Properties

Label 5225.2.a.r.1.8
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $1$
Dimension $15$
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: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 5 x^{14} - 11 x^{13} + 87 x^{12} - 4 x^{11} - 545 x^{10} + 431 x^{9} + 1480 x^{8} - 1763 x^{7} + \cdots + 15 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(0.692644\) 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)
 
\(f(q)\) \(=\) \(q-0.692644 q^{2} -1.14700 q^{3} -1.52024 q^{4} +0.794464 q^{6} -3.60442 q^{7} +2.43828 q^{8} -1.68438 q^{9} +O(q^{10})\) \(q-0.692644 q^{2} -1.14700 q^{3} -1.52024 q^{4} +0.794464 q^{6} -3.60442 q^{7} +2.43828 q^{8} -1.68438 q^{9} +1.00000 q^{11} +1.74372 q^{12} -7.02467 q^{13} +2.49658 q^{14} +1.35163 q^{16} -0.385149 q^{17} +1.16668 q^{18} +1.00000 q^{19} +4.13428 q^{21} -0.692644 q^{22} +4.88630 q^{23} -2.79671 q^{24} +4.86559 q^{26} +5.37300 q^{27} +5.47960 q^{28} -3.20619 q^{29} +4.99714 q^{31} -5.81275 q^{32} -1.14700 q^{33} +0.266771 q^{34} +2.56068 q^{36} +0.661585 q^{37} -0.692644 q^{38} +8.05731 q^{39} +3.92936 q^{41} -2.86358 q^{42} -1.76009 q^{43} -1.52024 q^{44} -3.38446 q^{46} +2.20788 q^{47} -1.55033 q^{48} +5.99184 q^{49} +0.441767 q^{51} +10.6792 q^{52} -4.15602 q^{53} -3.72158 q^{54} -8.78857 q^{56} -1.14700 q^{57} +2.22074 q^{58} +11.1047 q^{59} +7.70215 q^{61} -3.46124 q^{62} +6.07123 q^{63} +1.32290 q^{64} +0.794464 q^{66} +1.12280 q^{67} +0.585521 q^{68} -5.60460 q^{69} +13.3284 q^{71} -4.10699 q^{72} -5.81274 q^{73} -0.458242 q^{74} -1.52024 q^{76} -3.60442 q^{77} -5.58085 q^{78} -11.1857 q^{79} -1.10969 q^{81} -2.72165 q^{82} +6.08541 q^{83} -6.28512 q^{84} +1.21912 q^{86} +3.67750 q^{87} +2.43828 q^{88} -10.2064 q^{89} +25.3199 q^{91} -7.42837 q^{92} -5.73173 q^{93} -1.52927 q^{94} +6.66724 q^{96} +10.4502 q^{97} -4.15021 q^{98} -1.68438 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 5 q^{2} - 4 q^{3} + 17 q^{4} - q^{6} - 21 q^{7} - 9 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - 5 q^{2} - 4 q^{3} + 17 q^{4} - q^{6} - 21 q^{7} - 9 q^{8} + 15 q^{9} + 15 q^{11} - 11 q^{12} - 13 q^{13} + 9 q^{14} + 21 q^{16} - 17 q^{17} - 22 q^{18} + 15 q^{19} + 6 q^{21} - 5 q^{22} - 26 q^{23} + q^{24} + 3 q^{26} - q^{27} - 46 q^{28} + 9 q^{29} + 14 q^{31} - 18 q^{32} - 4 q^{33} - 13 q^{34} + 12 q^{36} - 9 q^{37} - 5 q^{38} - 22 q^{39} + 4 q^{41} + 6 q^{42} - 28 q^{43} + 17 q^{44} + 27 q^{46} - 14 q^{47} + 4 q^{48} + 32 q^{49} - 40 q^{51} - 14 q^{52} - 3 q^{53} - 39 q^{54} + 34 q^{56} - 4 q^{57} - 26 q^{58} + q^{59} + 2 q^{61} + 3 q^{62} - 45 q^{63} + 5 q^{64} - q^{66} - 37 q^{67} - 26 q^{68} - 7 q^{69} - 7 q^{71} - 16 q^{72} - 42 q^{73} - 43 q^{74} + 17 q^{76} - 21 q^{77} + 64 q^{78} - 10 q^{79} + 31 q^{81} - 22 q^{82} - 14 q^{83} - 32 q^{84} + 37 q^{86} - 29 q^{87} - 9 q^{88} + 15 q^{89} - 22 q^{91} - 26 q^{92} + 18 q^{93} - 44 q^{94} + 71 q^{96} - 8 q^{97} + 10 q^{98} + 15 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.692644 −0.489773 −0.244886 0.969552i \(-0.578751\pi\)
−0.244886 + 0.969552i \(0.578751\pi\)
\(3\) −1.14700 −0.662222 −0.331111 0.943592i \(-0.607424\pi\)
−0.331111 + 0.943592i \(0.607424\pi\)
\(4\) −1.52024 −0.760122
\(5\) 0 0
\(6\) 0.794464 0.324339
\(7\) −3.60442 −1.36234 −0.681171 0.732124i \(-0.738529\pi\)
−0.681171 + 0.732124i \(0.738529\pi\)
\(8\) 2.43828 0.862060
\(9\) −1.68438 −0.561462
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 1.74372 0.503370
\(13\) −7.02467 −1.94829 −0.974146 0.225919i \(-0.927462\pi\)
−0.974146 + 0.225919i \(0.927462\pi\)
\(14\) 2.49658 0.667239
\(15\) 0 0
\(16\) 1.35163 0.337909
\(17\) −0.385149 −0.0934123 −0.0467062 0.998909i \(-0.514872\pi\)
−0.0467062 + 0.998909i \(0.514872\pi\)
\(18\) 1.16668 0.274989
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 4.13428 0.902174
\(22\) −0.692644 −0.147672
\(23\) 4.88630 1.01886 0.509432 0.860511i \(-0.329856\pi\)
0.509432 + 0.860511i \(0.329856\pi\)
\(24\) −2.79671 −0.570876
\(25\) 0 0
\(26\) 4.86559 0.954221
\(27\) 5.37300 1.03403
\(28\) 5.47960 1.03555
\(29\) −3.20619 −0.595374 −0.297687 0.954664i \(-0.596215\pi\)
−0.297687 + 0.954664i \(0.596215\pi\)
\(30\) 0 0
\(31\) 4.99714 0.897513 0.448756 0.893654i \(-0.351867\pi\)
0.448756 + 0.893654i \(0.351867\pi\)
\(32\) −5.81275 −1.02756
\(33\) −1.14700 −0.199668
\(34\) 0.266771 0.0457508
\(35\) 0 0
\(36\) 2.56068 0.426780
\(37\) 0.661585 0.108764 0.0543819 0.998520i \(-0.482681\pi\)
0.0543819 + 0.998520i \(0.482681\pi\)
\(38\) −0.692644 −0.112362
\(39\) 8.05731 1.29020
\(40\) 0 0
\(41\) 3.92936 0.613663 0.306832 0.951764i \(-0.400731\pi\)
0.306832 + 0.951764i \(0.400731\pi\)
\(42\) −2.86358 −0.441860
\(43\) −1.76009 −0.268412 −0.134206 0.990953i \(-0.542848\pi\)
−0.134206 + 0.990953i \(0.542848\pi\)
\(44\) −1.52024 −0.229186
\(45\) 0 0
\(46\) −3.38446 −0.499012
\(47\) 2.20788 0.322052 0.161026 0.986950i \(-0.448520\pi\)
