Properties

Label 5239.2.a.l.1.4
Level $5239$
Weight $2$
Character 5239.1
Self dual yes
Analytic conductor $41.834$
Analytic rank $0$
Dimension $16$
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] = [5239,2,Mod(1,5239)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5239, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("5239.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5239 = 13^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5239.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: \(41.8336256189\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 17 x^{14} + 80 x^{13} + 98 x^{12} - 628 x^{11} - 158 x^{10} + 2458 x^{9} - 510 x^{8} - 5030 x^{7} + 2318 x^{6} + 5112 x^{5} - 3154 x^{4} - 2086 x^{3} + \cdots - 147 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 403)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.44015\) of defining polynomial
Character \(\chi\) \(=\) 5239.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.44015 q^{2} -2.79139 q^{3} +0.0740342 q^{4} +2.61757 q^{5} +4.02002 q^{6} +4.99660 q^{7} +2.77368 q^{8} +4.79185 q^{9} +O(q^{10})\) \(q-1.44015 q^{2} -2.79139 q^{3} +0.0740342 q^{4} +2.61757 q^{5} +4.02002 q^{6} +4.99660 q^{7} +2.77368 q^{8} +4.79185 q^{9} -3.76969 q^{10} +0.0546038 q^{11} -0.206658 q^{12} -7.19586 q^{14} -7.30664 q^{15} -4.14259 q^{16} +5.08006 q^{17} -6.90098 q^{18} -1.17518 q^{19} +0.193789 q^{20} -13.9475 q^{21} -0.0786376 q^{22} +7.81207 q^{23} -7.74242 q^{24} +1.85165 q^{25} -5.00174 q^{27} +0.369920 q^{28} -7.02474 q^{29} +10.5227 q^{30} -1.00000 q^{31} +0.418588 q^{32} -0.152420 q^{33} -7.31605 q^{34} +13.0789 q^{35} +0.354761 q^{36} +6.75099 q^{37} +1.69243 q^{38} +7.26030 q^{40} +1.55918 q^{41} +20.0864 q^{42} +3.02644 q^{43} +0.00404255 q^{44} +12.5430 q^{45} -11.2506 q^{46} +0.191499 q^{47} +11.5636 q^{48} +17.9661 q^{49} -2.66666 q^{50} -14.1804 q^{51} +8.59226 q^{53} +7.20326 q^{54} +0.142929 q^{55} +13.8590 q^{56} +3.28038 q^{57} +10.1167 q^{58} -2.91407 q^{59} -0.540942 q^{60} +5.96368 q^{61} +1.44015 q^{62} +23.9430 q^{63} +7.68234 q^{64} +0.219508 q^{66} -1.50321 q^{67} +0.376098 q^{68} -21.8065 q^{69} -18.8357 q^{70} -1.44037 q^{71} +13.2911 q^{72} +13.9550 q^{73} -9.72244 q^{74} -5.16869 q^{75} -0.0870034 q^{76} +0.272833 q^{77} +1.10819 q^{79} -10.8435 q^{80} -0.413740 q^{81} -2.24545 q^{82} -1.74923 q^{83} -1.03259 q^{84} +13.2974 q^{85} -4.35854 q^{86} +19.6088 q^{87} +0.151453 q^{88} +2.35995 q^{89} -18.0638 q^{90} +0.578360 q^{92} +2.79139 q^{93} -0.275787 q^{94} -3.07611 q^{95} -1.16844 q^{96} +12.1357 q^{97} -25.8738 q^{98} +0.261653 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} - 2 q^{3} + 18 q^{4} + 4 q^{5} + 6 q^{6} + 2 q^{7} + 12 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{2} - 2 q^{3} + 18 q^{4} + 4 q^{5} + 6 q^{6} + 2 q^{7} + 12 q^{8} + 10 q^{9} - 2 q^{10} + 14 q^{11} + 8 q^{12} - 8 q^{14} + 14 q^{16} + 4 q^{17} - 28 q^{18} + 22 q^{19} + 28 q^{20} + 12 q^{21} - 8 q^{22} + 4 q^{23} - 8 q^{24} - 2 q^{25} + 10 q^{27} + 16 q^{28} - 8 q^{29} - 20 q^{30} - 16 q^{31} + 48 q^{32} + 10 q^{33} + 8 q^{34} - 2 q^{35} + 22 q^{36} + 16 q^{37} - 6 q^{38} + 14 q^{40} + 44 q^{41} + 14 q^{42} + 16 q^{43} + 4 q^{44} + 56 q^{45} + 10 q^{47} + 32 q^{49} + 2 q^{50} - 6 q^{53} + 24 q^{54} + 22 q^{55} - 4 q^{56} - 8 q^{57} + 74 q^{58} + 2 q^{59} + 40 q^{60} + 8 q^{61} - 4 q^{62} + 56 q^{63} + 38 q^{64} - 34 q^{66} - 8 q^{67} + 32 q^{68} - 10 q^{69} - 108 q^{70} + 50 q^{71} - 44 q^{72} + 14 q^{73} + 8 q^{74} + 44 q^{76} + 16 q^{77} + 32 q^{79} + 68 q^{80} - 8 q^{81} - 6 q^{82} - 20 q^{83} + 136 q^{84} - 32 q^{85} + 8 q^{86} - 36 q^{87} - 40 q^{88} + 52 q^{89} - 34 q^{90} + 14 q^{92} + 2 q^{93} + 44 q^{94} - 2 q^{95} - 80 q^{96} + 18 q^{97} + 12 q^{98} + 38 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.44015 −1.01834 −0.509170 0.860666i \(-0.670048\pi\)
−0.509170 + 0.860666i \(0.670048\pi\)
\(3\) −2.79139 −1.61161 −0.805804 0.592182i \(-0.798267\pi\)
−0.805804 + 0.592182i \(0.798267\pi\)
\(4\) 0.0740342 0.0370171
\(5\) 2.61757 1.17061 0.585306 0.810813i \(-0.300975\pi\)
0.585306 + 0.810813i \(0.300975\pi\)
\(6\) 4.02002 1.64117
\(7\) 4.99660 1.88854 0.944269 0.329173i \(-0.106770\pi\)
0.944269 + 0.329173i \(0.106770\pi\)
\(8\) 2.77368 0.980644
\(9\) 4.79185 1.59728
\(10\) −3.76969 −1.19208
\(11\) 0.0546038 0.0164637 0.00823183 0.999966i \(-0.497380\pi\)
0.00823183 + 0.999966i \(0.497380\pi\)
\(12\) −0.206658 −0.0596571
\(13\) 0 0
\(14\) −7.19586 −1.92318
\(15\) −7.30664 −1.88657
\(16\) −4.14259 −1.03565
\(17\) 5.08006 1.23209 0.616047 0.787709i \(-0.288733\pi\)
0.616047 + 0.787709i \(0.288733\pi\)
\(18\) −6.90098 −1.62658
\(19\) −1.17518 −0.269604 −0.134802 0.990873i \(-0.543040\pi\)
