Properties

Label 5225.2.a.e.1.1
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 1.1
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)
 
\(f(q)\) \(=\) \(q-2.41421 q^{2} -2.00000 q^{3} +3.82843 q^{4} +4.82843 q^{6} +2.82843 q^{7} -4.41421 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.41421 q^{2} -2.00000 q^{3} +3.82843 q^{4} +4.82843 q^{6} +2.82843 q^{7} -4.41421 q^{8} +1.00000 q^{9} -1.00000 q^{11} -7.65685 q^{12} -6.82843 q^{13} -6.82843 q^{14} +3.00000 q^{16} +4.82843 q^{17} -2.41421 q^{18} -1.00000 q^{19} -5.65685 q^{21} +2.41421 q^{22} -4.00000 q^{23} +8.82843 q^{24} +16.4853 q^{26} +4.00000 q^{27} +10.8284 q^{28} +3.17157 q^{29} +1.17157 q^{31} +1.58579 q^{32} +2.00000 q^{33} -11.6569 q^{34} +3.82843 q^{36} +1.17157 q^{37} +2.41421 q^{38} +13.6569 q^{39} -8.82843 q^{41} +13.6569 q^{42} +6.82843 q^{43} -3.82843 q^{44} +9.65685 q^{46} -6.00000 q^{48} +1.00000 q^{49} -9.65685 q^{51} -26.1421 q^{52} +2.82843 q^{53} -9.65685 q^{54} -12.4853 q^{56} +2.00000 q^{57} -7.65685 q^{58} +12.4853 q^{59} -3.65685 q^{61} -2.82843 q^{62} +2.82843 q^{63} -9.82843 q^{64} -4.82843 q^{66} -0.343146 q^{67} +18.4853 q^{68} +8.00000 q^{69} +9.17157 q^{71} -4.41421 q^{72} +10.4853 q^{73} -2.82843 q^{74} -3.82843 q^{76} -2.82843 q^{77} -32.9706 q^{78} -12.0000 q^{79} -11.0000 q^{81} +21.3137 q^{82} -2.82843 q^{83} -21.6569 q^{84} -16.4853 q^{86} -6.34315 q^{87} +4.41421 q^{88} -13.3137 q^{89} -19.3137 q^{91} -15.3137 q^{92} -2.34315 q^{93} -3.17157 q^{96} -10.1421 q^{97} -2.41421 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} + 4 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.41421 −1.70711 −0.853553 0.521005i \(-0.825557\pi\)
−0.853553 + 0.521005i \(0.825557\pi\)
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 3.82843 1.91421
\(5\) 0 0
\(6\) 4.82843 1.97120
\(7\) 2.82843 1.06904 0.534522 0.845154i \(-0.320491\pi\)
0.534522 + 0.845154i \(0.320491\pi\)
\(8\) −4.41421 −1.56066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −7.65685 −2.21034
\(13\) −6.82843 −1.89386 −0.946932 0.321433i \(-0.895836\pi\)
−0.946932 + 0.321433i \(0.895836\pi\)
\(14\) −6.82843 −1.82497
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 4.82843 1.17107 0.585533 0.810649i \(-0.300885\pi\)
0.585533 + 0.810649i \(0.300885\pi\)
\(18\) −2.41421 −0.569036
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −5.65685 −1.23443
\(22\) 2.41421 0.514712
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 8.82843 1.80210
\(25\) 0 0
\(26\) 16.4853 3.23303
\(27\) 4.00000 0.769800
\(28\) 10.8284 2.04638
\(29\) 3.17157 0.588946 0.294473 0.955660i \(-0.404856\pi\)
0.294473 + 0.955660i \(0.404856\pi\)
\(30\) 0 0
\(31\) 1.17157 0.210421 0.105210 0.994450i \(-0.466448\pi\)
0.105210 + 0.994450i \(0.466448\pi\)
\(32\) 1.58579 0.280330
\(33\) 2.00000 0.348155
\(34\) −11.6569 −1.99913
\(35\) 0 0
\(36\) 3.82843 0.638071
\(37\) 1.17157 0.192605 0.0963027 0.995352i \(-0.469298\pi\)
0.0963027 + 0.995352i \(0.469298\pi\)
\(38\) 2.41421 0.391637
\(39\) 13.6569 2.18685
\(40\) 0 0
\(41\) −8.82843 −1.37877 −0.689384 0.724396i \(-0.742119\pi\)
−0.689384 + 0.724396i \(0.742119\pi\)
\(42\) 13.6569 2.10730
\(43\) 6.82843 1.04133 0.520663 0.853762i \(-0.325685\pi\)
0.520663 + 0.853762i \(0.325685\pi\)
\(44\) −3.82843 −0.577157
\(45\) 0 0
\(46\) 9.65685 1.42383
\(47\) 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
\(51\) −9.65685 −1.35223
\(52\) −26.1421 −3.62526
\(53\) 2.82843 0.388514 0.194257 0.980951i \(-0.437770\pi\)
0.194257 + 0.980951i \(0.437770\pi\)
\(54\) −9.65685 −1.31413
\(55\) 0 0
\(56\) −12.4853 −1.66842
\(57\) 2.00000 0.264906
\(58\) −7.65685 −1.00539
\(59\) 12.4853 1.62545 0.812723 0.582651i \(-0.197984\pi\)
0.812723 + 0.582651i \(0.197984\pi\)
\(60\) 0 0
\(61\) −3.65685 −0.468212 −0.234106 0.972211i \(-0.575216\pi\)
−0.234106 + 0.972211i \(0.575216\pi\)
\(62\) −2.82843 −0.359211
\(63\) 2.82843 0.356348
\(64\) −9.82843 −1.22855
\(65\) 0 0
\(66\) −4.82843 −0.594338
\(67\) −0.343146 −0.0419219 −0.0209610 0.999780i \(-0.506673\pi\)
−0.0209610 + 0.999780i \(0.506673\pi\)
\(68\) 18.4853 2.24167
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) 9.17157 1.08847 0.544233 0.838934i \(-0.316821\pi\)
0.544233 + 0.838934i \(0.316821\pi\)
\(72\) −4.41421 −0.520220
\(73\) 10.4853 1.22721 0.613605 0.789613i \(-0.289719\pi\)
0.613605 + 0.789613i \(0.289719\pi\)
\(74\) −2.82843 −0.328798
\(75\) 0 0
\(76\) −3.82843 −0.439151
\(77\) −2.82843 −0.322329
\(78\) −32.9706 −3.73318
\(79\) −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
