Properties

Label 825.2.a.f.1.1
Level $825$
Weight $2$
Character 825.1
Self dual yes
Analytic conductor $6.588$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [825,2,Mod(1,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, 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("825.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.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: \(6.58765816676\)
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} -0.414214 q^{6} -2.41421 q^{7} +1.58579 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} -0.414214 q^{6} -2.41421 q^{7} +1.58579 q^{8} +1.00000 q^{9} -1.00000 q^{11} -1.82843 q^{12} +2.82843 q^{13} +1.00000 q^{14} +3.00000 q^{16} -0.414214 q^{17} -0.414214 q^{18} +3.58579 q^{19} -2.41421 q^{21} +0.414214 q^{22} +1.00000 q^{23} +1.58579 q^{24} -1.17157 q^{26} +1.00000 q^{27} +4.41421 q^{28} +6.82843 q^{29} +8.48528 q^{31} -4.41421 q^{32} -1.00000 q^{33} +0.171573 q^{34} -1.82843 q^{36} -5.82843 q^{37} -1.48528 q^{38} +2.82843 q^{39} +8.89949 q^{41} +1.00000 q^{42} -0.343146 q^{43} +1.82843 q^{44} -0.414214 q^{46} +9.48528 q^{47} +3.00000 q^{48} -1.17157 q^{49} -0.414214 q^{51} -5.17157 q^{52} -3.65685 q^{53} -0.414214 q^{54} -3.82843 q^{56} +3.58579 q^{57} -2.82843 q^{58} +11.0000 q^{59} +3.17157 q^{61} -3.51472 q^{62} -2.41421 q^{63} -4.17157 q^{64} +0.414214 q^{66} -11.6569 q^{67} +0.757359 q^{68} +1.00000 q^{69} +2.17157 q^{71} +1.58579 q^{72} -3.17157 q^{73} +2.41421 q^{74} -6.55635 q^{76} +2.41421 q^{77} -1.17157 q^{78} +4.75736 q^{79} +1.00000 q^{81} -3.68629 q^{82} +12.4853 q^{83} +4.41421 q^{84} +0.142136 q^{86} +6.82843 q^{87} -1.58579 q^{88} -7.65685 q^{89} -6.82843 q^{91} -1.82843 q^{92} +8.48528 q^{93} -3.92893 q^{94} -4.41421 q^{96} +0.171573 q^{97} +0.485281 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{7} + 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{7} + 6 q^{8} + 2 q^{9} - 2 q^{11} + 2 q^{12} + 2 q^{14} + 6 q^{16} + 2 q^{17} + 2 q^{18} + 10 q^{19} - 2 q^{21} - 2 q^{22} + 2 q^{23} + 6 q^{24} - 8 q^{26} + 2 q^{27} + 6 q^{28} + 8 q^{29} - 6 q^{32} - 2 q^{33} + 6 q^{34} + 2 q^{36} - 6 q^{37} + 14 q^{38} - 2 q^{41} + 2 q^{42} - 12 q^{43} - 2 q^{44} + 2 q^{46} + 2 q^{47} + 6 q^{48} - 8 q^{49} + 2 q^{51} - 16 q^{52} + 4 q^{53} + 2 q^{54} - 2 q^{56} + 10 q^{57} + 22 q^{59} + 12 q^{61} - 24 q^{62} - 2 q^{63} - 14 q^{64} - 2 q^{66} - 12 q^{67} + 10 q^{68} + 2 q^{69} + 10 q^{71} + 6 q^{72} - 12 q^{73} + 2 q^{74} + 18 q^{76} + 2 q^{77} - 8 q^{78} + 18 q^{79} + 2 q^{81} - 30 q^{82} + 8 q^{83} + 6 q^{84} - 28 q^{86} + 8 q^{87} - 6 q^{88} - 4 q^{89} - 8 q^{91} + 2 q^{92} - 22 q^{94} - 6 q^{96} + 6 q^{97} - 16 q^{98} - 2 q^{99}+O(q^{100}) \) 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 \(0.546784\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.82843 −0.914214
\(5\) 0 0
\(6\) −0.414214 −0.169102
\(7\) −2.41421 −0.912487 −0.456243 0.889855i \(-0.650805\pi\)
−0.456243 + 0.889855i \(0.650805\pi\)
\(8\) 1.58579 0.560660
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −1.82843 −0.527821
\(13\) 2.82843 0.784465 0.392232 0.919866i \(-0.371703\pi\)
0.392232 + 0.919866i \(0.371703\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) −0.414214 −0.100462 −0.0502308 0.998738i \(-0.515996\pi\)
−0.0502308 + 0.998738i \(0.515996\pi\)
\(18\) −0.414214 −0.0976311
\(19\) 3.58579 0.822636 0.411318 0.911492i \(-0.365069\pi\)
0.411318 + 0.911492i \(0.365069\pi\)
\(20\) 0 0
\(21\) −2.41421 −0.526825
\(22\) 0.414214 0.0883106
\(23\) 1.00000 0.208514 0.104257 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 1.58579 0.323697
\(25\) 0 0
\(26\) −1.17157 −0.229764
\(27\) 1.00000 0.192450
\(28\) 4.41421 0.834208
\(29\) 6.82843 1.26801 0.634004 0.773330i \(-0.281410\pi\)
0.634004 + 0.773330i \(0.281410\pi\)
\(30\) 0 0
\(31\) 8.48528 1.52400 0.762001 0.647576i \(-0.224217\pi\)
0.762001 + 0.647576i \(0.224217\pi\)
\(32\) −4.41421 −0.780330
\(33\) −1.00000 −0.174078
\(34\) 0.171573 0.0294245
\(35\) 0 0
\(36\) −1.82843 −0.304738
\(37\) −5.82843 −0.958188 −0.479094 0.877764i \(-0.659035\pi\)
−0.479094 + 0.877764i \(0.659035\pi\)
\(38\) −1.48528 −0.240944
\(39\) 2.82843 0.452911
\(40\) 0 0
\(41\) 8.89949 1.38987 0.694934 0.719074i \(-0.255434\pi\)
0.694934 + 0.719074i \(0.255434\pi\)
\(42\) 1.00000 0.154303
\(43\) −0.343146 −0.0523292 −0.0261646 0.999658i \(-0.508329\pi\)
−0.0261646 + 0.999658i \(0.508329\pi\)
\(44\) 1.82843 0.275646
\(45\) 0 0
\(46\) −0.414214 −0.0610725
\(47\) 9.48528 1.38357 0.691785 0.722103i \(-0.256824\pi\)
0.691785 + 0.722103i \(0.256824\pi\)
\(48\) 3.00000 0.433013
\(49\) −1.17157 −0.167368
\(50\) 0 0
\(51\) −0.414214 −0.0580015
\(52\) −5.17157 −0.717168
\(53\) −3.65685 −0.502308 −0.251154 0.967947i \(-0.580810\pi\)
−0.251154 + 0.967947i \(0.580810\pi\)
\(54\) −0.414214 −0.0563673
\(55\) 0 0
\(56\) −3.82843 −0.511595
\(57\) 3.58579 0.474949
\(58\) −2.82843 −0.371391