−0.134802 + 0.990873i \(0.543040\pi\)
\(20\) 0.193789 0.0433326
\(21\) −13.9475 −3.04359
\(22\) −0.0786376 −0.0167656
\(23\) 7.81207 1.62893 0.814464 0.580213i \(-0.197031\pi\)
0.814464 + 0.580213i \(0.197031\pi\)
\(24\) −7.74242 −1.58041
\(25\) 1.85165 0.370331
\(26\) 0 0
\(27\) −5.00174 −0.962586
\(28\) 0.369920 0.0699082
\(29\) −7.02474 −1.30446 −0.652231 0.758020i \(-0.726167\pi\)
−0.652231 + 0.758020i \(0.726167\pi\)
\(30\) 10.5227 1.92117
\(31\) −1.00000 −0.179605
\(32\) 0.418588 0.0739966
\(33\) −0.152420 −0.0265330
\(34\) −7.31605 −1.25469
\(35\) 13.0789 2.21075
\(36\) 0.354761 0.0591268
\(37\) 6.75099 1.10986 0.554928 0.831899i \(-0.312746\pi\)
0.554928 + 0.831899i \(0.312746\pi\)
\(38\) 1.69243 0.274549
\(39\) 0 0
\(40\) 7.26030 1.14795
\(41\) 1.55918 0.243503 0.121751 0.992561i \(-0.461149\pi\)
0.121751 + 0.992561i \(0.461149\pi\)
\(42\) 20.0864 3.09941
\(43\) 3.02644 0.461528 0.230764 0.973010i \(-0.425877\pi\)
0.230764 + 0.973010i \(0.425877\pi\)
\(44\) 0.00404255 0.000609437 0
\(45\) 12.5430 1.86980
\(46\) −11.2506 −1.65880
\(47\) 0.191499 0.0279330 0.0139665 0.999902i \(-0.495554\pi\)
0.0139665 + 0.999902i \(0.495554\pi\)
\(48\) 11.5636 1.66906
\(49\) 17.9661 2.56658
\(50\) −2.66666 −0.377123
\(51\) −14.1804 −1.98565
\(52\) 0 0
\(53\) 8.59226 1.18024 0.590119 0.807317i \(-0.299081\pi\)
0.590119 + 0.807317i \(0.299081\pi\)
\(54\) 7.20326 0.980240
\(55\) 0.142929 0.0192725
\(56\) 13.8590 1.85199
\(57\) 3.28038 0.434497
\(58\) 10.1167 1.32839
\(59\) −2.91407 −0.379379 −0.189690 0.981844i \(-0.560748\pi\)
−0.189690 + 0.981844i \(0.560748\pi\)
\(60\) −0.540942 −0.0698353
\(61\) 5.96368 0.763571 0.381786 0.924251i \(-0.375309\pi\)
0.381786 + 0.924251i \(0.375309\pi\)
\(62\) 1.44015 0.182899
\(63\) 23.9430 3.01653
\(64\) 7.68234 0.960293
\(65\) 0 0
\(66\) 0.219508 0.0270196
\(67\) −1.50321 −0.183646 −0.0918231 0.995775i \(-0.529269\pi\)
−0.0918231 + 0.995775i \(0.529269\pi\)
\(68\) 0.376098 0.0456086
\(69\) −21.8065 −2.62520
\(70\) −18.8357 −2.25129
\(71\) −1.44037 −0.170940 −0.0854701 0.996341i \(-0.527239\pi\)
−0.0854701 + 0.996341i \(0.527239\pi\)
\(72\) 13.2911 1.56637
\(73\) 13.9550 1.63331 0.816656 0.577124i \(-0.195825\pi\)
0.816656 + 0.577124i \(0.195825\pi\)
\(74\) −9.72244 −1.13021
\(75\) −5.16869 −0.596829
\(76\) −0.0870034 −0.00997997
\(77\) 0.272833 0.0310922
\(78\) 0 0
\(79\) 1.10819 0.124681 0.0623404 0.998055i \(-0.480144\pi\)
0.0623404 + 0.998055i \(0.480144\pi\)
\(80\) −10.8435 −1.21234
\(81\) −0.413740 −0.0459711
\(82\) −2.24545 −0.247969
\(83\) −1.74923 −0.192003 −0.0960015 0.995381i \(-0.530605\pi\)
−0.0960015 + 0.995381i \(0.530605\pi\)
\(84\) −1.03259 −0.112665
\(85\) 13.2974 1.44230
\(86\) −4.35854 −0.469993
\(87\) 19.6088 2.10228
\(88\) 0.151453 0.0161450
\(89\) 2.35995 0.250154 0.125077 0.992147i \(-0.460082\pi\)
0.125077 + 0.992147i \(0.460082\pi\)
\(90\) −18.0638 −1.90409
\(91\) 0 0
\(92\) 0.578360 0.0602982
\(93\) 2.79139 0.289453
\(94\) −0.275787 −0.0284453
\(95\) −3.07611 −0.315602
\(96\) −1.16844 −0.119254
\(97\) 12.1357 1.23220 0.616098 0.787670i \(-0.288713\pi\)
0.616098 + 0.787670i \(0.288713\pi\)
\(98\) −25.8738 −2.61365
\(99\) 0.261653 0.0262971
\(100\) 0.137086 0.0137086
\(101\) −11.6128 −1.15552 −0.577758 0.816208i \(-0.696072\pi\)
−0.577758 + 0.816208i \(0.696072\pi\)
\(102\) 20.4219 2.02207
\(103\) 12.7238 1.25371 0.626857 0.779134i \(-0.284341\pi\)
0.626857 + 0.779134i \(0.284341\pi\)
\(104\) 0 0
\(105\) −36.5084 −3.56286
\(106\) −12.3741 −1.20188
\(107\) −2.95463 −0.285635 −0.142817 0.989749i \(-0.545616\pi\)
−0.142817 + 0.989749i \(0.545616\pi\)
\(108\) −0.370300 −0.0356321
\(109\) −19.9084 −1.90688 −0.953439 0.301587i \(-0.902484\pi\)
−0.953439 + 0.301587i \(0.902484\pi\)
\(110\) −0.205839 −0.0196260
\(111\) −18.8446 −1.78865
\(112\) −20.6989 −1.95586
\(113\) 0.926163 0.0871261 0.0435630 0.999051i \(-0.486129\pi\)
0.0435630 + 0.999051i \(0.486129\pi\)
\(114\) −4.72424 −0.442465
\(115\) 20.4486 1.90684
\(116\) −0.520071 −0.0482874
\(117\) 0 0
\(118\) 4.19670 0.386337
\(119\) 25.3830 2.32686
\(120\) −20.2663 −1.85005
\(121\) −10.9970 −0.999729
\(122\) −8.58860 −0.777575
\(123\) −4.35227 −0.392431
\(124\) −0.0740342 −0.00664847
\(125\) −8.24100 −0.737098
\(126\) −34.4815 −3.07185
\(127\) −16.3194 −1.44812 −0.724058 0.689739i \(-0.757725\pi\)
−0.724058 + 0.689739i \(0.757725\pi\)
\(128\) −11.9009 −1.05190
\(129\) −8.44798 −0.743803
\(130\) 0 0
\(131\) 15.1689 1.32532 0.662658 0.748922i \(-0.269428\pi\)
0.662658 + 0.748922i \(0.269428\pi\)
\(132\) −0.0112843 −0.000982174 0
\(133\) −5.87190 −0.509158
\(134\) 2.16485 0.187014
\(135\) −13.0924 −1.12681
\(136\) 14.0905 1.20825
\(137\) −11.4877 −0.981463 −0.490732 0.871311i \(-0.663270\pi\)
−0.490732 + 0.871311i \(0.663270\pi\)
