 0.939793i \(0.388983\pi\)
\(138\) −6.34315 −0.539964
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 23.3137 1.96337
\(142\) 6.14214 0.515437
\(143\) −6.82843 −0.571022
\(144\) 3.00000 0.250000
\(145\) 0 0
\(146\) −0.485281 −0.0401622
\(147\) −12.6274 −1.04149
\(148\) 15.5147 1.27530
\(149\) −8.34315 −0.683497 −0.341749 0.939791i \(-0.611019\pi\)
−0.341749 + 0.939791i \(0.611019\pi\)
\(150\) 0 0
\(151\) 10.3431 0.841713 0.420857 0.907127i \(-0.361730\pi\)
0.420857 + 0.907127i \(0.361730\pi\)
\(152\) −1.58579 −0.128624
\(153\) 6.82843 0.552046
\(154\) −0.343146 −0.0276515
\(155\) 0 0
\(156\) −24.9706 −1.99925
\(157\) 12.0000 0.957704 0.478852 0.877896i \(-0.341053\pi\)
0.478852 + 0.877896i \(0.341053\pi\)
\(158\) −2.34315 −0.186411
\(159\) 21.6569 1.71750
\(160\) 0 0
\(161\) −6.34315 −0.499910
\(162\) 4.55635 0.357981
\(163\) −2.00000 −0.156652 −0.0783260 0.996928i \(-0.524958\pi\)
−0.0783260 + 0.996928i \(0.524958\pi\)
\(164\) −11.8579 −0.925944
\(165\) 0 0
\(166\) 2.68629 0.208497
\(167\) 8.34315 0.645612 0.322806 0.946465i \(-0.395374\pi\)
0.322806 + 0.946465i \(0.395374\pi\)
\(168\) −2.62742 −0.202710
\(169\) 33.6274 2.58672
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 1.51472 0.115496
\(173\) 12.4853 0.949238 0.474619 0.880191i \(-0.342586\pi\)
0.474619 + 0.880191i \(0.342586\pi\)
\(174\) 4.00000 0.303239
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) −5.65685 −0.425195
\(178\) 1.79899 0.134840
\(179\) 16.4853 1.23217 0.616084 0.787681i \(-0.288718\pi\)
0.616084 + 0.787681i \(0.288718\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 2.34315 0.173686
\(183\) −4.00000 −0.295689
\(184\) 12.1421 0.895130
\(185\) 0 0
\(186\) 5.65685 0.414781
\(187\) −6.82843 −0.499344
\(188\) −21.3137 −1.55446
\(189\) 3.31371 0.241037
\(190\) 0 0
\(191\) 16.9706 1.22795 0.613973 0.789327i \(-0.289570\pi\)
0.613973 + 0.789327i \(0.289570\pi\)
\(192\) −8.34315 −0.602115
\(193\) −14.1421 −1.01797 −0.508987 0.860774i \(-0.669980\pi\)
−0.508987 + 0.860774i \(0.669980\pi\)
\(194\) −0.485281 −0.0348412
\(195\) 0 0
\(196\) 11.5442 0.824583
\(197\) −6.14214 −0.437609 −0.218805 0.975769i \(-0.570216\pi\)
−0.218805 + 0.975769i \(0.570216\pi\)
\(198\) 0.414214 0.0294369
\(199\) −5.65685 −0.401004 −0.200502 0.979693i \(-0.564257\pi\)
−0.200502 + 0.979693i \(0.564257\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) −3.85786 −0.271438
\(203\) 4.00000 0.280745
\(204\) −24.9706 −1.74829
\(205\) 0 0
\(206\) 1.79899 0.125342
\(207\) 7.65685 0.532188
\(208\) 20.4853 1.42040
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 15.3137 1.05424 0.527120 0.849791i \(-0.323272\pi\)
0.527120 + 0.849791i \(0.323272\pi\)
\(212\) −19.7990 −1.35980
\(213\) −29.6569 −2.03205
\(214\) −7.85786 −0.537153
\(215\) 0 0
\(216\) −6.34315 −0.431596
\(217\) 5.65685 0.384012
\(218\) −0.343146 −0.0232408
\(219\) 2.34315 0.158335
\(220\) 0 0
\(221\) 46.6274 3.13650
\(222\) 7.02944 0.471785
\(223\) 11.6569 0.780601 0.390300 0.920688i \(-0.372371\pi\)
0.390300 + 0.920688i \(0.372371\pi\)
\(224\) 3.65685 0.244334
\(225\) 0 0
\(226\) −4.20101 −0.279447
\(227\) 12.3431 0.819243 0.409622 0.912255i \(-0.365661\pi\)
0.409622 + 0.912255i \(0.365661\pi\)
\(228\) 3.65685 0.242181
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 1.65685 0.109013
\(232\) −7.65685 −0.502697
\(233\) −4.48528 −0.293841 −0.146920 0.989148i \(-0.546936\pi\)
−0.146920 + 0.989148i \(0.546936\pi\)
\(234\) −2.82843 −0.184900
\(235\) 0 0
\(236\) 5.17157 0.336641
\(237\) 11.3137 0.734904
\(238\) 2.34315 0.151884
\(239\) 4.68629 0.303131 0.151565 0.988447i \(-0.451569\pi\)
0.151565 + 0.988447i \(0.451569\pi\)
\(240\) 0 0
\(241\) 6.48528 0.417754 0.208877 0.977942i \(-0.433019\pi\)
0.208877 + 0.977942i \(0.433019\pi\)
\(242\) −0.414214 −0.0266267
