Properties

Label 825.2.a.j.1.3
Level $825$
Weight $2$
Character 825.1
Self dual yes
Analytic conductor $6.588$
Analytic rank $1$
Dimension $3$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.48119\) 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)
 
\(f(q)\) \(=\) \(q+1.48119 q^{2} -1.00000 q^{3} +0.193937 q^{4} -1.48119 q^{6} -1.19394 q^{7} -2.67513 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.48119 q^{2} -1.00000 q^{3} +0.193937 q^{4} -1.48119 q^{6} -1.19394 q^{7} -2.67513 q^{8} +1.00000 q^{9} +1.00000 q^{11} -0.193937 q^{12} -0.806063 q^{13} -1.76845 q^{14} -4.35026 q^{16} -3.76845 q^{17} +1.48119 q^{18} -5.35026 q^{19} +1.19394 q^{21} +1.48119 q^{22} -4.00000 q^{23} +2.67513 q^{24} -1.19394 q^{26} -1.00000 q^{27} -0.231548 q^{28} -4.31265 q^{29} +0.962389 q^{31} -1.09332 q^{32} -1.00000 q^{33} -5.58181 q^{34} +0.193937 q^{36} -1.61213 q^{37} -7.92478 q^{38} +0.806063 q^{39} +9.08840 q^{41} +1.76845 q^{42} -4.41819 q^{43} +0.193937 q^{44} -5.92478 q^{46} -12.3127 q^{47} +4.35026 q^{48} -5.57452 q^{49} +3.76845 q^{51} -0.156325 q^{52} -1.42548 q^{53} -1.48119 q^{54} +3.19394 q^{56} +5.35026 q^{57} -6.38787 q^{58} +13.2750 q^{59} -0.0752228 q^{61} +1.42548 q^{62} -1.19394 q^{63} +7.08110 q^{64} -1.48119 q^{66} +2.70052 q^{67} -0.730841 q^{68} +4.00000 q^{69} -14.0508 q^{71} -2.67513 q^{72} +10.7308 q^{73} -2.38787 q^{74} -1.03761 q^{76} -1.19394 q^{77} +1.19394 q^{78} +13.9756 q^{79} +1.00000 q^{81} +13.4617 q^{82} +9.89446 q^{83} +0.231548 q^{84} -6.54420 q^{86} +4.31265 q^{87} -2.67513 q^{88} +16.8872 q^{89} +0.962389 q^{91} -0.775746 q^{92} -0.962389 q^{93} -18.2374 q^{94} +1.09332 q^{96} -11.4763 q^{97} -8.25694 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} - 3 q^{3} + q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} - 3 q^{3} + q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 3 q^{9} + 3 q^{11} - q^{12} - 2 q^{13} + 6 q^{14} - 3 q^{16} - q^{18} - 6 q^{19} + 4 q^{21} - q^{22} - 12 q^{23} + 3 q^{24} - 4 q^{26} - 3 q^{27} - 12 q^{28} + 8 q^{29} - 8 q^{31} + 3 q^{32} - 3 q^{33} - 18 q^{34} + q^{36} - 4 q^{37} - 2 q^{38} + 2 q^{39} + 8 q^{41} - 6 q^{42} - 12 q^{43} + q^{44} + 4 q^{46} - 16 q^{47} + 3 q^{48} - 5 q^{49} + 10 q^{52} - 16 q^{53} + q^{54} + 10 q^{56} + 6 q^{57} - 20 q^{58} + 8 q^{59} - 22 q^{61} + 16 q^{62} - 4 q^{63} - 11 q^{64} + q^{66} - 12 q^{67} + 20 q^{68} + 12 q^{69} - 12 q^{71} - 3 q^{72} + 10 q^{73} - 8 q^{74} - 14 q^{76} - 4 q^{77} + 4 q^{78} - 10 q^{79} + 3 q^{81} - 4 q^{82} + 10 q^{83} + 12 q^{84} - 10 q^{86} - 8 q^{87} - 3 q^{88} + 18 q^{89} - 8 q^{91} - 4 q^{92} + 8 q^{93} - 12 q^{94} - 3 q^{96} - 16 q^{97} - 21 q^{98} + 3 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.48119 1.04736 0.523681 0.851914i \(-0.324558\pi\)
0.523681 + 0.851914i \(0.324558\pi\)
\(3\) −1.00000 −0.577350
\(4\) 0.193937 0.0969683
\(5\) 0 0
\(6\) −1.48119 −0.604695
\(7\) −1.19394 −0.451266 −0.225633 0.974212i \(-0.572445\pi\)
−0.225633 + 0.974212i \(0.572445\pi\)
\(8\) −2.67513 −0.945802
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) −0.193937 −0.0559847
\(13\) −0.806063 −0.223562 −0.111781 0.993733i \(-0.535655\pi\)
−0.111781 + 0.993733i \(0.535655\pi\)
\(14\) −1.76845 −0.472639
\(15\) 0 0
\(16\) −4.35026 −1.08757
\(17\) −3.76845 −0.913984 −0.456992 0.889471i \(-0.651073\pi\)
−0.456992 + 0.889471i \(0.651073\pi\)
\(18\) 1.48119 0.349121
\(19\) −5.35026 −1.22743 −0.613717 0.789526i \(-0.710326\pi\)
−0.613717 + 0.789526i \(0.710326\pi\)
\(20\) 0 0
\(21\) 1.19394 0.260538
\(22\) 1.48119 0.315792
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 2.67513 0.546059
\(25\) 0 0
\(26\) −1.19394 −0.234150
\(27\) −1.00000 −0.192450
\(28\) −0.231548 −0.0437585
\(29\) −4.31265 −0.800839 −0.400420 0.916332i \(-0.631136\pi\)
−0.400420 + 0.916332i \(0.631136\pi\)
\(30\) 0 0
\(31\) 0.962389 0.172850 0.0864250 0.996258i \(-0.472456\pi\)
0.0864250 + 0.996258i \(0.472456\pi\)
\(32\) −1.09332 −0.193274
\(33\) −1.00000 −0.174078
\(34\) −5.58181 −0.957272
\(35\) 0 0
\(36\) 0.193937 0.0323228
\(37\) −1.61213 −0.265032 −0.132516 0.991181i \(-0.542306\pi\)
−0.132516 + 0.991181i \(0.542306\pi\)
\(38\) −7.92478 −1.28557
\(39\) 0.806063 0.129073
\(40\) 0 0
\(41\) 9.08840 1.41937 0.709685 0.704520i \(-0.248837\pi\)
0.709685 + 0.704520i \(0.248837\pi\)
\(42\) 1.76845 0.272878
\(43\) −4.41819 −0.673768 −0.336884 0.941546i \(-0.609373\pi\)
−0.336884 + 0.941546i \(0.609373\pi\)
\(44\) 0.193937 0.0292370
\(45\) 0 0
\(46\) −5.92478 −0.873561
\(47\) −12.3127 −1.79598 −0.897992 0.440011i \(-0.854974\pi\)
−0.897992 + 0.440011i \(0.854974\pi\)
\(48\) 4.35026 0.627906
\(49\) −5.57452 −0.796359
\(50\) 0 0
\(51\) 3.76845 0.527689
\(52\) −0.156325 −0.0216784
\(53\) −1.42548 −0.195805 −0.0979027 0.995196i \(-0.531213\pi\)
−0.0979027 + 0.995196i \(0.531213\pi\)
\(54\) −1.48119 −0.201565
\(55\) 0 0
\(56\) 3.19394 0.426808
\(57\) 5.35026 0.708659
\(58\) −6.38787 −0.838769
