Properties

Label 5225.2.a.e.1.2
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $1$
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: \(1\)
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 1.2
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)
 
\(f(q)\) \(=\) \(q+0.414214 q^{2} -2.00000 q^{3} -1.82843 q^{4} -0.828427 q^{6} -2.82843 q^{7} -1.58579 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.414214 q^{2} -2.00000 q^{3} -1.82843 q^{4} -0.828427 q^{6} -2.82843 q^{7} -1.58579 q^{8} +1.00000 q^{9} -1.00000 q^{11} +3.65685 q^{12} -1.17157 q^{13} -1.17157 q^{14} +3.00000 q^{16} -0.828427 q^{17} +0.414214 q^{18} -1.00000 q^{19} +5.65685 q^{21} -0.414214 q^{22} -4.00000 q^{23} +3.17157 q^{24} -0.485281 q^{26} +4.00000 q^{27} +5.17157 q^{28} +8.82843 q^{29} +6.82843 q^{31} +4.41421 q^{32} +2.00000 q^{33} -0.343146 q^{34} -1.82843 q^{36} +6.82843 q^{37} -0.414214 q^{38} +2.34315 q^{39} -3.17157 q^{41} +2.34315 q^{42} +1.17157 q^{43} +1.82843 q^{44} -1.65685 q^{46} -6.00000 q^{48} +1.00000 q^{49} +1.65685 q^{51} +2.14214 q^{52} -2.82843 q^{53} +1.65685 q^{54} +4.48528 q^{56} +2.00000 q^{57} +3.65685 q^{58} -4.48528 q^{59} +7.65685 q^{61} +2.82843 q^{62} -2.82843 q^{63} -4.17157 q^{64} +0.828427 q^{66} -11.6569 q^{67} +1.51472 q^{68} +8.00000 q^{69} +14.8284 q^{71} -1.58579 q^{72} -6.48528 q^{73} +2.82843 q^{74} +1.82843 q^{76} +2.82843 q^{77} +0.970563 q^{78} -12.0000 q^{79} -11.0000 q^{81} -1.31371 q^{82} +2.82843 q^{83} -10.3431 q^{84} +0.485281 q^{86} -17.6569 q^{87} +1.58579 q^{88} +9.31371 q^{89} +3.31371 q^{91} +7.31371 q^{92} -13.6569 q^{93} -8.82843 q^{96} +18.1421 q^{97} +0.414214 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} + 4 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} + 4 q^{6} - 6 q^{8} + 2 q^{9} - 2 q^{11} - 4 q^{12} - 8 q^{13} - 8 q^{14} + 6 q^{16} + 4 q^{17} - 2 q^{18} - 2 q^{19} + 2 q^{22} - 8 q^{23} + 12 q^{24} + 16 q^{26} + 8 q^{27} + 16 q^{28} + 12 q^{29} + 8 q^{31} + 6 q^{32} + 4 q^{33} - 12 q^{34} + 2 q^{36} + 8 q^{37} + 2 q^{38} + 16 q^{39} - 12 q^{41} + 16 q^{42} + 8 q^{43} - 2 q^{44} + 8 q^{46} - 12 q^{48} + 2 q^{49} - 8 q^{51} - 24 q^{52} - 8 q^{54} - 8 q^{56} + 4 q^{57} - 4 q^{58} + 8 q^{59} + 4 q^{61} - 14 q^{64} - 4 q^{66} - 12 q^{67} + 20 q^{68} + 16 q^{69} + 24 q^{71} - 6 q^{72} + 4 q^{73} - 2 q^{76} - 32 q^{78} - 24 q^{79} - 22 q^{81} + 20 q^{82} - 32 q^{84} - 16 q^{86} - 24 q^{87} + 6 q^{88} - 4 q^{89} - 16 q^{91} - 8 q^{92} - 16 q^{93} - 12 q^{96} + 8 q^{97} - 2 q^{98} - 2 q^{99}+O(q^{100}) \) 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 \(-0.453216\pi\)
0.146447 + 0.989219i \(0.453216\pi\)
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) −1.82843 −0.914214
\(5\) 0 0
\(6\) −0.828427 −0.338204
\(7\) −2.82843 −1.06904 −0.534522 0.845154i \(-0.679509\pi\)
−0.534522 + 0.845154i \(0.679509\pi\)
\(8\) −1.58579 −0.560660
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 3.65685 1.05564
\(13\) −1.17157 −0.324936 −0.162468 0.986714i \(-0.551945\pi\)
−0.162468 + 0.986714i \(0.551945\pi\)
\(14\) −1.17157 −0.313116
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) −0.828427 −0.200923 −0.100462 0.994941i \(-0.532032\pi\)
−0.100462 + 0.994941i \(0.532032\pi\)
\(18\) 0.414214 0.0976311
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 5.65685 1.23443
\(22\) −0.414214 −0.0883106
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 3.17157 0.647395
\(25\) 0 0
\(26\) −0.485281 −0.0951715
\(27\) 4.00000 0.769800
\(28\) 5.17157 0.977335
\(29\) 8.82843 1.63940 0.819699 0.572795i \(-0.194141\pi\)
0.819699 + 0.572795i \(0.194141\pi\)
\(30\) 0 0
\(31\) 6.82843 1.22642 0.613211 0.789919i \(-0.289878\pi\)
0.613211 + 0.789919i \(0.289878\pi\)
\(32\) 4.41421 0.780330
\(33\) 2.00000 0.348155
\(34\) −0.343146 −0.0588490
\(35\) 0 0
\(36\) −1.82843 −0.304738
\(37\) 6.82843 1.12259 0.561293 0.827617i \(-0.310304\pi\)
0.561293 + 0.827617i \(0.310304\pi\)
\(38\) −0.414214 −0.0671943
\(39\) 2.34315 0.375204
\(40\) 0 0
\(41\) −3.17157 −0.495316 −0.247658 0.968847i \(-0.579661\pi\)
−0.247658 + 0.968847i \(0.579661\pi\)
\(42\) 2.34315 0.361555
\(43\) 1.17157 0.178663 0.0893316 0.996002i \(-0.471527\pi\)
0.0893316 + 0.996002i \(0.471527\pi\)
\(44\) 1.82843 0.275646
\(45\) 0 0
\(46\) −1.65685 −0.244290
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −6.00000 −0.866025
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 1.65685 0.232006
\(52\) 2.14214 0.297061
\(53\) −2.82843 −0.388514 −0.194257 0.980951i \(-0.562230\pi\)
−0.194257 + 0.980951i \(0.562230\pi\)
\(54\) 1.65685 0.225469
\(55\) 0 0
\(56\) 4.48528 0.599371
\(57\) 2.00000 0.264906
\(58\) 3.65685 0.480168
\(59\) −4.48528 −0.583934 −0.291967 0.956428i \(-0.594310\pi\)