\(59\) 11.0000 1.43208 0.716039 0.698060i \(-0.245953\pi\)
0.716039 + 0.698060i \(0.245953\pi\)
\(60\) 0 0
\(61\) 3.17157 0.406078 0.203039 0.979171i \(-0.434918\pi\)
0.203039 + 0.979171i \(0.434918\pi\)
\(62\) −3.51472 −0.446370
\(63\) −2.41421 −0.304162
\(64\) −4.17157 −0.521447
\(65\) 0 0
\(66\) 0.414214 0.0509862
\(67\) −11.6569 −1.42411 −0.712056 0.702123i \(-0.752236\pi\)
−0.712056 + 0.702123i \(0.752236\pi\)
\(68\) 0.757359 0.0918433
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 2.17157 0.257718 0.128859 0.991663i \(-0.458868\pi\)
0.128859 + 0.991663i \(0.458868\pi\)
\(72\) 1.58579 0.186887
\(73\) −3.17157 −0.371205 −0.185602 0.982625i \(-0.559424\pi\)
−0.185602 + 0.982625i \(0.559424\pi\)
\(74\) 2.41421 0.280647
\(75\) 0 0
\(76\) −6.55635 −0.752065
\(77\) 2.41421 0.275125
\(78\) −1.17157 −0.132655
\(79\) 4.75736 0.535245 0.267622 0.963524i \(-0.413762\pi\)
0.267622 + 0.963524i \(0.413762\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −3.68629 −0.407083
\(83\) 12.4853 1.37044 0.685219 0.728337i \(-0.259707\pi\)
0.685219 + 0.728337i \(0.259707\pi\)
\(84\) 4.41421 0.481630
\(85\) 0 0
\(86\) 0.142136 0.0153269
\(87\) 6.82843 0.732084
\(88\) −1.58579 −0.169045
\(89\) −7.65685 −0.811625 −0.405812 0.913956i \(-0.633011\pi\)
−0.405812 + 0.913956i \(0.633011\pi\)
\(90\) 0 0
\(91\) −6.82843 −0.715814
\(92\) −1.82843 −0.190627
\(93\) 8.48528 0.879883
\(94\) −3.92893 −0.405238
\(95\) 0 0
\(96\) −4.41421 −0.450524
\(97\) 0.171573 0.0174206 0.00871029 0.999962i \(-0.497227\pi\)
0.00871029 + 0.999962i \(0.497227\pi\)
\(98\) 0.485281 0.0490208
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −4.89949 −0.487518 −0.243759 0.969836i \(-0.578381\pi\)
−0.243759 + 0.969836i \(0.578381\pi\)
\(102\) 0.171573 0.0169882
\(103\) −2.34315 −0.230877 −0.115439 0.993315i \(-0.536827\pi\)
−0.115439 + 0.993315i \(0.536827\pi\)
\(104\) 4.48528 0.439818
\(105\) 0 0
\(106\) 1.51472 0.147122
\(107\) 17.3137 1.67378 0.836890 0.547372i \(-0.184372\pi\)
0.836890 + 0.547372i \(0.184372\pi\)
\(108\) −1.82843 −0.175940
\(109\) −17.3137 −1.65835 −0.829176 0.558987i \(-0.811190\pi\)
−0.829176 + 0.558987i \(0.811190\pi\)
\(110\) 0 0
\(111\) −5.82843 −0.553210
\(112\) −7.24264 −0.684365
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) −1.48528 −0.139109
\(115\) 0 0
\(116\) −12.4853 −1.15923
\(117\) 2.82843 0.261488
\(118\) −4.55635 −0.419446
\(119\) 1.00000 0.0916698
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −1.31371 −0.118938
\(123\) 8.89949 0.802440
\(124\) −15.5147 −1.39326
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) −1.24264 −0.110267 −0.0551333 0.998479i \(-0.517558\pi\)
−0.0551333 + 0.998479i \(0.517558\pi\)
\(128\) 10.5563 0.933058
\(129\) −0.343146 −0.0302123
\(130\) 0 0
\(131\) 1.17157 0.102361 0.0511804 0.998689i \(-0.483702\pi\)
0.0511804 + 0.998689i \(0.483702\pi\)
\(132\) 1.82843 0.159144
\(133\) −8.65685 −0.750644
\(134\) 4.82843 0.417113
\(135\) 0 0
\(136\) −0.656854 −0.0563248
\(137\) 16.1421 1.37912 0.689558 0.724231i \(-0.257805\pi\)
0.689558 + 0.724231i \(0.257805\pi\)
\(138\) −0.414214 −0.0352602
\(139\) 14.9706 1.26979 0.634893 0.772600i \(-0.281044\pi\)
0.634893 + 0.772600i \(0.281044\pi\)
\(140\) 0 0
\(141\) 9.48528 0.798805
\(142\) −0.899495 −0.0754839
\(143\) −2.82843 −0.236525
\(144\) 3.00000 0.250000
\(145\) 0 0
\(146\) 1.31371 0.108723
\(147\) −1.17157 −0.0966297
\(148\) 10.6569 0.875988
\(149\) −17.7279 −1.45233 −0.726164 0.687522i \(-0.758699\pi\)
−0.726164 + 0.687522i \(0.758699\pi\)
\(150\) 0 0
\(151\) 14.0000 1.13930 0.569652 0.821886i \(-0.307078\pi\)
0.569652 + 0.821886i \(0.307078\pi\)
\(152\) 5.68629 0.461219
\(153\) −0.414214 −0.0334872
\(154\) −1.00000 −0.0805823
\(155\) 0 0
\(156\) −5.17157 −0.414057
\(157\) 6.00000 0.478852 0.239426 0.970915i \(-0.423041\pi\)
0.239426 + 0.970915i \(0.423041\pi\)
\(158\) −1.97056 −0.156770
\(159\) −3.65685 −0.290007
\(160\) 0 0
\(161\) −2.41421 −0.190267
\(162\) −0.414214 −0.0325437
\(163\) −23.7990 −1.86408 −0.932040 0.362354i \(-0.881973\pi\)
−0.932040 + 0.362354i \(0.881973\pi\)
\(164\) −16.2721 −1.27064
\(165\) 0 0
\(166\) −5.17157 −0.401392
\(167\) −17.7990 −1.37733 −0.688664 0.725081i \(-0.741802\pi\)
−0.688664 + 0.725081i \(0.741802\pi\)
\(168\) −3.82843 −0.295370
\(169\) −5.00000 −0.384615
\(170\) 0 0
\(171\) 3.58579 0.274212
\(172\) 0.627417 0.0478401
\(173\) 18.5563 1.41081 0.705407 0.708803i \(-0.250764\pi\)
0.705407 + 0.708803i \(0.250764\pi\)
\(174\) −2.82843 −0.214423
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) 11.0000 0.826811
\(178\) 3.17157 0.237719
\(179\) −22.7990 −1.70408 −0.852038 0.523480i \(-0.824634\pi\)
−0.852038 + 0.523480i \(0.824634\pi\)
\(180\) 0 0
\(181\) 11.9706 0.889765 0.444882 0.895589i \(-0.353245\pi\)
0.444882 + 0.895589i \(0.353245\pi\)
\(182\) 2.82843 0.209657
\(183\) 3.17157 0.234449
\(184\) 1.58579 0.116906
\(185\) 0 0