\(59\) 13.2750 1.72826 0.864131 0.503266i \(-0.167868\pi\)
0.864131 + 0.503266i \(0.167868\pi\)
\(60\) 0 0
\(61\) −0.0752228 −0.00963129 −0.00481565 0.999988i \(-0.501533\pi\)
−0.00481565 + 0.999988i \(0.501533\pi\)
\(62\) 1.42548 0.181037
\(63\) −1.19394 −0.150422
\(64\) 7.08110 0.885138
\(65\) 0 0
\(66\) −1.48119 −0.182322
\(67\) 2.70052 0.329921 0.164961 0.986300i \(-0.447250\pi\)
0.164961 + 0.986300i \(0.447250\pi\)
\(68\) −0.730841 −0.0886274
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −14.0508 −1.66752 −0.833761 0.552126i \(-0.813817\pi\)
−0.833761 + 0.552126i \(0.813817\pi\)
\(72\) −2.67513 −0.315267
\(73\) 10.7308 1.25595 0.627975 0.778234i \(-0.283884\pi\)
0.627975 + 0.778234i \(0.283884\pi\)
\(74\) −2.38787 −0.277585
\(75\) 0 0
\(76\) −1.03761 −0.119022
\(77\) −1.19394 −0.136062
\(78\) 1.19394 0.135187
\(79\) 13.9756 1.57237 0.786187 0.617989i \(-0.212052\pi\)
0.786187 + 0.617989i \(0.212052\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 13.4617 1.48659
\(83\) 9.89446 1.08606 0.543029 0.839714i \(-0.317277\pi\)
0.543029 + 0.839714i \(0.317277\pi\)
\(84\) 0.231548 0.0252640
\(85\) 0 0
\(86\) −6.54420 −0.705679
\(87\) 4.31265 0.462365
\(88\) −2.67513 −0.285170
\(89\) 16.8872 1.79004 0.895018 0.446030i \(-0.147163\pi\)
0.895018 + 0.446030i \(0.147163\pi\)
\(90\) 0 0
\(91\) 0.962389 0.100886
\(92\) −0.775746 −0.0808771
\(93\) −0.962389 −0.0997950
\(94\) −18.2374 −1.88105
\(95\) 0 0
\(96\) 1.09332 0.111587
\(97\) −11.4763 −1.16524 −0.582619 0.812745i \(-0.697972\pi\)
−0.582619 + 0.812745i \(0.697972\pi\)
\(98\) −8.25694 −0.834077
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 10.7612 1.07078 0.535388 0.844606i \(-0.320166\pi\)
0.535388 + 0.844606i \(0.320166\pi\)
\(102\) 5.58181 0.552682
\(103\) −16.9380 −1.66895 −0.834473 0.551049i \(-0.814228\pi\)
−0.834473 + 0.551049i \(0.814228\pi\)
\(104\) 2.15633 0.211445
\(105\) 0 0
\(106\) −2.11142 −0.205079
\(107\) −8.28233 −0.800683 −0.400342 0.916366i \(-0.631109\pi\)
−0.400342 + 0.916366i \(0.631109\pi\)
\(108\) −0.193937 −0.0186616
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 1.61213 0.153016
\(112\) 5.19394 0.490781
\(113\) −2.26187 −0.212778 −0.106389 0.994325i \(-0.533929\pi\)
−0.106389 + 0.994325i \(0.533929\pi\)
\(114\) 7.92478 0.742223
\(115\) 0 0
\(116\) −0.836381 −0.0776560
\(117\) −0.806063 −0.0745206
\(118\) 19.6629 1.81012
\(119\) 4.49929 0.412449
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −0.111420 −0.0100875
\(123\) −9.08840 −0.819473
\(124\) 0.186642 0.0167610
\(125\) 0 0
\(126\) −1.76845 −0.157546
\(127\) 13.8192 1.22626 0.613130 0.789982i \(-0.289910\pi\)
0.613130 + 0.789982i \(0.289910\pi\)
\(128\) 12.6751 1.12033
\(129\) 4.41819 0.389000
\(130\) 0 0
\(131\) −5.92478 −0.517650 −0.258825 0.965924i \(-0.583335\pi\)
−0.258825 + 0.965924i \(0.583335\pi\)
\(132\) −0.193937 −0.0168800
\(133\) 6.38787 0.553899
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 10.0811 0.864447
\(137\) −3.35026 −0.286232 −0.143116 0.989706i \(-0.545712\pi\)
−0.143116 + 0.989706i \(0.545712\pi\)
\(138\) 5.92478 0.504351
\(139\) −21.1998 −1.79814 −0.899072 0.437800i \(-0.855758\pi\)
−0.899072 + 0.437800i \(0.855758\pi\)
\(140\) 0 0
\(141\) 12.3127 1.03691
\(142\) −20.8119 −1.74650
\(143\) −0.806063 −0.0674064
\(144\) −4.35026 −0.362522
\(145\) 0 0
\(146\) 15.8945 1.31543
\(147\) 5.57452 0.459778
\(148\) −0.312650 −0.0256997
\(149\) 6.38787 0.523315 0.261657 0.965161i \(-0.415731\pi\)
0.261657 + 0.965161i \(0.415731\pi\)
\(150\) 0 0
\(151\) −2.64974 −0.215633 −0.107816 0.994171i \(-0.534386\pi\)
−0.107816 + 0.994171i \(0.534386\pi\)
\(152\) 14.3127 1.16091
\(153\) −3.76845 −0.304661
\(154\) −1.76845 −0.142506
\(155\) 0 0
\(156\) 0.156325 0.0125160
\(157\) −1.61213 −0.128662 −0.0643309 0.997929i \(-0.520491\pi\)
−0.0643309 + 0.997929i \(0.520491\pi\)
\(158\) 20.7005 1.64685
\(159\) 1.42548 0.113048
\(160\) 0 0
\(161\) 4.77575 0.376382
\(162\) 1.48119 0.116374
\(163\) 0.312650 0.0244887 0.0122443 0.999925i \(-0.496102\pi\)
0.0122443 + 0.999925i \(0.496102\pi\)
\(164\) 1.76257 0.137634
\(165\) 0 0
\(166\) 14.6556 1.13750
\(167\) −0.493413 −0.0381815 −0.0190907 0.999818i \(-0.506077\pi\)
−0.0190907 + 0.999818i \(0.506077\pi\)
\(168\) −3.19394 −0.246418
\(169\) −12.3503 −0.950020
\(170\) 0 0
\(171\) −5.35026 −0.409145
\(172\) −0.856849 −0.0653341
\(173\) 23.3054 1.77187 0.885937 0.463806i \(-0.153517\pi\)
0.885937 + 0.463806i \(0.153517\pi\)
\(174\) 6.38787 0.484263
\(175\) 0 0
\(176\) −4.35026 −0.327913
\(177\) −13.2750 −0.997813
\(178\) 25.0132 1.87482
\(179\) −10.7005 −0.799795 −0.399897 0.916560i \(-0.630954\pi\)
−0.399897 + 0.916560i \(0.630954\pi\)
\(180\) 0 0
\(181\) −7.79877 −0.579678 −0.289839 0.957075i \(-0.593602\pi\)
−0.289839 + 0.957075i \(0.593602\pi\)
\(182\) 1.42548 0.105664
\(183\) 0.0752228 0.00556063
\(184\) 10.7005 0.788853
\(185\) 0 0
\(186\) −1.42548 −0.104522
\(187\) −3.76845 −0.275577