0.161026 + 0.986950i \(0.448520\pi\)
\(48\) −1.55033 −0.223771
\(49\) 5.99184 0.855978
\(50\) 0 0
\(51\) 0.441767 0.0618597
\(52\) 10.6792 1.48094
\(53\) −4.15602 −0.570873 −0.285437 0.958398i \(-0.592139\pi\)
−0.285437 + 0.958398i \(0.592139\pi\)
\(54\) −3.72158 −0.506442
\(55\) 0 0
\(56\) −8.78857 −1.17442
\(57\) −1.14700 −0.151924
\(58\) 2.22074 0.291598
\(59\) 11.1047 1.44571 0.722857 0.690998i \(-0.242829\pi\)
0.722857 + 0.690998i \(0.242829\pi\)
\(60\) 0 0
\(61\) 7.70215 0.986160 0.493080 0.869984i \(-0.335871\pi\)
0.493080 + 0.869984i \(0.335871\pi\)
\(62\) −3.46124 −0.439578
\(63\) 6.07123 0.764903
\(64\) 1.32290 0.165362
\(65\) 0 0
\(66\) 0.794464 0.0977918
\(67\) 1.12280 0.137172 0.0685859 0.997645i \(-0.478151\pi\)
0.0685859 + 0.997645i \(0.478151\pi\)
\(68\) 0.585521 0.0710048
\(69\) −5.60460 −0.674714
\(70\) 0 0
\(71\) 13.3284 1.58179 0.790893 0.611955i \(-0.209617\pi\)
0.790893 + 0.611955i \(0.209617\pi\)
\(72\) −4.10699 −0.484014
\(73\) −5.81274 −0.680330 −0.340165 0.940366i \(-0.610483\pi\)
−0.340165 + 0.940366i \(0.610483\pi\)
\(74\) −0.458242 −0.0532696
\(75\) 0 0
\(76\) −1.52024 −0.174384
\(77\) −3.60442 −0.410762
\(78\) −5.58085 −0.631906
\(79\) −11.1857 −1.25849 −0.629246 0.777207i \(-0.716636\pi\)
−0.629246 + 0.777207i \(0.716636\pi\)
\(80\) 0 0
\(81\) −1.10969 −0.123299
\(82\) −2.72165 −0.300556
\(83\) 6.08541 0.667961 0.333980 0.942580i \(-0.391608\pi\)
0.333980 + 0.942580i \(0.391608\pi\)
\(84\) −6.28512 −0.685762
\(85\) 0 0
\(86\) 1.21912 0.131461
\(87\) 3.67750 0.394270
\(88\) 2.43828 0.259921
\(89\) −10.2064 −1.08187 −0.540936 0.841064i \(-0.681930\pi\)
−0.540936 + 0.841064i \(0.681930\pi\)
\(90\) 0 0
\(91\) 25.3199 2.65424
\(92\) −7.42837 −0.774461
\(93\) −5.73173 −0.594353
\(94\) −1.52927 −0.157732
\(95\) 0 0
\(96\) 6.66724 0.680472
\(97\) 10.4502 1.06106 0.530529 0.847667i \(-0.321993\pi\)
0.530529 + 0.847667i \(0.321993\pi\)
\(98\) −4.15021 −0.419235
\(99\) −1.68438 −0.169287
\(100\) 0 0
\(101\) −12.1058 −1.20457 −0.602285 0.798281i \(-0.705743\pi\)
−0.602285 + 0.798281i \(0.705743\pi\)
\(102\) −0.305987 −0.0302972
\(103\) 11.9174 1.17426 0.587129 0.809494i \(-0.300258\pi\)
0.587129 + 0.809494i \(0.300258\pi\)
\(104\) −17.1281 −1.67955
\(105\) 0 0
\(106\) 2.87864 0.279598
\(107\) −9.16324 −0.885844 −0.442922 0.896560i \(-0.646058\pi\)
−0.442922 + 0.896560i \(0.646058\pi\)
\(108\) −8.16828 −0.785993
\(109\) 9.05838 0.867635 0.433817 0.901001i \(-0.357166\pi\)
0.433817 + 0.901001i \(0.357166\pi\)
\(110\) 0 0
\(111\) −0.758839 −0.0720258
\(112\) −4.87186 −0.460347
\(113\) −0.699650 −0.0658176 −0.0329088 0.999458i \(-0.510477\pi\)
−0.0329088 + 0.999458i \(0.510477\pi\)
\(114\) 0.794464 0.0744084
\(115\) 0 0
\(116\) 4.87419 0.452557
\(117\) 11.8322 1.09389
\(118\) −7.69162 −0.708071
\(119\) 1.38824 0.127260
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −5.33485 −0.482994
\(123\) −4.50699 −0.406381
\(124\) −7.59688 −0.682220
\(125\) 0 0
\(126\) −4.20520 −0.374629
\(127\) −18.6133 −1.65167 −0.825834 0.563914i \(-0.809295\pi\)
−0.825834 + 0.563914i \(0.809295\pi\)
\(128\) 10.7092 0.946569
\(129\) 2.01883 0.177748
\(130\) 0 0
\(131\) −8.82334 −0.770899 −0.385450 0.922729i \(-0.625954\pi\)
−0.385450 + 0.922729i \(0.625954\pi\)
\(132\) 1.74372 0.151772
\(133\) −3.60442 −0.312543
\(134\) −0.777700 −0.0671831
\(135\) 0 0
\(136\) −0.939099 −0.0805271
\(137\) 3.11018 0.265720 0.132860 0.991135i \(-0.457584\pi\)
0.132860 + 0.991135i \(0.457584\pi\)
\(138\) 3.88199 0.330457
\(139\) 7.07826 0.600370 0.300185 0.953881i \(-0.402952\pi\)
0.300185 + 0.953881i \(0.402952\pi\)
\(140\) 0 0
\(141\) −2.53244 −0.213270
\(142\) −9.23180 −0.774716
\(143\) −7.02467 −0.587432
\(144\) −2.27667 −0.189723
\(145\) 0 0
\(146\) 4.02616 0.333207
\(147\) −6.87266 −0.566848
\(148\) −1.00577 −0.0826738
\(149\) 2.35944 0.193293 0.0966466 0.995319i \(-0.469188\pi\)
0.0966466 + 0.995319i \(0.469188\pi\)
\(150\) 0 0
\(151\) 3.09275 0.251684 0.125842 0.992050i \(-0.459837\pi\)
0.125842 + 0.992050i \(0.459837\pi\)
\(152\) 2.43828 0.197770
\(153\) 0.648739 0.0524474
\(154\) 2.49658 0.201180
\(155\) 0 0
\(156\) −12.2491 −0.980712
\(157\) −10.8156 −0.863179 −0.431589 0.902070i \(-0.642047\pi\)
−0.431589 + 0.902070i \(0.642047\pi\)
\(158\) 7.74771 0.616375
\(159\) 4.76697 0.378045
\(160\) 0 0
\(161\) −17.6123 −1.38804
\(162\) 0.768622 0.0603886
\(163\) −3.37074 −0.264017 −0.132009 0.991249i \(-0.542143\pi\)
−0.132009 + 0.991249i \(0.542143\pi\)
\(164\) −5.97359 −0.466459
\(165\) 0 0
\(166\) −4.21502 −0.327149
\(167\) −8.37434 −0.648026 −0.324013 0.946053i \(-0.605032\pi\)
−0.324013 + 0.946053i \(0.605032\pi\)
\(168\) 10.0805 0.777728
\(169\) 36.3460 2.79584
\(170\) 0 0
\(171\) −1.68438 −0.128808
\(172\) 2.67577 0.204026
\(173\) −8.91885 −0.678088 −0.339044 0.940771i \(-0.610103\pi\)
−0.339044 + 0.940771i \(0.610103\pi\)