\(186\) −3.51472 −0.257712
\(187\) 0.414214 0.0302903
\(188\) −17.3431 −1.26488
\(189\) −2.41421 −0.175608
\(190\) 0 0
\(191\) 11.8284 0.855875 0.427937 0.903808i \(-0.359240\pi\)
0.427937 + 0.903808i \(0.359240\pi\)
\(192\) −4.17157 −0.301057
\(193\) −19.3137 −1.39023 −0.695116 0.718898i \(-0.744647\pi\)
−0.695116 + 0.718898i \(0.744647\pi\)
\(194\) −0.0710678 −0.00510237
\(195\) 0 0
\(196\) 2.14214 0.153010
\(197\) 13.2426 0.943499 0.471750 0.881733i \(-0.343623\pi\)
0.471750 + 0.881733i \(0.343623\pi\)
\(198\) 0.414214 0.0294369
\(199\) 5.17157 0.366603 0.183302 0.983057i \(-0.441322\pi\)
0.183302 + 0.983057i \(0.441322\pi\)
\(200\) 0 0
\(201\) −11.6569 −0.822211
\(202\) 2.02944 0.142791
\(203\) −16.4853 −1.15704
\(204\) 0.757359 0.0530258
\(205\) 0 0
\(206\) 0.970563 0.0676223
\(207\) 1.00000 0.0695048
\(208\) 8.48528 0.588348
\(209\) −3.58579 −0.248034
\(210\) 0 0
\(211\) 9.31371 0.641182 0.320591 0.947218i \(-0.396118\pi\)
0.320591 + 0.947218i \(0.396118\pi\)
\(212\) 6.68629 0.459216
\(213\) 2.17157 0.148794
\(214\) −7.17157 −0.490239
\(215\) 0 0
\(216\) 1.58579 0.107899
\(217\) −20.4853 −1.39063
\(218\) 7.17157 0.485720
\(219\) −3.17157 −0.214315
\(220\) 0 0
\(221\) −1.17157 −0.0788085
\(222\) 2.41421 0.162031
\(223\) −26.8284 −1.79656 −0.898282 0.439419i \(-0.855184\pi\)
−0.898282 + 0.439419i \(0.855184\pi\)
\(224\) 10.6569 0.712041
\(225\) 0 0
\(226\) 4.14214 0.275531
\(227\) −1.51472 −0.100535 −0.0502677 0.998736i \(-0.516007\pi\)
−0.0502677 + 0.998736i \(0.516007\pi\)
\(228\) −6.55635 −0.434205
\(229\) −19.4853 −1.28762 −0.643812 0.765184i \(-0.722648\pi\)
−0.643812 + 0.765184i \(0.722648\pi\)
\(230\) 0 0
\(231\) 2.41421 0.158844
\(232\) 10.8284 0.710921
\(233\) −14.5563 −0.953618 −0.476809 0.879007i \(-0.658207\pi\)
−0.476809 + 0.879007i \(0.658207\pi\)
\(234\) −1.17157 −0.0765881
\(235\) 0 0
\(236\) −20.1127 −1.30923
\(237\) 4.75736 0.309024
\(238\) −0.414214 −0.0268495
\(239\) 12.3431 0.798412 0.399206 0.916861i \(-0.369286\pi\)
0.399206 + 0.916861i \(0.369286\pi\)
\(240\) 0 0
\(241\) −14.1421 −0.910975 −0.455488 0.890242i \(-0.650535\pi\)
−0.455488 + 0.890242i \(0.650535\pi\)
\(242\) −0.414214 −0.0266267
\(243\) 1.00000 0.0641500
\(244\) −5.79899 −0.371242
\(245\) 0 0
\(246\) −3.68629 −0.235029
\(247\) 10.1421 0.645329
\(248\) 13.4558 0.854447
\(249\) 12.4853 0.791223
\(250\) 0 0
\(251\) −24.9706 −1.57613 −0.788064 0.615593i \(-0.788916\pi\)
−0.788064 + 0.615593i \(0.788916\pi\)
\(252\) 4.41421 0.278069
\(253\) −1.00000 −0.0628695
\(254\) 0.514719 0.0322963
\(255\) 0 0
\(256\) 3.97056 0.248160
\(257\) −13.3137 −0.830486 −0.415243 0.909710i \(-0.636304\pi\)
−0.415243 + 0.909710i \(0.636304\pi\)
\(258\) 0.142136 0.00884898
\(259\) 14.0711 0.874334
\(260\) 0 0
\(261\) 6.82843 0.422669
\(262\) −0.485281 −0.0299808
\(263\) −10.9706 −0.676474 −0.338237 0.941061i \(-0.609831\pi\)
−0.338237 + 0.941061i \(0.609831\pi\)
\(264\) −1.58579 −0.0975984
\(265\) 0 0
\(266\) 3.58579 0.219859
\(267\) −7.65685 −0.468592
\(268\) 21.3137 1.30194
\(269\) 23.7990 1.45105 0.725525 0.688196i \(-0.241597\pi\)
0.725525 + 0.688196i \(0.241597\pi\)
\(270\) 0 0
\(271\) −10.8995 −0.662097 −0.331049 0.943614i \(-0.607402\pi\)
−0.331049 + 0.943614i \(0.607402\pi\)
\(272\) −1.24264 −0.0753462
\(273\) −6.82843 −0.413275
\(274\) −6.68629 −0.403934
\(275\) 0 0
\(276\) −1.82843 −0.110058
\(277\) −0.828427 −0.0497754 −0.0248877 0.999690i \(-0.507923\pi\)
−0.0248877 + 0.999690i \(0.507923\pi\)
\(278\) −6.20101 −0.371912
\(279\) 8.48528 0.508001
\(280\) 0 0
\(281\) 17.9289 1.06955 0.534775 0.844994i \(-0.320396\pi\)
0.534775 + 0.844994i \(0.320396\pi\)
\(282\) −3.92893 −0.233965
\(283\) 18.8995 1.12346 0.561729 0.827321i \(-0.310136\pi\)
0.561729 + 0.827321i \(0.310136\pi\)
\(284\) −3.97056 −0.235610
\(285\) 0 0
\(286\) 1.17157 0.0692766
\(287\) −21.4853 −1.26824
\(288\) −4.41421 −0.260110
\(289\) −16.8284 −0.989907
\(290\) 0 0
\(291\) 0.171573 0.0100578
\(292\) 5.79899 0.339360
\(293\) 20.4142 1.19261 0.596306 0.802758i \(-0.296635\pi\)
0.596306 + 0.802758i \(0.296635\pi\)
\(294\) 0.485281 0.0283022
\(295\) 0 0
\(296\) −9.24264 −0.537218
\(297\) −1.00000 −0.0580259
\(298\) 7.34315 0.425377
\(299\) 2.82843 0.163572
\(300\) 0 0
\(301\) 0.828427 0.0477497
\(302\) −5.79899 −0.333694
\(303\) −4.89949 −0.281469
\(304\) 10.7574 0.616977
\(305\) 0 0
\(306\) 0.171573 0.00980817
\(307\) 29.3137 1.67302 0.836511 0.547950i \(-0.184592\pi\)
0.836511 + 0.547950i \(0.184592\pi\)
\(308\) −4.41421 −0.251523
\(309\) −2.34315 −0.133297
\(310\) 0 0
\(311\) 2.34315 0.132868 0.0664338 0.997791i \(-0.478838\pi\)
0.0664338 + 0.997791i \(0.478838\pi\)
\(312\) 4.48528 0.253929
\(313\) 1.14214 0.0645573 0.0322787 0.999479i \(-0.489724\pi\)
0.0322787 + 0.999479i \(0.489724\pi\)
\(314\) −2.48528 −0.140253
\(315\) 0 0
\(316\) −8.69848 −0.489328
\(317\) 25.1716 1.41378 0.706888 0.707325i \(-0.250098\pi\)
0.706888 + 0.707325i \(0.250098\pi\)