\(138\) 31.4047 2.67334
\(139\) −6.64791 −0.563869 −0.281934 0.959434i \(-0.590976\pi\)
−0.281934 + 0.959434i \(0.590976\pi\)
\(140\) 0.968289 0.0818354
\(141\) −0.534548 −0.0450170
\(142\) 2.07435 0.174075
\(143\) 0 0
\(144\) −19.8506 −1.65422
\(145\) −18.3877 −1.52702
\(146\) −20.0973 −1.66327
\(147\) −50.1502 −4.13632
\(148\) 0.499804 0.0410836
\(149\) 1.35792 0.111245 0.0556224 0.998452i \(-0.482286\pi\)
0.0556224 + 0.998452i \(0.482286\pi\)
\(150\) 7.44369 0.607775
\(151\) 12.4451 1.01277 0.506384 0.862308i \(-0.330982\pi\)
0.506384 + 0.862308i \(0.330982\pi\)
\(152\) −3.25957 −0.264386
\(153\) 24.3429 1.96800
\(154\) −0.392921 −0.0316625
\(155\) −2.61757 −0.210248
\(156\) 0 0
\(157\) 15.5894 1.24417 0.622086 0.782949i \(-0.286286\pi\)
0.622086 + 0.782949i \(0.286286\pi\)
\(158\) −1.59596 −0.126968
\(159\) −23.9843 −1.90208
\(160\) 1.09568 0.0866213
\(161\) 39.0338 3.07630
\(162\) 0.595848 0.0468143
\(163\) −2.52440 −0.197726 −0.0988632 0.995101i \(-0.531521\pi\)
−0.0988632 + 0.995101i \(0.531521\pi\)
\(164\) 0.115433 0.00901377
\(165\) −0.398970 −0.0310598
\(166\) 2.51916 0.195524
\(167\) 9.92382 0.767928 0.383964 0.923348i \(-0.374559\pi\)
0.383964 + 0.923348i \(0.374559\pi\)
\(168\) −38.6858 −2.98468
\(169\) 0 0
\(170\) −19.1502 −1.46876
\(171\) −5.63127 −0.430634
\(172\) 0.224060 0.0170844
\(173\) 20.3536 1.54745 0.773726 0.633520i \(-0.218391\pi\)
0.773726 + 0.633520i \(0.218391\pi\)
\(174\) −28.2396 −2.14084
\(175\) 9.25199 0.699385
\(176\) −0.226201 −0.0170505
\(177\) 8.13429 0.611411
\(178\) −3.39869 −0.254742
\(179\) −4.85781 −0.363090 −0.181545 0.983383i \(-0.558110\pi\)
−0.181545 + 0.983383i \(0.558110\pi\)
\(180\) 0.928610 0.0692145
\(181\) −6.82673 −0.507427 −0.253713 0.967279i \(-0.581652\pi\)
−0.253713 + 0.967279i \(0.581652\pi\)
\(182\) 0 0
\(183\) −16.6469 −1.23058
\(184\) 21.6682 1.59740
\(185\) 17.6712 1.29921
\(186\) −4.02002 −0.294762
\(187\) 0.277390 0.0202848
\(188\) 0.0141775 0.00103400
\(189\) −24.9917 −1.81788
\(190\) 4.43006 0.321390
\(191\) −13.8071 −0.999043 −0.499522 0.866301i \(-0.666491\pi\)
−0.499522 + 0.866301i \(0.666491\pi\)
\(192\) −21.4444 −1.54762
\(193\) 8.92165 0.642194 0.321097 0.947046i \(-0.395948\pi\)
0.321097 + 0.947046i \(0.395948\pi\)
\(194\) −17.4773 −1.25479
\(195\) 0 0
\(196\) 1.33010 0.0950073
\(197\) 8.13765 0.579783 0.289892 0.957059i \(-0.406381\pi\)
0.289892 + 0.957059i \(0.406381\pi\)
\(198\) −0.376820 −0.0267794
\(199\) −12.4940 −0.885678 −0.442839 0.896601i \(-0.646029\pi\)
−0.442839 + 0.896601i \(0.646029\pi\)
\(200\) 5.13590 0.363163
\(201\) 4.19604 0.295966
\(202\) 16.7242 1.17671
\(203\) −35.0999 −2.46353
\(204\) −1.04984 −0.0735032
\(205\) 4.08126 0.285047
\(206\) −18.3242 −1.27671
\(207\) 37.4342 2.60186
\(208\) 0 0
\(209\) −0.0641691 −0.00443867
\(210\) 52.5776 3.62820
\(211\) −20.8611 −1.43614 −0.718068 0.695973i \(-0.754973\pi\)
−0.718068 + 0.695973i \(0.754973\pi\)
\(212\) 0.636121 0.0436890
\(213\) 4.02063 0.275489
\(214\) 4.25511 0.290873
\(215\) 7.92192 0.540270
\(216\) −13.8732 −0.943954
\(217\) −4.99660 −0.339192
\(218\) 28.6711 1.94185
\(219\) −38.9539 −2.63226
\(220\) 0.0105816 0.000713414 0
\(221\) 0 0
\(222\) 27.1391 1.82146
\(223\) −18.6577 −1.24941 −0.624707 0.780859i \(-0.714782\pi\)
−0.624707 + 0.780859i \(0.714782\pi\)
\(224\) 2.09152 0.139746
\(225\) 8.87285 0.591523
\(226\) −1.33381 −0.0887240
\(227\) −27.9915 −1.85786 −0.928932 0.370251i \(-0.879272\pi\)
−0.928932 + 0.370251i \(0.879272\pi\)
\(228\) 0.242860 0.0160838
\(229\) −9.20943 −0.608576 −0.304288 0.952580i \(-0.598419\pi\)
−0.304288 + 0.952580i \(0.598419\pi\)
\(230\) −29.4491 −1.94181
\(231\) −0.761584 −0.0501085
\(232\) −19.4844 −1.27921
\(233\) −19.5339 −1.27971 −0.639854 0.768497i \(-0.721005\pi\)
−0.639854 + 0.768497i \(0.721005\pi\)
\(234\) 0 0
\(235\) 0.501261 0.0326987
\(236\) −0.215741 −0.0140435
\(237\) −3.09338 −0.200937
\(238\) −36.5554 −2.36953
\(239\) −16.9297 −1.09509 −0.547546 0.836776i \(-0.684438\pi\)
−0.547546 + 0.836776i \(0.684438\pi\)
\(240\) 30.2684 1.95382
\(241\) 14.6063 0.940875 0.470438 0.882433i \(-0.344096\pi\)
0.470438 + 0.882433i \(0.344096\pi\)
\(242\) 15.8374 1.01806
\(243\) 16.1601 1.03667
\(244\) 0.441516 0.0282652
\(245\) 47.0273 3.00447
\(246\) 6.26793 0.399629
\(247\) 0 0
\(248\) −2.77368 −0.176129
\(249\) 4.88278 0.309434
\(250\) 11.8683 0.750616
\(251\) 28.9232 1.82561 0.912807 0.408391i \(-0.133910\pi\)
0.912807 + 0.408391i \(0.133910\pi\)
\(252\) 1.77260 0.111663
\(253\) 0.426568 0.0268181
\(254\) 23.5025 1.47468
\(255\) −37.1182 −2.32443
\(256\) 1.77442 0.110901
\(257\) 19.4544 1.21353 0.606767 0.794880i \(-0.292466\pi\)
0.606767 + 0.794880i \(0.292466\pi\)
\(258\) 12.1664 0.757445
\(259\) 33.7320 2.09601