\(174\) −2.54720 −0.193103
\(175\) 0 0
\(176\) 1.35163 0.101883
\(177\) −12.7372 −0.957384
\(178\) 7.06936 0.529871
\(179\) 8.35181 0.624243 0.312122 0.950042i \(-0.398960\pi\)
0.312122 + 0.950042i \(0.398960\pi\)
\(180\) 0 0
\(181\) 23.1394 1.71994 0.859968 0.510349i \(-0.170484\pi\)
0.859968 + 0.510349i \(0.170484\pi\)
\(182\) −17.5376 −1.29998
\(183\) −8.83439 −0.653057
\(184\) 11.9141 0.878322
\(185\) 0 0
\(186\) 3.97005 0.291098
\(187\) −0.385149 −0.0281649
\(188\) −3.35651 −0.244799
\(189\) −19.3666 −1.40871
\(190\) 0 0
\(191\) 11.4744 0.830262 0.415131 0.909762i \(-0.363736\pi\)
0.415131 + 0.909762i \(0.363736\pi\)
\(192\) −1.51737 −0.109506
\(193\) 17.3482 1.24875 0.624374 0.781125i \(-0.285354\pi\)
0.624374 + 0.781125i \(0.285354\pi\)
\(194\) −7.23827 −0.519678
\(195\) 0 0
\(196\) −9.10907 −0.650648
\(197\) −13.6075 −0.969495 −0.484748 0.874654i \(-0.661089\pi\)
−0.484748 + 0.874654i \(0.661089\pi\)
\(198\) 1.16668 0.0829122
\(199\) 8.75803 0.620840 0.310420 0.950599i \(-0.399530\pi\)
0.310420 + 0.950599i \(0.399530\pi\)
\(200\) 0 0
\(201\) −1.28785 −0.0908383
\(202\) 8.38499 0.589966
\(203\) 11.5564 0.811103
\(204\) −0.671594 −0.0470210
\(205\) 0 0
\(206\) −8.25452 −0.575119
\(207\) −8.23040 −0.572053
\(208\) −9.49478 −0.658345
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 20.2964 1.39726 0.698631 0.715482i \(-0.253793\pi\)
0.698631 + 0.715482i \(0.253793\pi\)
\(212\) 6.31817 0.433933
\(213\) −15.2877 −1.04749
\(214\) 6.34686 0.433862
\(215\) 0 0
\(216\) 13.1009 0.891400
\(217\) −18.0118 −1.22272
\(218\) −6.27423 −0.424944
\(219\) 6.66723 0.450530
\(220\) 0 0
\(221\) 2.70554 0.181994
\(222\) 0.525605 0.0352763
\(223\) −19.8774 −1.33109 −0.665546 0.746357i \(-0.731801\pi\)
−0.665546 + 0.746357i \(0.731801\pi\)
\(224\) 20.9516 1.39989
\(225\) 0 0
\(226\) 0.484608 0.0322357
\(227\) 0.523563 0.0347501 0.0173751 0.999849i \(-0.494469\pi\)
0.0173751 + 0.999849i \(0.494469\pi\)
\(228\) 1.74372 0.115481
\(229\) 18.6726 1.23392 0.616960 0.786995i \(-0.288364\pi\)
0.616960 + 0.786995i \(0.288364\pi\)
\(230\) 0 0
\(231\) 4.13428 0.272016
\(232\) −7.81757 −0.513248
\(233\) 1.69912 0.111313 0.0556565 0.998450i \(-0.482275\pi\)
0.0556565 + 0.998450i \(0.482275\pi\)
\(234\) −8.19553 −0.535758
\(235\) 0 0
\(236\) −16.8819 −1.09892
\(237\) 12.8300 0.833401
\(238\) −0.961554 −0.0623283
\(239\) −29.7115 −1.92188 −0.960938 0.276763i \(-0.910738\pi\)
−0.960938 + 0.276763i \(0.910738\pi\)
\(240\) 0 0
\(241\) −16.6028 −1.06948 −0.534741 0.845016i \(-0.679591\pi\)
−0.534741 + 0.845016i \(0.679591\pi\)
\(242\) −0.692644 −0.0445248
\(243\) −14.8462 −0.952383
\(244\) −11.7092 −0.749602
\(245\) 0 0
\(246\) 3.12174 0.199035
\(247\) −7.02467 −0.446969
\(248\) 12.1844 0.773710
\(249\) −6.97998 −0.442339
\(250\) 0 0
\(251\) 13.7486 0.867807 0.433903 0.900959i \(-0.357136\pi\)
0.433903 + 0.900959i \(0.357136\pi\)
\(252\) −9.22976 −0.581420
\(253\) 4.88630 0.307199
\(254\) 12.8924 0.808942
\(255\) 0 0
\(256\) −10.0635 −0.628966
\(257\) −26.7962 −1.67150 −0.835752 0.549108i \(-0.814968\pi\)
−0.835752 + 0.549108i \(0.814968\pi\)
\(258\) −1.39833 −0.0870563
\(259\) −2.38463 −0.148174
\(260\) 0 0
\(261\) 5.40045 0.334280
\(262\) 6.11143 0.377566
\(263\) 17.2355 1.06279 0.531395 0.847124i \(-0.321668\pi\)
0.531395 + 0.847124i \(0.321668\pi\)
\(264\) −2.79671 −0.172125
\(265\) 0 0
\(266\) 2.49658 0.153075
\(267\) 11.7067 0.716439
\(268\) −1.70693 −0.104267
\(269\) −17.9313 −1.09329 −0.546646 0.837364i \(-0.684096\pi\)
−0.546646 + 0.837364i \(0.684096\pi\)
\(270\) 0 0
\(271\) −9.75162 −0.592369 −0.296184 0.955131i \(-0.595714\pi\)
−0.296184 + 0.955131i \(0.595714\pi\)
\(272\) −0.520580 −0.0315648
\(273\) −29.0419 −1.75770
\(274\) −2.15424 −0.130143
\(275\) 0 0
\(276\) 8.52036 0.512865
\(277\) 20.9151 1.25667 0.628334 0.777943i \(-0.283737\pi\)
0.628334 + 0.777943i \(0.283737\pi\)
\(278\) −4.90271 −0.294045
\(279\) −8.41711 −0.503919
\(280\) 0 0
\(281\) 1.52898 0.0912115 0.0456058 0.998960i \(-0.485478\pi\)
0.0456058 + 0.998960i \(0.485478\pi\)
\(282\) 1.75408 0.104454
\(283\) −7.92671 −0.471194 −0.235597 0.971851i \(-0.575705\pi\)
−0.235597 + 0.971851i \(0.575705\pi\)
\(284\) −20.2624 −1.20235
\(285\) 0 0
\(286\) 4.86559 0.287708
\(287\) −14.1631 −0.836019
\(288\) 9.79091 0.576935
\(289\) −16.8517 −0.991274
\(290\) 0 0
\(291\) −11.9864 −0.702656
\(292\) 8.83679 0.517134
\(293\) 4.47050 0.261170 0.130585 0.991437i \(-0.458315\pi\)
0.130585 + 0.991437i \(0.458315\pi\)
\(294\) 4.76030 0.277627
\(295\) 0 0
\(296\) 1.61313 0.0937610
\(297\) 5.37300 0.311773
\(298\) −1.63425 −0.0946698
\(299\) −34.3246 −1.98504
\(300\) 0 0
\(301\) 6.34412 0.365669
\(302\) −2.14217 −0.123268
\(303\) 13.8854 0.797693
\(304\) 1.35163 0.0775215
\(305\) 0 0
\(306\) −0.449345 −0.0256873
\(307\) −25.1091 −1.43305 −0.716525 0.697562i \(-0.754268\pi\)
−0.716525 + 0.697562i \(0.754268\pi\)