\(318\) 1.51472 0.0849412
\(319\) −6.82843 −0.382319
\(320\) 0 0
\(321\) 17.3137 0.966357
\(322\) 1.00000 0.0557278
\(323\) −1.48528 −0.0826433
\(324\) −1.82843 −0.101579
\(325\) 0 0
\(326\) 9.85786 0.545977
\(327\) −17.3137 −0.957450
\(328\) 14.1127 0.779243
\(329\) −22.8995 −1.26249
\(330\) 0 0
\(331\) −3.85786 −0.212047 −0.106024 0.994364i \(-0.533812\pi\)
−0.106024 + 0.994364i \(0.533812\pi\)
\(332\) −22.8284 −1.25287
\(333\) −5.82843 −0.319396
\(334\) 7.37258 0.403410
\(335\) 0 0
\(336\) −7.24264 −0.395118
\(337\) −24.1421 −1.31511 −0.657553 0.753408i \(-0.728408\pi\)
−0.657553 + 0.753408i \(0.728408\pi\)
\(338\) 2.07107 0.112651
\(339\) −10.0000 −0.543125
\(340\) 0 0
\(341\) −8.48528 −0.459504
\(342\) −1.48528 −0.0803148
\(343\) 19.7279 1.06521
\(344\) −0.544156 −0.0293389
\(345\) 0 0
\(346\) −7.68629 −0.413218
\(347\) −26.8284 −1.44023 −0.720113 0.693857i \(-0.755910\pi\)
−0.720113 + 0.693857i \(0.755910\pi\)
\(348\) −12.4853 −0.669281
\(349\) 14.4853 0.775379 0.387690 0.921790i \(-0.373273\pi\)
0.387690 + 0.921790i \(0.373273\pi\)
\(350\) 0 0
\(351\) 2.82843 0.150970
\(352\) 4.41421 0.235278
\(353\) 12.4853 0.664524 0.332262 0.943187i \(-0.392188\pi\)
0.332262 + 0.943187i \(0.392188\pi\)
\(354\) −4.55635 −0.242167
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) 1.00000 0.0529256
\(358\) 9.44365 0.499112
\(359\) 32.4853 1.71451 0.857254 0.514894i \(-0.172169\pi\)
0.857254 + 0.514894i \(0.172169\pi\)
\(360\) 0 0
\(361\) −6.14214 −0.323270
\(362\) −4.95837 −0.260606
\(363\) 1.00000 0.0524864
\(364\) 12.4853 0.654407
\(365\) 0 0
\(366\) −1.31371 −0.0686686
\(367\) −21.3137 −1.11257 −0.556283 0.830993i \(-0.687773\pi\)
−0.556283 + 0.830993i \(0.687773\pi\)
\(368\) 3.00000 0.156386
\(369\) 8.89949 0.463289
\(370\) 0 0
\(371\) 8.82843 0.458349
\(372\) −15.5147 −0.804401
\(373\) −12.3431 −0.639104 −0.319552 0.947569i \(-0.603532\pi\)
−0.319552 + 0.947569i \(0.603532\pi\)
\(374\) −0.171573 −0.00887182
\(375\) 0 0
\(376\) 15.0416 0.775713
\(377\) 19.3137 0.994707
\(378\) 1.00000 0.0514344
\(379\) 14.8284 0.761685 0.380843 0.924640i \(-0.375634\pi\)
0.380843 + 0.924640i \(0.375634\pi\)
\(380\) 0 0
\(381\) −1.24264 −0.0636624
\(382\) −4.89949 −0.250680
\(383\) 20.0000 1.02195 0.510976 0.859595i \(-0.329284\pi\)
0.510976 + 0.859595i \(0.329284\pi\)
\(384\) 10.5563 0.538701
\(385\) 0 0
\(386\) 8.00000 0.407189
\(387\) −0.343146 −0.0174431
\(388\) −0.313708 −0.0159261
\(389\) −6.34315 −0.321610 −0.160805 0.986986i \(-0.551409\pi\)
−0.160805 + 0.986986i \(0.551409\pi\)
\(390\) 0 0
\(391\) −0.414214 −0.0209477
\(392\) −1.85786 −0.0938363
\(393\) 1.17157 0.0590980
\(394\) −5.48528 −0.276344
\(395\) 0 0
\(396\) 1.82843 0.0918819
\(397\) 31.9411 1.60308 0.801540 0.597942i \(-0.204015\pi\)
0.801540 + 0.597942i \(0.204015\pi\)
\(398\) −2.14214 −0.107376
\(399\) −8.65685 −0.433385
\(400\) 0 0
\(401\) −7.79899 −0.389463 −0.194731 0.980857i \(-0.562384\pi\)
−0.194731 + 0.980857i \(0.562384\pi\)
\(402\) 4.82843 0.240820
\(403\) 24.0000 1.19553
\(404\) 8.95837 0.445696
\(405\) 0 0
\(406\) 6.82843 0.338889
\(407\) 5.82843 0.288904
\(408\) −0.656854 −0.0325191
\(409\) 24.1421 1.19375 0.596876 0.802334i \(-0.296408\pi\)
0.596876 + 0.802334i \(0.296408\pi\)
\(410\) 0 0
\(411\) 16.1421 0.796233
\(412\) 4.28427 0.211071
\(413\) −26.5563 −1.30675
\(414\) −0.414214 −0.0203575
\(415\) 0 0
\(416\) −12.4853 −0.612141
\(417\) 14.9706 0.733112
\(418\) 1.48528 0.0726475
\(419\) −8.51472 −0.415971 −0.207986 0.978132i \(-0.566691\pi\)
−0.207986 + 0.978132i \(0.566691\pi\)
\(420\) 0 0
\(421\) −27.0000 −1.31590 −0.657950 0.753062i \(-0.728576\pi\)
−0.657950 + 0.753062i \(0.728576\pi\)
\(422\) −3.85786 −0.187798
\(423\) 9.48528 0.461190
\(424\) −5.79899 −0.281624
\(425\) 0 0
\(426\) −0.899495 −0.0435807
\(427\) −7.65685 −0.370541
\(428\) −31.6569 −1.53019
\(429\) −2.82843 −0.136558
\(430\) 0 0
\(431\) 6.82843 0.328914 0.164457 0.986384i \(-0.447413\pi\)
0.164457 + 0.986384i \(0.447413\pi\)
\(432\) 3.00000 0.144338
\(433\) 9.31371 0.447588 0.223794 0.974636i \(-0.428156\pi\)
0.223794 + 0.974636i \(0.428156\pi\)
\(434\) 8.48528 0.407307
\(435\) 0 0
\(436\) 31.6569 1.51609
\(437\) 3.58579 0.171531
\(438\) 1.31371 0.0627714
\(439\) 27.7279 1.32338 0.661691 0.749777i \(-0.269839\pi\)
0.661691 + 0.749777i \(0.269839\pi\)
\(440\) 0 0
\(441\) −1.17157 −0.0557892
\(442\) 0.485281 0.0230825
\(443\) 25.9706 1.23390 0.616949 0.787003i \(-0.288368\pi\)
0.616949 + 0.787003i \(0.288368\pi\)
\(444\) 10.6569 0.505752
\(445\) 0 0
\(446\) 11.1127 0.526202
\(447\) −17.7279 −0.838502
\(448\) 10.0711 0.475813
\(449\) 10.4853 0.494831 0.247416 0.968909i \(-0.420419\pi\)
0.247416 + 0.968909i \(0.420419\pi\)
\(450\) 0 0
\(451\) −8.89949 −0.419061
\(452\) 18.2843 0.860020
\(453\) 14.0000 0.657777
\(454\) 0.627417 0.0294461
\(455\) 0 0
\(456\) 5.68629 0.266285
\(457\) 32.1421 1.50355 0.751773 0.659422i \(-0.229199\pi\)