\(243\) −10.0000 −0.641500
\(244\) 3.65685 0.234106
\(245\) 0 0
\(246\) −5.37258 −0.342543
\(247\) −6.82843 −0.434482
\(248\) −10.8284 −0.687606
\(249\) −12.9706 −0.821976
\(250\) 0 0
\(251\) −4.00000 −0.252478 −0.126239 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) 1.51472 0.0954183
\(253\) −7.65685 −0.481382
\(254\) −5.79899 −0.363861
\(255\) 0 0
\(256\) 3.97056 0.248160
\(257\) −18.1421 −1.13168 −0.565838 0.824517i \(-0.691447\pi\)
−0.565838 + 0.824517i \(0.691447\pi\)
\(258\) 0.686292 0.0427266
\(259\) 7.02944 0.436788
\(260\) 0 0
\(261\) −4.82843 −0.298872
\(262\) −3.02944 −0.187159
\(263\) −18.4853 −1.13985 −0.569926 0.821696i \(-0.693028\pi\)
−0.569926 + 0.821696i \(0.693028\pi\)
\(264\) −3.17157 −0.195197
\(265\) 0 0
\(266\) −0.343146 −0.0210396
\(267\) −8.68629 −0.531592
\(268\) −10.9706 −0.670134
\(269\) −22.9706 −1.40054 −0.700270 0.713878i \(-0.746937\pi\)
−0.700270 + 0.713878i \(0.746937\pi\)
\(270\) 0 0
\(271\) 11.3137 0.687259 0.343629 0.939105i \(-0.388344\pi\)
0.343629 + 0.939105i \(0.388344\pi\)
\(272\) 20.4853 1.24210
\(273\) −11.3137 −0.684737
\(274\) −3.31371 −0.200188
\(275\) 0 0
\(276\) −28.0000 −1.68540
\(277\) −18.8284 −1.13129 −0.565645 0.824649i \(-0.691373\pi\)
−0.565645 + 0.824649i \(0.691373\pi\)
\(278\) −1.65685 −0.0993715
\(279\) −6.82843 −0.408807
\(280\) 0 0
\(281\) −19.4558 −1.16064 −0.580319 0.814389i \(-0.697072\pi\)
−0.580319 + 0.814389i \(0.697072\pi\)
\(282\) −9.65685 −0.575057
\(283\) −6.48528 −0.385510 −0.192755 0.981247i \(-0.561742\pi\)
−0.192755 + 0.981247i \(0.561742\pi\)
\(284\) 27.1127 1.60884
\(285\) 0 0
\(286\) 2.82843 0.167248
\(287\) −5.37258 −0.317134
\(288\) −4.41421 −0.260110
\(289\) 29.6274 1.74279
\(290\) 0 0
\(291\) 2.34315 0.137358
\(292\) −2.14214 −0.125359
\(293\) −26.1421 −1.52724 −0.763620 0.645666i \(-0.776580\pi\)
−0.763620 + 0.645666i \(0.776580\pi\)
\(294\) 5.23045 0.305046
\(295\) 0 0
\(296\) −13.4558 −0.782105
\(297\) 4.00000 0.232104
\(298\) 3.45584 0.200192
\(299\) 52.2843 3.02368
\(300\) 0 0
\(301\) 0.686292 0.0395572
\(302\) −4.28427 −0.246532
\(303\) 18.6274 1.07012
\(304\) −3.00000 −0.172062
\(305\) 0 0
\(306\) −2.82843 −0.161690
\(307\) 26.9706 1.53929 0.769646 0.638471i \(-0.220433\pi\)
0.769646 + 0.638471i \(0.220433\pi\)
\(308\) −1.51472 −0.0863091
\(309\) −8.68629 −0.494146
\(310\) 0 0
\(311\) 0.970563 0.0550356 0.0275178 0.999621i \(-0.491240\pi\)
0.0275178 + 0.999621i \(0.491240\pi\)
\(312\) 21.6569 1.22608
\(313\) 16.9706 0.959233 0.479616 0.877478i \(-0.340776\pi\)
0.479616 + 0.877478i \(0.340776\pi\)
\(314\) −4.97056 −0.280505
\(315\) 0 0
\(316\) −10.3431 −0.581847
\(317\) −6.14214 −0.344977 −0.172488 0.985012i \(-0.555181\pi\)
−0.172488 + 0.985012i \(0.555181\pi\)
\(318\) −8.97056 −0.503044
\(319\) 4.82843 0.270340
\(320\) 0 0
\(321\) 37.9411 2.11767
\(322\) 2.62742 0.146420
\(323\) −6.82843 −0.379944
\(324\) 20.1127 1.11737
\(325\) 0 0
\(326\) 0.828427 0.0458823
\(327\) 1.65685 0.0916242
\(328\) 10.2843 0.567854
\(329\) −9.65685 −0.532400
\(330\) 0 0
\(331\) −6.14214 −0.337602 −0.168801 0.985650i \(-0.553990\pi\)
−0.168801 + 0.985650i \(0.553990\pi\)
\(332\) 11.8579 0.650785
\(333\) −8.48528 −0.464991
\(334\) −3.45584 −0.189095
\(335\) 0 0
\(336\) −4.97056 −0.271166
\(337\) 2.82843 0.154074 0.0770371 0.997028i \(-0.475454\pi\)
0.0770371 + 0.997028i \(0.475454\pi\)
\(338\) −13.9289 −0.757634
\(339\) 20.2843 1.10169
\(340\) 0 0
\(341\) 6.82843 0.369780
\(342\) 0.414214 0.0223981
\(343\) 11.0294 0.595534
\(344\) −1.31371 −0.0708304
\(345\) 0 0
\(346\) −5.17157 −0.278025
\(347\) 24.8284 1.33286 0.666430 0.745568i \(-0.267822\pi\)
0.666430 + 0.745568i \(0.267822\pi\)
\(348\) 17.6569 0.946507
\(349\) −27.6569 −1.48044 −0.740219 0.672366i \(-0.765278\pi\)
−0.740219 + 0.672366i \(0.765278\pi\)