\(308\) 5.47960 0.312229
\(309\) −13.6693 −0.777619
\(310\) 0 0
\(311\) −24.6576 −1.39821 −0.699103 0.715021i \(-0.746417\pi\)
−0.699103 + 0.715021i \(0.746417\pi\)
\(312\) 19.6459 1.11223
\(313\) −14.7748 −0.835123 −0.417562 0.908649i \(-0.637115\pi\)
−0.417562 + 0.908649i \(0.637115\pi\)
\(314\) 7.49136 0.422762
\(315\) 0 0
\(316\) 17.0050 0.956607
\(317\) 9.64687 0.541822 0.270911 0.962604i \(-0.412675\pi\)
0.270911 + 0.962604i \(0.412675\pi\)
\(318\) −3.30181 −0.185156
\(319\) −3.20619 −0.179512
\(320\) 0 0
\(321\) 10.5103 0.586626
\(322\) 12.1990 0.679825
\(323\) −0.385149 −0.0214303
\(324\) 1.68700 0.0937225
\(325\) 0 0
\(326\) 2.33472 0.129308
\(327\) −10.3900 −0.574567
\(328\) 9.58086 0.529015
\(329\) −7.95811 −0.438745
\(330\) 0 0
\(331\) −10.8051 −0.593903 −0.296952 0.954893i \(-0.595970\pi\)
−0.296952 + 0.954893i \(0.595970\pi\)
\(332\) −9.25132 −0.507732
\(333\) −1.11436 −0.0610667
\(334\) 5.80044 0.317386
\(335\) 0 0
\(336\) 5.58803 0.304852
\(337\) 35.0913 1.91154 0.955772 0.294110i \(-0.0950232\pi\)
0.955772 + 0.294110i \(0.0950232\pi\)
\(338\) −25.1748 −1.36933
\(339\) 0.802501 0.0435859
\(340\) 0 0
\(341\) 4.99714 0.270610
\(342\) 1.16668 0.0630867
\(343\) 3.63382 0.196208
\(344\) −4.29159 −0.231387
\(345\) 0 0
\(346\) 6.17758 0.332109
\(347\) −3.96060 −0.212616 −0.106308 0.994333i \(-0.533903\pi\)
−0.106308 + 0.994333i \(0.533903\pi\)
\(348\) −5.59071 −0.299693
\(349\) 16.1640 0.865241 0.432621 0.901576i \(-0.357589\pi\)
0.432621 + 0.901576i \(0.357589\pi\)
\(350\) 0 0
\(351\) −37.7436 −2.01460
\(352\) −5.81275 −0.309821
\(353\) −20.9960 −1.11751 −0.558753 0.829334i \(-0.688720\pi\)
−0.558753 + 0.829334i \(0.688720\pi\)
\(354\) 8.82231 0.468901
\(355\) 0 0
\(356\) 15.5162 0.822354
\(357\) −1.59231 −0.0842741
\(358\) −5.78483 −0.305738
\(359\) −27.1251 −1.43161 −0.715804 0.698301i \(-0.753940\pi\)
−0.715804 + 0.698301i \(0.753940\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −16.0273 −0.842378
\(363\) −1.14700 −0.0602020
\(364\) −38.4924 −2.01755
\(365\) 0 0
\(366\) 6.11908 0.319850
\(367\) 6.77494 0.353649 0.176824 0.984242i \(-0.443418\pi\)
0.176824 + 0.984242i \(0.443418\pi\)
\(368\) 6.60449 0.344283
\(369\) −6.61856 −0.344548
\(370\) 0 0
\(371\) 14.9800 0.777725
\(372\) 8.71364 0.451781
\(373\) −2.57955 −0.133564 −0.0667821 0.997768i \(-0.521273\pi\)
−0.0667821 + 0.997768i \(0.521273\pi\)
\(374\) 0.266771 0.0137944
\(375\) 0 0
\(376\) 5.38341 0.277628
\(377\) 22.5224 1.15996
\(378\) 13.4141 0.689948
\(379\) −8.19550 −0.420975 −0.210487 0.977597i \(-0.567505\pi\)
−0.210487 + 0.977597i \(0.567505\pi\)
\(380\) 0 0
\(381\) 21.3496 1.09377
\(382\) −7.94770 −0.406640
\(383\) −36.7852 −1.87963 −0.939817 0.341677i \(-0.889005\pi\)
−0.939817 + 0.341677i \(0.889005\pi\)
\(384\) −12.2835 −0.626839
\(385\) 0 0
\(386\) −12.0161 −0.611603
\(387\) 2.96468 0.150703
\(388\) −15.8869 −0.806534
\(389\) −20.6422 −1.04660 −0.523300 0.852149i \(-0.675299\pi\)
−0.523300 + 0.852149i \(0.675299\pi\)
\(390\) 0 0
\(391\) −1.88195 −0.0951744
\(392\) 14.6098 0.737905
\(393\) 10.1204 0.510507
\(394\) 9.42516 0.474832
\(395\) 0 0
\(396\) 2.56068 0.128679
\(397\) 15.3057 0.768172 0.384086 0.923297i \(-0.374517\pi\)
0.384086 + 0.923297i \(0.374517\pi\)
\(398\) −6.06619 −0.304071
\(399\) 4.13428 0.206973
\(400\) 0 0
\(401\) 26.2309 1.30991 0.654955 0.755668i \(-0.272687\pi\)
0.654955 + 0.755668i \(0.272687\pi\)
\(402\) 0.892024 0.0444901
\(403\) −35.1033 −1.74862
\(404\) 18.4037 0.915621
\(405\) 0 0
\(406\) −8.00450 −0.397257
\(407\) 0.661585 0.0327935
\(408\) 1.07715 0.0533268
\(409\) −4.93983 −0.244259 −0.122129 0.992514i \(-0.538972\pi\)
−0.122129 + 0.992514i \(0.538972\pi\)
\(410\) 0 0
\(411\) −3.56738 −0.175966
\(412\) −18.1174 −0.892579
\(413\) −40.0261 −1.96956
\(414\) 5.70074 0.280176
\(415\) 0 0
\(416\) 40.8326 2.00198
\(417\) −8.11878 −0.397579
\(418\) −0.692644 −0.0338783
\(419\) −8.64971 −0.422566 −0.211283 0.977425i \(-0.567764\pi\)
−0.211283 + 0.977425i \(0.567764\pi\)
\(420\) 0 0
\(421\) −29.7428 −1.44957 −0.724787 0.688973i \(-0.758062\pi\)
−0.724787 + 0.688973i \(0.758062\pi\)
\(422\) −14.0582 −0.684342
\(423\) −3.71891 −0.180820
\(424\) −10.1335 −0.492127
\(425\) 0 0
\(426\) 10.5889 0.513034
\(427\) −27.7618 −1.34349
\(428\) 13.9304 0.673350
\(429\) 8.05731 0.389011
\(430\) 0 0
\(431\) −32.4162 −1.56143 −0.780716 0.624885i \(-0.785146\pi\)
−0.780716 + 0.624885i \(0.785146\pi\)
\(432\) 7.26233 0.349409
\(433\) −33.9587 −1.63195 −0.815975 0.578087i \(-0.803799\pi\)
−0.815975 + 0.578087i \(0.803799\pi\)
\(434\) 12.4758 0.598855
\(435\) 0 0
\(436\) −13.7709 −0.659509
\(437\) 4.88630 0.233743
\(438\) −4.61802 −0.220657
\(439\) 14.4133 0.687907 0.343954 0.938987i \(-0.388234\pi\)
0.343954 + 0.938987i \(0.388234\pi\)
\(440\) 0 0
\(441\) −10.0926 −0.480599
\(442\) −1.87398 −0.0891360
\(443\) 36.0311 1.71189 0.855943 0.517069i \(-0.172977\pi\)