\(188\) −2.38787 −0.174154
\(189\) 1.19394 0.0868461
\(190\) 0 0
\(191\) 1.29948 0.0940268 0.0470134 0.998894i \(-0.485030\pi\)
0.0470134 + 0.998894i \(0.485030\pi\)
\(192\) −7.08110 −0.511035
\(193\) −8.59498 −0.618680 −0.309340 0.950951i \(-0.600108\pi\)
−0.309340 + 0.950951i \(0.600108\pi\)
\(194\) −16.9986 −1.22043
\(195\) 0 0
\(196\) −1.08110 −0.0772216
\(197\) 20.7064 1.47527 0.737635 0.675200i \(-0.235943\pi\)
0.737635 + 0.675200i \(0.235943\pi\)
\(198\) 1.48119 0.105264
\(199\) 5.55149 0.393535 0.196767 0.980450i \(-0.436956\pi\)
0.196767 + 0.980450i \(0.436956\pi\)
\(200\) 0 0
\(201\) −2.70052 −0.190480
\(202\) 15.9394 1.12149
\(203\) 5.14903 0.361391
\(204\) 0.730841 0.0511691
\(205\) 0 0
\(206\) −25.0884 −1.74799
\(207\) −4.00000 −0.278019
\(208\) 3.50659 0.243138
\(209\) −5.35026 −0.370085
\(210\) 0 0
\(211\) −18.4993 −1.27354 −0.636772 0.771052i \(-0.719731\pi\)
−0.636772 + 0.771052i \(0.719731\pi\)
\(212\) −0.276454 −0.0189869
\(213\) 14.0508 0.962744
\(214\) −12.2677 −0.838606
\(215\) 0 0
\(216\) 2.67513 0.182020
\(217\) −1.14903 −0.0780013
\(218\) −14.8119 −1.00319
\(219\) −10.7308 −0.725123
\(220\) 0 0
\(221\) 3.03761 0.204332
\(222\) 2.38787 0.160264
\(223\) −17.6121 −1.17940 −0.589698 0.807624i \(-0.700753\pi\)
−0.589698 + 0.807624i \(0.700753\pi\)
\(224\) 1.30536 0.0872178
\(225\) 0 0
\(226\) −3.35026 −0.222856
\(227\) 17.4314 1.15696 0.578480 0.815696i \(-0.303646\pi\)
0.578480 + 0.815696i \(0.303646\pi\)
\(228\) 1.03761 0.0687175
\(229\) 13.0738 0.863942 0.431971 0.901888i \(-0.357818\pi\)
0.431971 + 0.901888i \(0.357818\pi\)
\(230\) 0 0
\(231\) 1.19394 0.0785553
\(232\) 11.5369 0.757435
\(233\) 13.8437 0.906929 0.453465 0.891274i \(-0.350188\pi\)
0.453465 + 0.891274i \(0.350188\pi\)
\(234\) −1.19394 −0.0780501
\(235\) 0 0
\(236\) 2.57452 0.167587
\(237\) −13.9756 −0.907810
\(238\) 6.66433 0.431984
\(239\) −12.3733 −0.800361 −0.400181 0.916436i \(-0.631053\pi\)
−0.400181 + 0.916436i \(0.631053\pi\)
\(240\) 0 0
\(241\) −24.5501 −1.58141 −0.790705 0.612198i \(-0.790286\pi\)
−0.790705 + 0.612198i \(0.790286\pi\)
\(242\) 1.48119 0.0952148
\(243\) −1.00000 −0.0641500
\(244\) −0.0145884 −0.000933930 0
\(245\) 0 0
\(246\) −13.4617 −0.858285
\(247\) 4.31265 0.274407
\(248\) −2.57452 −0.163482
\(249\) −9.89446 −0.627036
\(250\) 0 0
\(251\) 13.9003 0.877382 0.438691 0.898638i \(-0.355442\pi\)
0.438691 + 0.898638i \(0.355442\pi\)
\(252\) −0.231548 −0.0145862
\(253\) −4.00000 −0.251478
\(254\) 20.4690 1.28434
\(255\) 0 0
\(256\) 4.61213 0.288258
\(257\) 18.8872 1.17815 0.589075 0.808079i \(-0.299492\pi\)
0.589075 + 0.808079i \(0.299492\pi\)
\(258\) 6.54420 0.407424
\(259\) 1.92478 0.119600
\(260\) 0 0
\(261\) −4.31265 −0.266946
\(262\) −8.77575 −0.542167
\(263\) −20.8061 −1.28296 −0.641478 0.767141i \(-0.721679\pi\)
−0.641478 + 0.767141i \(0.721679\pi\)
\(264\) 2.67513 0.164643
\(265\) 0 0
\(266\) 9.46168 0.580133
\(267\) −16.8872 −1.03348
\(268\) 0.523730 0.0319919
\(269\) −32.3996 −1.97544 −0.987720 0.156233i \(-0.950065\pi\)
−0.987720 + 0.156233i \(0.950065\pi\)
\(270\) 0 0
\(271\) −16.8265 −1.02214 −0.511069 0.859539i \(-0.670751\pi\)
−0.511069 + 0.859539i \(0.670751\pi\)
\(272\) 16.3938 0.994017
\(273\) −0.962389 −0.0582464
\(274\) −4.96239 −0.299789
\(275\) 0 0
\(276\) 0.775746 0.0466944
\(277\) −16.9076 −1.01588 −0.507941 0.861392i \(-0.669593\pi\)
−0.507941 + 0.861392i \(0.669593\pi\)
\(278\) −31.4010 −1.88331
\(279\) 0.962389 0.0576167
\(280\) 0 0
\(281\) 5.61213 0.334791 0.167396 0.985890i \(-0.446464\pi\)
0.167396 + 0.985890i \(0.446464\pi\)
\(282\) 18.2374 1.08602
\(283\) −5.81924 −0.345918 −0.172959 0.984929i \(-0.555333\pi\)
−0.172959 + 0.984929i \(0.555333\pi\)
\(284\) −2.72496 −0.161697
\(285\) 0 0
\(286\) −1.19394 −0.0705989
\(287\) −10.8510 −0.640512
\(288\) −1.09332 −0.0644246
\(289\) −2.79877 −0.164633
\(290\) 0 0
\(291\) 11.4763 0.672751
\(292\) 2.08110 0.121787
\(293\) 8.29218 0.484434 0.242217 0.970222i \(-0.422125\pi\)
0.242217 + 0.970222i \(0.422125\pi\)
\(294\) 8.25694 0.481555
\(295\) 0 0
\(296\) 4.31265 0.250668
\(297\) −1.00000 −0.0580259
\(298\) 9.46168 0.548100
\(299\) 3.22425 0.186463
\(300\) 0 0
\(301\) 5.27504 0.304048
\(302\) −3.92478 −0.225846
\(303\) −10.7612 −0.618212
\(304\) 23.2750 1.33492
\(305\) 0 0
\(306\) −5.58181 −0.319091
\(307\) −25.6688 −1.46500 −0.732498 0.680770i \(-0.761646\pi\)
−0.732498 + 0.680770i \(0.761646\pi\)
\(308\) −0.231548 −0.0131937
\(309\) 16.9380 0.963566
\(310\) 0 0
\(311\) −15.7235 −0.891601 −0.445800 0.895132i \(-0.647081\pi\)
−0.445800 + 0.895132i \(0.647081\pi\)
\(312\) −2.15633 −0.122078
\(313\) 26.8627 1.51837 0.759186 0.650874i \(-0.225597\pi\)
0.759186 + 0.650874i \(0.225597\pi\)
\(314\) −2.38787 −0.134755
\(315\) 0 0
\(316\) 2.71037 0.152470
\(317\) −0.710373 −0.0398985 −0.0199492 0.999801i \(-0.506350\pi\)
−0.0199492 + 0.999801i \(0.506350\pi\)
\(318\) 2.11142 0.118403
\(319\) −4.31265 −0.241462