0.751773 + 0.659422i \(0.229199\pi\)
\(458\) 8.07107 0.377136
\(459\) −0.414214 −0.0193338
\(460\) 0 0
\(461\) 40.7696 1.89883 0.949414 0.314028i \(-0.101679\pi\)
0.949414 + 0.314028i \(0.101679\pi\)
\(462\) −1.00000 −0.0465242
\(463\) 1.02944 0.0478420 0.0239210 0.999714i \(-0.492385\pi\)
0.0239210 + 0.999714i \(0.492385\pi\)
\(464\) 20.4853 0.951005
\(465\) 0 0
\(466\) 6.02944 0.279308
\(467\) −34.6274 −1.60237 −0.801183 0.598420i \(-0.795796\pi\)
−0.801183 + 0.598420i \(0.795796\pi\)
\(468\) −5.17157 −0.239056
\(469\) 28.1421 1.29948
\(470\) 0 0
\(471\) 6.00000 0.276465
\(472\) 17.4437 0.802909
\(473\) 0.343146 0.0157779
\(474\) −1.97056 −0.0905109
\(475\) 0 0
\(476\) −1.82843 −0.0838058
\(477\) −3.65685 −0.167436
\(478\) −5.11270 −0.233849
\(479\) −7.51472 −0.343356 −0.171678 0.985153i \(-0.554919\pi\)
−0.171678 + 0.985153i \(0.554919\pi\)
\(480\) 0 0
\(481\) −16.4853 −0.751664
\(482\) 5.85786 0.266818
\(483\) −2.41421 −0.109851
\(484\) −1.82843 −0.0831103
\(485\) 0 0
\(486\) −0.414214 −0.0187891
\(487\) 10.4853 0.475133 0.237567 0.971371i \(-0.423650\pi\)
0.237567 + 0.971371i \(0.423650\pi\)
\(488\) 5.02944 0.227672
\(489\) −23.7990 −1.07623
\(490\) 0 0
\(491\) 4.14214 0.186932 0.0934660 0.995622i \(-0.470205\pi\)
0.0934660 + 0.995622i \(0.470205\pi\)
\(492\) −16.2721 −0.733602
\(493\) −2.82843 −0.127386
\(494\) −4.20101 −0.189012
\(495\) 0 0
\(496\) 25.4558 1.14300
\(497\) −5.24264 −0.235165
\(498\) −5.17157 −0.231744
\(499\) 40.8284 1.82773 0.913866 0.406017i \(-0.133082\pi\)
0.913866 + 0.406017i \(0.133082\pi\)
\(500\) 0 0
\(501\) −17.7990 −0.795200
\(502\) 10.3431 0.461637
\(503\) 22.2843 0.993607 0.496803 0.867863i \(-0.334507\pi\)
0.496803 + 0.867863i \(0.334507\pi\)
\(504\) −3.82843 −0.170532
\(505\) 0 0
\(506\) 0.414214 0.0184140
\(507\) −5.00000 −0.222058
\(508\) 2.27208 0.100807
\(509\) −40.6274 −1.80078 −0.900389 0.435085i \(-0.856718\pi\)
−0.900389 + 0.435085i \(0.856718\pi\)
\(510\) 0 0
\(511\) 7.65685 0.338719
\(512\) −22.7574 −1.00574
\(513\) 3.58579 0.158316
\(514\) 5.51472 0.243244
\(515\) 0 0
\(516\) 0.627417 0.0276205
\(517\) −9.48528 −0.417162
\(518\) −5.82843 −0.256086
\(519\) 18.5563 0.814533
\(520\) 0 0
\(521\) −7.85786 −0.344259 −0.172130 0.985074i \(-0.555065\pi\)
−0.172130 + 0.985074i \(0.555065\pi\)
\(522\) −2.82843 −0.123797
\(523\) −0.213203 −0.00932274 −0.00466137 0.999989i \(-0.501484\pi\)
−0.00466137 + 0.999989i \(0.501484\pi\)
\(524\) −2.14214 −0.0935796
\(525\) 0 0
\(526\) 4.54416 0.198135
\(527\) −3.51472 −0.153104
\(528\) −3.00000 −0.130558
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) 11.0000 0.477359
\(532\) 15.8284 0.686249
\(533\) 25.1716 1.09030
\(534\) 3.17157 0.137247
\(535\) 0 0
\(536\) −18.4853 −0.798443
\(537\) −22.7990 −0.983849
\(538\) −9.85786 −0.425003
\(539\) 1.17157 0.0504632
\(540\) 0 0
\(541\) −11.3137 −0.486414 −0.243207 0.969974i \(-0.578199\pi\)
−0.243207 + 0.969974i \(0.578199\pi\)
\(542\) 4.51472 0.193924
\(543\) 11.9706 0.513706
\(544\) 1.82843 0.0783932
\(545\) 0 0
\(546\) 2.82843 0.121046
\(547\) −17.8701 −0.764068 −0.382034 0.924148i \(-0.624776\pi\)
−0.382034 + 0.924148i \(0.624776\pi\)
\(548\) −29.5147 −1.26081
\(549\) 3.17157 0.135359
\(550\) 0 0
\(551\) 24.4853 1.04311
\(552\) 1.58579 0.0674956
\(553\) −11.4853 −0.488404
\(554\) 0.343146 0.0145789
\(555\) 0 0
\(556\) −27.3726 −1.16086
\(557\) −10.8284 −0.458815 −0.229408 0.973330i \(-0.573679\pi\)
−0.229408 + 0.973330i \(0.573679\pi\)
\(558\) −3.51472 −0.148790
\(559\) −0.970563 −0.0410504
\(560\) 0 0
\(561\) 0.414214 0.0174881
\(562\) −7.42641 −0.313264
\(563\) −7.31371 −0.308236 −0.154118 0.988052i \(-0.549254\pi\)
−0.154118 + 0.988052i \(0.549254\pi\)
\(564\) −17.3431 −0.730278
\(565\) 0 0
\(566\) −7.82843 −0.329053
\(567\) −2.41421 −0.101387
\(568\) 3.44365 0.144492
\(569\) 6.75736 0.283283 0.141642 0.989918i \(-0.454762\pi\)
0.141642 + 0.989918i \(0.454762\pi\)
\(570\) 0 0
\(571\) −42.9706 −1.79826 −0.899131 0.437680i \(-0.855800\pi\)
−0.899131 + 0.437680i \(0.855800\pi\)
\(572\) 5.17157 0.216234
\(573\) 11.8284 0.494140
\(574\) 8.89949 0.371458
\(575\) 0 0
\(576\) −4.17157 −0.173816
\(577\) −9.97056 −0.415080 −0.207540 0.978227i \(-0.566546\pi\)
−0.207540 + 0.978227i \(0.566546\pi\)
\(578\) 6.97056 0.289937
\(579\) −19.3137 −0.802650
\(580\) 0 0
\(581\) −30.1421 −1.25051
\(582\) −0.0710678 −0.00294586
\(583\) 3.65685 0.151451
\(584\) −5.02944 −0.208120
\(585\) 0 0
\(586\) −8.45584 −0.349308
\(587\) −25.3431 −1.04602 −0.523012 0.852325i \(-0.675192\pi\)
−0.523012 + 0.852325i \(0.675192\pi\)
\(588\) 2.14214 0.0883402
\(589\) 30.4264 1.25370
\(590\) 0 0
\(591\) 13.2426 0.544729
\(592\) −17.4853 −0.718641
\(593\) −35.7990 −1.47009 −0.735044 0.678019i \(-0.762839\pi\)
−0.735044 + 0.678019i \(0.762839\pi\)
\(594\) 0.414214 0.0169954
\(595\) 0 0
\(596\) 32.4142 1.32774
\(597\) 5.17157 0.211658
\(598\) −1.17157 −0.0479092