\(82\) 21.3137 2.35371
\(83\) −2.82843 −0.310460 −0.155230 0.987878i \(-0.549612\pi\)
−0.155230 + 0.987878i \(0.549612\pi\)
\(84\) −21.6569 −2.36296
\(85\) 0 0
\(86\) −16.4853 −1.77765
\(87\) −6.34315 −0.680057
\(88\) 4.41421 0.470557
\(89\) −13.3137 −1.41125 −0.705625 0.708585i \(-0.749334\pi\)
−0.705625 + 0.708585i \(0.749334\pi\)
\(90\) 0 0
\(91\) −19.3137 −2.02463
\(92\) −15.3137 −1.59656
\(93\) −2.34315 −0.242973
\(94\) 0 0
\(95\) 0 0
\(96\) −3.17157 −0.323697
\(97\) −10.1421 −1.02978 −0.514889 0.857257i \(-0.672167\pi\)
−0.514889 + 0.857257i \(0.672167\pi\)
\(98\) −2.41421 −0.243872
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 7.65685 0.761885 0.380943 0.924599i \(-0.375599\pi\)
0.380943 + 0.924599i \(0.375599\pi\)
\(102\) 23.3137 2.30840
\(103\) −10.0000 −0.985329 −0.492665 0.870219i \(-0.663977\pi\)
−0.492665 + 0.870219i \(0.663977\pi\)
\(104\) 30.1421 2.95568
\(105\) 0 0
\(106\) −6.82843 −0.663235
\(107\) −13.3137 −1.28708 −0.643542 0.765410i \(-0.722536\pi\)
−0.643542 + 0.765410i \(0.722536\pi\)
\(108\) 15.3137 1.47356
\(109\) −0.828427 −0.0793489 −0.0396745 0.999213i \(-0.512632\pi\)
−0.0396745 + 0.999213i \(0.512632\pi\)
\(110\) 0 0
\(111\) −2.34315 −0.222402
\(112\) 8.48528 0.801784
\(113\) 12.4853 1.17452 0.587258 0.809400i \(-0.300207\pi\)
0.587258 + 0.809400i \(0.300207\pi\)
\(114\) −4.82843 −0.452224
\(115\) 0 0
\(116\) 12.1421 1.12737
\(117\) −6.82843 −0.631288
\(118\) −30.1421 −2.77481
\(119\) 13.6569 1.25192
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 8.82843 0.799288
\(123\) 17.6569 1.59206
\(124\) 4.48528 0.402790
\(125\) 0 0
\(126\) −6.82843 −0.608325
\(127\) −14.9706 −1.32842 −0.664211 0.747545i \(-0.731232\pi\)
−0.664211 + 0.747545i \(0.731232\pi\)
\(128\) 20.5563 1.81694
\(129\) −13.6569 −1.20242
\(130\) 0 0
\(131\) 9.65685 0.843723 0.421862 0.906660i \(-0.361377\pi\)
0.421862 + 0.906660i \(0.361377\pi\)
\(132\) 7.65685 0.666444
\(133\) −2.82843 −0.245256
\(134\) 0.828427 0.0715652
\(135\) 0 0
\(136\) −21.3137 −1.82764
\(137\) −9.31371 −0.795724 −0.397862 0.917445i \(-0.630248\pi\)
−0.397862 + 0.917445i \(0.630248\pi\)
\(138\) −19.3137 −1.64409
\(139\) 20.9706 1.77870 0.889350 0.457227i \(-0.151157\pi\)
0.889350 + 0.457227i \(0.151157\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −22.1421 −1.85813
\(143\) 6.82843 0.571022
\(144\) 3.00000 0.250000
\(145\) 0 0
\(146\) −25.3137 −2.09498
\(147\) −2.00000 −0.164957
\(148\) 4.48528 0.368688
\(149\) 7.65685 0.627274 0.313637 0.949543i \(-0.398453\pi\)
0.313637 + 0.949543i \(0.398453\pi\)
\(150\) 0 0
\(151\) 9.65685 0.785864 0.392932 0.919568i \(-0.371461\pi\)
0.392932 + 0.919568i \(0.371461\pi\)
\(152\) 4.41421 0.358040
\(153\) 4.82843 0.390355
\(154\) 6.82843 0.550250
\(155\) 0 0
\(156\) 52.2843 4.18609
\(157\) 17.3137 1.38178 0.690892 0.722958i \(-0.257218\pi\)
0.690892 + 0.722958i \(0.257218\pi\)
\(158\) 28.9706 2.30477
\(159\) −5.65685 −0.448618
\(160\) 0 0
\(161\) −11.3137 −0.891645
\(162\) 26.5563 2.08646
\(163\) −17.6569 −1.38299 −0.691496 0.722380i \(-0.743048\pi\)
−0.691496 + 0.722380i \(0.743048\pi\)
\(164\) −33.7990 −2.63926
\(165\) 0 0
\(166\) 6.82843 0.529989
\(167\) 14.0000 1.08335 0.541676 0.840587i \(-0.317790\pi\)
0.541676 + 0.840587i \(0.317790\pi\)
\(168\) 24.9706 1.92652
\(169\) 33.6274 2.58672
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 26.1421 1.99332
\(173\) −2.82843 −0.215041 −0.107521 0.994203i \(-0.534291\pi\)
−0.107521 + 0.994203i \(0.534291\pi\)
\(174\) 15.3137 1.16093
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) −24.9706 −1.87690
\(178\) 32.1421 2.40915
\(179\) 12.4853 0.933194 0.466597 0.884470i \(-0.345480\pi\)
0.466597 + 0.884470i \(0.345480\pi\)
\(180\) 0 0
\(181\) −5.31371 −0.394965 −0.197482 0.980306i \(-0.563277\pi\)
−0.197482 + 0.980306i \(0.563277\pi\)
\(182\) 46.6274 3.45625
\(183\) 7.31371 0.540645
\(184\) 17.6569 1.30168
\(185\) 0 0
\(186\) 5.65685 0.414781
\(187\) −4.82843 −0.353090
\(188\) 0 0
\(189\) 11.3137 0.822951
\(190\) 0 0
\(191\) 16.9706 1.22795 0.613973 0.789327i \(-0.289570\pi\)
0.613973 + 0.789327i \(0.289570\pi\)
\(192\) 19.6569 1.41861
\(193\) −15.7990 −1.13724 −0.568618 0.822602i \(-0.692522\pi\)
−0.568618 + 0.822602i \(0.692522\pi\)
\(194\) 24.4853 1.75794
\(195\) 0 0
\(196\) 3.82843 0.273459
\(197\) 13.7990 0.983137 0.491569 0.870839i \(-0.336424\pi\)
0.491569 + 0.870839i \(0.336424\pi\)
\(198\) 2.41421 0.171571
\(199\) 2.34315 0.166101 0.0830506 0.996545i \(-0.473534\pi\)