\(320\) 0 0
\(321\) 8.28233 0.462275
\(322\) 7.07381 0.394208
\(323\) 20.1622 1.12186
\(324\) 0.193937 0.0107743
\(325\) 0 0
\(326\) 0.463096 0.0256485
\(327\) 10.0000 0.553001
\(328\) −24.3127 −1.34244
\(329\) 14.7005 0.810466
\(330\) 0 0
\(331\) 0.962389 0.0528977 0.0264488 0.999650i \(-0.491580\pi\)
0.0264488 + 0.999650i \(0.491580\pi\)
\(332\) 1.91890 0.105313
\(333\) −1.61213 −0.0883440
\(334\) −0.730841 −0.0399898
\(335\) 0 0
\(336\) −5.19394 −0.283352
\(337\) −19.8192 −1.07962 −0.539811 0.841786i \(-0.681504\pi\)
−0.539811 + 0.841786i \(0.681504\pi\)
\(338\) −18.2931 −0.995015
\(339\) 2.26187 0.122848
\(340\) 0 0
\(341\) 0.962389 0.0521163
\(342\) −7.92478 −0.428523
\(343\) 15.0132 0.810635
\(344\) 11.8192 0.637251
\(345\) 0 0
\(346\) 34.5198 1.85579
\(347\) −6.20711 −0.333215 −0.166608 0.986023i \(-0.553281\pi\)
−0.166608 + 0.986023i \(0.553281\pi\)
\(348\) 0.836381 0.0448347
\(349\) 4.44851 0.238123 0.119062 0.992887i \(-0.462011\pi\)
0.119062 + 0.992887i \(0.462011\pi\)
\(350\) 0 0
\(351\) 0.806063 0.0430245
\(352\) −1.09332 −0.0582742
\(353\) −6.57452 −0.349926 −0.174963 0.984575i \(-0.555981\pi\)
−0.174963 + 0.984575i \(0.555981\pi\)
\(354\) −19.6629 −1.04507
\(355\) 0 0
\(356\) 3.27504 0.173577
\(357\) −4.49929 −0.238128
\(358\) −15.8496 −0.837675
\(359\) 8.62530 0.455226 0.227613 0.973752i \(-0.426908\pi\)
0.227613 + 0.973752i \(0.426908\pi\)
\(360\) 0 0
\(361\) 9.62530 0.506595
\(362\) −11.5515 −0.607133
\(363\) −1.00000 −0.0524864
\(364\) 0.186642 0.00978272
\(365\) 0 0
\(366\) 0.111420 0.00582399
\(367\) −23.0132 −1.20128 −0.600639 0.799520i \(-0.705087\pi\)
−0.600639 + 0.799520i \(0.705087\pi\)
\(368\) 17.4010 0.907092
\(369\) 9.08840 0.473123
\(370\) 0 0
\(371\) 1.70194 0.0883602
\(372\) −0.186642 −0.00967695
\(373\) −28.1925 −1.45975 −0.729877 0.683579i \(-0.760423\pi\)
−0.729877 + 0.683579i \(0.760423\pi\)
\(374\) −5.58181 −0.288629
\(375\) 0 0
\(376\) 32.9380 1.69865
\(377\) 3.47627 0.179037
\(378\) 1.76845 0.0909594
\(379\) 3.74798 0.192521 0.0962605 0.995356i \(-0.469312\pi\)
0.0962605 + 0.995356i \(0.469312\pi\)
\(380\) 0 0
\(381\) −13.8192 −0.707981
\(382\) 1.92478 0.0984802
\(383\) −1.76257 −0.0900632 −0.0450316 0.998986i \(-0.514339\pi\)
−0.0450316 + 0.998986i \(0.514339\pi\)
\(384\) −12.6751 −0.646825
\(385\) 0 0
\(386\) −12.7308 −0.647983
\(387\) −4.41819 −0.224589
\(388\) −2.22567 −0.112991
\(389\) 6.52373 0.330766 0.165383 0.986229i \(-0.447114\pi\)
0.165383 + 0.986229i \(0.447114\pi\)
\(390\) 0 0
\(391\) 15.0738 0.762315
\(392\) 14.9126 0.753198
\(393\) 5.92478 0.298865
\(394\) 30.6702 1.54514
\(395\) 0 0
\(396\) 0.193937 0.00974568
\(397\) 23.6991 1.18942 0.594712 0.803939i \(-0.297266\pi\)
0.594712 + 0.803939i \(0.297266\pi\)
\(398\) 8.22284 0.412174
\(399\) −6.38787 −0.319794
\(400\) 0 0
\(401\) 8.88717 0.443804 0.221902 0.975069i \(-0.428774\pi\)
0.221902 + 0.975069i \(0.428774\pi\)
\(402\) −4.00000 −0.199502
\(403\) −0.775746 −0.0386427
\(404\) 2.08698 0.103831
\(405\) 0 0
\(406\) 7.62672 0.378508
\(407\) −1.61213 −0.0799102
\(408\) −10.0811 −0.499089
\(409\) −4.85097 −0.239865 −0.119932 0.992782i \(-0.538268\pi\)
−0.119932 + 0.992782i \(0.538268\pi\)
\(410\) 0 0
\(411\) 3.35026 0.165256
\(412\) −3.28489 −0.161835
\(413\) −15.8496 −0.779906
\(414\) −5.92478 −0.291187
\(415\) 0 0
\(416\) 0.881286 0.0432086
\(417\) 21.1998 1.03816
\(418\) −7.92478 −0.387614
\(419\) −10.7005 −0.522755 −0.261377 0.965237i \(-0.584177\pi\)
−0.261377 + 0.965237i \(0.584177\pi\)
\(420\) 0 0
\(421\) −30.6009 −1.49139 −0.745697 0.666285i \(-0.767884\pi\)
−0.745697 + 0.666285i \(0.767884\pi\)
\(422\) −27.4010 −1.33386
\(423\) −12.3127 −0.598662
\(424\) 3.81336 0.185193
\(425\) 0 0
\(426\) 20.8119 1.00834
\(427\) 0.0898112 0.00434627
\(428\) −1.60625 −0.0776409
\(429\) 0.806063 0.0389171
\(430\) 0 0
\(431\) −5.92478 −0.285386 −0.142693 0.989767i \(-0.545576\pi\)
−0.142693 + 0.989767i \(0.545576\pi\)
\(432\) 4.35026 0.209302
\(433\) 16.0000 0.768911 0.384455 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) −1.70194 −0.0816956
\(435\) 0 0
\(436\) −1.93937 −0.0928788
\(437\) 21.4010 1.02375
\(438\) −15.8945 −0.759467
\(439\) 5.35026 0.255354 0.127677 0.991816i \(-0.459248\pi\)
0.127677 + 0.991816i \(0.459248\pi\)
\(440\) 0 0
\(441\) −5.57452 −0.265453
\(442\) 4.49929 0.214010
\(443\) −19.6873 −0.935374 −0.467687 0.883894i \(-0.654913\pi\)
−0.467687 + 0.883894i \(0.654913\pi\)
\(444\) 0.312650 0.0148377
\(445\) 0 0
\(446\) −26.0870 −1.23525
\(447\) −6.38787 −0.302136
\(448\) −8.45439 −0.399432
\(449\) −31.3357 −1.47882 −0.739411 0.673254i \(-0.764896\pi\)
−0.739411 + 0.673254i \(0.764896\pi\)
\(450\) 0 0
\(451\) 9.08840 0.427956
\(452\) −0.438658 −0.0206328
\(453\) 2.64974 0.124496
\(454\) 25.8192 1.21176
\(455\) 0 0
\(456\) −14.3127 −0.670251
\(457\) 37.5936 1.75855 0.879276 0.476312i \(-0.158027\pi\)
0.879276 + 0.476312i \(0.158027\pi\)
\(458\) 19.3649 0.904860