\(350\) 0 0
\(351\) −27.3137 −1.45790
\(352\) 4.41421 0.235278
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 2.34315 0.124537
\(355\) 0 0
\(356\) 7.94113 0.420879
\(357\) −11.3137 −0.598785
\(358\) −6.82843 −0.360894
\(359\) 16.9706 0.895672 0.447836 0.894116i \(-0.352195\pi\)
0.447836 + 0.894116i \(0.352195\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −5.79899 −0.304788
\(363\) 2.00000 0.104973
\(364\) 10.3431 0.542128
\(365\) 0 0
\(366\) 1.65685 0.0866052
\(367\) −20.3431 −1.06190 −0.530952 0.847402i \(-0.678165\pi\)
−0.530952 + 0.847402i \(0.678165\pi\)
\(368\) 22.9706 1.19742
\(369\) 6.48528 0.337610
\(370\) 0 0
\(371\) −8.97056 −0.465728
\(372\) 24.9706 1.29466
\(373\) −17.1716 −0.889110 −0.444555 0.895751i \(-0.646638\pi\)
−0.444555 + 0.895751i \(0.646638\pi\)
\(374\) 2.82843 0.146254
\(375\) 0 0
\(376\) 18.4853 0.953306
\(377\) −32.9706 −1.69807
\(378\) −1.37258 −0.0705981
\(379\) 10.8284 0.556219 0.278109 0.960549i \(-0.410292\pi\)
0.278109 + 0.960549i \(0.410292\pi\)
\(380\) 0 0
\(381\) 28.0000 1.43448
\(382\) −7.02944 −0.359657
\(383\) −26.9706 −1.37813 −0.689066 0.724699i \(-0.741979\pi\)
−0.689066 + 0.724699i \(0.741979\pi\)
\(384\) 21.1127 1.07740
\(385\) 0 0
\(386\) 5.85786 0.298157
\(387\) −0.828427 −0.0421113
\(388\) −2.14214 −0.108750
\(389\) 13.3137 0.675032 0.337516 0.941320i \(-0.390413\pi\)
0.337516 + 0.941320i \(0.390413\pi\)
\(390\) 0 0
\(391\) 52.2843 2.64413
\(392\) −10.0122 −0.505692
\(393\) 14.6274 0.737856
\(394\) 2.54416 0.128173
\(395\) 0 0
\(396\) 1.82843 0.0918819
\(397\) 7.31371 0.367065 0.183532 0.983014i \(-0.441247\pi\)
0.183532 + 0.983014i \(0.441247\pi\)
\(398\) 2.34315 0.117451
\(399\) 1.65685 0.0829465
\(400\) 0 0
\(401\) 5.31371 0.265354 0.132677 0.991159i \(-0.457643\pi\)
0.132677 + 0.991159i \(0.457643\pi\)
\(402\) −4.97056 −0.247909
\(403\) −46.6274 −2.32268
\(404\) −17.0294 −0.847246
\(405\) 0 0
\(406\) −1.65685 −0.0822283
\(407\) 8.48528 0.420600
\(408\) 21.6569 1.07217
\(409\) −24.8284 −1.22769 −0.613843 0.789428i \(-0.710377\pi\)
−0.613843 + 0.789428i \(0.710377\pi\)
\(410\) 0 0
\(411\) 16.0000 0.789222
\(412\) 7.94113 0.391231
\(413\) 2.34315 0.115299
\(414\) −3.17157 −0.155874
\(415\) 0 0
\(416\) −30.1421 −1.47784
\(417\) 8.00000 0.391762
\(418\) −0.414214 −0.0202598
\(419\) −20.9706 −1.02448 −0.512240 0.858843i \(-0.671184\pi\)
−0.512240 + 0.858843i \(0.671184\pi\)
\(420\) 0 0
\(421\) −26.9706 −1.31446 −0.657232 0.753688i \(-0.728273\pi\)
−0.657232 + 0.753688i \(0.728273\pi\)
\(422\) −6.34315 −0.308780
\(423\) 11.6569 0.566776
\(424\) 17.1716 0.833925
\(425\) 0 0
\(426\) 12.2843 0.595175
\(427\) 1.65685 0.0801808
\(428\) −34.6863 −1.67663
\(429\) −13.6569 −0.659359
\(430\) 0 0
\(431\) 6.62742 0.319231 0.159616 0.987179i \(-0.448974\pi\)
0.159616 + 0.987179i \(0.448974\pi\)
\(432\) −12.0000 −0.577350
\(433\) 1.17157 0.0563022 0.0281511 0.999604i \(-0.491038\pi\)
0.0281511 + 0.999604i \(0.491038\pi\)
\(434\) −2.34315 −0.112475
\(435\) 0 0
\(436\) −1.51472 −0.0725419
\(437\) −7.65685 −0.366277
\(438\) −0.970563 −0.0463753
\(439\) 18.3431 0.875471 0.437735 0.899104i \(-0.355781\pi\)
0.437735 + 0.899104i \(0.355781\pi\)
\(440\) 0 0
\(441\) −6.31371 −0.300653
\(442\) −19.3137 −0.918659
\(443\) 9.31371 0.442508 0.221254 0.975216i \(-0.428985\pi\)
0.221254 + 0.975216i \(0.428985\pi\)
\(444\) 31.0294 1.47259
\(445\) 0 0
\(446\) −4.82843 −0.228633
\(447\) −16.6863 −0.789235
\(448\) 3.45584 0.163273
\(449\) −2.97056 −0.140190 −0.0700948 0.997540i \(-0.522330\pi\)
−0.0700948 + 0.997540i \(0.522330\pi\)
\(450\) 0 0
\(451\) −6.48528 −0.305380
\(452\) −18.5442 −0.872244
\(453\) 20.6863 0.971927
\(454\) −5.11270 −0.239951
\(455\) 0 0
\(456\) −3.17157 −0.148523
\(457\) −15.7990 −0.739046 −0.369523 0.929222i \(-0.620479\pi\)