0.0830506 + 0.996545i \(0.473534\pi\)
\(200\) 0 0
\(201\) 0.686292 0.0484073
\(202\) −18.4853 −1.30062
\(203\) 8.97056 0.629610
\(204\) −36.9706 −2.58846
\(205\) 0 0
\(206\) 24.1421 1.68206
\(207\) −4.00000 −0.278019
\(208\) −20.4853 −1.42040
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) −24.0000 −1.65223 −0.826114 0.563503i \(-0.809453\pi\)
−0.826114 + 0.563503i \(0.809453\pi\)
\(212\) 10.8284 0.743699
\(213\) −18.3431 −1.25685
\(214\) 32.1421 2.19719
\(215\) 0 0
\(216\) −17.6569 −1.20140
\(217\) 3.31371 0.224949
\(218\) 2.00000 0.135457
\(219\) −20.9706 −1.41706
\(220\) 0 0
\(221\) −32.9706 −2.21784
\(222\) 5.65685 0.379663
\(223\) −2.00000 −0.133930 −0.0669650 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) 4.48528 0.299685
\(225\) 0 0
\(226\) −30.1421 −2.00503
\(227\) −21.3137 −1.41464 −0.707320 0.706893i \(-0.750096\pi\)
−0.707320 + 0.706893i \(0.750096\pi\)
\(228\) 7.65685 0.507088
\(229\) 16.6274 1.09877 0.549385 0.835569i \(-0.314862\pi\)
0.549385 + 0.835569i \(0.314862\pi\)
\(230\) 0 0
\(231\) 5.65685 0.372194
\(232\) −14.0000 −0.919145
\(233\) −10.4853 −0.686914 −0.343457 0.939168i \(-0.611598\pi\)
−0.343457 + 0.939168i \(0.611598\pi\)
\(234\) 16.4853 1.07768
\(235\) 0 0
\(236\) 47.7990 3.11145
\(237\) 24.0000 1.55897
\(238\) −32.9706 −2.13716
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −16.8284 −1.08401 −0.542007 0.840374i \(-0.682335\pi\)
−0.542007 + 0.840374i \(0.682335\pi\)
\(242\) −2.41421 −0.155192
\(243\) 10.0000 0.641500
\(244\) −14.0000 −0.896258
\(245\) 0 0
\(246\) −42.6274 −2.71782
\(247\) 6.82843 0.434482
\(248\) −5.17157 −0.328395
\(249\) 5.65685 0.358489
\(250\) 0 0
\(251\) 7.31371 0.461637 0.230819 0.972997i \(-0.425860\pi\)
0.230819 + 0.972997i \(0.425860\pi\)
\(252\) 10.8284 0.682127
\(253\) 4.00000 0.251478
\(254\) 36.1421 2.26776
\(255\) 0 0
\(256\) −29.9706 −1.87316
\(257\) −1.85786 −0.115890 −0.0579452 0.998320i \(-0.518455\pi\)
−0.0579452 + 0.998320i \(0.518455\pi\)
\(258\) 32.9706 2.05266
\(259\) 3.31371 0.205904
\(260\) 0 0
\(261\) 3.17157 0.196315
\(262\) −23.3137 −1.44033
\(263\) 12.4853 0.769875 0.384938 0.922943i \(-0.374223\pi\)
0.384938 + 0.922943i \(0.374223\pi\)
\(264\) −8.82843 −0.543352
\(265\) 0 0
\(266\) 6.82843 0.418678
\(267\) 26.6274 1.62957
\(268\) −1.31371 −0.0802475
\(269\) −23.6569 −1.44238 −0.721192 0.692735i \(-0.756405\pi\)
−0.721192 + 0.692735i \(0.756405\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) 14.4853 0.878299
\(273\) 38.6274 2.33784
\(274\) 22.4853 1.35839
\(275\) 0 0
\(276\) 30.6274 1.84355
\(277\) −8.14214 −0.489214 −0.244607 0.969622i \(-0.578659\pi\)
−0.244607 + 0.969622i \(0.578659\pi\)
\(278\) −50.6274 −3.03643
\(279\) 1.17157 0.0701402
\(280\) 0 0
\(281\) 10.4853 0.625499 0.312750 0.949836i \(-0.398750\pi\)
0.312750 + 0.949836i \(0.398750\pi\)
\(282\) 0 0
\(283\) 16.4853 0.979948 0.489974 0.871737i \(-0.337006\pi\)
0.489974 + 0.871737i \(0.337006\pi\)
\(284\) 35.1127 2.08356
\(285\) 0 0
\(286\) −16.4853 −0.974795
\(287\) −24.9706 −1.47397
\(288\) 1.58579 0.0934434
\(289\) 6.31371 0.371395
\(290\) 0 0
\(291\) 20.2843 1.18909
\(292\) 40.1421 2.34914
\(293\) 5.17157 0.302127 0.151063 0.988524i \(-0.451730\pi\)
0.151063 + 0.988524i \(0.451730\pi\)
\(294\) 4.82843 0.281600
\(295\) 0 0
\(296\) −5.17157 −0.300592
\(297\) −4.00000 −0.232104
\(298\) −18.4853 −1.07082
\(299\) 27.3137 1.57959
\(300\) 0 0
\(301\) 19.3137 1.11322
\(302\) −23.3137 −1.34155
\(303\) −15.3137 −0.879750
\(304\) −3.00000 −0.172062
\(305\) 0 0
\(306\) −11.6569 −0.666378
\(307\) 33.3137 1.90131 0.950657 0.310244i \(-0.100411\pi\)
0.950657 + 0.310244i \(0.100411\pi\)
\(308\) −10.8284 −0.617007
\(309\) 20.0000 1.13776
\(310\) 0 0
\(311\) 13.6569 0.774409 0.387205 0.921994i \(-0.373441\pi\)
0.387205 + 0.921994i \(0.373441\pi\)
\(312\) −60.2843 −3.41292
\(313\) −26.9706 −1.52447 −0.762233 0.647303i \(-0.775897\pi\)
−0.762233 + 0.647303i \(0.775897\pi\)
\(314\) −41.7990 −2.35885
\(315\) 0 0
\(316\) −45.9411 −2.58439
\(317\) 15.7990 0.887360 0.443680 0.896185i \(-0.353673\pi\)
0.443680 + 0.896185i \(0.353673\pi\)
\(318\) 13.6569 0.765838
\(319\) −3.17157 −0.177574
\(320\) 0 0
\(321\) 26.6274 1.48620
\(322\) 27.3137 1.52213
\(323\) −4.82843 −0.268661
\(324\) −42.1127 −2.33959
\(325\) 0 0
\(326\) 42.6274 2.36091
\(327\) 1.65685 0.0916242
\(328\) 38.9706 2.15179
\(329\) 0 0
\(330\) 0 0
\(331\) −23.7990 −1.30811 −0.654055 0.756447i \(-0.726934\pi\)
−0.654055 + 0.756447i \(0.726934\pi\)
\(332\) −10.8284 −0.594287
\(333\) 1.17157 0.0642018