−0.291967 + 0.956428i \(0.594310\pi\)
\(60\) 0 0
\(61\) 7.65685 0.980360 0.490180 0.871621i \(-0.336931\pi\)
0.490180 + 0.871621i \(0.336931\pi\)
\(62\) 2.82843 0.359211
\(63\) −2.82843 −0.356348
\(64\) −4.17157 −0.521447
\(65\) 0 0
\(66\) 0.828427 0.101972
\(67\) −11.6569 −1.42411 −0.712056 0.702123i \(-0.752236\pi\)
−0.712056 + 0.702123i \(0.752236\pi\)
\(68\) 1.51472 0.183687
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) 14.8284 1.75981 0.879905 0.475149i \(-0.157606\pi\)
0.879905 + 0.475149i \(0.157606\pi\)
\(72\) −1.58579 −0.186887
\(73\) −6.48528 −0.759045 −0.379522 0.925183i \(-0.623912\pi\)
−0.379522 + 0.925183i \(0.623912\pi\)
\(74\) 2.82843 0.328798
\(75\) 0 0
\(76\) 1.82843 0.209735
\(77\) 2.82843 0.322329
\(78\) 0.970563 0.109895
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) −1.31371 −0.145075
\(83\) 2.82843 0.310460 0.155230 0.987878i \(-0.450388\pi\)
0.155230 + 0.987878i \(0.450388\pi\)
\(84\) −10.3431 −1.12853
\(85\) 0 0
\(86\) 0.485281 0.0523292
\(87\) −17.6569 −1.89301
\(88\) 1.58579 0.169045
\(89\) 9.31371 0.987251 0.493626 0.869675i \(-0.335671\pi\)
0.493626 + 0.869675i \(0.335671\pi\)
\(90\) 0 0
\(91\) 3.31371 0.347371
\(92\) 7.31371 0.762507
\(93\) −13.6569 −1.41615
\(94\) 0 0
\(95\) 0 0
\(96\) −8.82843 −0.901048
\(97\) 18.1421 1.84205 0.921027 0.389498i \(-0.127351\pi\)
0.921027 + 0.389498i \(0.127351\pi\)
\(98\) 0.414214 0.0418419
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −3.65685 −0.363871 −0.181935 0.983311i \(-0.558236\pi\)
−0.181935 + 0.983311i \(0.558236\pi\)
\(102\) 0.686292 0.0679530
\(103\) −10.0000 −0.985329 −0.492665 0.870219i \(-0.663977\pi\)
−0.492665 + 0.870219i \(0.663977\pi\)
\(104\) 1.85786 0.182179
\(105\) 0 0
\(106\) −1.17157 −0.113793
\(107\) 9.31371 0.900390 0.450195 0.892930i \(-0.351354\pi\)
0.450195 + 0.892930i \(0.351354\pi\)
\(108\) −7.31371 −0.703762
\(109\) 4.82843 0.462479 0.231240 0.972897i \(-0.425722\pi\)
0.231240 + 0.972897i \(0.425722\pi\)
\(110\) 0 0
\(111\) −13.6569 −1.29625
\(112\) −8.48528 −0.801784
\(113\) −4.48528 −0.421940 −0.210970 0.977493i \(-0.567662\pi\)
−0.210970 + 0.977493i \(0.567662\pi\)
\(114\) 0.828427 0.0775893
\(115\) 0 0
\(116\) −16.1421 −1.49876
\(117\) −1.17157 −0.108312
\(118\) −1.85786 −0.171030
\(119\) 2.34315 0.214796
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 3.17157 0.287141
\(123\) 6.34315 0.571942
\(124\) −12.4853 −1.12121
\(125\) 0 0
\(126\) −1.17157 −0.104372
\(127\) 18.9706 1.68337 0.841683 0.539973i \(-0.181565\pi\)
0.841683 + 0.539973i \(0.181565\pi\)
\(128\) −10.5563 −0.933058
\(129\) −2.34315 −0.206302
\(130\) 0 0
\(131\) −1.65685 −0.144760 −0.0723800 0.997377i \(-0.523059\pi\)
−0.0723800 + 0.997377i \(0.523059\pi\)
\(132\) −3.65685 −0.318288
\(133\) 2.82843 0.245256
\(134\) −4.82843 −0.417113
\(135\) 0 0
\(136\) 1.31371 0.112650
\(137\) 13.3137 1.13747 0.568733 0.822522i \(-0.307434\pi\)
0.568733 + 0.822522i \(0.307434\pi\)
\(138\) 3.31371 0.282082
\(139\) −12.9706 −1.10015 −0.550074 0.835116i \(-0.685401\pi\)
−0.550074 + 0.835116i \(0.685401\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 6.14214 0.515437
\(143\) 1.17157 0.0979718
\(144\) 3.00000 0.250000
\(145\) 0 0
\(146\) −2.68629 −0.222319
\(147\) −2.00000 −0.164957
\(148\) −12.4853 −1.02628
\(149\) −3.65685 −0.299581 −0.149791 0.988718i \(-0.547860\pi\)
−0.149791 + 0.988718i \(0.547860\pi\)
\(150\) 0 0
\(151\) −1.65685 −0.134833 −0.0674164 0.997725i \(-0.521476\pi\)
−0.0674164 + 0.997725i \(0.521476\pi\)
\(152\) 1.58579 0.128624
\(153\) −0.828427 −0.0669744
\(154\) 1.17157 0.0944080
\(155\) 0 0
\(156\) −4.28427 −0.343016
\(157\) −5.31371 −0.424080 −0.212040 0.977261i \(-0.568011\pi\)
−0.212040 + 0.977261i \(0.568011\pi\)
\(158\) −4.97056 −0.395437
\(159\) 5.65685 0.448618
\(160\) 0 0
\(161\) 11.3137 0.891645
\(162\) −4.55635 −0.357981
\(163\) −6.34315 −0.496834 −0.248417 0.968653i \(-0.579910\pi\)
−0.248417 + 0.968653i \(0.579910\pi\)
\(164\) 5.79899 0.452825
\(165\) 0 0
\(166\) 1.17157 0.0909317
\(167\) 14.0000 1.08335 0.541676 0.840587i \(-0.317790\pi\)
0.541676 + 0.840587i \(0.317790\pi\)
\(168\) −8.97056 −0.692094
\(169\) −11.6274 −0.894417
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) −2.14214 −0.163336
\(173\) 2.82843 0.215041 0.107521 0.994203i \(-0.465709\pi\)
0.107521 + 0.994203i \(0.465709\pi\)
\(174\) −7.31371 −0.554451
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) 8.97056 0.674269
\(178\) 3.85786 0.289159
\(179\) −4.48528 −0.335246 −0.167623 0.985851i \(-0.553609\pi\)
−0.167623 + 0.985851i \(0.553609\pi\)
\(180\) 0 0
\(181\) 17.3137 1.28692 0.643459 0.765481i \(-0.277499\pi\)
0.643459 + 0.765481i \(0.277499\pi\)
\(182\) 1.37258 0.101743