\(459\) 3.76845 0.175896
\(460\) 0 0
\(461\) 17.9854 0.837664 0.418832 0.908064i \(-0.362440\pi\)
0.418832 + 0.908064i \(0.362440\pi\)
\(462\) 1.76845 0.0822758
\(463\) −39.0132 −1.81310 −0.906548 0.422103i \(-0.861292\pi\)
−0.906548 + 0.422103i \(0.861292\pi\)
\(464\) 18.7612 0.870965
\(465\) 0 0
\(466\) 20.5052 0.949884
\(467\) −14.5501 −0.673297 −0.336649 0.941630i \(-0.609293\pi\)
−0.336649 + 0.941630i \(0.609293\pi\)
\(468\) −0.156325 −0.00722613
\(469\) −3.22425 −0.148882
\(470\) 0 0
\(471\) 1.61213 0.0742829
\(472\) −35.5125 −1.63459
\(473\) −4.41819 −0.203149
\(474\) −20.7005 −0.950807
\(475\) 0 0
\(476\) 0.872577 0.0399945
\(477\) −1.42548 −0.0652685
\(478\) −18.3272 −0.838268
\(479\) 28.6253 1.30792 0.653962 0.756528i \(-0.273106\pi\)
0.653962 + 0.756528i \(0.273106\pi\)
\(480\) 0 0
\(481\) 1.29948 0.0592510
\(482\) −36.3634 −1.65631
\(483\) −4.77575 −0.217304
\(484\) 0.193937 0.00881530
\(485\) 0 0
\(486\) −1.48119 −0.0671883
\(487\) −1.44992 −0.0657022 −0.0328511 0.999460i \(-0.510459\pi\)
−0.0328511 + 0.999460i \(0.510459\pi\)
\(488\) 0.201231 0.00910929
\(489\) −0.312650 −0.0141385
\(490\) 0 0
\(491\) −8.00000 −0.361035 −0.180517 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) −1.76257 −0.0794629
\(493\) 16.2520 0.731954
\(494\) 6.38787 0.287404
\(495\) 0 0
\(496\) −4.18664 −0.187986
\(497\) 16.7757 0.752495
\(498\) −14.6556 −0.656734
\(499\) 30.7005 1.37434 0.687172 0.726495i \(-0.258852\pi\)
0.687172 + 0.726495i \(0.258852\pi\)
\(500\) 0 0
\(501\) 0.493413 0.0220441
\(502\) 20.5891 0.918937
\(503\) 19.7586 0.880993 0.440496 0.897754i \(-0.354802\pi\)
0.440496 + 0.897754i \(0.354802\pi\)
\(504\) 3.19394 0.142269
\(505\) 0 0
\(506\) −5.92478 −0.263388
\(507\) 12.3503 0.548494
\(508\) 2.68006 0.118908
\(509\) 22.1016 0.979635 0.489817 0.871825i \(-0.337063\pi\)
0.489817 + 0.871825i \(0.337063\pi\)
\(510\) 0 0
\(511\) −12.8119 −0.566767
\(512\) −18.5188 −0.818423
\(513\) 5.35026 0.236220
\(514\) 27.9756 1.23395
\(515\) 0 0
\(516\) 0.856849 0.0377207
\(517\) −12.3127 −0.541510
\(518\) 2.85097 0.125264
\(519\) −23.3054 −1.02299
\(520\) 0 0
\(521\) −22.8119 −0.999409 −0.499705 0.866196i \(-0.666558\pi\)
−0.499705 + 0.866196i \(0.666558\pi\)
\(522\) −6.38787 −0.279590
\(523\) 12.2677 0.536431 0.268216 0.963359i \(-0.413566\pi\)
0.268216 + 0.963359i \(0.413566\pi\)
\(524\) −1.14903 −0.0501957
\(525\) 0 0
\(526\) −30.8178 −1.34372
\(527\) −3.62672 −0.157982
\(528\) 4.35026 0.189321
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 13.2750 0.576088
\(532\) 1.23884 0.0537106
\(533\) −7.32582 −0.317317
\(534\) −25.0132 −1.08243
\(535\) 0 0
\(536\) −7.22425 −0.312040
\(537\) 10.7005 0.461762
\(538\) −47.9902 −2.06900
\(539\) −5.57452 −0.240111
\(540\) 0 0
\(541\) −5.22425 −0.224608 −0.112304 0.993674i \(-0.535823\pi\)
−0.112304 + 0.993674i \(0.535823\pi\)
\(542\) −24.9234 −1.07055
\(543\) 7.79877 0.334677
\(544\) 4.12013 0.176649
\(545\) 0 0
\(546\) −1.42548 −0.0610051
\(547\) 17.9697 0.768328 0.384164 0.923265i \(-0.374490\pi\)
0.384164 + 0.923265i \(0.374490\pi\)
\(548\) −0.649738 −0.0277554
\(549\) −0.0752228 −0.00321043
\(550\) 0 0
\(551\) 23.0738 0.982977
\(552\) −10.7005 −0.455445
\(553\) −16.6859 −0.709558
\(554\) −25.0435 −1.06400
\(555\) 0 0
\(556\) −4.11142 −0.174363
\(557\) −15.8700 −0.672434 −0.336217 0.941784i \(-0.609148\pi\)
−0.336217 + 0.941784i \(0.609148\pi\)
\(558\) 1.42548 0.0603456
\(559\) 3.56134 0.150629
\(560\) 0 0
\(561\) 3.76845 0.159104
\(562\) 8.31265 0.350648
\(563\) −31.6688 −1.33468 −0.667340 0.744753i \(-0.732567\pi\)
−0.667340 + 0.744753i \(0.732567\pi\)
\(564\) 2.38787 0.100548
\(565\) 0 0
\(566\) −8.61942 −0.362301
\(567\) −1.19394 −0.0501406
\(568\) 37.5877 1.57714
\(569\) −24.3127 −1.01924 −0.509620 0.860400i \(-0.670214\pi\)
−0.509620 + 0.860400i \(0.670214\pi\)
\(570\) 0 0
\(571\) 8.05079 0.336915 0.168457 0.985709i \(-0.446121\pi\)
0.168457 + 0.985709i \(0.446121\pi\)
\(572\) −0.156325 −0.00653628
\(573\) −1.29948 −0.0542864
\(574\) −16.0724 −0.670849
\(575\) 0 0
\(576\) 7.08110 0.295046
\(577\) 44.5355 1.85404 0.927018 0.375016i \(-0.122363\pi\)
0.927018 + 0.375016i \(0.122363\pi\)
\(578\) −4.14552 −0.172431
\(579\) 8.59498 0.357195
\(580\) 0 0
\(581\) −11.8134 −0.490101
\(582\) 16.9986 0.704614
\(583\) −1.42548 −0.0590375
\(584\) −28.7064 −1.18788
\(585\) 0 0
\(586\) 12.2823 0.507379
\(587\) 15.4763 0.638774 0.319387 0.947624i \(-0.396523\pi\)
0.319387 + 0.947624i \(0.396523\pi\)
\(588\) 1.08110 0.0445839
\(589\) −5.14903 −0.212162
\(590\) 0 0
\(591\) −20.7064 −0.851748
\(592\) 7.01317 0.288240
\(593\) 9.53102 0.391392 0.195696 0.980665i \(-0.437303\pi\)
0.195696 + 0.980665i \(0.437303\pi\)
\(594\) −1.48119 −0.0607741
\(595\) 0 0
\(596\) 1.23884 0.0507450
\(597\) −5.55149 −0.227207
\(598\) 4.77575 0.195295
\(599\) −25.5515 −1.04401 −0.522003 0.852944i \(-0.674815\pi\)
−0.522003 + 0.852944i \(0.674815\pi\)
\(600\) 0 0