\(334\) −33.7990 −1.84940
\(335\) 0 0
\(336\) −16.9706 −0.925820
\(337\) 11.5147 0.627247 0.313623 0.949547i \(-0.398457\pi\)
0.313623 + 0.949547i \(0.398457\pi\)
\(338\) −81.1838 −4.41581
\(339\) −24.9706 −1.35621
\(340\) 0 0
\(341\) −1.17157 −0.0634442
\(342\) 2.41421 0.130546
\(343\) −16.9706 −0.916324
\(344\) −30.1421 −1.62516
\(345\) 0 0
\(346\) 6.82843 0.367099
\(347\) 31.1127 1.67022 0.835109 0.550085i \(-0.185405\pi\)
0.835109 + 0.550085i \(0.185405\pi\)
\(348\) −24.2843 −1.30177
\(349\) −24.6274 −1.31828 −0.659138 0.752022i \(-0.729079\pi\)
−0.659138 + 0.752022i \(0.729079\pi\)
\(350\) 0 0
\(351\) −27.3137 −1.45790
\(352\) −1.58579 −0.0845227
\(353\) 10.0000 0.532246 0.266123 0.963939i \(-0.414257\pi\)
0.266123 + 0.963939i \(0.414257\pi\)
\(354\) 60.2843 3.20407
\(355\) 0 0
\(356\) −50.9706 −2.70143
\(357\) −27.3137 −1.44559
\(358\) −30.1421 −1.59306
\(359\) 19.3137 1.01934 0.509669 0.860370i \(-0.329768\pi\)
0.509669 + 0.860370i \(0.329768\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 12.8284 0.674247
\(363\) −2.00000 −0.104973
\(364\) −73.9411 −3.87557
\(365\) 0 0
\(366\) −17.6569 −0.922939
\(367\) −35.3137 −1.84336 −0.921680 0.387950i \(-0.873183\pi\)
−0.921680 + 0.387950i \(0.873183\pi\)
\(368\) −12.0000 −0.625543
\(369\) −8.82843 −0.459590
\(370\) 0 0
\(371\) 8.00000 0.415339
\(372\) −8.97056 −0.465102
\(373\) 13.1716 0.681998 0.340999 0.940064i \(-0.389235\pi\)
0.340999 + 0.940064i \(0.389235\pi\)
\(374\) 11.6569 0.602762
\(375\) 0 0
\(376\) 0 0
\(377\) −21.6569 −1.11538
\(378\) −27.3137 −1.40487
\(379\) −18.8284 −0.967151 −0.483576 0.875303i \(-0.660662\pi\)
−0.483576 + 0.875303i \(0.660662\pi\)
\(380\) 0 0
\(381\) 29.9411 1.53393
\(382\) −40.9706 −2.09624
\(383\) 13.3137 0.680299 0.340149 0.940371i \(-0.389522\pi\)
0.340149 + 0.940371i \(0.389522\pi\)
\(384\) −41.1127 −2.09802
\(385\) 0 0
\(386\) 38.1421 1.94138
\(387\) 6.82843 0.347108
\(388\) −38.8284 −1.97121
\(389\) −16.6274 −0.843044 −0.421522 0.906818i \(-0.638504\pi\)
−0.421522 + 0.906818i \(0.638504\pi\)
\(390\) 0 0
\(391\) −19.3137 −0.976736
\(392\) −4.41421 −0.222951
\(393\) −19.3137 −0.974248
\(394\) −33.3137 −1.67832
\(395\) 0 0
\(396\) −3.82843 −0.192386
\(397\) −13.3137 −0.668196 −0.334098 0.942538i \(-0.608432\pi\)
−0.334098 + 0.942538i \(0.608432\pi\)
\(398\) −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 \(0.351624\pi\)
\(402\) −1.65685 −0.0826364
\(403\) −8.00000 −0.398508
\(404\) 29.3137 1.45841
\(405\) 0 0
\(406\) −21.6569 −1.07481
\(407\) −1.17157 −0.0580727
\(408\) 42.6274 2.11037
\(409\) −7.17157 −0.354611 −0.177306 0.984156i \(-0.556738\pi\)
−0.177306 + 0.984156i \(0.556738\pi\)
\(410\) 0 0
\(411\) 18.6274 0.918823
\(412\) −38.2843 −1.88613
\(413\) 35.3137 1.73767
\(414\) 9.65685 0.474608
\(415\) 0 0
\(416\) −10.8284 −0.530907
\(417\) −41.9411 −2.05387
\(418\) −2.41421 −0.118083
\(419\) −12.9706 −0.633653 −0.316827 0.948483i \(-0.602617\pi\)
−0.316827 + 0.948483i \(0.602617\pi\)
\(420\) 0 0
\(421\) −10.6863 −0.520818 −0.260409 0.965498i \(-0.583857\pi\)
−0.260409 + 0.965498i \(0.583857\pi\)
\(422\) 57.9411 2.82053
\(423\) 0 0
\(424\) −12.4853 −0.606339
\(425\) 0 0
\(426\) 44.2843 2.14558
\(427\) −10.3431 −0.500540
\(428\) −50.9706 −2.46376
\(429\) −13.6569 −0.659359
\(430\) 0 0
\(431\) −26.6274 −1.28260 −0.641299 0.767291i \(-0.721604\pi\)
−0.641299 + 0.767291i \(0.721604\pi\)
\(432\) 12.0000 0.577350
\(433\) −37.4558 −1.80001 −0.900006 0.435876i \(-0.856438\pi\)
−0.900006 + 0.435876i \(0.856438\pi\)
\(434\) −8.00000 −0.384012
\(435\) 0 0
\(436\) −3.17157 −0.151891
\(437\) 4.00000 0.191346
\(438\) 50.6274 2.41907
\(439\) 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
\(442\) 79.5980 3.78609
\(443\) −26.3431 −1.25160 −0.625800 0.779983i \(-0.715228\pi\)
−0.625800 + 0.779983i \(0.715228\pi\)
\(444\) −8.97056 −0.425724
\(445\) 0 0
\(446\) 4.82843 0.228633
\(447\) −15.3137 −0.724314
\(448\) −27.7990 −1.31338
\(449\) 23.6569 1.11644 0.558218 0.829694i \(-0.311485\pi\)
0.558218 + 0.829694i \(0.311485\pi\)
\(450\) 0 0
\(451\) 8.82843 0.415714
\(452\) 47.7990 2.24828
\(453\) −19.3137 −0.907437
\(454\) 51.4558 2.41494
\(455\) 0 0
\(456\) −8.82843 −0.413429
\(457\) 23.4558 1.09722 0.548609 0.836079i \(-0.315158\pi\)
0.548609 + 0.836079i \(0.315158\pi\)
\(458\) −40.1421 −1.87572
\(459\) 19.3137 0.901487
\(460\) 0 0
\(461\) 38.9706 1.81504 0.907520 0.420009i \(-0.137973\pi\)
0.907520 + 0.420009i \(0.137973\pi\)
\(462\) −13.6569 −0.635374
\(463\) −26.6274 −1.23748 −0.618741 0.785595i \(-0.712357\pi\)