−0.369523 + 0.929222i \(0.620479\pi\)
\(458\) 9.11270 0.425808
\(459\) −27.3137 −1.27489
\(460\) 0 0
\(461\) −22.2843 −1.03788 −0.518941 0.854810i \(-0.673674\pi\)
−0.518941 + 0.854810i \(0.673674\pi\)
\(462\) −0.686292 −0.0319292
\(463\) 30.2843 1.40743 0.703715 0.710483i \(-0.251523\pi\)
0.703715 + 0.710483i \(0.251523\pi\)
\(464\) −14.4853 −0.672462
\(465\) 0 0
\(466\) 1.85786 0.0860639
\(467\) 4.34315 0.200977 0.100488 0.994938i \(-0.467959\pi\)
0.100488 + 0.994938i \(0.467959\pi\)
\(468\) −12.4853 −0.577132
\(469\) −4.97056 −0.229519
\(470\) 0 0
\(471\) 24.0000 1.10586
\(472\) −4.48528 −0.206452
\(473\) 0.828427 0.0380911
\(474\) −4.68629 −0.215248
\(475\) 0 0
\(476\) 10.3431 0.474077
\(477\) 10.8284 0.495800
\(478\) −1.94113 −0.0887850
\(479\) −3.31371 −0.151407 −0.0757036 0.997130i \(-0.524120\pi\)
−0.0757036 + 0.997130i \(0.524120\pi\)
\(480\) 0 0
\(481\) −57.9411 −2.64189
\(482\) −2.68629 −0.122357
\(483\) −12.6863 −0.577246
\(484\) −1.82843 −0.0831103
\(485\) 0 0
\(486\) 4.14214 0.187891
\(487\) 15.6569 0.709480 0.354740 0.934965i \(-0.384569\pi\)
0.354740 + 0.934965i \(0.384569\pi\)
\(488\) −3.17157 −0.143570
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) −36.9706 −1.66846 −0.834229 0.551418i \(-0.814087\pi\)
−0.834229 + 0.551418i \(0.814087\pi\)
\(492\) −23.7157 −1.06919
\(493\) −32.9706 −1.48492
\(494\) 2.82843 0.127257
\(495\) 0 0
\(496\) −20.4853 −0.919816
\(497\) 12.2843 0.551025
\(498\) 5.37258 0.240751
\(499\) 0.686292 0.0307226 0.0153613 0.999882i \(-0.495110\pi\)
0.0153613 + 0.999882i \(0.495110\pi\)
\(500\) 0 0
\(501\) 16.6863 0.745489
\(502\) 1.65685 0.0739490
\(503\) 41.7990 1.86372 0.931862 0.362812i \(-0.118183\pi\)
0.931862 + 0.362812i \(0.118183\pi\)
\(504\) −1.31371 −0.0585172
\(505\) 0 0
\(506\) 3.17157 0.140994
\(507\) 67.2548 2.98689
\(508\) −25.5980 −1.13573
\(509\) 31.6569 1.40317 0.701583 0.712588i \(-0.252477\pi\)
0.701583 + 0.712588i \(0.252477\pi\)
\(510\) 0 0
\(511\) −0.970563 −0.0429352
\(512\) −22.7574 −1.00574
\(513\) 4.00000 0.176604
\(514\) 7.51472 0.331460
\(515\) 0 0
\(516\) 3.02944 0.133364
\(517\) −11.6569 −0.512668
\(518\) −2.91169 −0.127932
\(519\) 24.9706 1.09609
\(520\) 0 0
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) 2.00000 0.0875376
\(523\) 6.97056 0.304801 0.152401 0.988319i \(-0.451300\pi\)
0.152401 + 0.988319i \(0.451300\pi\)
\(524\) −13.3726 −0.584184
\(525\) 0 0
\(526\) 7.65685 0.333855
\(527\) −46.6274 −2.03112
\(528\) −6.00000 −0.261116
\(529\) 35.6274 1.54902
\(530\) 0 0
\(531\) −2.82843 −0.122743
\(532\) −1.51472 −0.0656714
\(533\) 44.2843 1.91816
\(534\) 3.59798 0.155700
\(535\) 0 0
\(536\) 9.51472 0.410973
\(537\) 32.9706 1.42278
\(538\) 9.51472 0.410209
\(539\) 6.31371 0.271951
\(540\) 0 0
\(541\) −12.3431 −0.530673 −0.265337 0.964156i \(-0.585483\pi\)
−0.265337 + 0.964156i \(0.585483\pi\)
\(542\) −4.68629 −0.201293
\(543\) 28.0000 1.20160
\(544\) −30.1421 −1.29233
\(545\) 0 0
\(546\) 4.68629 0.200555
\(547\) −34.2843 −1.46589 −0.732945 0.680288i \(-0.761855\pi\)
−0.732945 + 0.680288i \(0.761855\pi\)
\(548\) −14.6274 −0.624852
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 4.82843 0.205698
\(552\) 24.2843 1.03361
\(553\) −4.68629 −0.199281
\(554\) 7.79899 0.331347
\(555\) 0 0
\(556\) −7.31371 −0.310170
\(557\) 23.1127 0.979316 0.489658 0.871914i \(-0.337122\pi\)
0.489658 + 0.871914i \(0.337122\pi\)
\(558\) 2.82843 0.119737
\(559\) −5.65685 −0.239259
\(560\) 0 0
\(561\) −13.6569 −0.576593
\(562\) 8.05887 0.339943
\(563\) −18.9706 −0.799514 −0.399757 0.916621i \(-0.630905\pi\)
−0.399757 + 0.916621i \(0.630905\pi\)
\(564\) −42.6274 −1.79494
\(565\) 0 0
\(566\) 2.68629 0.112913
\(567\) 9.11270 0.382697
\(568\) −23.5147 −0.986656
\(569\) −32.8284 −1.37624 −0.688120 0.725597i \(-0.741564\pi\)