\(599\) 13.6863 0.559207 0.279603 0.960116i \(-0.409797\pi\)
0.279603 + 0.960116i \(0.409797\pi\)
\(600\) 0 0
\(601\) −9.17157 −0.374116 −0.187058 0.982349i \(-0.559895\pi\)
−0.187058 + 0.982349i \(0.559895\pi\)
\(602\) −0.343146 −0.0139856
\(603\) −11.6569 −0.474704
\(604\) −25.5980 −1.04157
\(605\) 0 0
\(606\) 2.02944 0.0824403
\(607\) 30.9706 1.25706 0.628528 0.777787i \(-0.283658\pi\)
0.628528 + 0.777787i \(0.283658\pi\)
\(608\) −15.8284 −0.641927
\(609\) −16.4853 −0.668017
\(610\) 0 0
\(611\) 26.8284 1.08536
\(612\) 0.757359 0.0306144
\(613\) −42.0000 −1.69636 −0.848182 0.529705i \(-0.822303\pi\)
−0.848182 + 0.529705i \(0.822303\pi\)
\(614\) −12.1421 −0.490017
\(615\) 0 0
\(616\) 3.82843 0.154252
\(617\) −38.1421 −1.53554 −0.767772 0.640723i \(-0.778635\pi\)
−0.767772 + 0.640723i \(0.778635\pi\)
\(618\) 0.970563 0.0390418
\(619\) −6.62742 −0.266378 −0.133189 0.991091i \(-0.542522\pi\)
−0.133189 + 0.991091i \(0.542522\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) −0.970563 −0.0389160
\(623\) 18.4853 0.740597
\(624\) 8.48528 0.339683
\(625\) 0 0
\(626\) −0.473088 −0.0189084
\(627\) −3.58579 −0.143203
\(628\) −10.9706 −0.437773
\(629\) 2.41421 0.0962610
\(630\) 0 0
\(631\) −18.6274 −0.741546 −0.370773 0.928724i \(-0.620907\pi\)
−0.370773 + 0.928724i \(0.620907\pi\)
\(632\) 7.54416 0.300090
\(633\) 9.31371 0.370187
\(634\) −10.4264 −0.414086
\(635\) 0 0
\(636\) 6.68629 0.265129
\(637\) −3.31371 −0.131294
\(638\) 2.82843 0.111979
\(639\) 2.17157 0.0859061
\(640\) 0 0
\(641\) −25.5147 −1.00777 −0.503885 0.863771i \(-0.668097\pi\)
−0.503885 + 0.863771i \(0.668097\pi\)
\(642\) −7.17157 −0.283039
\(643\) 0.970563 0.0382753 0.0191376 0.999817i \(-0.493908\pi\)
0.0191376 + 0.999817i \(0.493908\pi\)
\(644\) 4.41421 0.173944
\(645\) 0 0
\(646\) 0.615224 0.0242057
\(647\) 28.6569 1.12662 0.563309 0.826247i \(-0.309528\pi\)
0.563309 + 0.826247i \(0.309528\pi\)
\(648\) 1.58579 0.0622956
\(649\) −11.0000 −0.431788
\(650\) 0 0
\(651\) −20.4853 −0.802881
\(652\) 43.5147 1.70417
\(653\) −23.5147 −0.920202 −0.460101 0.887867i \(-0.652187\pi\)
−0.460101 + 0.887867i \(0.652187\pi\)
\(654\) 7.17157 0.280431
\(655\) 0 0
\(656\) 26.6985 1.04240
\(657\) −3.17157 −0.123735
\(658\) 9.48528 0.369775
\(659\) −47.1127 −1.83525 −0.917625 0.397447i \(-0.869896\pi\)
−0.917625 + 0.397447i \(0.869896\pi\)
\(660\) 0 0
\(661\) −23.3431 −0.907943 −0.453972 0.891016i \(-0.649993\pi\)
−0.453972 + 0.891016i \(0.649993\pi\)
\(662\) 1.59798 0.0621072
\(663\) −1.17157 −0.0455001
\(664\) 19.7990 0.768350
\(665\) 0 0
\(666\) 2.41421 0.0935489
\(667\) 6.82843 0.264398
\(668\) 32.5442 1.25917
\(669\) −26.8284 −1.03725
\(670\) 0 0
\(671\) −3.17157 −0.122437
\(672\) 10.6569 0.411097
\(673\) 0.343146 0.0132273 0.00661365 0.999978i \(-0.497895\pi\)
0.00661365 + 0.999978i \(0.497895\pi\)
\(674\) 10.0000 0.385186
\(675\) 0 0
\(676\) 9.14214 0.351621
\(677\) 23.5147 0.903744 0.451872 0.892083i \(-0.350756\pi\)
0.451872 + 0.892083i \(0.350756\pi\)
\(678\) 4.14214 0.159078
\(679\) −0.414214 −0.0158961
\(680\) 0 0
\(681\) −1.51472 −0.0580441
\(682\) 3.51472 0.134586
\(683\) 12.5147 0.478862 0.239431 0.970913i \(-0.423039\pi\)
0.239431 + 0.970913i \(0.423039\pi\)
\(684\) −6.55635 −0.250688
\(685\) 0 0
\(686\) −8.17157 −0.311992
\(687\) −19.4853 −0.743410
\(688\) −1.02944 −0.0392469
\(689\) −10.3431 −0.394042
\(690\) 0 0
\(691\) −35.4558 −1.34880 −0.674402 0.738364i \(-0.735598\pi\)
−0.674402 + 0.738364i \(0.735598\pi\)
\(692\) −33.9289 −1.28978
\(693\) 2.41421 0.0917084
\(694\) 11.1127 0.421832
\(695\) 0 0
\(696\) 10.8284 0.410450
\(697\) −3.68629 −0.139628
\(698\) −6.00000 −0.227103
\(699\) −14.5563 −0.550572
\(700\) 0 0
\(701\) −41.3848 −1.56308 −0.781541 0.623854i \(-0.785566\pi\)
−0.781541 + 0.623854i \(0.785566\pi\)
\(702\) −1.17157 −0.0442182
\(703\) −20.8995 −0.788239
\(704\) 4.17157 0.157222
\(705\) 0 0
\(706\) −5.17157 −0.194635
\(707\) 11.8284 0.444854
\(708\) −20.1127 −0.755881
\(709\) −29.1421 −1.09446 −0.547228 0.836984i \(-0.684317\pi\)
−0.547228 + 0.836984i \(0.684317\pi\)
\(710\) 0 0
\(711\) 4.75736 0.178415
\(712\) −12.1421 −0.455046
\(713\) 8.48528 0.317776
\(714\) −0.414214 −0.0155016
\(715\) 0 0
\(716\) 41.6863 1.55789
\(717\) 12.3431 0.460963
\(718\) −13.4558 −0.502168
\(719\) 9.65685 0.360140 0.180070 0.983654i \(-0.442368\pi\)
0.180070 + 0.983654i \(0.442368\pi\)
\(720\) 0 0
\(721\) 5.65685 0.210672
\(722\) 2.54416 0.0946837
\(723\) −14.1421 −0.525952
\(724\) −21.8873 −0.813435
\(725\) 0 0
\(726\) −0.414214 −0.0153729
\(727\) 16.9706 0.629403 0.314702 0.949191i \(-0.398096\pi\)
0.314702 + 0.949191i \(0.398096\pi\)
\(728\) −10.8284 −0.401328
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0.142136 0.00525708
\(732\) −5.79899 −0.214337
\(733\) 32.1421 1.18720 0.593598 0.804761i \(-0.297707\pi\)
0.593598 + 0.804761i \(0.297707\pi\)
\(734\) 8.82843 0.325863
\(735\) 0 0