−0.618741 + 0.785595i \(0.712357\pi\)
\(464\) 9.51472 0.441710
\(465\) 0 0
\(466\) 25.3137 1.17263
\(467\) −24.0000 −1.11059 −0.555294 0.831654i \(-0.687394\pi\)
−0.555294 + 0.831654i \(0.687394\pi\)
\(468\) −26.1421 −1.20842
\(469\) −0.970563 −0.0448164
\(470\) 0 0
\(471\) −34.6274 −1.59555
\(472\) −55.1127 −2.53677
\(473\) −6.82843 −0.313971
\(474\) −57.9411 −2.66132
\(475\) 0 0
\(476\) 52.2843 2.39645
\(477\) 2.82843 0.129505
\(478\) 19.3137 0.883388
\(479\) −24.9706 −1.14093 −0.570467 0.821320i \(-0.693238\pi\)
−0.570467 + 0.821320i \(0.693238\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) 40.6274 1.85053
\(483\) 22.6274 1.02958
\(484\) 3.82843 0.174019
\(485\) 0 0
\(486\) −24.1421 −1.09511
\(487\) 9.31371 0.422044 0.211022 0.977481i \(-0.432321\pi\)
0.211022 + 0.977481i \(0.432321\pi\)
\(488\) 16.1421 0.730720
\(489\) 35.3137 1.59694
\(490\) 0 0
\(491\) −23.3137 −1.05213 −0.526066 0.850443i \(-0.676334\pi\)
−0.526066 + 0.850443i \(0.676334\pi\)
\(492\) 67.5980 3.04755
\(493\) 15.3137 0.689695
\(494\) −16.4853 −0.741708
\(495\) 0 0
\(496\) 3.51472 0.157816
\(497\) 25.9411 1.16362
\(498\) −13.6569 −0.611978
\(499\) −34.6274 −1.55014 −0.775068 0.631878i \(-0.782284\pi\)
−0.775068 + 0.631878i \(0.782284\pi\)
\(500\) 0 0
\(501\) −28.0000 −1.25095
\(502\) −17.6569 −0.788064
\(503\) −7.79899 −0.347740 −0.173870 0.984769i \(-0.555627\pi\)
−0.173870 + 0.984769i \(0.555627\pi\)
\(504\) −12.4853 −0.556139
\(505\) 0 0
\(506\) −9.65685 −0.429300
\(507\) −67.2548 −2.98689
\(508\) −57.3137 −2.54288
\(509\) 2.00000 0.0886484 0.0443242 0.999017i \(-0.485887\pi\)
0.0443242 + 0.999017i \(0.485887\pi\)
\(510\) 0 0
\(511\) 29.6569 1.31194
\(512\) 31.2426 1.38074
\(513\) −4.00000 −0.176604
\(514\) 4.48528 0.197837
\(515\) 0 0
\(516\) −52.2843 −2.30169
\(517\) 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 \(0.541958\pi\)
\(522\) −7.65685 −0.335131
\(523\) −27.6569 −1.20935 −0.604675 0.796472i \(-0.706697\pi\)
−0.604675 + 0.796472i \(0.706697\pi\)
\(524\) 36.9706 1.61507
\(525\) 0 0
\(526\) −30.1421 −1.31426
\(527\) 5.65685 0.246416
\(528\) 6.00000 0.261116
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 12.4853 0.541815
\(532\) −10.8284 −0.469472
\(533\) 60.2843 2.61120
\(534\) −64.2843 −2.78185
\(535\) 0 0
\(536\) 1.51472 0.0654259
\(537\) −24.9706 −1.07756
\(538\) 57.1127 2.46230
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −39.9411 −1.71720 −0.858602 0.512644i \(-0.828666\pi\)
−0.858602 + 0.512644i \(0.828666\pi\)
\(542\) 57.9411 2.48878
\(543\) 10.6274 0.456066
\(544\) 7.65685 0.328285
\(545\) 0 0
\(546\) −93.2548 −3.99094
\(547\) 34.2843 1.46589 0.732945 0.680288i \(-0.238145\pi\)
0.732945 + 0.680288i \(0.238145\pi\)
\(548\) −35.6569 −1.52319
\(549\) −3.65685 −0.156071
\(550\) 0 0
\(551\) −3.17157 −0.135114
\(552\) −35.3137 −1.50305
\(553\) −33.9411 −1.44332
\(554\) 19.6569 0.835140
\(555\) 0 0
\(556\) 80.2843 3.40481
\(557\) −35.4558 −1.50231 −0.751156 0.660125i \(-0.770503\pi\)
−0.751156 + 0.660125i \(0.770503\pi\)
\(558\) −2.82843 −0.119737
\(559\) −46.6274 −1.97213
\(560\) 0 0
\(561\) 9.65685 0.407713
\(562\) −25.3137 −1.06779
\(563\) 12.3431 0.520202 0.260101 0.965581i \(-0.416244\pi\)
0.260101 + 0.965581i \(0.416244\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −39.7990 −1.67288
\(567\) −31.1127 −1.30661
\(568\) −40.4853 −1.69872
\(569\) 20.1421 0.844402 0.422201 0.906502i \(-0.361258\pi\)
0.422201 + 0.906502i \(0.361258\pi\)
\(570\) 0 0
\(571\) −30.3431 −1.26982 −0.634911 0.772586i \(-0.718963\pi\)
−0.634911 + 0.772586i \(0.718963\pi\)
\(572\) 26.1421 1.09306
\(573\) −33.9411 −1.41791
\(574\) 60.2843 2.51622
\(575\) 0 0
\(576\) −9.82843 −0.409518
\(577\) 17.3137 0.720779 0.360390 0.932802i \(-0.382644\pi\)
0.360390 + 0.932802i \(0.382644\pi\)
\(578\) −15.2426 −0.634010
\(579\) 31.5980 1.31317
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) −48.9706 −2.02990
\(583\) −2.82843 −0.117141
\(584\) −46.2843 −1.91526
\(585\) 0 0
\(586\) −12.4853 −0.515762
\(587\) −32.0000 −1.32078 −0.660391 0.750922i \(-0.729609\pi\)
−0.660391 + 0.750922i \(0.729609\pi\)
\(588\) −7.65685 −0.315763
\(589\) −1.17157 −0.0482738
\(590\) 0 0
\(591\) −27.5980 −1.13523
\(592\) 3.51472 0.144454
\(593\) −36.8284 −1.51236 −0.756181 0.654362i \(-0.772937\pi\)
−0.756181 + 0.654362i \(0.772937\pi\)
\(594\) 9.65685 0.396226
\(595\) 0 0
\(596\) 29.3137 1.20074
\(597\) −4.68629 −0.191797
\(598\) −65.9411 −2.69653
\(599\) −27.7990 −1.13584 −0.567918 0.823085i \(-0.692251\pi\)