0.855943 + 0.517069i \(0.172977\pi\)
\(444\) 1.15362 0.0547485
\(445\) 0 0
\(446\) 13.7680 0.651933
\(447\) −2.70629 −0.128003
\(448\) −4.76828 −0.225280
\(449\) −10.3140 −0.486745 −0.243373 0.969933i \(-0.578254\pi\)
−0.243373 + 0.969933i \(0.578254\pi\)
\(450\) 0 0
\(451\) 3.92936 0.185026
\(452\) 1.06364 0.0500294
\(453\) −3.54739 −0.166671
\(454\) −0.362643 −0.0170197
\(455\) 0 0
\(456\) −2.79671 −0.130968
\(457\) 5.58176 0.261104 0.130552 0.991441i \(-0.458325\pi\)
0.130552 + 0.991441i \(0.458325\pi\)
\(458\) −12.9334 −0.604340
\(459\) −2.06941 −0.0965916
\(460\) 0 0
\(461\) 16.9193 0.788008 0.394004 0.919109i \(-0.371090\pi\)
0.394004 + 0.919109i \(0.371090\pi\)
\(462\) −2.86358 −0.133226
\(463\) −0.827191 −0.0384428 −0.0192214 0.999815i \(-0.506119\pi\)
−0.0192214 + 0.999815i \(0.506119\pi\)
\(464\) −4.33359 −0.201182
\(465\) 0 0
\(466\) −1.17688 −0.0545181
\(467\) 38.7142 1.79148 0.895739 0.444580i \(-0.146647\pi\)
0.895739 + 0.444580i \(0.146647\pi\)
\(468\) −17.9879 −0.831491
\(469\) −4.04704 −0.186875
\(470\) 0 0
\(471\) 12.4055 0.571616
\(472\) 27.0764 1.24629
\(473\) −1.76009 −0.0809292
\(474\) −8.88665 −0.408177
\(475\) 0 0
\(476\) −2.11046 −0.0967329
\(477\) 7.00034 0.320523
\(478\) 20.5795 0.941283
\(479\) 12.1245 0.553984 0.276992 0.960872i \(-0.410662\pi\)
0.276992 + 0.960872i \(0.410662\pi\)
\(480\) 0 0
\(481\) −4.64741 −0.211904
\(482\) 11.4998 0.523804
\(483\) 20.2013 0.919192
\(484\) −1.52024 −0.0691020
\(485\) 0 0
\(486\) 10.2831 0.466452
\(487\) 8.72457 0.395348 0.197674 0.980268i \(-0.436661\pi\)
0.197674 + 0.980268i \(0.436661\pi\)
\(488\) 18.7800 0.850129
\(489\) 3.86625 0.174838
\(490\) 0 0
\(491\) −42.1520 −1.90229 −0.951147 0.308737i \(-0.900094\pi\)
−0.951147 + 0.308737i \(0.900094\pi\)
\(492\) 6.85172 0.308900
\(493\) 1.23486 0.0556153
\(494\) 4.86559 0.218913
\(495\) 0 0
\(496\) 6.75431 0.303277
\(497\) −48.0410 −2.15493
\(498\) 4.83464 0.216645
\(499\) 18.9525 0.848431 0.424215 0.905561i \(-0.360550\pi\)
0.424215 + 0.905561i \(0.360550\pi\)
\(500\) 0 0
\(501\) 9.60539 0.429137
\(502\) −9.52291 −0.425028
\(503\) 25.2153 1.12430 0.562148 0.827036i \(-0.309975\pi\)
0.562148 + 0.827036i \(0.309975\pi\)
\(504\) 14.8033 0.659393
\(505\) 0 0
\(506\) −3.38446 −0.150458
\(507\) −41.6889 −1.85147
\(508\) 28.2968 1.25547
\(509\) −15.3133 −0.678751 −0.339375 0.940651i \(-0.610216\pi\)
−0.339375 + 0.940651i \(0.610216\pi\)
\(510\) 0 0
\(511\) 20.9516 0.926843
\(512\) −14.4480 −0.638518
\(513\) 5.37300 0.237224
\(514\) 18.5602 0.818657
\(515\) 0 0
\(516\) −3.06912 −0.135110
\(517\) 2.20788 0.0971022
\(518\) 1.65170 0.0725714
\(519\) 10.2299 0.449045
\(520\) 0 0
\(521\) −6.62604 −0.290292 −0.145146 0.989410i \(-0.546365\pi\)
−0.145146 + 0.989410i \(0.546365\pi\)
\(522\) −3.74059 −0.163721
\(523\) 10.2228 0.447014 0.223507 0.974702i \(-0.428250\pi\)
0.223507 + 0.974702i \(0.428250\pi\)
\(524\) 13.4136 0.585978
\(525\) 0 0
\(526\) −11.9381 −0.520526
\(527\) −1.92464 −0.0838388
\(528\) −1.55033 −0.0674694
\(529\) 0.875897 0.0380825
\(530\) 0 0
\(531\) −18.7047 −0.811713
\(532\) 5.47960 0.237571
\(533\) −27.6025 −1.19560
\(534\) −8.10858 −0.350893
\(535\) 0 0
\(536\) 2.73770 0.118250
\(537\) −9.57954 −0.413388
\(538\) 12.4200 0.535465
\(539\) 5.99184 0.258087
\(540\) 0 0
\(541\) −11.5547 −0.496777 −0.248388 0.968661i \(-0.579901\pi\)
−0.248388 + 0.968661i \(0.579901\pi\)
\(542\) 6.75440 0.290126
\(543\) −26.5409 −1.13898
\(544\) 2.23877 0.0959867
\(545\) 0 0
\(546\) 20.1157 0.860873
\(547\) 2.09444 0.0895518 0.0447759 0.998997i \(-0.485743\pi\)
0.0447759 + 0.998997i \(0.485743\pi\)
\(548\) −4.72823 −0.201980
\(549\) −12.9734 −0.553691
\(550\) 0 0
\(551\) −3.20619 −0.136588
\(552\) −13.6655 −0.581644
\(553\) 40.3180 1.71450
\(554\) −14.4867 −0.615482
\(555\) 0 0
\(556\) −10.7607 −0.456355
\(557\) −4.21753 −0.178703 −0.0893513 0.996000i \(-0.528479\pi\)
−0.0893513 + 0.996000i \(0.528479\pi\)
\(558\) 5.83006 0.246806
\(559\) 12.3641 0.522945
\(560\) 0 0
\(561\) 0.441767 0.0186514
\(562\) −1.05904 −0.0446729
\(563\) −8.55912 −0.360724 −0.180362 0.983600i \(-0.557727\pi\)
−0.180362 + 0.983600i \(0.557727\pi\)
\(564\) 3.84993 0.162111
\(565\) 0 0
\(566\) 5.49038 0.230778
\(567\) 3.99980 0.167976
\(568\) 32.4982 1.36359
\(569\) 8.10736 0.339878 0.169939 0.985455i \(-0.445643\pi\)
0.169939 + 0.985455i \(0.445643\pi\)
\(570\) 0 0
\(571\) 8.88932 0.372006 0.186003 0.982549i \(-0.440447\pi\)
0.186003 + 0.982549i \(0.440447\pi\)
\(572\) 10.6792 0.446520
\(573\) −13.1612 −0.549818
\(574\) 9.80996 0.409460
\(575\) 0 0
\(576\) −2.22827 −0.0928445
\(577\) −26.4638 −1.10170 −0.550852 0.834603i \(-0.685697\pi\)
−0.550852 + 0.834603i \(0.685697\pi\)
\(578\) 11.6722 0.485499
\(579\) −19.8984 −0.826949
\(580\) 0 0
\(581\) −21.9344 −0.909992
\(582\) 8.30232 0.344142
\(583\) −4.15602 −0.172125
\(584\) −14.1731 −0.586486