\(183\) −15.3137 −1.13202
\(184\) 6.34315 0.467623
\(185\) 0 0
\(186\) −5.65685 −0.414781
\(187\) 0.828427 0.0605806
\(188\) 0 0
\(189\) −11.3137 −0.822951
\(190\) 0 0
\(191\) −16.9706 −1.22795 −0.613973 0.789327i \(-0.710430\pi\)
−0.613973 + 0.789327i \(0.710430\pi\)
\(192\) 8.34315 0.602115
\(193\) 23.7990 1.71309 0.856544 0.516073i \(-0.172607\pi\)
0.856544 + 0.516073i \(0.172607\pi\)
\(194\) 7.51472 0.539525
\(195\) 0 0
\(196\) −1.82843 −0.130602
\(197\) −25.7990 −1.83810 −0.919051 0.394139i \(-0.871043\pi\)
−0.919051 + 0.394139i \(0.871043\pi\)
\(198\) −0.414214 −0.0294369
\(199\) 13.6569 0.968109 0.484054 0.875038i \(-0.339164\pi\)
0.484054 + 0.875038i \(0.339164\pi\)
\(200\) 0 0
\(201\) 23.3137 1.64442
\(202\) −1.51472 −0.106575
\(203\) −24.9706 −1.75259
\(204\) −3.02944 −0.212103
\(205\) 0 0
\(206\) −4.14214 −0.288596
\(207\) −4.00000 −0.278019
\(208\) −3.51472 −0.243702
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) −24.0000 −1.65223 −0.826114 0.563503i \(-0.809453\pi\)
−0.826114 + 0.563503i \(0.809453\pi\)
\(212\) 5.17157 0.355185
\(213\) −29.6569 −2.03205
\(214\) 3.85786 0.263718
\(215\) 0 0
\(216\) −6.34315 −0.431596
\(217\) −19.3137 −1.31110
\(218\) 2.00000 0.135457
\(219\) 12.9706 0.876469
\(220\) 0 0
\(221\) 0.970563 0.0652871
\(222\) −5.65685 −0.379663
\(223\) −2.00000 −0.133930 −0.0669650 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) −12.4853 −0.834208
\(225\) 0 0
\(226\) −1.85786 −0.123583
\(227\) 1.31371 0.0871939 0.0435969 0.999049i \(-0.486118\pi\)
0.0435969 + 0.999049i \(0.486118\pi\)
\(228\) −3.65685 −0.242181
\(229\) −28.6274 −1.89175 −0.945876 0.324527i \(-0.894795\pi\)
−0.945876 + 0.324527i \(0.894795\pi\)
\(230\) 0 0
\(231\) −5.65685 −0.372194
\(232\) −14.0000 −0.919145
\(233\) 6.48528 0.424865 0.212432 0.977176i \(-0.431861\pi\)
0.212432 + 0.977176i \(0.431861\pi\)
\(234\) −0.485281 −0.0317238
\(235\) 0 0
\(236\) 8.20101 0.533840
\(237\) 24.0000 1.55897
\(238\) 0.970563 0.0629122
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −11.1716 −0.719624 −0.359812 0.933025i \(-0.617159\pi\)
−0.359812 + 0.933025i \(0.617159\pi\)
\(242\) 0.414214 0.0266267
\(243\) 10.0000 0.641500
\(244\) −14.0000 −0.896258
\(245\) 0 0
\(246\) 2.62742 0.167518
\(247\) 1.17157 0.0745454
\(248\) −10.8284 −0.687606
\(249\) −5.65685 −0.358489
\(250\) 0 0
\(251\) −15.3137 −0.966593 −0.483296 0.875457i \(-0.660561\pi\)
−0.483296 + 0.875457i \(0.660561\pi\)
\(252\) 5.17157 0.325778
\(253\) 4.00000 0.251478
\(254\) 7.85786 0.493046
\(255\) 0 0
\(256\) 3.97056 0.248160
\(257\) −30.1421 −1.88021 −0.940107 0.340878i \(-0.889276\pi\)
−0.940107 + 0.340878i \(0.889276\pi\)
\(258\) −0.970563 −0.0604246
\(259\) −19.3137 −1.20010
\(260\) 0 0
\(261\) 8.82843 0.546466
\(262\) −0.686292 −0.0423992
\(263\) −4.48528 −0.276574 −0.138287 0.990392i \(-0.544160\pi\)
−0.138287 + 0.990392i \(0.544160\pi\)
\(264\) −3.17157 −0.195197
\(265\) 0 0
\(266\) 1.17157 0.0718337
\(267\) −18.6274 −1.13998
\(268\) 21.3137 1.30194
\(269\) −12.3431 −0.752575 −0.376287 0.926503i \(-0.622799\pi\)
−0.376287 + 0.926503i \(0.622799\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) −2.48528 −0.150692
\(273\) −6.62742 −0.401110
\(274\) 5.51472 0.333156
\(275\) 0 0
\(276\) −14.6274 −0.880467
\(277\) 20.1421 1.21022 0.605112 0.796140i \(-0.293128\pi\)
0.605112 + 0.796140i \(0.293128\pi\)
\(278\) −5.37258 −0.322226
\(279\) 6.82843 0.408807
\(280\) 0 0
\(281\) −6.48528 −0.386879 −0.193440 0.981112i \(-0.561964\pi\)
−0.193440 + 0.981112i \(0.561964\pi\)
\(282\) 0 0
\(283\) −0.485281 −0.0288470 −0.0144235 0.999896i \(-0.504591\pi\)
−0.0144235 + 0.999896i \(0.504591\pi\)
\(284\) −27.1127 −1.60884
\(285\) 0 0
\(286\) 0.485281 0.0286953
\(287\) 8.97056 0.529516
\(288\) 4.41421 0.260110
\(289\) −16.3137 −0.959630
\(290\) 0 0
\(291\) −36.2843 −2.12702
\(292\) 11.8579 0.693929
\(293\) 10.8284 0.632603 0.316302 0.948659i \(-0.397559\pi\)
0.316302 + 0.948659i \(0.397559\pi\)
\(294\) −0.828427 −0.0483149
\(295\) 0 0
\(296\) −10.8284 −0.629390
\(297\) −4.00000 −0.232104
\(298\) −1.51472 −0.0877453
\(299\) 4.68629 0.271015
\(300\) 0 0
\(301\) −3.31371 −0.190999
\(302\) −0.686292 −0.0394916
\(303\) 7.31371 0.420162
\(304\) −3.00000 −0.172062
\(305\) 0 0
\(306\) −0.343146 −0.0196163
\(307\) 10.6863 0.609899 0.304949 0.952369i \(-0.401360\pi\)
0.304949 + 0.952369i \(0.401360\pi\)
\(308\) −5.17157 −0.294678
\(309\) 20.0000 1.13776
\(310\) 0 0
\(311\) 2.34315 0.132868 0.0664338 0.997791i \(-0.478838\pi\)
0.0664338 + 0.997791i \(0.478838\pi\)
\(312\) −3.71573 −0.210362
\(313\) 6.97056 0.394000 0.197000 0.980404i \(-0.436880\pi\)
0.197000 + 0.980404i \(0.436880\pi\)
\(314\) −2.20101 −0.124210
\(315\) 0 0