−0.688120 + 0.725597i \(0.741564\pi\)
\(570\) 0 0
\(571\) −24.2843 −1.01627 −0.508133 0.861279i \(-0.669664\pi\)
−0.508133 + 0.861279i \(0.669664\pi\)
\(572\) 12.4853 0.522036
\(573\) 33.9411 1.41791
\(574\) 2.22540 0.0928863
\(575\) 0 0
\(576\) −4.17157 −0.173816
\(577\) 32.0000 1.33218 0.666089 0.745873i \(-0.267967\pi\)
0.666089 + 0.745873i \(0.267967\pi\)
\(578\) −12.2721 −0.510451
\(579\) −28.2843 −1.17545
\(580\) 0 0
\(581\) 5.37258 0.222892
\(582\) −0.970563 −0.0402311
\(583\) −10.8284 −0.448468
\(584\) 1.85786 0.0768790
\(585\) 0 0
\(586\) 10.8284 0.447318
\(587\) 22.2843 0.919770 0.459885 0.887978i \(-0.347891\pi\)
0.459885 + 0.887978i \(0.347891\pi\)
\(588\) 23.0883 0.952146
\(589\) 6.82843 0.281360
\(590\) 0 0
\(591\) −12.2843 −0.505307
\(592\) −25.4558 −1.04623
\(593\) −5.45584 −0.224045 −0.112022 0.993706i \(-0.535733\pi\)
−0.112022 + 0.993706i \(0.535733\pi\)
\(594\) −1.65685 −0.0679816
\(595\) 0 0
\(596\) 15.2548 0.624862
\(597\) −11.3137 −0.463039
\(598\) −21.6569 −0.885615
\(599\) 30.8284 1.25962 0.629808 0.776751i \(-0.283134\pi\)
0.629808 + 0.776751i \(0.283134\pi\)
\(600\) 0 0
\(601\) −25.1127 −1.02437 −0.512184 0.858876i \(-0.671163\pi\)
−0.512184 + 0.858876i \(0.671163\pi\)
\(602\) −0.284271 −0.0115860
\(603\) 6.00000 0.244339
\(604\) −18.9117 −0.769506
\(605\) 0 0
\(606\) −7.71573 −0.313430
\(607\) −28.3431 −1.15041 −0.575206 0.818008i \(-0.695078\pi\)
−0.575206 + 0.818008i \(0.695078\pi\)
\(608\) 4.41421 0.179020
\(609\) 8.00000 0.324176
\(610\) 0 0
\(611\) 79.5980 3.22019
\(612\) −12.4853 −0.504688
\(613\) −13.1716 −0.531995 −0.265997 0.963974i \(-0.585701\pi\)
−0.265997 + 0.963974i \(0.585701\pi\)
\(614\) −11.1716 −0.450848
\(615\) 0 0
\(616\) 1.31371 0.0529308
\(617\) 21.6569 0.871872 0.435936 0.899978i \(-0.356417\pi\)
0.435936 + 0.899978i \(0.356417\pi\)
\(618\) 3.59798 0.144732
\(619\) −26.6274 −1.07025 −0.535123 0.844774i \(-0.679735\pi\)
−0.535123 + 0.844774i \(0.679735\pi\)
\(620\) 0 0
\(621\) −30.6274 −1.22904
\(622\) −0.402020 −0.0161195
\(623\) 3.59798 0.144150
\(624\) 40.9706 1.64014
\(625\) 0 0
\(626\) −7.02944 −0.280953
\(627\) 2.00000 0.0798723
\(628\) −21.9411 −0.875546
\(629\) −57.9411 −2.31026
\(630\) 0 0
\(631\) 32.9706 1.31254 0.656269 0.754527i \(-0.272134\pi\)
0.656269 + 0.754527i \(0.272134\pi\)
\(632\) 8.97056 0.356830
\(633\) 30.6274 1.21733
\(634\) 2.54416 0.101041
\(635\) 0 0
\(636\) −39.5980 −1.57016
\(637\) −43.1127 −1.70819
\(638\) −2.00000 −0.0791808
\(639\) −14.8284 −0.586604
\(640\) 0 0
\(641\) 7.65685 0.302428 0.151214 0.988501i \(-0.451682\pi\)
0.151214 + 0.988501i \(0.451682\pi\)
\(642\) −15.7157 −0.620250
\(643\) 47.9411 1.89061 0.945307 0.326183i \(-0.105762\pi\)
0.945307 + 0.326183i \(0.105762\pi\)
\(644\) 11.5980 0.457024
\(645\) 0 0
\(646\) 2.82843 0.111283
\(647\) 2.68629 0.105609 0.0528045 0.998605i \(-0.483184\pi\)
0.0528045 + 0.998605i \(0.483184\pi\)
\(648\) −17.4437 −0.685251
\(649\) 2.82843 0.111025
\(650\) 0 0
\(651\) 11.3137 0.443419
\(652\) 3.65685 0.143213
\(653\) −34.6274 −1.35508 −0.677538 0.735488i \(-0.736953\pi\)
−0.677538 + 0.735488i \(0.736953\pi\)
\(654\) −0.686292 −0.0268361
\(655\) 0 0
\(656\) 19.4558 0.759623
\(657\) 1.17157 0.0457074
\(658\) 4.00000 0.155936
\(659\) 26.6274 1.03726 0.518628 0.855000i \(-0.326443\pi\)
0.518628 + 0.855000i \(0.326443\pi\)
\(660\) 0 0
\(661\) −5.31371 −0.206679 −0.103340 0.994646i \(-0.532953\pi\)
−0.103340 + 0.994646i \(0.532953\pi\)
\(662\) 2.54416 0.0988814
\(663\) 93.2548 3.62172
\(664\) −10.2843 −0.399107
\(665\) 0 0
\(666\) 3.51472 0.136193
\(667\) −36.9706 −1.43151
\(668\) −15.2548 −0.590227
\(669\) 23.3137 0.901360
\(670\) 0 0
\(671\) 2.00000 0.0772091
\(672\) 7.31371 0.282132
\(673\) 22.1421 0.853517 0.426758 0.904366i \(-0.359656\pi\)
0.426758 + 0.904366i \(0.359656\pi\)