\(601\) 12.0263 0.490565 0.245282 0.969452i \(-0.421119\pi\)
0.245282 + 0.969452i \(0.421119\pi\)
\(602\) 7.81336 0.318449
\(603\) 2.70052 0.109974
\(604\) −0.513881 −0.0209095
\(605\) 0 0
\(606\) −15.9394 −0.647492
\(607\) 6.86670 0.278711 0.139355 0.990242i \(-0.455497\pi\)
0.139355 + 0.990242i \(0.455497\pi\)
\(608\) 5.84955 0.237231
\(609\) −5.14903 −0.208649
\(610\) 0 0
\(611\) 9.92478 0.401514
\(612\) −0.730841 −0.0295425
\(613\) 7.25457 0.293009 0.146505 0.989210i \(-0.453198\pi\)
0.146505 + 0.989210i \(0.453198\pi\)
\(614\) −38.0205 −1.53438
\(615\) 0 0
\(616\) 3.19394 0.128687
\(617\) −38.3634 −1.54445 −0.772227 0.635347i \(-0.780857\pi\)
−0.772227 + 0.635347i \(0.780857\pi\)
\(618\) 25.0884 1.00920
\(619\) −29.6893 −1.19331 −0.596656 0.802497i \(-0.703504\pi\)
−0.596656 + 0.802497i \(0.703504\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) −23.2896 −0.933829
\(623\) −20.1622 −0.807782
\(624\) −3.50659 −0.140376
\(625\) 0 0
\(626\) 39.7889 1.59029
\(627\) 5.35026 0.213669
\(628\) −0.312650 −0.0124761
\(629\) 6.07522 0.242235
\(630\) 0 0
\(631\) −19.6991 −0.784209 −0.392105 0.919921i \(-0.628253\pi\)
−0.392105 + 0.919921i \(0.628253\pi\)
\(632\) −37.3865 −1.48715
\(633\) 18.4993 0.735281
\(634\) −1.05220 −0.0417882
\(635\) 0 0
\(636\) 0.276454 0.0109621
\(637\) 4.49341 0.178036
\(638\) −6.38787 −0.252898
\(639\) −14.0508 −0.555840
\(640\) 0 0
\(641\) −31.4372 −1.24170 −0.620848 0.783931i \(-0.713212\pi\)
−0.620848 + 0.783931i \(0.713212\pi\)
\(642\) 12.2677 0.484169
\(643\) 34.4894 1.36013 0.680065 0.733151i \(-0.261951\pi\)
0.680065 + 0.733151i \(0.261951\pi\)
\(644\) 0.926192 0.0364971
\(645\) 0 0
\(646\) 29.8641 1.17499
\(647\) 5.61213 0.220635 0.110318 0.993896i \(-0.464813\pi\)
0.110318 + 0.993896i \(0.464813\pi\)
\(648\) −2.67513 −0.105089
\(649\) 13.2750 0.521091
\(650\) 0 0
\(651\) 1.14903 0.0450341
\(652\) 0.0606343 0.00237462
\(653\) −4.06537 −0.159090 −0.0795452 0.996831i \(-0.525347\pi\)
−0.0795452 + 0.996831i \(0.525347\pi\)
\(654\) 14.8119 0.579193
\(655\) 0 0
\(656\) −39.5369 −1.54366
\(657\) 10.7308 0.418650
\(658\) 21.7743 0.848852
\(659\) 11.3747 0.443095 0.221548 0.975150i \(-0.428889\pi\)
0.221548 + 0.975150i \(0.428889\pi\)
\(660\) 0 0
\(661\) −42.4749 −1.65208 −0.826040 0.563611i \(-0.809412\pi\)
−0.826040 + 0.563611i \(0.809412\pi\)
\(662\) 1.42548 0.0554030
\(663\) −3.03761 −0.117971
\(664\) −26.4690 −1.02720
\(665\) 0 0
\(666\) −2.38787 −0.0925282
\(667\) 17.2506 0.667946
\(668\) −0.0956908 −0.00370239
\(669\) 17.6121 0.680924
\(670\) 0 0
\(671\) −0.0752228 −0.00290394
\(672\) −1.30536 −0.0503552
\(673\) 14.8813 0.573631 0.286816 0.957986i \(-0.407403\pi\)
0.286816 + 0.957986i \(0.407403\pi\)
\(674\) −29.3561 −1.13076
\(675\) 0 0
\(676\) −2.39517 −0.0921218
\(677\) 27.7685 1.06723 0.533614 0.845728i \(-0.320833\pi\)
0.533614 + 0.845728i \(0.320833\pi\)
\(678\) 3.35026 0.128666
\(679\) 13.7019 0.525832
\(680\) 0 0
\(681\) −17.4314 −0.667971
\(682\) 1.42548 0.0545846
\(683\) 25.4617 0.974264 0.487132 0.873328i \(-0.338043\pi\)
0.487132 + 0.873328i \(0.338043\pi\)
\(684\) −1.03761 −0.0396741
\(685\) 0 0
\(686\) 22.2374 0.849029
\(687\) −13.0738 −0.498797
\(688\) 19.2203 0.732766
\(689\) 1.14903 0.0437746
\(690\) 0 0
\(691\) 43.6991 1.66239 0.831196 0.555979i \(-0.187657\pi\)
0.831196 + 0.555979i \(0.187657\pi\)
\(692\) 4.51976 0.171816
\(693\) −1.19394 −0.0453539
\(694\) −9.19394 −0.348997
\(695\) 0 0
\(696\) −11.5369 −0.437305
\(697\) −34.2492 −1.29728
\(698\) 6.58910 0.249401
\(699\) −13.8437 −0.523616
\(700\) 0 0
\(701\) 7.01317 0.264884 0.132442 0.991191i \(-0.457718\pi\)
0.132442 + 0.991191i \(0.457718\pi\)
\(702\) 1.19394 0.0450622
\(703\) 8.62530 0.325309
\(704\) 7.08110 0.266879
\(705\) 0 0
\(706\) −9.73813 −0.366500
\(707\) −12.8481 −0.483204
\(708\) −2.57452 −0.0967562
\(709\) 45.6747 1.71535 0.857674 0.514194i \(-0.171909\pi\)
0.857674 + 0.514194i \(0.171909\pi\)
\(710\) 0 0
\(711\) 13.9756 0.524125
\(712\) −45.1754 −1.69302
\(713\) −3.84955 −0.144167
\(714\) −6.66433 −0.249406
\(715\) 0 0
\(716\) −2.07522 −0.0775547
\(717\) 12.3733 0.462089
\(718\) 12.7757 0.476787
\(719\) 16.2520 0.606098 0.303049 0.952975i \(-0.401995\pi\)
0.303049 + 0.952975i \(0.401995\pi\)
\(720\) 0 0
\(721\) 20.2228 0.753138
\(722\) 14.2569 0.530588
\(723\) 24.5501 0.913027
\(724\) −1.51247 −0.0562104
\(725\) 0 0
\(726\) −1.48119 −0.0549723
\(727\) −15.2243 −0.564636 −0.282318 0.959321i \(-0.591103\pi\)
−0.282318 + 0.959321i \(0.591103\pi\)
\(728\) −2.57452 −0.0954179
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 16.6497 0.615813
\(732\) 0.0145884 0.000539205 0
\(733\) 43.5066 1.60695 0.803476 0.595337i \(-0.202981\pi\)
0.803476 + 0.595337i \(0.202981\pi\)
\(734\) −34.0870 −1.25817
\(735\) 0 0
\(736\) 4.37328 0.161201
\(737\) 2.70052 0.0994751
\(738\) 13.4617 0.495531
\(739\) −7.02302 −0.258346 −0.129173 0.991622i \(-0.541232\pi\)
−0.129173 + 0.991622i \(0.541232\pi\)
\(740\) 0 0