\(316\) 21.9411 1.23428
\(317\) −23.7990 −1.33668 −0.668342 0.743854i \(-0.732996\pi\)
−0.668342 + 0.743854i \(0.732996\pi\)
\(318\) 2.34315 0.131397
\(319\) −8.82843 −0.494297
\(320\) 0 0
\(321\) −18.6274 −1.03968
\(322\) 4.68629 0.261157
\(323\) 0.828427 0.0460949
\(324\) 20.1127 1.11737
\(325\) 0 0
\(326\) −2.62742 −0.145519
\(327\) −9.65685 −0.534025
\(328\) 5.02944 0.277704
\(329\) 0 0
\(330\) 0 0
\(331\) 15.7990 0.868391 0.434196 0.900819i \(-0.357033\pi\)
0.434196 + 0.900819i \(0.357033\pi\)
\(332\) −5.17157 −0.283827
\(333\) 6.82843 0.374196
\(334\) 5.79899 0.317307
\(335\) 0 0
\(336\) 16.9706 0.925820
\(337\) 28.4853 1.55169 0.775846 0.630922i \(-0.217323\pi\)
0.775846 + 0.630922i \(0.217323\pi\)
\(338\) −4.81623 −0.261969
\(339\) 8.97056 0.487214
\(340\) 0 0
\(341\) −6.82843 −0.369780
\(342\) −0.414214 −0.0223981
\(343\) 16.9706 0.916324
\(344\) −1.85786 −0.100169
\(345\) 0 0
\(346\) 1.17157 0.0629841
\(347\) −31.1127 −1.67022 −0.835109 0.550085i \(-0.814595\pi\)
−0.835109 + 0.550085i \(0.814595\pi\)
\(348\) 32.2843 1.73062
\(349\) 20.6274 1.10416 0.552080 0.833791i \(-0.313834\pi\)
0.552080 + 0.833791i \(0.313834\pi\)
\(350\) 0 0
\(351\) −4.68629 −0.250136
\(352\) −4.41421 −0.235278
\(353\) 10.0000 0.532246 0.266123 0.963939i \(-0.414257\pi\)
0.266123 + 0.963939i \(0.414257\pi\)
\(354\) 3.71573 0.197489
\(355\) 0 0
\(356\) −17.0294 −0.902558
\(357\) −4.68629 −0.248025
\(358\) −1.85786 −0.0981912
\(359\) −3.31371 −0.174891 −0.0874454 0.996169i \(-0.527870\pi\)
−0.0874454 + 0.996169i \(0.527870\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 7.17157 0.376930
\(363\) −2.00000 −0.104973
\(364\) −6.05887 −0.317571
\(365\) 0 0
\(366\) −6.34315 −0.331562
\(367\) −12.6863 −0.662219 −0.331110 0.943592i \(-0.607423\pi\)
−0.331110 + 0.943592i \(0.607423\pi\)
\(368\) −12.0000 −0.625543
\(369\) −3.17157 −0.165105
\(370\) 0 0
\(371\) 8.00000 0.415339
\(372\) 24.9706 1.29466
\(373\) 18.8284 0.974899 0.487450 0.873151i \(-0.337927\pi\)
0.487450 + 0.873151i \(0.337927\pi\)
\(374\) 0.343146 0.0177436
\(375\) 0 0
\(376\) 0 0
\(377\) −10.3431 −0.532699
\(378\) −4.68629 −0.241037
\(379\) −13.1716 −0.676578 −0.338289 0.941042i \(-0.609848\pi\)
−0.338289 + 0.941042i \(0.609848\pi\)
\(380\) 0 0
\(381\) −37.9411 −1.94378
\(382\) −7.02944 −0.359657
\(383\) −9.31371 −0.475908 −0.237954 0.971276i \(-0.576477\pi\)
−0.237954 + 0.971276i \(0.576477\pi\)
\(384\) 21.1127 1.07740
\(385\) 0 0
\(386\) 9.85786 0.501752
\(387\) 1.17157 0.0595544
\(388\) −33.1716 −1.68403
\(389\) 28.6274 1.45147 0.725734 0.687976i \(-0.241500\pi\)
0.725734 + 0.687976i \(0.241500\pi\)
\(390\) 0 0
\(391\) 3.31371 0.167581
\(392\) −1.58579 −0.0800943
\(393\) 3.31371 0.167154
\(394\) −10.6863 −0.538368
\(395\) 0 0
\(396\) 1.82843 0.0918819
\(397\) 9.31371 0.467442 0.233721 0.972304i \(-0.424910\pi\)
0.233721 + 0.972304i \(0.424910\pi\)
\(398\) 5.65685 0.283552
\(399\) −5.65685 −0.283197
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 9.65685 0.481640
\(403\) −8.00000 −0.398508
\(404\) 6.68629 0.332655
\(405\) 0 0
\(406\) −10.3431 −0.513322
\(407\) −6.82843 −0.338473
\(408\) −2.62742 −0.130077
\(409\) −12.8284 −0.634325 −0.317162 0.948371i \(-0.602730\pi\)
−0.317162 + 0.948371i \(0.602730\pi\)
\(410\) 0 0
\(411\) −26.6274 −1.31343
\(412\) 18.2843 0.900801
\(413\) 12.6863 0.624252
\(414\) −1.65685 −0.0814299
\(415\) 0 0
\(416\) −5.17157 −0.253557
\(417\) 25.9411 1.27034
\(418\) 0.414214 0.0202598
\(419\) 20.9706 1.02448 0.512240 0.858843i \(-0.328816\pi\)
0.512240 + 0.858843i \(0.328816\pi\)
\(420\) 0 0
\(421\) −33.3137 −1.62361 −0.811805 0.583928i \(-0.801515\pi\)
−0.811805 + 0.583928i \(0.801515\pi\)
\(422\) −9.94113 −0.483926
\(423\) 0 0
\(424\) 4.48528 0.217825
\(425\) 0 0
\(426\) −12.2843 −0.595175
\(427\) −21.6569 −1.04805
\(428\) −17.0294 −0.823149
\(429\) −2.34315 −0.113128
\(430\) 0 0
\(431\) 18.6274 0.897251 0.448626 0.893720i \(-0.351914\pi\)
0.448626 + 0.893720i \(0.351914\pi\)
\(432\) 12.0000 0.577350
\(433\) 13.4558 0.646647 0.323323 0.946289i \(-0.395200\pi\)
0.323323 + 0.946289i \(0.395200\pi\)
\(434\) −8.00000 −0.384012
\(435\) 0 0
\(436\) −8.82843 −0.422805
\(437\) 4.00000 0.191346
\(438\) 5.37258 0.256712
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0.402020 0.0191222
\(443\) −37.6569 −1.78913 −0.894566 0.446937i \(-0.852515\pi\)
−0.894566 + 0.446937i \(0.852515\pi\)
\(444\) 24.9706 1.18505
\(445\) 0 0
\(446\) −0.828427 −0.0392272
\(447\) 7.31371 0.345927
\(448\) 11.7990 0.557450
\(449\) 12.3431 0.582509 0.291255 0.956646i \(-0.405927\pi\)
0.291255 + 0.956646i \(0.405927\pi\)
\(450\) 0 0
\(451\) 3.17157 0.149344
\(452\) 8.20101 0.385743
\(453\) 3.31371 0.155692
\(454\) 0.544156 0.0255385