−0.567918 + 0.823085i \(0.692251\pi\)
\(600\) 0 0
\(601\) 36.8284 1.50226 0.751131 0.660153i \(-0.229508\pi\)
0.751131 + 0.660153i \(0.229508\pi\)
\(602\) −46.6274 −1.90039
\(603\) −0.343146 −0.0139740
\(604\) 36.9706 1.50431
\(605\) 0 0
\(606\) 36.9706 1.50183
\(607\) 6.97056 0.282926 0.141463 0.989944i \(-0.454819\pi\)
0.141463 + 0.989944i \(0.454819\pi\)
\(608\) −1.58579 −0.0643121
\(609\) −17.9411 −0.727011
\(610\) 0 0
\(611\) 0 0
\(612\) 18.4853 0.747223
\(613\) 28.1421 1.13665 0.568325 0.822804i \(-0.307592\pi\)
0.568325 + 0.822804i \(0.307592\pi\)
\(614\) −80.4264 −3.24575
\(615\) 0 0
\(616\) 12.4853 0.503046
\(617\) −22.9706 −0.924760 −0.462380 0.886682i \(-0.653004\pi\)
−0.462380 + 0.886682i \(0.653004\pi\)
\(618\) −48.2843 −1.94228
\(619\) 47.3137 1.90170 0.950849 0.309654i \(-0.100213\pi\)
0.950849 + 0.309654i \(0.100213\pi\)
\(620\) 0 0
\(621\) −16.0000 −0.642058
\(622\) −32.9706 −1.32200
\(623\) −37.6569 −1.50869
\(624\) 40.9706 1.64014
\(625\) 0 0
\(626\) 65.1127 2.60243
\(627\) −2.00000 −0.0798723
\(628\) 66.2843 2.64503
\(629\) 5.65685 0.225554
\(630\) 0 0
\(631\) −5.65685 −0.225196 −0.112598 0.993641i \(-0.535917\pi\)
−0.112598 + 0.993641i \(0.535917\pi\)
\(632\) 52.9706 2.10706
\(633\) 48.0000 1.90783
\(634\) −38.1421 −1.51482
\(635\) 0 0
\(636\) −21.6569 −0.858750
\(637\) −6.82843 −0.270552
\(638\) 7.65685 0.303138
\(639\) 9.17157 0.362822
\(640\) 0 0
\(641\) −19.9411 −0.787627 −0.393814 0.919190i \(-0.628844\pi\)
−0.393814 + 0.919190i \(0.628844\pi\)
\(642\) −64.2843 −2.53710
\(643\) 21.6569 0.854063 0.427031 0.904237i \(-0.359559\pi\)
0.427031 + 0.904237i \(0.359559\pi\)
\(644\) −43.3137 −1.70680
\(645\) 0 0
\(646\) 11.6569 0.458633
\(647\) −20.9706 −0.824438 −0.412219 0.911085i \(-0.635246\pi\)
−0.412219 + 0.911085i \(0.635246\pi\)
\(648\) 48.5563 1.90747
\(649\) −12.4853 −0.490090
\(650\) 0 0
\(651\) −6.62742 −0.259749
\(652\) −67.5980 −2.64734
\(653\) 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
\(656\) −26.4853 −1.03408
\(657\) 10.4853 0.409070
\(658\) 0 0
\(659\) 7.02944 0.273828 0.136914 0.990583i \(-0.456282\pi\)
0.136914 + 0.990583i \(0.456282\pi\)
\(660\) 0 0
\(661\) 42.9706 1.67136 0.835681 0.549216i \(-0.185073\pi\)
0.835681 + 0.549216i \(0.185073\pi\)
\(662\) 57.4558 2.23308
\(663\) 65.9411 2.56094
\(664\) 12.4853 0.484523
\(665\) 0 0
\(666\) −2.82843 −0.109599
\(667\) −12.6863 −0.491215
\(668\) 53.5980 2.07377
\(669\) 4.00000 0.154649
\(670\) 0 0
\(671\) 3.65685 0.141171
\(672\) −8.97056 −0.346047
\(673\) 33.4558 1.28963 0.644814 0.764340i \(-0.276935\pi\)
0.644814 + 0.764340i \(0.276935\pi\)
\(674\) −27.7990 −1.07078
\(675\) 0 0
\(676\) 128.740 4.95154
\(677\) 1.85786 0.0714035 0.0357018 0.999362i \(-0.488633\pi\)
0.0357018 + 0.999362i \(0.488633\pi\)
\(678\) 60.2843 2.31520
\(679\) −28.6863 −1.10088
\(680\) 0 0
\(681\) 42.6274 1.63349
\(682\) 2.82843 0.108306
\(683\) 5.31371 0.203323 0.101662 0.994819i \(-0.467584\pi\)
0.101662 + 0.994819i \(0.467584\pi\)
\(684\) −3.82843 −0.146384
\(685\) 0 0
\(686\) 40.9706 1.56426
\(687\) −33.2548 −1.26875
\(688\) 20.4853 0.780994
\(689\) −19.3137 −0.735794
\(690\) 0 0
\(691\) 10.6274 0.404286 0.202143 0.979356i \(-0.435209\pi\)
0.202143 + 0.979356i \(0.435209\pi\)
\(692\) −10.8284 −0.411635
\(693\) −2.82843 −0.107443
\(694\) −75.1127 −2.85124
\(695\) 0 0
\(696\) 28.0000 1.06134
\(697\) −42.6274 −1.61463
\(698\) 59.4558 2.25044
\(699\) 20.9706 0.793180
\(700\) 0 0
\(701\) −39.9411 −1.50856 −0.754278 0.656555i \(-0.772013\pi\)
−0.754278 + 0.656555i \(0.772013\pi\)
\(702\) 65.9411 2.48879
\(703\) −1.17157 −0.0441867
\(704\) 9.82843 0.370423
\(705\) 0 0
\(706\) −24.1421 −0.908601
\(707\) 21.6569 0.814490
\(708\) −95.5980 −3.59279
\(709\) −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
\(712\) 58.7696 2.20248
\(713\) −4.68629 −0.175503
\(714\) 65.9411 2.46778
\(715\) 0 0
\(716\) 47.7990 1.78633
\(717\) 16.0000 0.597531
\(718\) −46.6274 −1.74012
\(719\) −12.6863 −0.473119 −0.236559 0.971617i \(-0.576020\pi\)
−0.236559 + 0.971617i \(0.576020\pi\)
\(720\) 0 0
\(721\) −28.2843 −1.05336
\(722\) −2.41421 −0.0898477
\(723\) 33.6569 1.25171
\(724\) −20.3431 −0.756047
\(725\) 0 0
\(726\) 4.82843 0.179200
\(727\) −24.9706 −0.926107 −0.463053 0.886330i \(-0.653246\pi\)
−0.463053 + 0.886330i \(0.653246\pi\)
\(728\) 85.2548 3.15975
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 32.9706 1.21946
\(732\) 28.0000 1.03491
\(733\) −6.48528 −0.239539 −0.119770 0.992802i \(-0.538216\pi\)
−0.119770 + 0.992802i \(0.538216\pi\)