\(741\) −4.31265 −0.158429
\(742\) 2.52090 0.0925452
\(743\) 2.94192 0.107929 0.0539643 0.998543i \(-0.482814\pi\)
0.0539643 + 0.998543i \(0.482814\pi\)
\(744\) 2.57452 0.0943863
\(745\) 0 0
\(746\) −41.7586 −1.52889
\(747\) 9.89446 0.362019
\(748\) −0.730841 −0.0267222
\(749\) 9.88858 0.361321
\(750\) 0 0
\(751\) 24.1016 0.879479 0.439739 0.898125i \(-0.355071\pi\)
0.439739 + 0.898125i \(0.355071\pi\)
\(752\) 53.5633 1.95325
\(753\) −13.9003 −0.506557
\(754\) 5.14903 0.187517
\(755\) 0 0
\(756\) 0.231548 0.00842132
\(757\) 16.3127 0.592893 0.296447 0.955049i \(-0.404198\pi\)
0.296447 + 0.955049i \(0.404198\pi\)
\(758\) 5.55149 0.201639
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) −5.08840 −0.184454 −0.0922271 0.995738i \(-0.529399\pi\)
−0.0922271 + 0.995738i \(0.529399\pi\)
\(762\) −20.4690 −0.741513
\(763\) 11.9394 0.432234
\(764\) 0.252016 0.00911762
\(765\) 0 0
\(766\) −2.61071 −0.0943289
\(767\) −10.7005 −0.386374
\(768\) −4.61213 −0.166426
\(769\) 2.10157 0.0757846 0.0378923 0.999282i \(-0.487936\pi\)
0.0378923 + 0.999282i \(0.487936\pi\)
\(770\) 0 0
\(771\) −18.8872 −0.680205
\(772\) −1.66688 −0.0599924
\(773\) −46.0625 −1.65675 −0.828377 0.560171i \(-0.810736\pi\)
−0.828377 + 0.560171i \(0.810736\pi\)
\(774\) −6.54420 −0.235226
\(775\) 0 0
\(776\) 30.7005 1.10208
\(777\) −1.92478 −0.0690510
\(778\) 9.66291 0.346432
\(779\) −48.6253 −1.74218
\(780\) 0 0
\(781\) −14.0508 −0.502777
\(782\) 22.3272 0.798420
\(783\) 4.31265 0.154122
\(784\) 24.2506 0.866093
\(785\) 0 0
\(786\) 8.77575 0.313021
\(787\) 26.0303 0.927881 0.463940 0.885866i \(-0.346435\pi\)
0.463940 + 0.885866i \(0.346435\pi\)
\(788\) 4.01573 0.143054
\(789\) 20.8061 0.740715
\(790\) 0 0
\(791\) 2.70052 0.0960196
\(792\) −2.67513 −0.0950566
\(793\) 0.0606343 0.00215319
\(794\) 35.1030 1.24576
\(795\) 0 0
\(796\) 1.07664 0.0381604
\(797\) 29.9902 1.06231 0.531153 0.847276i \(-0.321759\pi\)
0.531153 + 0.847276i \(0.321759\pi\)
\(798\) −9.46168 −0.334940
\(799\) 46.3996 1.64150
\(800\) 0 0
\(801\) 16.8872 0.596679
\(802\) 13.1636 0.464824
\(803\) 10.7308 0.378683
\(804\) −0.523730 −0.0184705
\(805\) 0 0
\(806\) −1.14903 −0.0404729
\(807\) 32.3996 1.14052
\(808\) −28.7875 −1.01274
\(809\) 39.7597 1.39788 0.698939 0.715181i \(-0.253656\pi\)
0.698939 + 0.715181i \(0.253656\pi\)
\(810\) 0 0
\(811\) −10.2765 −0.360855 −0.180428 0.983588i \(-0.557748\pi\)
−0.180428 + 0.983588i \(0.557748\pi\)
\(812\) 0.998585 0.0350435
\(813\) 16.8265 0.590132
\(814\) −2.38787 −0.0836949
\(815\) 0 0
\(816\) −16.3938 −0.573896
\(817\) 23.6385 0.827006
\(818\) −7.18523 −0.251226
\(819\) 0.962389 0.0336286
\(820\) 0 0
\(821\) 50.0870 1.74805 0.874024 0.485883i \(-0.161502\pi\)
0.874024 + 0.485883i \(0.161502\pi\)
\(822\) 4.96239 0.173083
\(823\) −29.7137 −1.03575 −0.517877 0.855455i \(-0.673278\pi\)
−0.517877 + 0.855455i \(0.673278\pi\)
\(824\) 45.3112 1.57849
\(825\) 0 0
\(826\) −23.4763 −0.816844
\(827\) 2.41819 0.0840887 0.0420444 0.999116i \(-0.486613\pi\)
0.0420444 + 0.999116i \(0.486613\pi\)
\(828\) −0.775746 −0.0269590
\(829\) −23.2750 −0.808376 −0.404188 0.914676i \(-0.632446\pi\)
−0.404188 + 0.914676i \(0.632446\pi\)
\(830\) 0 0
\(831\) 16.9076 0.586519
\(832\) −5.70782 −0.197883
\(833\) 21.0073 0.727860
\(834\) 31.4010 1.08733
\(835\) 0 0
\(836\) −1.03761 −0.0358865
\(837\) −0.962389 −0.0332650
\(838\) −15.8496 −0.547514
\(839\) −18.8265 −0.649964 −0.324982 0.945720i \(-0.605358\pi\)
−0.324982 + 0.945720i \(0.605358\pi\)
\(840\) 0 0
\(841\) −10.4010 −0.358657
\(842\) −45.3258 −1.56203
\(843\) −5.61213 −0.193292
\(844\) −3.58769 −0.123493
\(845\) 0 0
\(846\) −18.2374 −0.627016
\(847\) −1.19394 −0.0410241
\(848\) 6.20123 0.212951
\(849\) 5.81924 0.199716
\(850\) 0 0
\(851\) 6.44851 0.221052
\(852\) 2.72496 0.0933556
\(853\) 24.9076 0.852821 0.426411 0.904530i \(-0.359778\pi\)
0.426411 + 0.904530i \(0.359778\pi\)
\(854\) 0.133028 0.00455212
\(855\) 0 0
\(856\) 22.1563 0.757288
\(857\) 7.60625 0.259824 0.129912 0.991525i \(-0.458530\pi\)
0.129912 + 0.991525i \(0.458530\pi\)
\(858\) 1.19394 0.0407603
\(859\) −2.14060 −0.0730362 −0.0365181 0.999333i \(-0.511627\pi\)
−0.0365181 + 0.999333i \(0.511627\pi\)
\(860\) 0 0
\(861\) 10.8510 0.369800
\(862\) −8.77575 −0.298903
\(863\) −29.3112 −0.997766 −0.498883 0.866669i \(-0.666256\pi\)
−0.498883 + 0.866669i \(0.666256\pi\)
\(864\) 1.09332 0.0371955
\(865\) 0 0
\(866\) 23.6991 0.805329
\(867\) 2.79877 0.0950512
\(868\) −0.222839 −0.00756365
\(869\) 13.9756 0.474088
\(870\) 0 0
\(871\) −2.17679 −0.0737578
\(872\) 26.7513 0.905914
\(873\) −11.4763 −0.388413
\(874\) 31.6991 1.07224
\(875\) 0 0
\(876\) −2.08110 −0.0703139
\(877\) −31.5183 −1.06430 −0.532149 0.846650i \(-0.678616\pi\)
−0.532149 + 0.846650i \(0.678616\pi\)
\(878\) 7.92478 0.267448
\(879\) −8.29218 −0.279688
\(880\) 0 0
\(881\) 10.6253 0.357975 0.178988 0.983851i \(-0.442718\pi\)
0.178988 + 0.983851i \(0.442718\pi\)
\(882\) −8.25694 −0.278026