\(674\) −1.17157 −0.0451273
\(675\) 0 0
\(676\) −61.4853 −2.36482
\(677\) 19.5147 0.750012 0.375006 0.927022i \(-0.377641\pi\)
0.375006 + 0.927022i \(0.377641\pi\)
\(678\) −8.40202 −0.322678
\(679\) −0.970563 −0.0372468
\(680\) 0 0
\(681\) 24.6863 0.945981
\(682\) −2.82843 −0.108306
\(683\) 16.6274 0.636230 0.318115 0.948052i \(-0.396950\pi\)
0.318115 + 0.948052i \(0.396950\pi\)
\(684\) 1.82843 0.0699117
\(685\) 0 0
\(686\) −4.56854 −0.174428
\(687\) −44.0000 −1.67870
\(688\) −2.48528 −0.0947505
\(689\) 73.9411 2.81693
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) −22.8284 −0.867807
\(693\) 0.828427 0.0314693
\(694\) −10.2843 −0.390386
\(695\) 0 0
\(696\) −15.3137 −0.580465
\(697\) 44.2843 1.67739
\(698\) 11.4558 0.433610
\(699\) −8.97056 −0.339298
\(700\) 0 0
\(701\) 34.2843 1.29490 0.647450 0.762108i \(-0.275836\pi\)
0.647450 + 0.762108i \(0.275836\pi\)
\(702\) 11.3137 0.427008
\(703\) 8.48528 0.320028
\(704\) 4.17157 0.157222
\(705\) 0 0
\(706\) 0 0
\(707\) −7.71573 −0.290180
\(708\) 10.3431 0.388719
\(709\) 6.68629 0.251109 0.125554 0.992087i \(-0.459929\pi\)
0.125554 + 0.992087i \(0.459929\pi\)
\(710\) 0 0
\(711\) 5.65685 0.212149
\(712\) −6.88730 −0.258113
\(713\) −52.2843 −1.95806
\(714\) 4.68629 0.175380
\(715\) 0 0
\(716\) −30.1421 −1.12646
\(717\) 9.37258 0.350026
\(718\) −7.02944 −0.262336
\(719\) 14.6274 0.545511 0.272755 0.962083i \(-0.412065\pi\)
0.272755 + 0.962083i \(0.412065\pi\)
\(720\) 0 0
\(721\) 3.59798 0.133996
\(722\) −0.414214 −0.0154154
\(723\) 12.9706 0.482380
\(724\) −25.5980 −0.951341
\(725\) 0 0
\(726\) −0.828427 −0.0307458
\(727\) −13.3137 −0.493778 −0.246889 0.969044i \(-0.579408\pi\)
−0.246889 + 0.969044i \(0.579408\pi\)
\(728\) −8.97056 −0.332471
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −5.65685 −0.209226
\(732\) 7.31371 0.270322
\(733\) −8.48528 −0.313411 −0.156706 0.987645i \(-0.550087\pi\)
−0.156706 + 0.987645i \(0.550087\pi\)
\(734\) 8.42641 0.311024
\(735\) 0 0
\(736\) −33.7990 −1.24585
\(737\) −6.00000 −0.221013
\(738\) −2.68629 −0.0988838
\(739\) −44.0000 −1.61857 −0.809283 0.587419i \(-0.800144\pi\)
−0.809283 + 0.587419i \(0.800144\pi\)
\(740\) 0 0
\(741\) −13.6569 −0.501697
\(742\) 3.71573 0.136409
\(743\) 10.9706 0.402471 0.201235 0.979543i \(-0.435504\pi\)
0.201235 + 0.979543i \(0.435504\pi\)
\(744\) −21.6569 −0.793979
\(745\) 0 0
\(746\) 7.11270 0.260414
\(747\) −6.48528 −0.237284
\(748\) 12.4853 0.456507
\(749\) −15.7157 −0.574240
\(750\) 0 0
\(751\) 36.4853 1.33137 0.665683 0.746234i \(-0.268140\pi\)
0.665683 + 0.746234i \(0.268140\pi\)
\(752\) 34.9706 1.27525
\(753\) −8.00000 −0.291536
\(754\) 13.6569 0.497353
\(755\) 0 0
\(756\) −6.05887 −0.220359
\(757\) 20.0000 0.726912 0.363456 0.931611i \(-0.381597\pi\)
0.363456 + 0.931611i \(0.381597\pi\)
\(758\) −4.48528 −0.162913
\(759\) −15.3137 −0.555852
\(760\) 0 0
\(761\) −47.9411 −1.73786 −0.868932 0.494931i \(-0.835193\pi\)
−0.868932 + 0.494931i \(0.835193\pi\)
\(762\) −11.5980 −0.420150
\(763\) −0.686292 −0.0248454
\(764\) −31.0294 −1.12261
\(765\) 0 0
\(766\) 11.1716 0.403645
\(767\) −19.3137 −0.697378
\(768\) 7.94113 0.286551
\(769\) 8.34315 0.300862 0.150431 0.988621i \(-0.451934\pi\)
0.150431 + 0.988621i \(0.451934\pi\)
\(770\) 0 0
\(771\) −36.2843 −1.30675
\(772\) 25.8579 0.930645
\(773\) −5.17157 −0.186009 −0.0930043 0.995666i \(-0.529647\pi\)
−0.0930043 + 0.995666i \(0.529647\pi\)
\(774\) 0.343146 0.0123341
\(775\) 0 0
\(776\) 1.85786 0.0666934
\(777\) 14.0589 0.504359
\(778\) −5.51472 −0.197712
\(779\) −6.48528 −0.232359
\(780\) 0 0
\(781\) 14.8284 0.530603
\(782\) −21.6569 −0.774448
\(783\) 19.3137 0.690216
\(784\) −18.9411 −0.676469
\(785\) 0 0
\(786\) −6.05887 −0.216113
\(787\) 2.97056 0.105889 0.0529446 0.998597i \(-0.483139\pi\)
0.0529446 + 0.998597i \(0.483139\pi\)