\(734\) 85.2548 3.14681
\(735\) 0 0
\(736\) −6.34315 −0.233811
\(737\) 0.343146 0.0126399
\(738\) 21.3137 0.784568
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 0 0
\(741\) −13.6569 −0.501697
\(742\) −19.3137 −0.709029
\(743\) −27.6569 −1.01463 −0.507316 0.861760i \(-0.669362\pi\)
−0.507316 + 0.861760i \(0.669362\pi\)
\(744\) 10.3431 0.379198
\(745\) 0 0
\(746\) −31.7990 −1.16424
\(747\) −2.82843 −0.103487
\(748\) −18.4853 −0.675889
\(749\) −37.6569 −1.37595
\(750\) 0 0
\(751\) −51.7990 −1.89017 −0.945086 0.326822i \(-0.894022\pi\)
−0.945086 + 0.326822i \(0.894022\pi\)
\(752\) 0 0
\(753\) −14.6274 −0.533053
\(754\) 52.2843 1.90408
\(755\) 0 0
\(756\) 43.3137 1.57530
\(757\) 6.68629 0.243017 0.121509 0.992590i \(-0.461227\pi\)
0.121509 + 0.992590i \(0.461227\pi\)
\(758\) 45.4558 1.65103
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) 48.9117 1.77305 0.886524 0.462683i \(-0.153113\pi\)
0.886524 + 0.462683i \(0.153113\pi\)
\(762\) −72.2843 −2.61858
\(763\) −2.34315 −0.0848276
\(764\) 64.9706 2.35055
\(765\) 0 0
\(766\) −32.1421 −1.16134
\(767\) −85.2548 −3.07837
\(768\) 59.9411 2.16294
\(769\) −1.02944 −0.0371225 −0.0185612 0.999828i \(-0.505909\pi\)
−0.0185612 + 0.999828i \(0.505909\pi\)
\(770\) 0 0
\(771\) 3.71573 0.133819
\(772\) −60.4853 −2.17691
\(773\) 31.7990 1.14373 0.571865 0.820348i \(-0.306220\pi\)
0.571865 + 0.820348i \(0.306220\pi\)
\(774\) −16.4853 −0.592551
\(775\) 0 0
\(776\) 44.7696 1.60713
\(777\) −6.62742 −0.237757
\(778\) 40.1421 1.43917
\(779\) 8.82843 0.316311
\(780\) 0 0
\(781\) −9.17157 −0.328185
\(782\) 46.6274 1.66739
\(783\) 12.6863 0.453371
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) 46.6274 1.66314
\(787\) 14.9706 0.533643 0.266821 0.963746i \(-0.414027\pi\)
0.266821 + 0.963746i \(0.414027\pi\)
\(788\) 52.8284 1.88193
\(789\) −24.9706 −0.888976
\(790\) 0 0
\(791\) 35.3137 1.25561
\(792\) 4.41421 0.156852
\(793\) 24.9706 0.886731
\(794\) 32.1421 1.14068
\(795\) 0 0
\(796\) 8.97056 0.317953
\(797\) 10.8284 0.383563 0.191781 0.981438i \(-0.438574\pi\)
0.191781 + 0.981438i \(0.438574\pi\)
\(798\) −13.6569 −0.483447
\(799\) 0 0
\(800\) 0 0
\(801\) −13.3137 −0.470417
\(802\) −43.4558 −1.53448
\(803\) −10.4853 −0.370018
\(804\) 2.62742 0.0926619
\(805\) 0 0
\(806\) 19.3137 0.680296
\(807\) 47.3137 1.66552
\(808\) −33.7990 −1.18904
\(809\) −43.6569 −1.53489 −0.767447 0.641113i \(-0.778473\pi\)
−0.767447 + 0.641113i \(0.778473\pi\)
\(810\) 0 0
\(811\) −7.31371 −0.256819 −0.128410 0.991721i \(-0.540987\pi\)
−0.128410 + 0.991721i \(0.540987\pi\)
\(812\) 34.3431 1.20521
\(813\) 48.0000 1.68343
\(814\) 2.82843 0.0991363
\(815\) 0 0
\(816\) −28.9706 −1.01417
\(817\) −6.82843 −0.238896
\(818\) 17.3137 0.605360
\(819\) −19.3137 −0.674876
\(820\) 0 0
\(821\) 44.6274 1.55751 0.778754 0.627330i \(-0.215852\pi\)
0.778754 + 0.627330i \(0.215852\pi\)
\(822\) −44.9706 −1.56853
\(823\) −5.65685 −0.197186 −0.0985928 0.995128i \(-0.531434\pi\)
−0.0985928 + 0.995128i \(0.531434\pi\)
\(824\) 44.1421 1.53776
\(825\) 0 0
\(826\) −85.2548 −2.96640
\(827\) −35.6569 −1.23991 −0.619955 0.784637i \(-0.712849\pi\)
−0.619955 + 0.784637i \(0.712849\pi\)
\(828\) −15.3137 −0.532188
\(829\) −26.9706 −0.936726 −0.468363 0.883536i \(-0.655156\pi\)
−0.468363 + 0.883536i \(0.655156\pi\)
\(830\) 0 0
\(831\) 16.2843 0.564895
\(832\) 67.1127 2.32671
\(833\) 4.82843 0.167295
\(834\) 101.255 3.50617
\(835\) 0 0
\(836\) 3.82843 0.132409
\(837\) 4.68629 0.161982
\(838\) 31.3137 1.08171
\(839\) −7.51472 −0.259437 −0.129718 0.991551i \(-0.541407\pi\)
−0.129718 + 0.991551i \(0.541407\pi\)
\(840\) 0 0
\(841\) −18.9411 −0.653142
\(842\) 25.7990 0.889092
\(843\) −20.9706 −0.722265
\(844\) −91.8823 −3.16272
\(845\) 0 0
\(846\) 0 0
\(847\) 2.82843 0.0971859
\(848\) 8.48528 0.291386
\(849\) −32.9706 −1.13155
\(850\) 0 0
\(851\) −4.68629 −0.160644
\(852\) −70.2254 −2.40588
\(853\) −42.7696 −1.46440 −0.732201 0.681089i \(-0.761507\pi\)
−0.732201 + 0.681089i \(0.761507\pi\)
\(854\) 24.9706 0.854475
\(855\) 0 0
\(856\) 58.7696 2.00870
\(857\) 9.45584 0.323005 0.161503 0.986872i \(-0.448366\pi\)
0.161503 + 0.986872i \(0.448366\pi\)
\(858\) 32.9706 1.12560
\(859\) 31.3137 1.06841 0.534205 0.845355i \(-0.320611\pi\)
0.534205 + 0.845355i \(0.320611\pi\)
\(860\) 0 0
\(861\) 49.9411 1.70199
\(862\) 64.2843 2.18953
\(863\) −13.0294 −0.443527 −0.221764 0.975100i \(-0.571181\pi\)
−0.221764 + 0.975100i \(0.571181\pi\)
\(864\) 6.34315 0.215798
\(865\) 0 0
\(866\) 90.4264 3.07281
\(867\) −12.6274 −0.428850