\(883\) 6.29806 0.211947 0.105973 0.994369i \(-0.466204\pi\)
0.105973 + 0.994369i \(0.466204\pi\)
\(884\) 0.589104 0.0198137
\(885\) 0 0
\(886\) −29.1608 −0.979676
\(887\) −29.1187 −0.977711 −0.488855 0.872365i \(-0.662585\pi\)
−0.488855 + 0.872365i \(0.662585\pi\)
\(888\) −4.31265 −0.144723
\(889\) −16.4993 −0.553369
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) −3.41564 −0.114364
\(893\) 65.8759 2.20445
\(894\) −9.46168 −0.316446
\(895\) 0 0
\(896\) −15.1333 −0.505568
\(897\) −3.22425 −0.107655
\(898\) −46.4142 −1.54886
\(899\) −4.15045 −0.138425
\(900\) 0 0
\(901\) 5.37187 0.178963
\(902\) 13.4617 0.448225
\(903\) −5.27504 −0.175542
\(904\) 6.05079 0.201246
\(905\) 0 0
\(906\) 3.92478 0.130392
\(907\) 57.3112 1.90299 0.951494 0.307667i \(-0.0995482\pi\)
0.951494 + 0.307667i \(0.0995482\pi\)
\(908\) 3.38058 0.112188
\(909\) 10.7612 0.356925
\(910\) 0 0
\(911\) −5.52232 −0.182962 −0.0914812 0.995807i \(-0.529160\pi\)
−0.0914812 + 0.995807i \(0.529160\pi\)
\(912\) −23.2750 −0.770714
\(913\) 9.89446 0.327459
\(914\) 55.6834 1.84184
\(915\) 0 0
\(916\) 2.53549 0.0837749
\(917\) 7.07381 0.233598
\(918\) 5.58181 0.184227
\(919\) −5.75272 −0.189765 −0.0948824 0.995488i \(-0.530248\pi\)
−0.0948824 + 0.995488i \(0.530248\pi\)
\(920\) 0 0
\(921\) 25.6688 0.845815
\(922\) 26.6399 0.877338
\(923\) 11.3258 0.372794
\(924\) 0.231548 0.00761737
\(925\) 0 0
\(926\) −57.7861 −1.89897
\(927\) −16.9380 −0.556315
\(928\) 4.71511 0.154781
\(929\) 41.4274 1.35919 0.679594 0.733588i \(-0.262156\pi\)
0.679594 + 0.733588i \(0.262156\pi\)
\(930\) 0 0
\(931\) 29.8251 0.977479
\(932\) 2.68479 0.0879434
\(933\) 15.7235 0.514766
\(934\) −21.5515 −0.705186
\(935\) 0 0
\(936\) 2.15633 0.0704817
\(937\) 14.9222 0.487488 0.243744 0.969840i \(-0.421624\pi\)
0.243744 + 0.969840i \(0.421624\pi\)
\(938\) −4.77575 −0.155934
\(939\) −26.8627 −0.876632
\(940\) 0 0
\(941\) 57.0395 1.85944 0.929718 0.368273i \(-0.120051\pi\)
0.929718 + 0.368273i \(0.120051\pi\)
\(942\) 2.38787 0.0778011
\(943\) −36.3536 −1.18384
\(944\) −57.7499 −1.87960
\(945\) 0 0
\(946\) −6.54420 −0.212770
\(947\) −30.2374 −0.982584 −0.491292 0.870995i \(-0.663475\pi\)
−0.491292 + 0.870995i \(0.663475\pi\)
\(948\) −2.71037 −0.0880288
\(949\) −8.64974 −0.280782
\(950\) 0 0
\(951\) 0.710373 0.0230354
\(952\) −12.0362 −0.390095
\(953\) 46.2697 1.49882 0.749411 0.662106i \(-0.230337\pi\)
0.749411 + 0.662106i \(0.230337\pi\)
\(954\) −2.11142 −0.0683597
\(955\) 0 0
\(956\) −2.39963 −0.0776097
\(957\) 4.31265 0.139408
\(958\) 42.3996 1.36987
\(959\) 4.00000 0.129167
\(960\) 0 0
\(961\) −30.0738 −0.970123
\(962\) 1.92478 0.0620573
\(963\) −8.28233 −0.266894
\(964\) −4.76116 −0.153347
\(965\) 0 0
\(966\) −7.07381 −0.227596
\(967\) −49.5790 −1.59435 −0.797176 0.603747i \(-0.793674\pi\)
−0.797176 + 0.603747i \(0.793674\pi\)
\(968\) −2.67513 −0.0859820
\(969\) −20.1622 −0.647703
\(970\) 0 0
\(971\) −3.12268 −0.100212 −0.0501058 0.998744i \(-0.515956\pi\)
−0.0501058 + 0.998744i \(0.515956\pi\)
\(972\) −0.193937 −0.00622052
\(973\) 25.3112 0.811441
\(974\) −2.14762 −0.0688141
\(975\) 0 0
\(976\) 0.327239 0.0104747
\(977\) −5.15377 −0.164884 −0.0824419 0.996596i \(-0.526272\pi\)
−0.0824419 + 0.996596i \(0.526272\pi\)
\(978\) −0.463096 −0.0148082
\(979\) 16.8872 0.539716
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) −11.8496 −0.378134
\(983\) −14.8627 −0.474047 −0.237024 0.971504i \(-0.576172\pi\)
−0.237024 + 0.971504i \(0.576172\pi\)
\(984\) 24.3127 0.775059
\(985\) 0 0
\(986\) 24.0724 0.766621
\(987\) −14.7005 −0.467923
\(988\) 0.836381 0.0266088
\(989\) 17.6728 0.561961
\(990\) 0 0
\(991\) 4.43866 0.140999 0.0704993 0.997512i \(-0.477541\pi\)
0.0704993 + 0.997512i \(0.477541\pi\)
\(992\) −1.05220 −0.0334074
\(993\) −0.962389 −0.0305405
\(994\) 24.8481 0.788135
\(995\) 0 0
\(996\) −1.91890 −0.0608026
\(997\) 38.8324 1.22983 0.614917 0.788592i \(-0.289189\pi\)
0.614917 + 0.788592i \(0.289189\pi\)
\(998\) 45.4734 1.43944
\(999\) 1.61213 0.0510054
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.j.1.3 3
3.2 odd 2 2475.2.a.bc.1.1 3
5.2 odd 4 165.2.c.b.34.5 yes 6
5.3 odd 4 165.2.c.b.34.2 6
5.4 even 2 825.2.a.l.1.1 3
11.10 odd 2 9075.2.a.ch.1.1 3
15.2 even 4 495.2.c.e.199.2 6
15.8 even 4 495.2.c.e.199.5 6
15.14 odd 2 2475.2.a.ba.1.3 3
20.3 even 4 2640.2.d.h.529.4 6
20.7 even 4 2640.2.d.h.529.1 6
55.32 even 4 1815.2.c.e.364.2 6
55.43 even 4 1815.2.c.e.364.5 6
55.54 odd 2 9075.2.a.cg.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.c.b.34.2 6 5.3 odd 4
165.2.c.b.34.5 yes 6 5.2 odd 4
495.2.c.e.199.2 6 15.2 even 4
495.2.c.e.199.5 6 15.8 even 4
825.2.a.j.1.3 3 1.1 even 1 trivial
825.2.a.l.1.1 3 5.4 even 2
1815.2.c.e.364.2 6 55.32 even 4
1815.2.c.e.364.5 6 55.43 even 4
2475.2.a.ba.1.3 3 15.14 odd 2
2475.2.a.bc.1.1 3 3.2 odd 2
2640.2.d.h.529.1 6 20.7 even 4
2640.2.d.h.529.4 6 20.3 even 4
9075.2.a.cg.1.3 3 55.54 odd 2
9075.2.a.ch.1.1 3 11.10 odd 2