\(585\) 0 0
\(586\) −3.09647 −0.127914
\(587\) 11.2994 0.466376 0.233188 0.972432i \(-0.425084\pi\)
0.233188 + 0.972432i \(0.425084\pi\)
\(588\) 10.4481 0.430874
\(589\) 4.99714 0.205904
\(590\) 0 0
\(591\) 15.6079 0.642021
\(592\) 0.894220 0.0367522
\(593\) 47.7149 1.95942 0.979708 0.200432i \(-0.0642346\pi\)
0.979708 + 0.200432i \(0.0642346\pi\)
\(594\) −3.72158 −0.152698
\(595\) 0 0
\(596\) −3.58693 −0.146927
\(597\) −10.0455 −0.411134
\(598\) 23.7747 0.972221
\(599\) −3.68890 −0.150724 −0.0753622 0.997156i \(-0.524011\pi\)
−0.0753622 + 0.997156i \(0.524011\pi\)
\(600\) 0 0
\(601\) −20.9917 −0.856268 −0.428134 0.903715i \(-0.640829\pi\)
−0.428134 + 0.903715i \(0.640829\pi\)
\(602\) −4.39421 −0.179095
\(603\) −1.89123 −0.0770167
\(604\) −4.70174 −0.191311
\(605\) 0 0
\(606\) −9.61761 −0.390689
\(607\) −33.2697 −1.35037 −0.675187 0.737647i \(-0.735937\pi\)
−0.675187 + 0.737647i \(0.735937\pi\)
\(608\) −5.81275 −0.235738
\(609\) −13.2553 −0.537131
\(610\) 0 0
\(611\) −15.5096 −0.627451
\(612\) −0.986242 −0.0398665
\(613\) −39.8232 −1.60844 −0.804222 0.594330i \(-0.797417\pi\)
−0.804222 + 0.594330i \(0.797417\pi\)
\(614\) 17.3916 0.701869
\(615\) 0 0
\(616\) −8.78857 −0.354101
\(617\) −23.3355 −0.939454 −0.469727 0.882812i \(-0.655648\pi\)
−0.469727 + 0.882812i \(0.655648\pi\)
\(618\) 9.46795 0.380857
\(619\) −18.4550 −0.741771 −0.370886 0.928679i \(-0.620946\pi\)
−0.370886 + 0.928679i \(0.620946\pi\)
\(620\) 0 0
\(621\) 26.2541 1.05354
\(622\) 17.0790 0.684804
\(623\) 36.7880 1.47388
\(624\) 10.8905 0.435970
\(625\) 0 0
\(626\) 10.2337 0.409021
\(627\) −1.14700 −0.0458069
\(628\) 16.4424 0.656122
\(629\) −0.254809 −0.0101599
\(630\) 0 0
\(631\) −29.2519 −1.16450 −0.582249 0.813010i \(-0.697827\pi\)
−0.582249 + 0.813010i \(0.697827\pi\)
\(632\) −27.2738 −1.08490
\(633\) −23.2800 −0.925299
\(634\) −6.68184 −0.265370
\(635\) 0 0
\(636\) −7.24695 −0.287360
\(637\) −42.0907 −1.66769
\(638\) 2.22074 0.0879201
\(639\) −22.4501 −0.888112
\(640\) 0 0
\(641\) 25.9437 1.02471 0.512357 0.858772i \(-0.328772\pi\)
0.512357 + 0.858772i \(0.328772\pi\)
\(642\) −7.27987 −0.287313
\(643\) −41.8049 −1.64863 −0.824313 0.566135i \(-0.808438\pi\)
−0.824313 + 0.566135i \(0.808438\pi\)
\(644\) 26.7750 1.05508
\(645\) 0 0
\(646\) 0.266771 0.0104960
\(647\) 17.2410 0.677815 0.338907 0.940820i \(-0.389943\pi\)
0.338907 + 0.940820i \(0.389943\pi\)
\(648\) −2.70574 −0.106291
\(649\) 11.1047 0.435899
\(650\) 0 0
\(651\) 20.6596 0.809713
\(652\) 5.12436 0.200685
\(653\) 23.0285 0.901174 0.450587 0.892733i \(-0.351215\pi\)
0.450587 + 0.892733i \(0.351215\pi\)
\(654\) 7.19655 0.281408
\(655\) 0 0
\(656\) 5.31106 0.207362
\(657\) 9.79090 0.381979
\(658\) 5.51213 0.214885
\(659\) 43.1431 1.68062 0.840309 0.542108i \(-0.182374\pi\)
0.840309 + 0.542108i \(0.182374\pi\)
\(660\) 0 0
\(661\) −23.0871 −0.897985 −0.448993 0.893535i \(-0.648217\pi\)
−0.448993 + 0.893535i \(0.648217\pi\)
\(662\) 7.48410 0.290878
\(663\) −3.10326 −0.120521
\(664\) 14.8379 0.575823
\(665\) 0 0
\(666\) 0.771856 0.0299088
\(667\) −15.6664 −0.606605
\(668\) 12.7311 0.492579
\(669\) 22.7995 0.881479
\(670\) 0 0
\(671\) 7.70215 0.297338
\(672\) −24.0315 −0.927037
\(673\) 42.0990 1.62280 0.811398 0.584494i \(-0.198707\pi\)
0.811398 + 0.584494i \(0.198707\pi\)
\(674\) −24.3057 −0.936222
\(675\) 0 0
\(676\) −55.2547 −2.12518
\(677\) 51.2657 1.97030 0.985151 0.171693i \(-0.0549236\pi\)
0.985151 + 0.171693i \(0.0549236\pi\)
\(678\) −0.555847 −0.0213472
\(679\) −37.6670 −1.44552
\(680\) 0 0
\(681\) −0.600529 −0.0230123
\(682\) −3.46124 −0.132538
\(683\) −21.0036 −0.803680 −0.401840 0.915710i \(-0.631629\pi\)
−0.401840 + 0.915710i \(0.631629\pi\)
\(684\) 2.56068 0.0979099
\(685\) 0 0
\(686\) −2.51694 −0.0960972
\(687\) −21.4175 −0.817129
\(688\) −2.37900 −0.0906987
\(689\) 29.1947 1.11223
\(690\) 0 0
\(691\) −43.6608 −1.66094 −0.830468 0.557066i \(-0.811927\pi\)
−0.830468 + 0.557066i \(0.811927\pi\)
\(692\) 13.5588 0.515430
\(693\) 6.07123 0.230627
\(694\) 2.74329 0.104134
\(695\) 0 0
\(696\) 8.96677 0.339884
\(697\) −1.51339 −0.0573237
\(698\) −11.1959 −0.423772
\(699\) −1.94889 −0.0737139
\(700\) 0 0
\(701\) 1.36357 0.0515013 0.0257506 0.999668i \(-0.491802\pi\)
0.0257506 + 0.999668i \(0.491802\pi\)
\(702\) 26.1428 0.986698
\(703\) 0.661585 0.0249521
\(704\) 1.32290 0.0498585
\(705\) 0 0
\(706\) 14.5428 0.547324
\(707\) 43.6343 1.64104
\(708\) 19.3636 0.727729
\(709\) 24.2648 0.911285 0.455642 0.890163i \(-0.349410\pi\)
0.455642 + 0.890163i \(0.349410\pi\)
\(710\) 0 0
\(711\) 18.8410 0.706594
\(712\) −24.8859 −0.932638
\(713\) 24.4175 0.914443
\(714\) 1.10291 0.0412752
\(715\) 0 0
\(716\) −12.6968 −0.474501
\(717\) 34.0792 1.27271
\(718\) 18.7880 0.701163
\(719\) 18.2569 0.680867 0.340434 0.940269i \(-0.389426\pi\)
0.340434 + 0.940269i \(0.389426\pi\)
\(720\) 0 0
\(721\) −42.9553 −1.59974