\(455\) 0 0
\(456\) −3.17157 −0.148523
\(457\) −27.4558 −1.28433 −0.642165 0.766566i \(-0.721964\pi\)
−0.642165 + 0.766566i \(0.721964\pi\)
\(458\) −11.8579 −0.554082
\(459\) −3.31371 −0.154671
\(460\) 0 0
\(461\) 5.02944 0.234244 0.117122 0.993118i \(-0.462633\pi\)
0.117122 + 0.993118i \(0.462633\pi\)
\(462\) −2.34315 −0.109013
\(463\) 18.6274 0.865689 0.432845 0.901468i \(-0.357510\pi\)
0.432845 + 0.901468i \(0.357510\pi\)
\(464\) 26.4853 1.22955
\(465\) 0 0
\(466\) 2.68629 0.124440
\(467\) −24.0000 −1.11059 −0.555294 0.831654i \(-0.687394\pi\)
−0.555294 + 0.831654i \(0.687394\pi\)
\(468\) 2.14214 0.0990203
\(469\) 32.9706 1.52244
\(470\) 0 0
\(471\) 10.6274 0.489686
\(472\) 7.11270 0.327388
\(473\) −1.17157 −0.0538690
\(474\) 9.94113 0.456611
\(475\) 0 0
\(476\) −4.28427 −0.196369
\(477\) −2.82843 −0.129505
\(478\) −3.31371 −0.151565
\(479\) 8.97056 0.409875 0.204938 0.978775i \(-0.434301\pi\)
0.204938 + 0.978775i \(0.434301\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) −4.62742 −0.210773
\(483\) −22.6274 −1.02958
\(484\) −1.82843 −0.0831103
\(485\) 0 0
\(486\) 4.14214 0.187891
\(487\) −13.3137 −0.603302 −0.301651 0.953418i \(-0.597538\pi\)
−0.301651 + 0.953418i \(0.597538\pi\)
\(488\) −12.1421 −0.549649
\(489\) 12.6863 0.573694
\(490\) 0 0
\(491\) −0.686292 −0.0309719 −0.0154860 0.999880i \(-0.504930\pi\)
−0.0154860 + 0.999880i \(0.504930\pi\)
\(492\) −11.5980 −0.522877
\(493\) −7.31371 −0.329393
\(494\) 0.485281 0.0218338
\(495\) 0 0
\(496\) 20.4853 0.919816
\(497\) −41.9411 −1.88132
\(498\) −2.34315 −0.104999
\(499\) 10.6274 0.475749 0.237874 0.971296i \(-0.423549\pi\)
0.237874 + 0.971296i \(0.423549\pi\)
\(500\) 0 0
\(501\) −28.0000 −1.25095
\(502\) −6.34315 −0.283108
\(503\) 31.7990 1.41785 0.708923 0.705285i \(-0.249181\pi\)
0.708923 + 0.705285i \(0.249181\pi\)
\(504\) 4.48528 0.199790
\(505\) 0 0
\(506\) 1.65685 0.0736562
\(507\) 23.2548 1.03278
\(508\) −34.6863 −1.53896
\(509\) 2.00000 0.0886484 0.0443242 0.999017i \(-0.485887\pi\)
0.0443242 + 0.999017i \(0.485887\pi\)
\(510\) 0 0
\(511\) 18.3431 0.811453
\(512\) 22.7574 1.00574
\(513\) −4.00000 −0.176604
\(514\) −12.4853 −0.550702
\(515\) 0 0
\(516\) 4.28427 0.188605
\(517\) 0 0
\(518\) −8.00000 −0.351500
\(519\) −5.65685 −0.248308
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 3.65685 0.160056
\(523\) −16.3431 −0.714636 −0.357318 0.933983i \(-0.616309\pi\)
−0.357318 + 0.933983i \(0.616309\pi\)
\(524\) 3.02944 0.132342
\(525\) 0 0
\(526\) −1.85786 −0.0810067
\(527\) −5.65685 −0.246416
\(528\) 6.00000 0.261116
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −4.48528 −0.194645
\(532\) −5.17157 −0.224216
\(533\) 3.71573 0.160946
\(534\) −7.71573 −0.333892
\(535\) 0 0
\(536\) 18.4853 0.798443
\(537\) 8.97056 0.387108
\(538\) −5.11270 −0.220424
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) 27.9411 1.20128 0.600641 0.799519i \(-0.294912\pi\)
0.600641 + 0.799519i \(0.294912\pi\)
\(542\) −9.94113 −0.427008
\(543\) −34.6274 −1.48600
\(544\) −3.65685 −0.156786
\(545\) 0 0
\(546\) −2.74517 −0.117482
\(547\) −22.2843 −0.952807 −0.476403 0.879227i \(-0.658060\pi\)
−0.476403 + 0.879227i \(0.658060\pi\)
\(548\) −24.3431 −1.03989
\(549\) 7.65685 0.326787
\(550\) 0 0
\(551\) −8.82843 −0.376104
\(552\) −12.6863 −0.539964
\(553\) 33.9411 1.44332
\(554\) 8.34315 0.354466
\(555\) 0 0
\(556\) 23.7157 1.00577
\(557\) 15.4558 0.654885 0.327443 0.944871i \(-0.393813\pi\)
0.327443 + 0.944871i \(0.393813\pi\)
\(558\) 2.82843 0.119737
\(559\) −1.37258 −0.0580541
\(560\) 0 0
\(561\) −1.65685 −0.0699524
\(562\) −2.68629 −0.113314
\(563\) 23.6569 0.997018 0.498509 0.866885i \(-0.333881\pi\)
0.498509 + 0.866885i \(0.333881\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −0.201010 −0.00844909
\(567\) 31.1127 1.30661
\(568\) −23.5147 −0.986656
\(569\) −8.14214 −0.341336 −0.170668 0.985329i \(-0.554593\pi\)
−0.170668 + 0.985329i \(0.554593\pi\)
\(570\) 0 0
\(571\) −41.6569 −1.74329 −0.871643 0.490142i \(-0.836945\pi\)
−0.871643 + 0.490142i \(0.836945\pi\)
\(572\) −2.14214 −0.0895672
\(573\) 33.9411 1.41791
\(574\) 3.71573 0.155092
\(575\) 0 0
\(576\) −4.17157 −0.173816
\(577\) −5.31371 −0.221213 −0.110606 0.993864i \(-0.535279\pi\)
−0.110606 + 0.993864i \(0.535279\pi\)
\(578\) −6.75736 −0.281069
\(579\) −47.5980 −1.97810
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) −15.0294 −0.622990
\(583\) 2.82843 0.117141
\(584\) 10.2843 0.425566
\(585\) 0 0
\(586\) 4.48528 0.185285
\(587\) −32.0000 −1.32078 −0.660391 0.750922i \(-0.729609\pi\)
−0.660391 + 0.750922i \(0.729609\pi\)
\(588\) 3.65685 0.150806
\(589\) −6.82843 −0.281360
\(590\) 0 0
\(591\) 51.5980 2.12246
\(592\) 20.4853 0.841940