\(260\) 0 0
\(261\) −33.6615 −2.08359
\(262\) −21.8456 −1.34962
\(263\) 20.2310 1.24750 0.623750 0.781624i \(-0.285608\pi\)
0.623750 + 0.781624i \(0.285608\pi\)
\(264\) −0.422765 −0.0260194
\(265\) 22.4908 1.38160
\(266\) 8.45642 0.518496
\(267\) −6.58754 −0.403151
\(268\) −0.111289 −0.00679805
\(269\) −8.30420 −0.506316 −0.253158 0.967425i \(-0.581469\pi\)
−0.253158 + 0.967425i \(0.581469\pi\)
\(270\) 18.8550 1.14748
\(271\) −14.7486 −0.895912 −0.447956 0.894056i \(-0.647848\pi\)
−0.447956 + 0.894056i \(0.647848\pi\)
\(272\) −21.0446 −1.27601
\(273\) 0 0
\(274\) 16.5441 0.999464
\(275\) 0.101107 0.00609700
\(276\) −1.61443 −0.0971772
\(277\) 15.4228 0.926664 0.463332 0.886185i \(-0.346654\pi\)
0.463332 + 0.886185i \(0.346654\pi\)
\(278\) 9.57400 0.574210
\(279\) −4.79185 −0.286880
\(280\) 36.2768 2.16795
\(281\) 17.2595 1.02962 0.514808 0.857306i \(-0.327863\pi\)
0.514808 + 0.857306i \(0.327863\pi\)
\(282\) 0.769829 0.0458427
\(283\) −12.2230 −0.726583 −0.363291 0.931676i \(-0.618347\pi\)
−0.363291 + 0.931676i \(0.618347\pi\)
\(284\) −0.106636 −0.00632771
\(285\) 8.58661 0.508627
\(286\) 0 0
\(287\) 7.79060 0.459865
\(288\) 2.00581 0.118194
\(289\) 8.80698 0.518057
\(290\) 26.4811 1.55502
\(291\) −33.8755 −1.98582
\(292\) 1.03315 0.0604605
\(293\) −11.5516 −0.674850 −0.337425 0.941352i \(-0.609556\pi\)
−0.337425 + 0.941352i \(0.609556\pi\)
\(294\) 72.2239 4.21218
\(295\) −7.62776 −0.444106
\(296\) 18.7251 1.08837
\(297\) −0.273114 −0.0158477
\(298\) −1.95560 −0.113285
\(299\) 0 0
\(300\) −0.382660 −0.0220929
\(301\) 15.1219 0.871614
\(302\) −17.9228 −1.03134
\(303\) 32.4158 1.86224
\(304\) 4.86828 0.279215
\(305\) 15.6103 0.893845
\(306\) −35.0574 −2.00410
\(307\) −22.1226 −1.26260 −0.631302 0.775537i \(-0.717479\pi\)
−0.631302 + 0.775537i \(0.717479\pi\)
\(308\) 0.0201990 0.00115095
\(309\) −35.5171 −2.02050
\(310\) 3.76969 0.214104
\(311\) 17.1273 0.971203 0.485601 0.874180i \(-0.338601\pi\)
0.485601 + 0.874180i \(0.338601\pi\)
\(312\) 0 0
\(313\) −28.9822 −1.63817 −0.819086 0.573670i \(-0.805519\pi\)
−0.819086 + 0.573670i \(0.805519\pi\)
\(314\) −22.4511 −1.26699
\(315\) 62.6723 3.53118
\(316\) 0.0820438 0.00461532
\(317\) 6.22426 0.349590 0.174795 0.984605i \(-0.444074\pi\)
0.174795 + 0.984605i \(0.444074\pi\)
\(318\) 34.5410 1.93697
\(319\) −0.383577 −0.0214762
\(320\) 20.1090 1.12413
\(321\) 8.24751 0.460331
\(322\) −56.2146 −3.13272
\(323\) −5.96997 −0.332178
\(324\) −0.0306309 −0.00170172
\(325\) 0 0
\(326\) 3.63552 0.201353
\(327\) 55.5720 3.07314
\(328\) 4.32467 0.238790
\(329\) 0.956844 0.0527525
\(330\) 0.574577 0.0316294
\(331\) 5.28437 0.290455 0.145228 0.989398i \(-0.453609\pi\)
0.145228 + 0.989398i \(0.453609\pi\)
\(332\) −0.129503 −0.00710740
\(333\) 32.3497 1.77275
\(334\) −14.2918 −0.782012
\(335\) −3.93475 −0.214978
\(336\) 57.7786 3.15208
\(337\) 7.23845 0.394304 0.197152 0.980373i \(-0.436831\pi\)
0.197152 + 0.980373i \(0.436831\pi\)
\(338\) 0 0
\(339\) −2.58528 −0.140413
\(340\) 0.984462 0.0533899
\(341\) −0.0546038 −0.00295696
\(342\) 8.10988 0.438532
\(343\) 54.7930 2.95855
\(344\) 8.39439 0.452595
\(345\) −57.0800 −3.07308
\(346\) −29.3122 −1.57583
\(347\) 33.0880 1.77626 0.888128 0.459597i \(-0.152006\pi\)
0.888128 + 0.459597i \(0.152006\pi\)
\(348\) 1.45172 0.0778204
\(349\) −6.94086 −0.371536 −0.185768 0.982594i \(-0.559477\pi\)
−0.185768 + 0.982594i \(0.559477\pi\)
\(350\) −13.3243 −0.712211
\(351\) 0 0
\(352\) 0.0228565 0.00121825
\(353\) 30.0813 1.60107 0.800534 0.599288i \(-0.204550\pi\)
0.800534 + 0.599288i \(0.204550\pi\)
\(354\) −11.7146 −0.622624
\(355\) −3.77026 −0.200105
\(356\) 0.174717 0.00925999
\(357\) −70.8539 −3.74999
\(358\) 6.99598 0.369749
\(359\) −3.73997 −0.197388 −0.0986940 0.995118i \(-0.531466\pi\)
−0.0986940 + 0.995118i \(0.531466\pi\)
\(360\) 34.7902 1.83361
\(361\) −17.6190 −0.927314
\(362\) 9.83152 0.516733
\(363\) 30.6969 1.61117
\(364\) 0 0
\(365\) 36.5282 1.91197
\(366\) 23.9741 1.25315
\(367\) 4.57778 0.238958 0.119479 0.992837i \(-0.461878\pi\)
0.119479 + 0.992837i \(0.461878\pi\)
\(368\) −32.3622 −1.68700
\(369\) 7.47135 0.388943
\(370\) −25.4491 −1.32304
\(371\) 42.9321 2.22892
\(372\) 0.206658 0.0107147
\(373\) −11.9473 −0.618607 −0.309303 0.950963i \(-0.600096\pi\)
−0.309303 + 0.950963i \(0.600096\pi\)
\(374\) −0.399484 −0.0206568
\(375\) 23.0038 1.18791
\(376\) 0.531157 0.0273923
\(377\) 0 0
\(378\) 35.9919 1.85122
\(379\) −0.620093 −0.0318521 −0.0159260 0.999873i \(-0.505070\pi\)
−0.0159260 + 0.999873i \(0.505070\pi\)
\(380\) −0.227737 −0.0116827
\(381\) 45.5539 2.33380
\(382\) 19.8842 1.01737
\(383\) −27.8945 −1.42534 −0.712671 0.701498i \(-0.752515\pi\)
−0.712671 + 0.701498i \(0.752515\pi\)
\(384\) 33.2201 1.69525