\(788\) 11.2304 0.400068
\(789\) −36.9706 −1.31619
\(790\) 0 0
\(791\) −8.40202 −0.298741
\(792\) −1.58579 −0.0563485
\(793\) −13.6569 −0.484969
\(794\) −3.02944 −0.107511
\(795\) 0 0
\(796\) 10.3431 0.366603
\(797\) −10.8284 −0.383563 −0.191781 0.981438i \(-0.561426\pi\)
−0.191781 + 0.981438i \(0.561426\pi\)
\(798\) −0.686292 −0.0242945
\(799\) 79.5980 2.81597
\(800\) 0 0
\(801\) −4.34315 −0.153458
\(802\) −2.20101 −0.0777204
\(803\) −1.17157 −0.0413439
\(804\) −21.9411 −0.773804
\(805\) 0 0
\(806\) 19.3137 0.680296
\(807\) −45.9411 −1.61720
\(808\) 14.7696 0.519591
\(809\) −53.3137 −1.87441 −0.937205 0.348779i \(-0.886596\pi\)
−0.937205 + 0.348779i \(0.886596\pi\)
\(810\) 0 0
\(811\) −16.2843 −0.571818 −0.285909 0.958257i \(-0.592296\pi\)
−0.285909 + 0.958257i \(0.592296\pi\)
\(812\) −7.31371 −0.256661
\(813\) 22.6274 0.793578
\(814\) −3.51472 −0.123191
\(815\) 0 0
\(816\) 40.9706 1.43426
\(817\) 0.828427 0.0289830
\(818\) 10.2843 0.359581
\(819\) −5.65685 −0.197666
\(820\) 0 0
\(821\) −51.9411 −1.81276 −0.906379 0.422466i \(-0.861165\pi\)
−0.906379 + 0.422466i \(0.861165\pi\)
\(822\) −6.62742 −0.231158
\(823\) −9.31371 −0.324655 −0.162328 0.986737i \(-0.551900\pi\)
−0.162328 + 0.986737i \(0.551900\pi\)
\(824\) −6.88730 −0.239931
\(825\) 0 0
\(826\) −0.970563 −0.0337702
\(827\) 14.6863 0.510692 0.255346 0.966850i \(-0.417811\pi\)
0.255346 + 0.966850i \(0.417811\pi\)
\(828\) −14.0000 −0.486534
\(829\) 52.9117 1.83770 0.918849 0.394608i \(-0.129120\pi\)
0.918849 + 0.394608i \(0.129120\pi\)
\(830\) 0 0
\(831\) −37.6569 −1.30630
\(832\) −28.4853 −0.987549
\(833\) −43.1127 −1.49377
\(834\) −3.31371 −0.114744
\(835\) 0 0
\(836\) −1.82843 −0.0632375
\(837\) 27.3137 0.944100
\(838\) 8.68629 0.300063
\(839\) 7.79899 0.269251 0.134626 0.990897i \(-0.457017\pi\)
0.134626 + 0.990897i \(0.457017\pi\)
\(840\) 0 0
\(841\) −5.68629 −0.196079
\(842\) 11.1716 0.384998
\(843\) −38.9117 −1.34019
\(844\) −28.0000 −0.963800
\(845\) 0 0
\(846\) −4.82843 −0.166005
\(847\) −0.828427 −0.0284651
\(848\) 32.4853 1.11555
\(849\) −12.9706 −0.445149
\(850\) 0 0
\(851\) −64.9706 −2.22716
\(852\) 54.2254 1.85773
\(853\) 56.4853 1.93402 0.967010 0.254740i \(-0.0819899\pi\)
0.967010 + 0.254740i \(0.0819899\pi\)
\(854\) −0.686292 −0.0234844
\(855\) 0 0
\(856\) 30.0833 1.02822
\(857\) 48.0833 1.64249 0.821246 0.570574i \(-0.193279\pi\)
0.821246 + 0.570574i \(0.193279\pi\)
\(858\) 5.65685 0.193122
\(859\) −47.3137 −1.61432 −0.807161 0.590331i \(-0.798997\pi\)
−0.807161 + 0.590331i \(0.798997\pi\)
\(860\) 0 0
\(861\) −10.7452 −0.366194
\(862\) −2.74517 −0.0935007
\(863\) −42.9706 −1.46273 −0.731367 0.681984i \(-0.761118\pi\)
−0.731367 + 0.681984i \(0.761118\pi\)
\(864\) 17.6569 0.600698
\(865\) 0 0
\(866\) −0.485281 −0.0164905
\(867\) 59.2548 2.01240
\(868\) −10.3431 −0.351069
\(869\) −5.65685 −0.191896
\(870\) 0 0
\(871\) 40.9706 1.38823
\(872\) 1.31371 0.0444878
\(873\) 1.17157 0.0396517
\(874\) 3.17157 0.107280
\(875\) 0 0
\(876\) −4.28427 −0.144752
\(877\) −2.14214 −0.0723348 −0.0361674 0.999346i \(-0.511515\pi\)
−0.0361674 + 0.999346i \(0.511515\pi\)
\(878\) −7.59798 −0.256419
\(879\) −52.2843 −1.76350
\(880\) 0 0
\(881\) −31.9411 −1.07612 −0.538062 0.842905i \(-0.680843\pi\)
−0.538062 + 0.842905i \(0.680843\pi\)
\(882\) 2.61522 0.0880592
\(883\) 33.3137 1.12110 0.560548 0.828122i \(-0.310591\pi\)
0.560548 + 0.828122i \(0.310591\pi\)
\(884\) −85.2548 −2.86743
\(885\) 0 0
\(886\) −3.85786 −0.129607
\(887\) 15.9411 0.535251 0.267625 0.963523i \(-0.413761\pi\)
0.267625 + 0.963523i \(0.413761\pi\)
\(888\) −26.9117 −0.903097
\(889\) −11.5980 −0.388984
\(890\) 0 0
\(891\) 11.0000 0.368514
\(892\) −21.3137 −0.713636
\(893\) −11.6569 −0.390082
\(894\) 6.91169 0.231161
\(895\) 0 0
\(896\) −8.74517 −0.292155