\(593\) −31.1716 −1.28006 −0.640032 0.768349i \(-0.721079\pi\)
−0.640032 + 0.768349i \(0.721079\pi\)
\(594\) −1.65685 −0.0679816
\(595\) 0 0
\(596\) 6.68629 0.273881
\(597\) −27.3137 −1.11788
\(598\) 1.94113 0.0793785
\(599\) 11.7990 0.482094 0.241047 0.970513i \(-0.422509\pi\)
0.241047 + 0.970513i \(0.422509\pi\)
\(600\) 0 0
\(601\) 31.1716 1.27151 0.635757 0.771889i \(-0.280688\pi\)
0.635757 + 0.771889i \(0.280688\pi\)
\(602\) −1.37258 −0.0559423
\(603\) −11.6569 −0.474704
\(604\) 3.02944 0.123266
\(605\) 0 0
\(606\) 3.02944 0.123062
\(607\) −26.9706 −1.09470 −0.547351 0.836903i \(-0.684364\pi\)
−0.547351 + 0.836903i \(0.684364\pi\)
\(608\) −4.41421 −0.179020
\(609\) 49.9411 2.02372
\(610\) 0 0
\(611\) 0 0
\(612\) 1.51472 0.0612289
\(613\) −0.142136 −0.00574080 −0.00287040 0.999996i \(-0.500914\pi\)
−0.00287040 + 0.999996i \(0.500914\pi\)
\(614\) 4.42641 0.178635
\(615\) 0 0
\(616\) −4.48528 −0.180717
\(617\) 10.9706 0.441658 0.220829 0.975313i \(-0.429124\pi\)
0.220829 + 0.975313i \(0.429124\pi\)
\(618\) 8.28427 0.333242
\(619\) 24.6863 0.992226 0.496113 0.868258i \(-0.334760\pi\)
0.496113 + 0.868258i \(0.334760\pi\)
\(620\) 0 0
\(621\) −16.0000 −0.642058
\(622\) 0.970563 0.0389160
\(623\) −26.3431 −1.05542
\(624\) 7.02944 0.281403
\(625\) 0 0
\(626\) 2.88730 0.115400
\(627\) −2.00000 −0.0798723
\(628\) 9.71573 0.387700
\(629\) −5.65685 −0.225554
\(630\) 0 0
\(631\) 5.65685 0.225196 0.112598 0.993641i \(-0.464083\pi\)
0.112598 + 0.993641i \(0.464083\pi\)
\(632\) 19.0294 0.756950
\(633\) 48.0000 1.90783
\(634\) −9.85786 −0.391506
\(635\) 0 0
\(636\) −10.3431 −0.410132
\(637\) −1.17157 −0.0464194
\(638\) −3.65685 −0.144776
\(639\) 14.8284 0.586604
\(640\) 0 0
\(641\) 47.9411 1.89356 0.946780 0.321880i \(-0.104315\pi\)
0.946780 + 0.321880i \(0.104315\pi\)
\(642\) −7.71573 −0.304516
\(643\) 10.3431 0.407894 0.203947 0.978982i \(-0.434623\pi\)
0.203947 + 0.978982i \(0.434623\pi\)
\(644\) −20.6863 −0.815154
\(645\) 0 0
\(646\) 0.343146 0.0135009
\(647\) 12.9706 0.509925 0.254963 0.966951i \(-0.417937\pi\)
0.254963 + 0.966951i \(0.417937\pi\)
\(648\) 17.4437 0.685251
\(649\) 4.48528 0.176063
\(650\) 0 0
\(651\) 38.6274 1.51393
\(652\) 11.5980 0.454212
\(653\) 14.0000 0.547862 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) −4.00000 −0.156412
\(655\) 0 0
\(656\) −9.51472 −0.371487
\(657\) −6.48528 −0.253015
\(658\) 0 0
\(659\) 40.9706 1.59599 0.797993 0.602666i \(-0.205895\pi\)
0.797993 + 0.602666i \(0.205895\pi\)
\(660\) 0 0
\(661\) 9.02944 0.351204 0.175602 0.984461i \(-0.443813\pi\)
0.175602 + 0.984461i \(0.443813\pi\)
\(662\) 6.54416 0.254346
\(663\) −1.94113 −0.0753871
\(664\) −4.48528 −0.174063
\(665\) 0 0
\(666\) 2.82843 0.109599
\(667\) −35.3137 −1.36735
\(668\) −25.5980 −0.990416
\(669\) 4.00000 0.154649
\(670\) 0 0
\(671\) −7.65685 −0.295590
\(672\) 24.9706 0.963260
\(673\) −17.4558 −0.672873 −0.336437 0.941706i \(-0.609222\pi\)
−0.336437 + 0.941706i \(0.609222\pi\)
\(674\) 11.7990 0.454480
\(675\) 0 0
\(676\) 21.2599 0.817688
\(677\) 30.1421 1.15846 0.579228 0.815165i \(-0.303354\pi\)
0.579228 + 0.815165i \(0.303354\pi\)
\(678\) 3.71573 0.142702
\(679\) −51.3137 −1.96924
\(680\) 0 0
\(681\) −2.62742 −0.100683
\(682\) −2.82843 −0.108306
\(683\) −17.3137 −0.662491 −0.331245 0.943545i \(-0.607469\pi\)
−0.331245 + 0.943545i \(0.607469\pi\)
\(684\) 1.82843 0.0699117
\(685\) 0 0
\(686\) 7.02944 0.268385
\(687\) 57.2548 2.18441
\(688\) 3.51472 0.133997
\(689\) 3.31371 0.126242
\(690\) 0 0
\(691\) −34.6274 −1.31729 −0.658645 0.752454i \(-0.728870\pi\)
−0.658645 + 0.752454i \(0.728870\pi\)
\(692\) −5.17157 −0.196594
\(693\) 2.82843 0.107443
\(694\) −12.8873 −0.489195
\(695\) 0 0
\(696\) 28.0000 1.06134
\(697\) 2.62742 0.0995205
\(698\) 8.54416 0.323401
\(699\) −12.9706 −0.490592
\(700\) 0 0
\(701\) 27.9411 1.05532 0.527661 0.849455i \(-0.323069\pi\)
0.527661 + 0.849455i \(0.323069\pi\)
\(702\) −1.94113 −0.0732631
\(703\) −6.82843 −0.257539
\(704\) 4.17157 0.157222
\(705\) 0 0
\(706\) 4.14214 0.155891
\(707\) 10.3431 0.388994
\(708\) −16.4020 −0.616426
\(709\) −42.0000 −1.57734 −0.788672 0.614815i \(-0.789231\pi\)
−0.788672 + 0.614815i \(0.789231\pi\)
\(710\) 0 0
\(711\) −12.0000 −0.450035
\(712\) −14.7696 −0.553512
\(713\) −27.3137 −1.02291
\(714\) −1.94113 −0.0726448
\(715\) 0 0
\(716\) 8.20101 0.306486
\(717\) 16.0000 0.597531
\(718\) −1.37258 −0.0512243
\(719\) −35.3137 −1.31698 −0.658490 0.752590i \(-0.728804\pi\)
−0.658490 + 0.752590i \(0.728804\pi\)
\(720\) 0 0
\(721\) 28.2843 1.05336
\(722\) 0.414214 0.0154154
\(723\) 22.3431 0.830951
\(724\) −31.6569 −1.17652
\(725\) 0 0
\(726\) −0.828427 −0.0307458
\(727\) 8.97056 0.332700 0.166350 0.986067i \(-0.446802\pi\)
0.166350 + 0.986067i \(0.446802\pi\)