\(385\) 0.714160 0.0363969
\(386\) −12.8485 −0.653972
\(387\) 14.5023 0.737191
\(388\) 0.898459 0.0456123
\(389\) −12.8614 −0.652097 −0.326048 0.945353i \(-0.605717\pi\)
−0.326048 + 0.945353i \(0.605717\pi\)
\(390\) 0 0
\(391\) 39.6858 2.00699
\(392\) 49.8321 2.51690
\(393\) −42.3424 −2.13589
\(394\) −11.7194 −0.590417
\(395\) 2.90075 0.145953
\(396\) 0.0193713 0.000973443 0
\(397\) −24.6396 −1.23663 −0.618314 0.785931i \(-0.712184\pi\)
−0.618314 + 0.785931i \(0.712184\pi\)
\(398\) 17.9933 0.901922
\(399\) 16.3907 0.820564
\(400\) −7.67064 −0.383532
\(401\) 20.3020 1.01383 0.506917 0.861995i \(-0.330785\pi\)
0.506917 + 0.861995i \(0.330785\pi\)
\(402\) −6.04293 −0.301394
\(403\) 0 0
\(404\) −0.859744 −0.0427739
\(405\) −1.08299 −0.0538143
\(406\) 50.5491 2.50871
\(407\) 0.368629 0.0182723
\(408\) −39.3319 −1.94722
\(409\) 7.67595 0.379551 0.189776 0.981827i \(-0.439224\pi\)
0.189776 + 0.981827i \(0.439224\pi\)
\(410\) −5.87762 −0.290275
\(411\) 32.0667 1.58174
\(412\) 0.941997 0.0464089
\(413\) −14.5604 −0.716472
\(414\) −53.9110 −2.64958
\(415\) −4.57873 −0.224761
\(416\) 0 0
\(417\) 18.5569 0.908736
\(418\) 0.0924132 0.00452008
\(419\) −24.4074 −1.19238 −0.596189 0.802844i \(-0.703319\pi\)
−0.596189 + 0.802844i \(0.703319\pi\)
\(420\) −2.70287 −0.131887
\(421\) −7.83588 −0.381898 −0.190949 0.981600i \(-0.561156\pi\)
−0.190949 + 0.981600i \(0.561156\pi\)
\(422\) 30.0431 1.46248
\(423\) 0.917633 0.0446169
\(424\) 23.8322 1.15739
\(425\) 9.40651 0.456283
\(426\) −5.79031 −0.280541
\(427\) 29.7982 1.44203
\(428\) −0.218744 −0.0105734
\(429\) 0 0
\(430\) −11.4088 −0.550179
\(431\) −7.77841 −0.374673 −0.187337 0.982296i \(-0.559985\pi\)
−0.187337 + 0.982296i \(0.559985\pi\)
\(432\) 20.7202 0.996899
\(433\) −26.6655 −1.28146 −0.640732 0.767765i \(-0.721369\pi\)
−0.640732 + 0.767765i \(0.721369\pi\)
\(434\) 7.19586 0.345413
\(435\) 51.3273 2.46096
\(436\) −1.47390 −0.0705871
\(437\) −9.18057 −0.439166
\(438\) 56.0995 2.68054
\(439\) −14.4283 −0.688626 −0.344313 0.938855i \(-0.611888\pi\)
−0.344313 + 0.938855i \(0.611888\pi\)
\(440\) 0.396439 0.0188995
\(441\) 86.0906 4.09955
\(442\) 0 0
\(443\) 5.19341 0.246746 0.123373 0.992360i \(-0.460629\pi\)
0.123373 + 0.992360i \(0.460629\pi\)
\(444\) −1.39515 −0.0662108
\(445\) 6.17733 0.292834
\(446\) 26.8699 1.27233
\(447\) −3.79047 −0.179283
\(448\) 38.3856 1.81355
\(449\) 14.9729 0.706615 0.353308 0.935507i \(-0.385057\pi\)
0.353308 + 0.935507i \(0.385057\pi\)
\(450\) −12.7782 −0.602372
\(451\) 0.0851371 0.00400895
\(452\) 0.0685677 0.00322516
\(453\) −34.7391 −1.63218
\(454\) 40.3120 1.89194
\(455\) 0 0
\(456\) 9.09872 0.426087
\(457\) 42.0543 1.96722 0.983609 0.180316i \(-0.0577121\pi\)
0.983609 + 0.180316i \(0.0577121\pi\)
\(458\) 13.2630 0.619738
\(459\) −25.4091 −1.18600
\(460\) 1.51390 0.0705858
\(461\) 9.31370 0.433782 0.216891 0.976196i \(-0.430408\pi\)
0.216891 + 0.976196i \(0.430408\pi\)
\(462\) 1.09680 0.0510275
\(463\) 10.3203 0.479625 0.239813 0.970819i \(-0.422914\pi\)
0.239813 + 0.970819i \(0.422914\pi\)
\(464\) 29.1006 1.35096
\(465\) 7.30664 0.338838
\(466\) 28.1318 1.30318
\(467\) 0.661824 0.0306255 0.0153128 0.999883i \(-0.495126\pi\)
0.0153128 + 0.999883i \(0.495126\pi\)
\(468\) 0 0
\(469\) −7.51094 −0.346823
\(470\) −0.721891 −0.0332984
\(471\) −43.5161 −2.00512
\(472\) −8.08269 −0.372036
\(473\) 0.165255 0.00759844
\(474\) 4.45493 0.204622
\(475\) −2.17602 −0.0998428
\(476\) 1.87921 0.0861336
\(477\) 41.1728 1.88517
\(478\) 24.3813 1.11518
\(479\) −3.63440 −0.166060 −0.0830300 0.996547i \(-0.526460\pi\)
−0.0830300 + 0.996547i \(0.526460\pi\)
\(480\) −3.05847 −0.139600
\(481\) 0 0
\(482\) −21.0353 −0.958131
\(483\) −108.959 −4.95778
\(484\) −0.814156 −0.0370071
\(485\) 31.7661 1.44242
\(486\) −23.2730 −1.05569
\(487\) −31.1232 −1.41033 −0.705163 0.709045i \(-0.749126\pi\)
−0.705163 + 0.709045i \(0.749126\pi\)
\(488\) 16.5413 0.748792
\(489\) 7.04659 0.318658
\(490\) −67.7265 −3.05957
\(491\) −17.9607 −0.810555 −0.405277 0.914194i \(-0.632825\pi\)
−0.405277 + 0.914194i \(0.632825\pi\)
\(492\) −0.322217 −0.0145267
\(493\) −35.6861 −1.60722
\(494\) 0 0
\(495\) 0.684894 0.0307837
\(496\) 4.14259 0.186008
\(497\) −7.19695 −0.322827
\(498\) −7.03194 −0.315109
\(499\) 8.12403 0.363681 0.181841 0.983328i \(-0.441794\pi\)
0.181841 + 0.983328i \(0.441794\pi\)
\(500\) −0.610116 −0.0272852
\(501\) −27.7012 −1.23760
\(502\) −41.6537 −1.85910
\(503\) −38.6811 −1.72471 −0.862354 0.506307i \(-0.831010\pi\)
−0.862354 + 0.506307i \(0.831010\pi\)
\(504\) 66.4102 2.95814
\(505\) −30.3973 −1.35266
\(506\) −0.614323 −0.0273100
\(507\) 0 0
\(508\) −1.20820 −0.0536051
\(509\) −16.1272 −0.714826 −0.357413 0.933946i \(-0.616341\pi\)
−0.357413 + 0.933946i \(0.616341\pi\)