\(722\) −0.692644 −0.0257775
\(723\) 19.0435 0.708235
\(724\) −35.1775 −1.30736
\(725\) 0 0
\(726\) 0.794464 0.0294853
\(727\) 34.8231 1.29152 0.645759 0.763541i \(-0.276541\pi\)
0.645759 + 0.763541i \(0.276541\pi\)
\(728\) 61.7368 2.28812
\(729\) 20.3577 0.753989
\(730\) 0 0
\(731\) 0.677898 0.0250730
\(732\) 13.4304 0.496403
\(733\) 15.1353 0.559036 0.279518 0.960141i \(-0.409825\pi\)
0.279518 + 0.960141i \(0.409825\pi\)
\(734\) −4.69262 −0.173208
\(735\) 0 0
\(736\) −28.4028 −1.04694
\(737\) 1.12280 0.0413589
\(738\) 4.58430 0.168750
\(739\) −25.7057 −0.945600 −0.472800 0.881170i \(-0.656757\pi\)
−0.472800 + 0.881170i \(0.656757\pi\)
\(740\) 0 0
\(741\) 8.05731 0.295993
\(742\) −10.3758 −0.380909
\(743\) 25.9533 0.952136 0.476068 0.879409i \(-0.342062\pi\)
0.476068 + 0.879409i \(0.342062\pi\)
\(744\) −13.9755 −0.512368
\(745\) 0 0
\(746\) 1.78671 0.0654162
\(747\) −10.2502 −0.375034
\(748\) 0.585521 0.0214088
\(749\) 33.0282 1.20682
\(750\) 0 0
\(751\) −26.5830 −0.970026 −0.485013 0.874507i \(-0.661185\pi\)
−0.485013 + 0.874507i \(0.661185\pi\)
\(752\) 2.98424 0.108824
\(753\) −15.7697 −0.574681
\(754\) −15.6000 −0.568118
\(755\) 0 0
\(756\) 29.4419 1.07079
\(757\) 0.372319 0.0135322 0.00676608 0.999977i \(-0.497846\pi\)
0.00676608 + 0.999977i \(0.497846\pi\)
\(758\) 5.67656 0.206182
\(759\) −5.60460 −0.203434
\(760\) 0 0
\(761\) −12.7607 −0.462574 −0.231287 0.972886i \(-0.574294\pi\)
−0.231287 + 0.972886i \(0.574294\pi\)
\(762\) −14.7876 −0.535699
\(763\) −32.6502 −1.18202
\(764\) −17.4440 −0.631100
\(765\) 0 0
\(766\) 25.4790 0.920594
\(767\) −78.0071 −2.81667
\(768\) 11.5428 0.416515
\(769\) 46.2851 1.66908 0.834542 0.550944i \(-0.185732\pi\)
0.834542 + 0.550944i \(0.185732\pi\)
\(770\) 0 0
\(771\) 30.7354 1.10691
\(772\) −26.3735 −0.949202
\(773\) 12.5661 0.451973 0.225986 0.974130i \(-0.427439\pi\)
0.225986 + 0.974130i \(0.427439\pi\)
\(774\) −2.05346 −0.0738102
\(775\) 0 0
\(776\) 25.4805 0.914696
\(777\) 2.73518 0.0981239
\(778\) 14.2977 0.512596
\(779\) 3.92936 0.140784
\(780\) 0 0
\(781\) 13.3284 0.476926
\(782\) 1.30352 0.0466138
\(783\) −17.2268 −0.615637
\(784\) 8.09878 0.289242
\(785\) 0 0
\(786\) −7.00983 −0.250032
\(787\) 14.6706 0.522952 0.261476 0.965210i \(-0.415791\pi\)
0.261476 + 0.965210i \(0.415791\pi\)
\(788\) 20.6867 0.736935
\(789\) −19.7692 −0.703803
\(790\) 0 0
\(791\) 2.52183 0.0896661
\(792\) −4.10699 −0.145936
\(793\) −54.1051 −1.92133
\(794\) −10.6014 −0.376230
\(795\) 0 0
\(796\) −13.3144 −0.471915
\(797\) 37.6027 1.33196 0.665978 0.745972i \(-0.268015\pi\)
0.665978 + 0.745972i \(0.268015\pi\)
\(798\) −2.86358 −0.101370
\(799\) −0.850361 −0.0300836
\(800\) 0 0
\(801\) 17.1914 0.607429
\(802\) −18.1687 −0.641559
\(803\) −5.81274 −0.205127
\(804\) 1.95785 0.0690482
\(805\) 0 0
\(806\) 24.3140 0.856426
\(807\) 20.5673 0.724002
\(808\) −29.5172 −1.03841
\(809\) 5.32635 0.187264 0.0936322 0.995607i \(-0.470152\pi\)
0.0936322 + 0.995607i \(0.470152\pi\)
\(810\) 0 0
\(811\) 1.82625 0.0641282 0.0320641 0.999486i \(-0.489792\pi\)
0.0320641 + 0.999486i \(0.489792\pi\)
\(812\) −17.5686 −0.616538
\(813\) 11.1851 0.392280
\(814\) −0.458242 −0.0160614
\(815\) 0 0
\(816\) 0.597107 0.0209029
\(817\) −1.76009 −0.0615779
\(818\) 3.42154 0.119631
\(819\) −42.6484 −1.49025
\(820\) 0 0
\(821\) −54.3594 −1.89716 −0.948579 0.316541i \(-0.897479\pi\)
−0.948579 + 0.316541i \(0.897479\pi\)
\(822\) 2.47092 0.0861833
\(823\) 46.5557 1.62283 0.811415 0.584471i \(-0.198698\pi\)
0.811415 + 0.584471i \(0.198698\pi\)
\(824\) 29.0579 1.01228
\(825\) 0 0
\(826\) 27.7238 0.964636
\(827\) −1.34120 −0.0466380 −0.0233190 0.999728i \(-0.507423\pi\)
−0.0233190 + 0.999728i \(0.507423\pi\)
\(828\) 12.5122 0.434830
\(829\) −42.8307 −1.48757 −0.743785 0.668419i \(-0.766972\pi\)
−0.743785 + 0.668419i \(0.766972\pi\)
\(830\) 0 0
\(831\) −23.9897 −0.832194
\(832\) −9.29291 −0.322174
\(833\) −2.30775 −0.0799589
\(834\) 5.62342 0.194723
\(835\) 0 0
\(836\) −1.52024 −0.0525788
\(837\) 26.8496 0.928060
\(838\) 5.99117 0.206961
\(839\) 8.45704 0.291970 0.145985 0.989287i \(-0.453365\pi\)
0.145985 + 0.989287i \(0.453365\pi\)
\(840\) 0 0
\(841\) −18.7204 −0.645530
\(842\) 20.6011 0.709962
\(843\) −1.75375 −0.0604023
\(844\) −30.8555 −1.06209
\(845\) 0 0
\(846\) 2.57588 0.0885606
\(847\) −3.60442 −0.123849
\(848\) −5.61742 −0.192903
\(849\) 9.09196 0.312035
\(850\) 0 0
\(851\) 3.23270 0.110815
\(852\) 23.2410 0.796223
\(853\) −2.07765 −0.0711372 −0.0355686 0.999367i \(-0.511324\pi\)
−0.0355686 + 0.999367i \(0.511324\pi\)
\(854\) 19.2290 0.658004
\(855\) 0 0
\(856\) −22.3425 −0.763651
\(857\) 10.3114 0.352230 0.176115 0.984370i \(-0.443647\pi\)
0.176115 + 0.984370i \(0.443647\pi\)
\(858\) −5.58085 −0.190527
\(859\) 8.80362 0.300376 0.150188 0.988657i \(-0.452012\pi\)
0.150188 + 0.988657i \(0.452012\pi\)
\(860\) 0 0
\(861\) 16.2451 0.553631