\(728\) −5.25483 −0.194757
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −0.970563 −0.0358976
\(732\) 28.0000 1.03491
\(733\) 10.4853 0.387283 0.193641 0.981072i \(-0.437970\pi\)
0.193641 + 0.981072i \(0.437970\pi\)
\(734\) −5.25483 −0.193959
\(735\) 0 0
\(736\) −17.6569 −0.650840
\(737\) 11.6569 0.429386
\(738\) −1.31371 −0.0483583
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 0 0
\(741\) −2.34315 −0.0860776
\(742\) 3.31371 0.121650
\(743\) −16.3431 −0.599572 −0.299786 0.954006i \(-0.596915\pi\)
−0.299786 + 0.954006i \(0.596915\pi\)
\(744\) 21.6569 0.793979
\(745\) 0 0
\(746\) 7.79899 0.285541
\(747\) 2.82843 0.103487
\(748\) −1.51472 −0.0553836
\(749\) −26.3431 −0.962558
\(750\) 0 0
\(751\) −12.2010 −0.445221 −0.222611 0.974907i \(-0.571458\pi\)
−0.222611 + 0.974907i \(0.571458\pi\)
\(752\) 0 0
\(753\) 30.6274 1.11613
\(754\) −4.28427 −0.156024
\(755\) 0 0
\(756\) 20.6863 0.752353
\(757\) 29.3137 1.06542 0.532712 0.846296i \(-0.321173\pi\)
0.532712 + 0.846296i \(0.321173\pi\)
\(758\) −5.45584 −0.198165
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) −52.9117 −1.91805 −0.959024 0.283326i \(-0.908562\pi\)
−0.959024 + 0.283326i \(0.908562\pi\)
\(762\) −15.7157 −0.569321
\(763\) −13.6569 −0.494411
\(764\) 31.0294 1.12261
\(765\) 0 0
\(766\) −3.85786 −0.139390
\(767\) 5.25483 0.189741
\(768\) −7.94113 −0.286551
\(769\) −34.9706 −1.26107 −0.630535 0.776161i \(-0.717165\pi\)
−0.630535 + 0.776161i \(0.717165\pi\)
\(770\) 0 0
\(771\) 60.2843 2.17108
\(772\) −43.5147 −1.56613
\(773\) −7.79899 −0.280510 −0.140255 0.990115i \(-0.544792\pi\)
−0.140255 + 0.990115i \(0.544792\pi\)
\(774\) 0.485281 0.0174431
\(775\) 0 0
\(776\) −28.7696 −1.03277
\(777\) 38.6274 1.38575
\(778\) 11.8579 0.425125
\(779\) 3.17157 0.113633
\(780\) 0 0
\(781\) −14.8284 −0.530603
\(782\) 1.37258 0.0490835
\(783\) 35.3137 1.26201
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) 1.37258 0.0489584
\(787\) −18.9706 −0.676228 −0.338114 0.941105i \(-0.609789\pi\)
−0.338114 + 0.941105i \(0.609789\pi\)
\(788\) 47.1716 1.68042
\(789\) 8.97056 0.319360
\(790\) 0 0
\(791\) 12.6863 0.451073
\(792\) 1.58579 0.0563485
\(793\) −8.97056 −0.318554
\(794\) 3.85786 0.136910
\(795\) 0 0
\(796\) −24.9706 −0.885058
\(797\) 5.17157 0.183187 0.0915933 0.995797i \(-0.470804\pi\)
0.0915933 + 0.995797i \(0.470804\pi\)
\(798\) −2.34315 −0.0829465
\(799\) 0 0
\(800\) 0 0
\(801\) 9.31371 0.329084
\(802\) 7.45584 0.263275
\(803\) 6.48528 0.228861
\(804\) −42.6274 −1.50335
\(805\) 0 0
\(806\) −3.31371 −0.116720
\(807\) 24.6863 0.868999
\(808\) 5.79899 0.204008
\(809\) −32.3431 −1.13712 −0.568562 0.822640i \(-0.692500\pi\)
−0.568562 + 0.822640i \(0.692500\pi\)
\(810\) 0 0
\(811\) 15.3137 0.537737 0.268869 0.963177i \(-0.413350\pi\)
0.268869 + 0.963177i \(0.413350\pi\)
\(812\) 45.6569 1.60224
\(813\) 48.0000 1.68343
\(814\) −2.82843 −0.0991363
\(815\) 0 0
\(816\) 4.97056 0.174005
\(817\) −1.17157 −0.0409881
\(818\) −5.31371 −0.185789
\(819\) 3.31371 0.115790
\(820\) 0 0
\(821\) −0.627417 −0.0218970 −0.0109485 0.999940i \(-0.503485\pi\)
−0.0109485 + 0.999940i \(0.503485\pi\)
\(822\) −11.0294 −0.384696
\(823\) 5.65685 0.197186 0.0985928 0.995128i \(-0.468566\pi\)
0.0985928 + 0.995128i \(0.468566\pi\)
\(824\) 15.8579 0.552435
\(825\) 0 0
\(826\) 5.25483 0.182839
\(827\) −24.3431 −0.846494 −0.423247 0.906014i \(-0.639110\pi\)
−0.423247 + 0.906014i \(0.639110\pi\)
\(828\) 7.31371 0.254169
\(829\) 6.97056 0.242098 0.121049 0.992647i \(-0.461374\pi\)
0.121049 + 0.992647i \(0.461374\pi\)
\(830\) 0 0
\(831\) −40.2843 −1.39745
\(832\) 4.88730 0.169437
\(833\) −0.828427 −0.0287033
\(834\) 10.7452 0.372075
\(835\) 0 0
\(836\) −1.82843 −0.0632375
\(837\) 27.3137 0.944100
\(838\) 8.68629 0.300063
\(839\) −24.4853 −0.845326 −0.422663 0.906287i \(-0.638905\pi\)
−0.422663 + 0.906287i \(0.638905\pi\)
\(840\) 0 0
\(841\) 48.9411 1.68763
\(842\) −13.7990 −0.475545
\(843\) 12.9706 0.446730
\(844\) 43.8823 1.51049
\(845\) 0 0
\(846\) 0 0
\(847\) −2.82843 −0.0971859
\(848\) −8.48528 −0.291386
\(849\) 0.970563 0.0333096
\(850\) 0 0
\(851\) −27.3137 −0.936302
\(852\) 54.2254 1.85773
\(853\) 30.7696 1.05353 0.526765 0.850011i \(-0.323405\pi\)
0.526765 + 0.850011i \(0.323405\pi\)
\(854\) −8.97056 −0.306966
\(855\) 0 0
\(856\) −14.7696 −0.504813
\(857\) −41.4558 −1.41610 −0.708052 0.706160i \(-0.750426\pi\)
−0.708052 + 0.706160i \(0.750426\pi\)
\(858\) −0.970563 −0.0331345
\(859\) 8.68629 0.296372 0.148186 0.988959i \(-0.452657\pi\)
0.148186 + 0.988959i \(0.452657\pi\)
\(860\) 0 0
\(861\) −17.9411 −0.611432
\(862\) 7.71573 0.262799
\(863\) −46.9706 −1.59890 −0.799448 0.600735i \(-0.794875\pi\)
−0.799448 + 0.600735i \(0.794875\pi\)