\(510\) 53.4558 2.36706
\(511\) 69.7278 3.08458
\(512\) 21.2464 0.938967
\(513\) 5.87794 0.259517
\(514\) −28.0173 −1.23579
\(515\) 33.3054 1.46761
\(516\) −0.625439 −0.0275334
\(517\) 0.0104566 0.000459879 0
\(518\) −48.5792 −2.13445
\(519\) −56.8147 −2.49389
\(520\) 0 0
\(521\) 5.58342 0.244614 0.122307 0.992492i \(-0.460971\pi\)
0.122307 + 0.992492i \(0.460971\pi\)
\(522\) 48.4776 2.12181
\(523\) −24.8140 −1.08504 −0.542520 0.840043i \(-0.682530\pi\)
−0.542520 + 0.840043i \(0.682530\pi\)
\(524\) 1.12302 0.0490594
\(525\) −25.8259 −1.12713
\(526\) −29.1358 −1.27038
\(527\) −5.08006 −0.221291
\(528\) 0.631414 0.0274788
\(529\) 38.0284 1.65341
\(530\) −32.3901 −1.40694
\(531\) −13.9638 −0.605976
\(532\) −0.434721 −0.0188476
\(533\) 0 0
\(534\) 9.48705 0.410545
\(535\) −7.73394 −0.334367
\(536\) −4.16942 −0.180092
\(537\) 13.5600 0.585159
\(538\) 11.9593 0.515602
\(539\) 0.981014 0.0422553
\(540\) −0.969285 −0.0417114
\(541\) 30.2928 1.30239 0.651195 0.758911i \(-0.274268\pi\)
0.651195 + 0.758911i \(0.274268\pi\)
\(542\) 21.2402 0.912344
\(543\) 19.0560 0.817773
\(544\) 2.12645 0.0911709
\(545\) −52.1115 −2.23221
\(546\) 0 0
\(547\) −0.782832 −0.0334715 −0.0167357 0.999860i \(-0.505327\pi\)
−0.0167357 + 0.999860i \(0.505327\pi\)
\(548\) −0.850485 −0.0363309
\(549\) 28.5770 1.21964
\(550\) −0.145610 −0.00620882
\(551\) 8.25532 0.351688
\(552\) −60.4843 −2.57438
\(553\) 5.53717 0.235465
\(554\) −22.2111 −0.943659
\(555\) −49.3271 −2.09382
\(556\) −0.492173 −0.0208728
\(557\) 12.1105 0.513140 0.256570 0.966526i \(-0.417408\pi\)
0.256570 + 0.966526i \(0.417408\pi\)
\(558\) 6.90098 0.292142
\(559\) 0 0
\(560\) −54.1807 −2.28955
\(561\) −0.774304 −0.0326911
\(562\) −24.8563 −1.04850
\(563\) −11.9218 −0.502446 −0.251223 0.967929i \(-0.580833\pi\)
−0.251223 + 0.967929i \(0.580833\pi\)
\(564\) −0.0395748 −0.00166640
\(565\) 2.42429 0.101991
\(566\) 17.6030 0.739908
\(567\) −2.06730 −0.0868183
\(568\) −3.99512 −0.167632
\(569\) 14.4513 0.605831 0.302916 0.953017i \(-0.402040\pi\)
0.302916 + 0.953017i \(0.402040\pi\)
\(570\) −12.3660 −0.517955
\(571\) 22.3002 0.933233 0.466616 0.884460i \(-0.345473\pi\)
0.466616 + 0.884460i \(0.345473\pi\)
\(572\) 0 0
\(573\) 38.5408 1.61007
\(574\) −11.2196 −0.468299
\(575\) 14.4653 0.603243
\(576\) 36.8126 1.53386
\(577\) −34.1283 −1.42078 −0.710390 0.703808i \(-0.751481\pi\)
−0.710390 + 0.703808i \(0.751481\pi\)
\(578\) −12.6834 −0.527559
\(579\) −24.9038 −1.03497
\(580\) −1.36132 −0.0565258
\(581\) −8.74022 −0.362605
\(582\) 48.7858 2.02224
\(583\) 0.469169 0.0194310
\(584\) 38.7068 1.60170
\(585\) 0 0
\(586\) 16.6360 0.687227
\(587\) −17.1675 −0.708579 −0.354290 0.935136i \(-0.615277\pi\)
−0.354290 + 0.935136i \(0.615277\pi\)
\(588\) −3.71283 −0.153115
\(589\) 1.17518 0.0484224
\(590\) 10.9851 0.452251
\(591\) −22.7153 −0.934384
\(592\) −27.9666 −1.14942
\(593\) 23.3023 0.956909 0.478455 0.878112i \(-0.341197\pi\)
0.478455 + 0.878112i \(0.341197\pi\)
\(594\) 0.393325 0.0161383
\(595\) 66.4418 2.72385
\(596\) 0.100532 0.00411796
\(597\) 34.8757 1.42737
\(598\) 0 0
\(599\) −12.7509 −0.520986 −0.260493 0.965476i \(-0.583885\pi\)
−0.260493 + 0.965476i \(0.583885\pi\)
\(600\) −14.3363 −0.585277
\(601\) 21.0313 0.857886 0.428943 0.903332i \(-0.358886\pi\)
0.428943 + 0.903332i \(0.358886\pi\)
\(602\) −21.7779 −0.887600
\(603\) −7.20315 −0.293335
\(604\) 0.921363 0.0374897
\(605\) −28.7854 −1.17029
\(606\) −46.6837 −1.89639
\(607\) −28.5285 −1.15793 −0.578967 0.815351i \(-0.696544\pi\)
−0.578967 + 0.815351i \(0.696544\pi\)
\(608\) −0.491915 −0.0199498
\(609\) 97.9773 3.97024
\(610\) −22.4812 −0.910238
\(611\) 0 0
\(612\) 1.80220 0.0728498
\(613\) 20.0076 0.808099 0.404050 0.914737i \(-0.367602\pi\)
0.404050 + 0.914737i \(0.367602\pi\)
\(614\) 31.8599 1.28576
\(615\) −11.3924 −0.459385
\(616\) 0.756753 0.0304904
\(617\) 5.04682 0.203177 0.101589 0.994826i \(-0.467607\pi\)
0.101589 + 0.994826i \(0.467607\pi\)
\(618\) 51.1500 2.05755
\(619\) 27.5782 1.10846 0.554231 0.832363i \(-0.313013\pi\)
0.554231 + 0.832363i \(0.313013\pi\)
\(620\) −0.193789 −0.00778277
\(621\) −39.0740 −1.56798
\(622\) −24.6660 −0.989015
\(623\) 11.7917 0.472426
\(624\) 0 0
\(625\) −30.8296 −1.23319
\(626\) 41.7388 1.66822
\(627\) 0.179121 0.00715340
\(628\) 1.15415 0.0460556
\(629\) 34.2954 1.36745
\(630\) −90.2576 −3.59595
\(631\) −46.3626 −1.84566 −0.922832 0.385202i \(-0.874132\pi\)
−0.922832 + 0.385202i \(0.874132\pi\)
\(632\) 3.07376 0.122268
\(633\) 58.2314 2.31449
\(634\) −8.96388 −0.356001
\(635\) −42.7172 −1.69518
\(636\) −1.77566 −0.0704095
\(637\) 0 0
\(638\) 0.552409 0.0218701
\(639\) −6.90202 −0.273040
\(640\) −31.1514 −1.23137
\(641\) −16.3261 −0.644843 −0.322422 0.946596i \(-0.604497\pi\)