\(736\) −4.41421 −0.162710
\(737\) 11.6569 0.429386
\(738\) −3.68629 −0.135694
\(739\) 20.4142 0.750949 0.375474 0.926833i \(-0.377480\pi\)
0.375474 + 0.926833i \(0.377480\pi\)
\(740\) 0 0
\(741\) 10.1421 0.372581
\(742\) −3.65685 −0.134247
\(743\) −31.1127 −1.14141 −0.570707 0.821154i \(-0.693331\pi\)
−0.570707 + 0.821154i \(0.693331\pi\)
\(744\) 13.4558 0.493315
\(745\) 0 0
\(746\) 5.11270 0.187189
\(747\) 12.4853 0.456813
\(748\) −0.757359 −0.0276918
\(749\) −41.7990 −1.52730
\(750\) 0 0
\(751\) 31.5980 1.15303 0.576513 0.817088i \(-0.304413\pi\)
0.576513 + 0.817088i \(0.304413\pi\)
\(752\) 28.4558 1.03768
\(753\) −24.9706 −0.909978
\(754\) −8.00000 −0.291343
\(755\) 0 0
\(756\) 4.41421 0.160543
\(757\) −10.6863 −0.388400 −0.194200 0.980962i \(-0.562211\pi\)
−0.194200 + 0.980962i \(0.562211\pi\)
\(758\) −6.14214 −0.223092
\(759\) −1.00000 −0.0362977
\(760\) 0 0
\(761\) −5.17157 −0.187469 −0.0937347 0.995597i \(-0.529881\pi\)
−0.0937347 + 0.995597i \(0.529881\pi\)
\(762\) 0.514719 0.0186463
\(763\) 41.7990 1.51323
\(764\) −21.6274 −0.782452
\(765\) 0 0
\(766\) −8.28427 −0.299323
\(767\) 31.1127 1.12341
\(768\) 3.97056 0.143275
\(769\) −7.65685 −0.276113 −0.138057 0.990424i \(-0.544086\pi\)
−0.138057 + 0.990424i \(0.544086\pi\)
\(770\) 0 0
\(771\) −13.3137 −0.479481
\(772\) 35.3137 1.27097
\(773\) −0.828427 −0.0297965 −0.0148982 0.999889i \(-0.504742\pi\)
−0.0148982 + 0.999889i \(0.504742\pi\)
\(774\) 0.142136 0.00510896
\(775\) 0 0
\(776\) 0.272078 0.00976703
\(777\) 14.0711 0.504797
\(778\) 2.62742 0.0941975
\(779\) 31.9117 1.14335
\(780\) 0 0
\(781\) −2.17157 −0.0777050
\(782\) 0.171573 0.00613543
\(783\) 6.82843 0.244028
\(784\) −3.51472 −0.125526
\(785\) 0 0
\(786\) −0.485281 −0.0173094
\(787\) −7.92893 −0.282636 −0.141318 0.989964i \(-0.545134\pi\)
−0.141318 + 0.989964i \(0.545134\pi\)
\(788\) −24.2132 −0.862560
\(789\) −10.9706 −0.390562
\(790\) 0 0
\(791\) 24.1421 0.858396
\(792\) −1.58579 −0.0563485
\(793\) 8.97056 0.318554
\(794\) −13.2304 −0.469531
\(795\) 0 0
\(796\) −9.45584 −0.335154
\(797\) 40.9706 1.45125 0.725626 0.688089i \(-0.241550\pi\)
0.725626 + 0.688089i \(0.241550\pi\)
\(798\) 3.58579 0.126935
\(799\) −3.92893 −0.138996
\(800\) 0 0
\(801\) −7.65685 −0.270542
\(802\) 3.23045 0.114071
\(803\) 3.17157 0.111922
\(804\) 21.3137 0.751677
\(805\) 0 0
\(806\) −9.94113 −0.350161
\(807\) 23.7990 0.837764
\(808\) −7.76955 −0.273332
\(809\) 7.72792 0.271699 0.135850 0.990729i \(-0.456624\pi\)
0.135850 + 0.990729i \(0.456624\pi\)
\(810\) 0 0
\(811\) 10.2132 0.358634 0.179317 0.983791i \(-0.442611\pi\)
0.179317 + 0.983791i \(0.442611\pi\)
\(812\) 30.1421 1.05778
\(813\) −10.8995 −0.382262
\(814\) −2.41421 −0.0846181
\(815\) 0 0
\(816\) −1.24264 −0.0435011
\(817\) −1.23045 −0.0430479
\(818\) −10.0000 −0.349642
\(819\) −6.82843 −0.238605
\(820\) 0 0
\(821\) 15.7990 0.551389 0.275694 0.961245i \(-0.411092\pi\)
0.275694 + 0.961245i \(0.411092\pi\)
\(822\) −6.68629 −0.233211
\(823\) −14.9706 −0.521841 −0.260921 0.965360i \(-0.584026\pi\)
−0.260921 + 0.965360i \(0.584026\pi\)
\(824\) −3.71573 −0.129444
\(825\) 0 0
\(826\) 11.0000 0.382739
\(827\) −12.6863 −0.441146 −0.220573 0.975371i \(-0.570793\pi\)
−0.220573 + 0.975371i \(0.570793\pi\)
\(828\) −1.82843 −0.0635422
\(829\) −47.9411 −1.66506 −0.832532 0.553977i \(-0.813110\pi\)
−0.832532 + 0.553977i \(0.813110\pi\)
\(830\) 0 0
\(831\) −0.828427 −0.0287378
\(832\) −11.7990 −0.409056
\(833\) 0.485281 0.0168140
\(834\) −6.20101 −0.214723
\(835\) 0 0
\(836\) 6.55635 0.226756
\(837\) 8.48528 0.293294
\(838\) 3.52691 0.121835
\(839\) −47.3137 −1.63345 −0.816725 0.577027i \(-0.804213\pi\)
−0.816725 + 0.577027i \(0.804213\pi\)
\(840\) 0 0
\(841\) 17.6274 0.607842
\(842\) 11.1838 0.385418
\(843\) 17.9289 0.617505
\(844\) −17.0294 −0.586177
\(845\) 0 0
\(846\) −3.92893 −0.135079
\(847\) −2.41421 −0.0829534
\(848\) −10.9706 −0.376731
\(849\) 18.8995 0.648629
\(850\) 0 0
\(851\) −5.82843 −0.199796
\(852\) −3.97056 −0.136029
\(853\) −19.1716 −0.656422 −0.328211 0.944604i \(-0.606446\pi\)
−0.328211 + 0.944604i \(0.606446\pi\)
\(854\) 3.17157 0.108529
\(855\) 0 0
\(856\) 27.4558 0.938421
\(857\) −36.6985 −1.25360 −0.626798 0.779182i \(-0.715635\pi\)
−0.626798 + 0.779182i \(0.715635\pi\)
\(858\) 1.17157 0.0399968
\(859\) −20.4853 −0.698949 −0.349474 0.936946i \(-0.613640\pi\)
−0.349474 + 0.936946i \(0.613640\pi\)
\(860\) 0 0
\(861\) −21.4853 −0.732216
\(862\) −2.82843 −0.0963366
\(863\) −28.6863 −0.976493 −0.488246 0.872706i \(-0.662363\pi\)
−0.488246 + 0.872706i \(0.662363\pi\)
\(864\) −4.41421 −0.150175
\(865\) 0 0
\(866\) −3.85786 −0.131096
\(867\) −16.8284 −0.571523
\(868\) 37.4558 1.27133
\(869\) −4.75736 −0.161382
\(870\) 0 0
\(871\) −32.9706 −1.11716
\(872\) −27.4558 −0.929772
\(873\) 0.171573 0.00580686
\(874\) −1.48528 −0.0502404
\(875\) 0 0
\(876\) 5.79899 0.195930
\(877\) 15.1127 0.510320 0.255160 0.966899i \(-0.417872\pi\)