\(868\) 12.6863 0.430601
\(869\) 12.0000 0.407072
\(870\) 0 0
\(871\) 2.34315 0.0793945
\(872\) 3.65685 0.123837
\(873\) −10.1421 −0.343259
\(874\) −9.65685 −0.326648
\(875\) 0 0
\(876\) −80.2843 −2.71255
\(877\) −15.5147 −0.523895 −0.261947 0.965082i \(-0.584365\pi\)
−0.261947 + 0.965082i \(0.584365\pi\)
\(878\) −48.2843 −1.62952
\(879\) −10.3431 −0.348866
\(880\) 0 0
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) −2.41421 −0.0812908
\(883\) −48.2843 −1.62490 −0.812448 0.583034i \(-0.801865\pi\)
−0.812448 + 0.583034i \(0.801865\pi\)
\(884\) −126.225 −4.24542
\(885\) 0 0
\(886\) 63.5980 2.13662
\(887\) −25.3137 −0.849951 −0.424976 0.905205i \(-0.639717\pi\)
−0.424976 + 0.905205i \(0.639717\pi\)
\(888\) 10.3431 0.347093
\(889\) −42.3431 −1.42014
\(890\) 0 0
\(891\) 11.0000 0.368514
\(892\) −7.65685 −0.256370
\(893\) 0 0
\(894\) 36.9706 1.23648
\(895\) 0 0
\(896\) 58.1421 1.94239
\(897\) −54.6274 −1.82396
\(898\) −57.1127 −1.90588
\(899\) 3.71573 0.123926
\(900\) 0 0
\(901\) 13.6569 0.454976
\(902\) −21.3137 −0.709669
\(903\) −38.6274 −1.28544
\(904\) −55.1127 −1.83302
\(905\) 0 0
\(906\) 46.6274 1.54909
\(907\) 3.65685 0.121424 0.0607119 0.998155i \(-0.480663\pi\)
0.0607119 + 0.998155i \(0.480663\pi\)
\(908\) −81.5980 −2.70792
\(909\) 7.65685 0.253962
\(910\) 0 0
\(911\) 17.4558 0.578338 0.289169 0.957278i \(-0.406621\pi\)
0.289169 + 0.957278i \(0.406621\pi\)
\(912\) 6.00000 0.198680
\(913\) 2.82843 0.0936073
\(914\) −56.6274 −1.87307
\(915\) 0 0
\(916\) 63.6569 2.10328
\(917\) 27.3137 0.901978
\(918\) −46.6274 −1.53893
\(919\) −0.970563 −0.0320159 −0.0160080 0.999872i \(-0.505096\pi\)
−0.0160080 + 0.999872i \(0.505096\pi\)
\(920\) 0 0
\(921\) −66.6274 −2.19545
\(922\) −94.0833 −3.09847
\(923\) −62.6274 −2.06141
\(924\) 21.6569 0.712458
\(925\) 0 0
\(926\) 64.2843 2.11251
\(927\) −10.0000 −0.328443
\(928\) 5.02944 0.165099
\(929\) −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
\(932\) −40.1421 −1.31490
\(933\) −27.3137 −0.894211
\(934\) 57.9411 1.89589
\(935\) 0 0
\(936\) 30.1421 0.985227
\(937\) 17.5147 0.572181 0.286090 0.958203i \(-0.407644\pi\)
0.286090 + 0.958203i \(0.407644\pi\)
\(938\) 2.34315 0.0765064
\(939\) 53.9411 1.76030
\(940\) 0 0
\(941\) −4.14214 −0.135030 −0.0675149 0.997718i \(-0.521507\pi\)
−0.0675149 + 0.997718i \(0.521507\pi\)
\(942\) 83.5980 2.72377
\(943\) 35.3137 1.14997
\(944\) 37.4558 1.21908
\(945\) 0 0
\(946\) 16.4853 0.535983
\(947\) −19.3137 −0.627611 −0.313806 0.949487i \(-0.601604\pi\)
−0.313806 + 0.949487i \(0.601604\pi\)
\(948\) 91.8823 2.98420
\(949\) −71.5980 −2.32417
\(950\) 0 0
\(951\) −31.5980 −1.02463
\(952\) −60.2843 −1.95382
\(953\) 38.1421 1.23554 0.617772 0.786357i \(-0.288035\pi\)
0.617772 + 0.786357i \(0.288035\pi\)
\(954\) −6.82843 −0.221078
\(955\) 0 0
\(956\) −30.6274 −0.990561
\(957\) 6.34315 0.205045
\(958\) 60.2843 1.94770
\(959\) −26.3431 −0.850665
\(960\) 0 0
\(961\) −29.6274 −0.955723
\(962\) 19.3137 0.622699
\(963\) −13.3137 −0.429028
\(964\) −64.4264 −2.07503
\(965\) 0 0
\(966\) −54.6274 −1.75761
\(967\) 24.7696 0.796535 0.398268 0.917269i \(-0.369612\pi\)
0.398268 + 0.917269i \(0.369612\pi\)
\(968\) −4.41421 −0.141878
\(969\) 9.65685 0.310223
\(970\) 0 0
\(971\) 7.51472 0.241159 0.120579 0.992704i \(-0.461525\pi\)
0.120579 + 0.992704i \(0.461525\pi\)
\(972\) 38.2843 1.22797
\(973\) 59.3137 1.90151
\(974\) −22.4853 −0.720475
\(975\) 0 0
\(976\) −10.9706 −0.351159
\(977\) −21.1716 −0.677339 −0.338669 0.940905i \(-0.609977\pi\)
−0.338669 + 0.940905i \(0.609977\pi\)
\(978\) −85.2548 −2.72615
\(979\) 13.3137 0.425508
\(980\) 0 0
\(981\) −0.828427 −0.0264496
\(982\) 56.2843 1.79610
\(983\) −51.6569 −1.64760 −0.823799 0.566882i \(-0.808149\pi\)
−0.823799 + 0.566882i \(0.808149\pi\)
\(984\) −77.9411 −2.48467
\(985\) 0 0
\(986\) −36.9706 −1.17738
\(987\) 0 0
\(988\) 26.1421 0.831692
\(989\) −27.3137 −0.868525
\(990\) 0 0
\(991\) −40.0833 −1.27329 −0.636643 0.771158i \(-0.719678\pi\)
−0.636643 + 0.771158i \(0.719678\pi\)
\(992\) 1.85786 0.0589873
\(993\) 47.5980 1.51048
\(994\) −62.6274 −1.98642
\(995\) 0 0
\(996\) 21.6569 0.686224
\(997\) 4.14214 0.131183 0.0655914 0.997847i \(-0.479107\pi\)
0.0655914 + 0.997847i \(0.479107\pi\)
\(998\) 83.5980 2.64625
\(999\) 4.68629 0.148268
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.1.1 2
5.4 even 2 1045.2.a.c.1.2 2
15.14 odd 2 9405.2.a.o.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.c.1.2 2 5.4 even 2
5225.2.a.e.1.1 2 1.1 even 1 trivial
9405.2.a.o.1.1 2 15.14 odd 2