−0.322422 + 0.946596i \(0.604497\pi\)
\(642\) −11.8777 −0.468774
\(643\) −6.80196 −0.268243 −0.134122 0.990965i \(-0.542821\pi\)
−0.134122 + 0.990965i \(0.542821\pi\)
\(644\) 2.88984 0.113876
\(645\) −22.1131 −0.870704
\(646\) 8.59766 0.338270
\(647\) 12.7863 0.502681 0.251340 0.967899i \(-0.419129\pi\)
0.251340 + 0.967899i \(0.419129\pi\)
\(648\) −1.14758 −0.0450813
\(649\) −0.159119 −0.00624597
\(650\) 0 0
\(651\) 13.9475 0.546644
\(652\) −0.186892 −0.00731926
\(653\) 31.3311 1.22608 0.613041 0.790051i \(-0.289946\pi\)
0.613041 + 0.790051i \(0.289946\pi\)
\(654\) −80.0321 −3.12950
\(655\) 39.7057 1.55143
\(656\) −6.45904 −0.252183
\(657\) 66.8704 2.60886
\(658\) −1.37800 −0.0537200
\(659\) −18.7431 −0.730127 −0.365064 0.930983i \(-0.618953\pi\)
−0.365064 + 0.930983i \(0.618953\pi\)
\(660\) −0.0295374 −0.00114974
\(661\) −13.7249 −0.533836 −0.266918 0.963719i \(-0.586005\pi\)
−0.266918 + 0.963719i \(0.586005\pi\)
\(662\) −7.61029 −0.295782
\(663\) 0 0
\(664\) −4.85181 −0.188287
\(665\) −15.3701 −0.596026
\(666\) −46.5885 −1.80527
\(667\) −54.8778 −2.12488
\(668\) 0.734702 0.0284265
\(669\) 52.0810 2.01357
\(670\) 5.66663 0.218921
\(671\) 0.325639 0.0125712
\(672\) −5.83824 −0.225215
\(673\) −15.0786 −0.581237 −0.290619 0.956839i \(-0.593861\pi\)
−0.290619 + 0.956839i \(0.593861\pi\)
\(674\) −10.4245 −0.401535
\(675\) −9.26150 −0.356475
\(676\) 0 0
\(677\) −26.8652 −1.03251 −0.516257 0.856434i \(-0.672675\pi\)
−0.516257 + 0.856434i \(0.672675\pi\)
\(678\) 3.72319 0.142988
\(679\) 60.6374 2.32705
\(680\) 36.8827 1.41439
\(681\) 78.1352 2.99415
\(682\) 0.0786376 0.00301119
\(683\) 10.6702 0.408285 0.204143 0.978941i \(-0.434559\pi\)
0.204143 + 0.978941i \(0.434559\pi\)
\(684\) −0.416907 −0.0159408
\(685\) −30.0699 −1.14891
\(686\) −78.9102 −3.01281
\(687\) 25.7071 0.980787
\(688\) −12.5373 −0.477980
\(689\) 0 0
\(690\) 82.2038 3.12945
\(691\) 0.654117 0.0248838 0.0124419 0.999923i \(-0.496040\pi\)
0.0124419 + 0.999923i \(0.496040\pi\)
\(692\) 1.50686 0.0572822
\(693\) 1.30738 0.0496631
\(694\) −47.6517 −1.80883
\(695\) −17.4014 −0.660071
\(696\) 54.3885 2.06159
\(697\) 7.92072 0.300019
\(698\) 9.99588 0.378350
\(699\) 54.5267 2.06239
\(700\) 0.684964 0.0258892
\(701\) −20.2045 −0.763112 −0.381556 0.924346i \(-0.624612\pi\)
−0.381556 + 0.924346i \(0.624612\pi\)
\(702\) 0 0
\(703\) −7.93361 −0.299222
\(704\) 0.419485 0.0158099
\(705\) −1.39921 −0.0526974
\(706\) −43.3216 −1.63043
\(707\) −58.0246 −2.18224
\(708\) 0.602216 0.0226327
\(709\) −48.7025 −1.82906 −0.914530 0.404518i \(-0.867439\pi\)
−0.914530 + 0.404518i \(0.867439\pi\)
\(710\) 5.42974 0.203775
\(711\) 5.31026 0.199150
\(712\) 6.54575 0.245313
\(713\) −7.81207 −0.292564
\(714\) 102.040 3.81876
\(715\) 0 0
\(716\) −0.359644 −0.0134405
\(717\) 47.2574 1.76486
\(718\) 5.38612 0.201008
\(719\) −16.7247 −0.623725 −0.311862 0.950127i \(-0.600953\pi\)
−0.311862 + 0.950127i \(0.600953\pi\)
\(720\) −51.9604 −1.93645
\(721\) 63.5758 2.36769
\(722\) 25.3740 0.944321
\(723\) −40.7719 −1.51632
\(724\) −0.505411 −0.0187835
\(725\) −13.0074 −0.483083
\(726\) −44.2082 −1.64072
\(727\) 20.1667 0.747940 0.373970 0.927441i \(-0.377996\pi\)
0.373970 + 0.927441i \(0.377996\pi\)
\(728\) 0 0
\(729\) −43.8680 −1.62474
\(730\) −52.6062 −1.94704
\(731\) 15.3745 0.568647
\(732\) −1.23244 −0.0455524
\(733\) −6.69141 −0.247153 −0.123576 0.992335i \(-0.539436\pi\)
−0.123576 + 0.992335i \(0.539436\pi\)
\(734\) −6.59270 −0.243341
\(735\) −131.272 −4.84203
\(736\) 3.27004 0.120535
\(737\) −0.0820809 −0.00302349
\(738\) −10.7599 −0.396076
\(739\) 14.7948 0.544235 0.272118 0.962264i \(-0.412276\pi\)
0.272118 + 0.962264i \(0.412276\pi\)
\(740\) 1.30827 0.0480930
\(741\) 0 0
\(742\) −61.8287 −2.26980
\(743\) −12.2435 −0.449170 −0.224585 0.974455i \(-0.572103\pi\)
−0.224585 + 0.974455i \(0.572103\pi\)
\(744\) 7.74242 0.283851
\(745\) 3.55444 0.130224
\(746\) 17.2059 0.629952
\(747\) −8.38205 −0.306683
\(748\) 0.0205364 0.000750884 0
\(749\) −14.7631 −0.539432
\(750\) −33.1290 −1.20970
\(751\) −22.4866 −0.820548 −0.410274 0.911962i \(-0.634567\pi\)
−0.410274 + 0.911962i \(0.634567\pi\)
\(752\) −0.793301 −0.0289287
\(753\) −80.7358 −2.94218
\(754\) 0 0
\(755\) 32.5759 1.18556
\(756\) −1.85024 −0.0672927
\(757\) 3.82393 0.138983 0.0694915 0.997583i \(-0.477862\pi\)
0.0694915 + 0.997583i \(0.477862\pi\)
\(758\) 0.893028 0.0324362
\(759\) −1.19072 −0.0432203
\(760\) −8.53214 −0.309493
\(761\) 33.7254 1.22254 0.611272 0.791420i \(-0.290658\pi\)
0.611272 + 0.791420i \(0.290658\pi\)
\(762\) −65.6045 −2.37660
\(763\) −99.4743 −3.60121
\(764\) −1.02219 −0.0369817
\(765\) 63.7190 2.30377
\(766\) 40.1723 1.45148
\(767\) 0 0
\(768\) −4.95308 −0.178729
\(769\) 12.5287 0.451795 0.225897 0.974151i \(-0.427469\pi\)