0.255160 + 0.966899i \(0.417872\pi\)
\(878\) −11.4853 −0.387609
\(879\) 20.4142 0.688554
\(880\) 0 0
\(881\) −24.0000 −0.808581 −0.404290 0.914631i \(-0.632481\pi\)
−0.404290 + 0.914631i \(0.632481\pi\)
\(882\) 0.485281 0.0163403
\(883\) 38.6274 1.29992 0.649958 0.759970i \(-0.274786\pi\)
0.649958 + 0.759970i \(0.274786\pi\)
\(884\) 2.14214 0.0720478
\(885\) 0 0
\(886\) −10.7574 −0.361401
\(887\) −22.1421 −0.743460 −0.371730 0.928341i \(-0.621235\pi\)
−0.371730 + 0.928341i \(0.621235\pi\)
\(888\) −9.24264 −0.310163
\(889\) 3.00000 0.100617
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 49.0538 1.64244
\(893\) 34.0122 1.13817
\(894\) 7.34315 0.245592
\(895\) 0 0
\(896\) −25.4853 −0.851403
\(897\) 2.82843 0.0944384
\(898\) −4.34315 −0.144933
\(899\) 57.9411 1.93244
\(900\) 0 0
\(901\) 1.51472 0.0504626
\(902\) 3.68629 0.122740
\(903\) 0.828427 0.0275683
\(904\) −15.8579 −0.527425
\(905\) 0 0
\(906\) −5.79899 −0.192659
\(907\) −2.48528 −0.0825224 −0.0412612 0.999148i \(-0.513138\pi\)
−0.0412612 + 0.999148i \(0.513138\pi\)
\(908\) 2.76955 0.0919108
\(909\) −4.89949 −0.162506
\(910\) 0 0
\(911\) 28.5147 0.944735 0.472367 0.881402i \(-0.343400\pi\)
0.472367 + 0.881402i \(0.343400\pi\)
\(912\) 10.7574 0.356212
\(913\) −12.4853 −0.413203
\(914\) −13.3137 −0.440378
\(915\) 0 0
\(916\) 35.6274 1.17716
\(917\) −2.82843 −0.0934029
\(918\) 0.171573 0.00566275
\(919\) 1.78680 0.0589410 0.0294705 0.999566i \(-0.490618\pi\)
0.0294705 + 0.999566i \(0.490618\pi\)
\(920\) 0 0
\(921\) 29.3137 0.965920
\(922\) −16.8873 −0.556154
\(923\) 6.14214 0.202171
\(924\) −4.41421 −0.145217
\(925\) 0 0
\(926\) −0.426407 −0.0140126
\(927\) −2.34315 −0.0769590
\(928\) −30.1421 −0.989464
\(929\) −13.7990 −0.452730 −0.226365 0.974043i \(-0.572684\pi\)
−0.226365 + 0.974043i \(0.572684\pi\)
\(930\) 0 0
\(931\) −4.20101 −0.137683
\(932\) 26.6152 0.871811
\(933\) 2.34315 0.0767111
\(934\) 14.3431 0.469322
\(935\) 0 0
\(936\) 4.48528 0.146606
\(937\) −16.0000 −0.522697 −0.261349 0.965244i \(-0.584167\pi\)
−0.261349 + 0.965244i \(0.584167\pi\)
\(938\) −11.6569 −0.380610
\(939\) 1.14214 0.0372722
\(940\) 0 0
\(941\) −22.8995 −0.746502 −0.373251 0.927730i \(-0.621757\pi\)
−0.373251 + 0.927730i \(0.621757\pi\)
\(942\) −2.48528 −0.0809748
\(943\) 8.89949 0.289807
\(944\) 33.0000 1.07406
\(945\) 0 0
\(946\) −0.142136 −0.00462123
\(947\) −36.7990 −1.19581 −0.597903 0.801568i \(-0.703999\pi\)
−0.597903 + 0.801568i \(0.703999\pi\)
\(948\) −8.69848 −0.282514
\(949\) −8.97056 −0.291197
\(950\) 0 0
\(951\) 25.1716 0.816244
\(952\) 1.58579 0.0513956
\(953\) −29.0416 −0.940751 −0.470375 0.882466i \(-0.655881\pi\)
−0.470375 + 0.882466i \(0.655881\pi\)
\(954\) 1.51472 0.0490408
\(955\) 0 0
\(956\) −22.5685 −0.729919
\(957\) −6.82843 −0.220732
\(958\) 3.11270 0.100567
\(959\) −38.9706 −1.25843
\(960\) 0 0
\(961\) 41.0000 1.32258
\(962\) 6.82843 0.220157
\(963\) 17.3137 0.557926
\(964\) 25.8579 0.832826
\(965\) 0 0
\(966\) 1.00000 0.0321745
\(967\) 38.0000 1.22200 0.610999 0.791632i \(-0.290768\pi\)
0.610999 + 0.791632i \(0.290768\pi\)
\(968\) 1.58579 0.0509691
\(969\) −1.48528 −0.0477141
\(970\) 0 0
\(971\) −50.3137 −1.61464 −0.807322 0.590111i \(-0.799084\pi\)
−0.807322 + 0.590111i \(0.799084\pi\)
\(972\) −1.82843 −0.0586468
\(973\) −36.1421 −1.15866
\(974\) −4.34315 −0.139163
\(975\) 0 0
\(976\) 9.51472 0.304559
\(977\) 60.5685 1.93776 0.968880 0.247532i \(-0.0796196\pi\)
0.968880 + 0.247532i \(0.0796196\pi\)
\(978\) 9.85786 0.315220
\(979\) 7.65685 0.244714
\(980\) 0 0
\(981\) −17.3137 −0.552784
\(982\) −1.71573 −0.0547511
\(983\) 51.2843 1.63571 0.817857 0.575421i \(-0.195162\pi\)
0.817857 + 0.575421i \(0.195162\pi\)
\(984\) 14.1127 0.449896
\(985\) 0 0
\(986\) 1.17157 0.0373105
\(987\) −22.8995 −0.728899
\(988\) −18.5442 −0.589968
\(989\) −0.343146 −0.0109114
\(990\) 0 0
\(991\) −42.2843 −1.34320 −0.671602 0.740912i \(-0.734394\pi\)
−0.671602 + 0.740912i \(0.734394\pi\)
\(992\) −37.4558 −1.18922
\(993\) −3.85786 −0.122426
\(994\) 2.17157 0.0688781
\(995\) 0 0
\(996\) −22.8284 −0.723346
\(997\) −47.2548 −1.49658 −0.748288 0.663374i \(-0.769124\pi\)
−0.748288 + 0.663374i \(0.769124\pi\)
\(998\) −16.9117 −0.535330
\(999\) −5.82843 −0.184403
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 825.2.a.f.1.1 yes 2
3.2 odd 2 2475.2.a.l.1.2 2
5.2 odd 4 825.2.c.d.199.2 4
5.3 odd 4 825.2.c.d.199.3 4
5.4 even 2 825.2.a.d.1.2 2
11.10 odd 2 9075.2.a.w.1.2 2
15.2 even 4 2475.2.c.o.199.3 4
15.8 even 4 2475.2.c.o.199.2 4
15.14 odd 2 2475.2.a.w.1.1 2
55.54 odd 2 9075.2.a.ca.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.2.a.d.1.2 2 5.4 even 2
825.2.a.f.1.1 yes 2 1.1 even 1 trivial
825.2.c.d.199.2 4 5.2 odd 4
825.2.c.d.199.3 4 5.3 odd 4
2475.2.a.l.1.2 2 3.2 odd 2
2475.2.a.w.1.1 2 15.14 odd 2
2475.2.c.o.199.2 4 15.8 even 4
2475.2.c.o.199.3 4 15.2 even 4
9075.2.a.w.1.2 2 11.10 odd 2
9075.2.a.ca.1.1 2 55.54 odd 2