\(864\) 17.6569 0.600698
\(865\) 0 0
\(866\) 5.57359 0.189398
\(867\) 32.6274 1.10809
\(868\) 35.3137 1.19863
\(869\) 12.0000 0.407072
\(870\) 0 0
\(871\) 13.6569 0.462745
\(872\) −7.65685 −0.259294
\(873\) 18.1421 0.614018
\(874\) 1.65685 0.0560439
\(875\) 0 0
\(876\) −23.7157 −0.801280
\(877\) −32.4853 −1.09695 −0.548475 0.836167i \(-0.684791\pi\)
−0.548475 + 0.836167i \(0.684791\pi\)
\(878\) 8.28427 0.279581
\(879\) −21.6569 −0.730468
\(880\) 0 0
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 0.414214 0.0139473
\(883\) 8.28427 0.278788 0.139394 0.990237i \(-0.455485\pi\)
0.139394 + 0.990237i \(0.455485\pi\)
\(884\) −1.77460 −0.0596864
\(885\) 0 0
\(886\) −15.5980 −0.524024
\(887\) −2.68629 −0.0901968 −0.0450984 0.998983i \(-0.514360\pi\)
−0.0450984 + 0.998983i \(0.514360\pi\)
\(888\) 21.6569 0.726756
\(889\) −53.6569 −1.79959
\(890\) 0 0
\(891\) 11.0000 0.368514
\(892\) 3.65685 0.122441
\(893\) 0 0
\(894\) 3.02944 0.101320
\(895\) 0 0
\(896\) 29.8579 0.997481
\(897\) −9.37258 −0.312941
\(898\) 5.11270 0.170613
\(899\) 60.2843 2.01059
\(900\) 0 0
\(901\) 2.34315 0.0780615
\(902\) 1.31371 0.0437417
\(903\) 6.62742 0.220547
\(904\) 7.11270 0.236565
\(905\) 0 0
\(906\) 1.37258 0.0456010
\(907\) −7.65685 −0.254242 −0.127121 0.991887i \(-0.540574\pi\)
−0.127121 + 0.991887i \(0.540574\pi\)
\(908\) −2.40202 −0.0797138
\(909\) −3.65685 −0.121290
\(910\) 0 0
\(911\) −33.4558 −1.10844 −0.554221 0.832370i \(-0.686984\pi\)
−0.554221 + 0.832370i \(0.686984\pi\)
\(912\) 6.00000 0.198680
\(913\) −2.82843 −0.0936073
\(914\) −11.3726 −0.376172
\(915\) 0 0
\(916\) 52.3431 1.72947
\(917\) 4.68629 0.154755
\(918\) −1.37258 −0.0453020
\(919\) 32.9706 1.08760 0.543799 0.839215i \(-0.316985\pi\)
0.543799 + 0.839215i \(0.316985\pi\)
\(920\) 0 0
\(921\) −21.3726 −0.704251
\(922\) 2.08326 0.0686086
\(923\) −17.3726 −0.571826
\(924\) 10.3431 0.340265
\(925\) 0 0
\(926\) 7.71573 0.253555
\(927\) −10.0000 −0.328443
\(928\) 38.9706 1.27927
\(929\) −46.0000 −1.50921 −0.754606 0.656179i \(-0.772172\pi\)
−0.754606 + 0.656179i \(0.772172\pi\)
\(930\) 0 0
\(931\) −1.00000 −0.0327737
\(932\) −11.8579 −0.388417
\(933\) −4.68629 −0.153422
\(934\) −9.94113 −0.325284
\(935\) 0 0
\(936\) 1.85786 0.0607262
\(937\) 34.4853 1.12659 0.563293 0.826258i \(-0.309534\pi\)
0.563293 + 0.826258i \(0.309534\pi\)
\(938\) 13.6569 0.445912
\(939\) −13.9411 −0.454951
\(940\) 0 0
\(941\) 24.1421 0.787011 0.393506 0.919322i \(-0.371262\pi\)
0.393506 + 0.919322i \(0.371262\pi\)
\(942\) 4.40202 0.143426
\(943\) 12.6863 0.413122
\(944\) −13.4558 −0.437950
\(945\) 0 0
\(946\) −0.485281 −0.0157779
\(947\) 3.31371 0.107681 0.0538405 0.998550i \(-0.482854\pi\)
0.0538405 + 0.998550i \(0.482854\pi\)
\(948\) −43.8823 −1.42523
\(949\) 7.59798 0.246641
\(950\) 0 0
\(951\) 47.5980 1.54347
\(952\) −3.71573 −0.120427
\(953\) 9.85786 0.319328 0.159664 0.987171i \(-0.448959\pi\)
0.159664 + 0.987171i \(0.448959\pi\)
\(954\) −1.17157 −0.0379311
\(955\) 0 0
\(956\) 14.6274 0.473084
\(957\) 17.6569 0.570765
\(958\) 3.71573 0.120050
\(959\) −37.6569 −1.21600
\(960\) 0 0
\(961\) 15.6274 0.504110
\(962\) −3.31371 −0.106838
\(963\) 9.31371 0.300130
\(964\) 20.4264 0.657890
\(965\) 0 0
\(966\) −9.37258 −0.301558
\(967\) −48.7696 −1.56832 −0.784162 0.620557i \(-0.786907\pi\)
−0.784162 + 0.620557i \(0.786907\pi\)
\(968\) −1.58579 −0.0509691
\(969\) −1.65685 −0.0532258
\(970\) 0 0
\(971\) 24.4853 0.785770 0.392885 0.919588i \(-0.371477\pi\)
0.392885 + 0.919588i \(0.371477\pi\)
\(972\) −18.2843 −0.586468
\(973\) 36.6863 1.17611
\(974\) −5.51472 −0.176703
\(975\) 0 0
\(976\) 22.9706 0.735270
\(977\) −26.8284 −0.858317 −0.429159 0.903229i \(-0.641190\pi\)
−0.429159 + 0.903229i \(0.641190\pi\)
\(978\) 5.25483 0.168031
\(979\) −9.31371 −0.297667
\(980\) 0 0
\(981\) 4.82843 0.154160
\(982\) −0.284271 −0.00907146
\(983\) −40.3431 −1.28675 −0.643373 0.765553i \(-0.722466\pi\)
−0.643373 + 0.765553i \(0.722466\pi\)
\(984\) −10.0589 −0.320665
\(985\) 0 0
\(986\) −3.02944 −0.0964769
\(987\) 0 0
\(988\) −2.14214 −0.0681504
\(989\) −4.68629 −0.149015
\(990\) 0 0
\(991\) 56.0833 1.78154 0.890772 0.454451i \(-0.150165\pi\)
0.890772 + 0.454451i \(0.150165\pi\)
\(992\) 30.1421 0.957014
\(993\) −31.5980 −1.00273
\(994\) −17.3726 −0.551025
\(995\) 0 0
\(996\) 10.3431 0.327735
\(997\) −24.1421 −0.764589 −0.382295 0.924041i \(-0.624866\pi\)
−0.382295 + 0.924041i \(0.624866\pi\)
\(998\) 4.40202 0.139344
\(999\) 27.3137 0.864167
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.e.1.2 2
5.4 even 2 1045.2.a.c.1.1 2
15.14 odd 2 9405.2.a.o.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.c.1.1 2 5.4 even 2
5225.2.a.e.1.2 2 1.1 even 1 trivial
9405.2.a.o.1.2 2 15.14 odd 2