\(897\) 104.569 3.49144
\(898\) 1.23045 0.0410606
\(899\) 32.9706 1.09963
\(900\) 0 0
\(901\) 73.9411 2.46334
\(902\) 2.68629 0.0894437
\(903\) 1.37258 0.0456767
\(904\) 16.0833 0.534921
\(905\) 0 0
\(906\) −8.56854 −0.284671
\(907\) 28.6274 0.950558 0.475279 0.879835i \(-0.342347\pi\)
0.475279 + 0.879835i \(0.342347\pi\)
\(908\) −22.5685 −0.748963
\(909\) 9.31371 0.308916
\(910\) 0 0
\(911\) 25.1716 0.833971 0.416986 0.908913i \(-0.363087\pi\)
0.416986 + 0.908913i \(0.363087\pi\)
\(912\) −6.00000 −0.198680
\(913\) 6.48528 0.214631
\(914\) 6.54416 0.216461
\(915\) 0 0
\(916\) 40.2254 1.32908
\(917\) −6.05887 −0.200082
\(918\) 11.3137 0.373408
\(919\) −48.9706 −1.61539 −0.807695 0.589601i \(-0.799285\pi\)
−0.807695 + 0.589601i \(0.799285\pi\)
\(920\) 0 0
\(921\) 53.9411 1.77742
\(922\) 9.23045 0.303989
\(923\) −101.255 −3.33284
\(924\) −3.02944 −0.0996612
\(925\) 0 0
\(926\) −12.5442 −0.412227
\(927\) −4.34315 −0.142648
\(928\) 21.3137 0.699657
\(929\) 55.9411 1.83537 0.917684 0.397310i \(-0.130056\pi\)
0.917684 + 0.397310i \(0.130056\pi\)
\(930\) 0 0
\(931\) 6.31371 0.206923
\(932\) 8.20101 0.268633
\(933\) 1.94113 0.0635496
\(934\) −1.79899 −0.0588647
\(935\) 0 0
\(936\) 10.8284 0.353938
\(937\) −19.1127 −0.624385 −0.312192 0.950019i \(-0.601063\pi\)
−0.312192 + 0.950019i \(0.601063\pi\)
\(938\) 2.05887 0.0672246
\(939\) 33.9411 1.10763
\(940\) 0 0
\(941\) −23.8579 −0.777744 −0.388872 0.921292i \(-0.627135\pi\)
−0.388872 + 0.921292i \(0.627135\pi\)
\(942\) −9.94113 −0.323899
\(943\) 49.6569 1.61705
\(944\) −8.48528 −0.276172
\(945\) 0 0
\(946\) −0.343146 −0.0111566
\(947\) −58.2843 −1.89398 −0.946992 0.321257i \(-0.895895\pi\)
−0.946992 + 0.321257i \(0.895895\pi\)
\(948\) −20.6863 −0.671860
\(949\) 8.00000 0.259691
\(950\) 0 0
\(951\) −12.2843 −0.398345
\(952\) −8.97056 −0.290738
\(953\) 39.1127 1.26698 0.633492 0.773749i \(-0.281621\pi\)
0.633492 + 0.773749i \(0.281621\pi\)
\(954\) −4.48528 −0.145216
\(955\) 0 0
\(956\) −8.56854 −0.277126
\(957\) 9.65685 0.312162
\(958\) 1.37258 0.0443461
\(959\) −6.62742 −0.214010
\(960\) 0 0
\(961\) 15.6274 0.504110
\(962\) 24.0000 0.773791
\(963\) 18.9706 0.611318
\(964\) −11.8579 −0.381916
\(965\) 0 0
\(966\) 5.25483 0.169072
\(967\) −12.1421 −0.390465 −0.195232 0.980757i \(-0.562546\pi\)
−0.195232 + 0.980757i \(0.562546\pi\)
\(968\) 1.58579 0.0509691
\(969\) −13.6569 −0.438721
\(970\) 0 0
\(971\) −15.1127 −0.484990 −0.242495 0.970153i \(-0.577966\pi\)
−0.242495 + 0.970153i \(0.577966\pi\)
\(972\) 18.2843 0.586468
\(973\) −3.31371 −0.106233
\(974\) −6.48528 −0.207802
\(975\) 0 0
\(976\) −6.00000 −0.192055
\(977\) −7.79899 −0.249512 −0.124756 0.992187i \(-0.539815\pi\)
−0.124756 + 0.992187i \(0.539815\pi\)
\(978\) 1.65685 0.0529804
\(979\) 4.34315 0.138808
\(980\) 0 0
\(981\) 0.828427 0.0264496
\(982\) 15.3137 0.488680
\(983\) 8.34315 0.266105 0.133053 0.991109i \(-0.457522\pi\)
0.133053 + 0.991109i \(0.457522\pi\)
\(984\) 20.5685 0.655701
\(985\) 0 0
\(986\) 13.6569 0.434923
\(987\) −19.3137 −0.614762
\(988\) 12.4853 0.397210
\(989\) −6.34315 −0.201700
\(990\) 0 0
\(991\) 22.8284 0.725169 0.362584 0.931951i \(-0.381894\pi\)
0.362584 + 0.931951i \(0.381894\pi\)
\(992\) 30.1421 0.957014
\(993\) −12.2843 −0.389830
\(994\) −5.08831 −0.161391
\(995\) 0 0
\(996\) 23.7157 0.751462
\(997\) −16.4853 −0.522094 −0.261047 0.965326i \(-0.584068\pi\)
−0.261047 + 0.965326i \(0.584068\pi\)
\(998\) −0.284271 −0.00899845
\(999\) 33.9411 1.07385
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 5225.2.a.g.1.1 2
5.2 odd 4 1045.2.b.a.419.2 4
5.3 odd 4 1045.2.b.a.419.3 yes 4
5.4 even 2 5225.2.a.d.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.b.a.419.2 4 5.2 odd 4
1045.2.b.a.419.3 yes 4 5.3 odd 4
5225.2.a.d.1.2 2 5.4 even 2
5225.2.a.g.1.1 2 1.1 even 1 trivial