\(862\) 22.4529 0.764748
\(863\) 7.62460 0.259544 0.129772 0.991544i \(-0.458575\pi\)
0.129772 + 0.991544i \(0.458575\pi\)
\(864\) −31.2319 −1.06253
\(865\) 0 0
\(866\) 23.5213 0.799285
\(867\) 19.3289 0.656444
\(868\) 27.3823 0.929417
\(869\) −11.1857 −0.379449
\(870\) 0 0
\(871\) −7.88730 −0.267251
\(872\) 22.0868 0.747954
\(873\) −17.6022 −0.595743
\(874\) −3.38446 −0.114481
\(875\) 0 0
\(876\) −10.1358 −0.342458
\(877\) 34.3933 1.16138 0.580689 0.814125i \(-0.302783\pi\)
0.580689 + 0.814125i \(0.302783\pi\)
\(878\) −9.98325 −0.336919
\(879\) −5.12768 −0.172952
\(880\) 0 0
\(881\) 8.09345 0.272675 0.136338 0.990662i \(-0.456467\pi\)
0.136338 + 0.990662i \(0.456467\pi\)
\(882\) 6.99056 0.235384
\(883\) −26.3719 −0.887486 −0.443743 0.896154i \(-0.646350\pi\)
−0.443743 + 0.896154i \(0.646350\pi\)
\(884\) −4.11309 −0.138338
\(885\) 0 0
\(886\) −24.9567 −0.838436
\(887\) 35.2984 1.18520 0.592602 0.805496i \(-0.298101\pi\)
0.592602 + 0.805496i \(0.298101\pi\)
\(888\) −1.85026 −0.0620906
\(889\) 67.0903 2.25014
\(890\) 0 0
\(891\) −1.10969 −0.0371761
\(892\) 30.2186 1.01179
\(893\) 2.20788 0.0738837
\(894\) 1.87449 0.0626925
\(895\) 0 0
\(896\) −38.6005 −1.28955
\(897\) 39.3704 1.31454
\(898\) 7.14389 0.238395
\(899\) −16.0218 −0.534356
\(900\) 0 0
\(901\) 1.60069 0.0533266
\(902\) −2.72165 −0.0906209
\(903\) −7.27672 −0.242154
\(904\) −1.70594 −0.0567387
\(905\) 0 0
\(906\) 2.45708 0.0816310
\(907\) 57.0934 1.89576 0.947878 0.318634i \(-0.103224\pi\)
0.947878 + 0.318634i \(0.103224\pi\)
\(908\) −0.795945 −0.0264144
\(909\) 20.3908 0.676320
\(910\) 0 0
\(911\) −18.2973 −0.606216 −0.303108 0.952956i \(-0.598024\pi\)
−0.303108 + 0.952956i \(0.598024\pi\)
\(912\) −1.55033 −0.0513365
\(913\) 6.08541 0.201398
\(914\) −3.86617 −0.127882
\(915\) 0 0
\(916\) −28.3869 −0.937930
\(917\) 31.8030 1.05023
\(918\) 1.43336 0.0473079
\(919\) −53.1018 −1.75167 −0.875833 0.482614i \(-0.839688\pi\)
−0.875833 + 0.482614i \(0.839688\pi\)
\(920\) 0 0
\(921\) 28.8001 0.948997
\(922\) −11.7190 −0.385945
\(923\) −93.6273 −3.08178
\(924\) −6.28512 −0.206765
\(925\) 0 0
\(926\) 0.572948 0.0188282
\(927\) −20.0735 −0.659300
\(928\) 18.6368 0.611782
\(929\) 28.2532 0.926957 0.463479 0.886108i \(-0.346601\pi\)
0.463479 + 0.886108i \(0.346601\pi\)
\(930\) 0 0
\(931\) 5.99184 0.196375
\(932\) −2.58308 −0.0846114
\(933\) 28.2824 0.925923
\(934\) −26.8151 −0.877418
\(935\) 0 0
\(936\) 28.8503 0.943000
\(937\) 15.6921 0.512637 0.256319 0.966592i \(-0.417490\pi\)
0.256319 + 0.966592i \(0.417490\pi\)
\(938\) 2.80316 0.0915264
\(939\) 16.9468 0.553037
\(940\) 0 0
\(941\) 16.4701 0.536910 0.268455 0.963292i \(-0.413487\pi\)
0.268455 + 0.963292i \(0.413487\pi\)
\(942\) −8.59260 −0.279962
\(943\) 19.2000 0.625239
\(944\) 15.0095 0.488519
\(945\) 0 0
\(946\) 1.21912 0.0396370
\(947\) −26.2431 −0.852785 −0.426392 0.904538i \(-0.640216\pi\)
−0.426392 + 0.904538i \(0.640216\pi\)
\(948\) −19.5048 −0.633487
\(949\) 40.8326 1.32548
\(950\) 0 0
\(951\) −11.0650 −0.358807
\(952\) 3.38491 0.109705
\(953\) 37.0207 1.19922 0.599610 0.800293i \(-0.295323\pi\)
0.599610 + 0.800293i \(0.295323\pi\)
\(954\) −4.84874 −0.156984
\(955\) 0 0
\(956\) 45.1687 1.46086
\(957\) 3.67750 0.118877
\(958\) −8.39798 −0.271326
\(959\) −11.2104 −0.362002
\(960\) 0 0
\(961\) −6.02858 −0.194470
\(962\) 3.21900 0.103785
\(963\) 15.4344 0.497367
\(964\) 25.2404 0.812938
\(965\) 0 0
\(966\) −13.9923 −0.450195
\(967\) −52.9984 −1.70431 −0.852157 0.523286i \(-0.824706\pi\)
−0.852157 + 0.523286i \(0.824706\pi\)
\(968\) 2.43828 0.0783691
\(969\) 0.441767 0.0141916
\(970\) 0 0
\(971\) 16.5465 0.531003 0.265501 0.964110i \(-0.414463\pi\)
0.265501 + 0.964110i \(0.414463\pi\)
\(972\) 22.5698 0.723928
\(973\) −25.5130 −0.817910
\(974\) −6.04302 −0.193631
\(975\) 0 0
\(976\) 10.4105 0.333232
\(977\) 7.03173 0.224965 0.112482 0.993654i \(-0.464120\pi\)
0.112482 + 0.993654i \(0.464120\pi\)
\(978\) −2.67793 −0.0856309
\(979\) −10.2064 −0.326196
\(980\) 0 0
\(981\) −15.2578 −0.487144
\(982\) 29.1963 0.931693
\(983\) 5.31776 0.169610 0.0848052 0.996398i \(-0.472973\pi\)
0.0848052 + 0.996398i \(0.472973\pi\)
\(984\) −10.9893 −0.350325
\(985\) 0 0
\(986\) −0.855317 −0.0272389
\(987\) 9.12797 0.290547
\(988\) 10.6792 0.339751
\(989\) −8.60034 −0.273475
\(990\) 0 0
\(991\) −33.9658 −1.07896 −0.539480 0.841998i \(-0.681379\pi\)
−0.539480 + 0.841998i \(0.681379\pi\)
\(992\) −29.0471 −0.922247
\(993\) 12.3935 0.393296
\(994\) 33.2753 1.05543
\(995\) 0 0
\(996\) 10.6113 0.336231
\(997\) −48.4773 −1.53529 −0.767645 0.640875i \(-0.778572\pi\)
−0.767645 + 0.640875i \(0.778572\pi\)
\(998\) −13.1273 −0.415538
\(999\) 3.55470 0.112466
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.r.1.8 15
5.4 even 2 5225.2.a.y.1.8 yes 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5225.2.a.r.1.8 15 1.1 even 1 trivial
5225.2.a.y.1.8 yes 15 5.4 even 2