0.225897 + 0.974151i \(0.427469\pi\)
\(770\) −1.02850 −0.0370645
\(771\) −54.3048 −1.95574
\(772\) 0.660507 0.0237722
\(773\) 23.6248 0.849723 0.424862 0.905258i \(-0.360323\pi\)
0.424862 + 0.905258i \(0.360323\pi\)
\(774\) −20.8854 −0.750711
\(775\) −1.85165 −0.0665134
\(776\) 33.6606 1.20835
\(777\) −94.1592 −3.37794
\(778\) 18.5223 0.664056
\(779\) −1.83231 −0.0656494
\(780\) 0 0
\(781\) −0.0786495 −0.00281430
\(782\) −57.1535 −2.04380
\(783\) 35.1359 1.25566
\(784\) −74.4260 −2.65807
\(785\) 40.8063 1.45644
\(786\) 60.9794 2.17506
\(787\) 1.20685 0.0430194 0.0215097 0.999769i \(-0.493153\pi\)
0.0215097 + 0.999769i \(0.493153\pi\)
\(788\) 0.602464 0.0214619
\(789\) −56.4727 −2.01048
\(790\) −4.17752 −0.148630
\(791\) 4.62767 0.164541
\(792\) 0.725742 0.0257881
\(793\) 0 0
\(794\) 35.4848 1.25931
\(795\) −62.7806 −2.22660
\(796\) −0.924985 −0.0327852
\(797\) 28.1207 0.996085 0.498042 0.867153i \(-0.334052\pi\)
0.498042 + 0.867153i \(0.334052\pi\)
\(798\) −23.6051 −0.835613
\(799\) 0.972825 0.0344161
\(800\) 0.775081 0.0274032
\(801\) 11.3085 0.399567
\(802\) −29.2380 −1.03243
\(803\) 0.761997 0.0268903
\(804\) 0.310650 0.0109558
\(805\) 102.174 3.60115
\(806\) 0 0
\(807\) 23.1802 0.815983
\(808\) −32.2102 −1.13315
\(809\) −32.4182 −1.13976 −0.569881 0.821727i \(-0.693011\pi\)
−0.569881 + 0.821727i \(0.693011\pi\)
\(810\) 1.55967 0.0548013
\(811\) 15.1013 0.530279 0.265139 0.964210i \(-0.414582\pi\)
0.265139 + 0.964210i \(0.414582\pi\)
\(812\) −2.59859 −0.0911926
\(813\) 41.1690 1.44386
\(814\) −0.530882 −0.0186074
\(815\) −6.60779 −0.231461
\(816\) 58.7436 2.05644
\(817\) −3.55661 −0.124430
\(818\) −11.0545 −0.386513
\(819\) 0 0
\(820\) 0.302153 0.0105516
\(821\) 7.53751 0.263061 0.131531 0.991312i \(-0.458011\pi\)
0.131531 + 0.991312i \(0.458011\pi\)
\(822\) −46.1809 −1.61074
\(823\) 20.7098 0.721897 0.360949 0.932586i \(-0.382453\pi\)
0.360949 + 0.932586i \(0.382453\pi\)
\(824\) 35.2918 1.22945
\(825\) −0.282230 −0.00982598
\(826\) 20.9692 0.729613
\(827\) −22.6064 −0.786101 −0.393051 0.919517i \(-0.628580\pi\)
−0.393051 + 0.919517i \(0.628580\pi\)
\(828\) 2.77141 0.0963133
\(829\) −50.3014 −1.74704 −0.873520 0.486789i \(-0.838168\pi\)
−0.873520 + 0.486789i \(0.838168\pi\)
\(830\) 6.59406 0.228883
\(831\) −43.0509 −1.49342
\(832\) 0 0
\(833\) 91.2686 3.16227
\(834\) −26.7247 −0.925402
\(835\) 25.9763 0.898945
\(836\) −0.00475071 −0.000164307 0
\(837\) 5.00174 0.172885
\(838\) 35.1503 1.21425
\(839\) 27.1797 0.938346 0.469173 0.883106i \(-0.344552\pi\)
0.469173 + 0.883106i \(0.344552\pi\)
\(840\) −101.263 −3.49389
\(841\) 20.3470 0.701620
\(842\) 11.2849 0.388902
\(843\) −48.1779 −1.65934
\(844\) −1.54443 −0.0531616
\(845\) 0 0
\(846\) −1.32153 −0.0454351
\(847\) −54.9478 −1.88803
\(848\) −35.5942 −1.22231
\(849\) 34.1192 1.17097
\(850\) −13.5468 −0.464651
\(851\) 52.7392 1.80788
\(852\) 0.297664 0.0101978
\(853\) −7.69689 −0.263536 −0.131768 0.991281i \(-0.542065\pi\)
−0.131768 + 0.991281i \(0.542065\pi\)
\(854\) −42.9138 −1.46848
\(855\) −14.7402 −0.504105
\(856\) −8.19520 −0.280106
\(857\) 53.2084 1.81756 0.908782 0.417272i \(-0.137014\pi\)
0.908782 + 0.417272i \(0.137014\pi\)
\(858\) 0 0
\(859\) −28.0111 −0.955728 −0.477864 0.878434i \(-0.658589\pi\)
−0.477864 + 0.878434i \(0.658589\pi\)
\(860\) 0.586493 0.0199992
\(861\) −21.7466 −0.741122
\(862\) 11.2021 0.381545
\(863\) 19.3779 0.659630 0.329815 0.944046i \(-0.393014\pi\)
0.329815 + 0.944046i \(0.393014\pi\)
\(864\) −2.09367 −0.0712281
\(865\) 53.2768 1.81146
\(866\) 38.4024 1.30497
\(867\) −24.5837 −0.834906
\(868\) −0.369920 −0.0125559
\(869\) 0.0605112 0.00205270
\(870\) −73.9190 −2.50609
\(871\) 0 0
\(872\) −55.2195 −1.86997
\(873\) 58.1525 1.96817
\(874\) 13.2214 0.447221
\(875\) −41.1770 −1.39204
\(876\) −2.88392 −0.0974387
\(877\) 6.54713 0.221081 0.110540 0.993872i \(-0.464742\pi\)
0.110540 + 0.993872i \(0.464742\pi\)
\(878\) 20.7790 0.701256
\(879\) 32.2449 1.08759
\(880\) −0.592096 −0.0199595
\(881\) 50.1477 1.68952 0.844760 0.535145i \(-0.179743\pi\)
0.844760 + 0.535145i \(0.179743\pi\)
\(882\) −123.983 −4.17474
\(883\) 6.21470 0.209141 0.104571 0.994517i \(-0.466653\pi\)
0.104571 + 0.994517i \(0.466653\pi\)
\(884\) 0 0
\(885\) 21.2920 0.715724
\(886\) −7.47929 −0.251272
\(887\) −0.519213 −0.0174335 −0.00871673 0.999962i \(-0.502775\pi\)
−0.00871673 + 0.999962i \(0.502775\pi\)
\(888\) −52.2690 −1.75403
\(889\) −81.5418 −2.73482
\(890\) −8.89629 −0.298204
\(891\) −0.0225918 −0.000756853 0
\(892\) −1.38131 −0.0462497
\(893\) −0.225045 −0.00753085
\(894\) 5.45885 0.182571
\(895\) −12.7156 −0.425037
\(896\) −59.4641 −1.98656
\(897\) 0 0
\(898\) −21.5632 −0.719575
\(899\) 7.02474 0.234288
\(900\) 0.656894 0.0218965
\(901\) 43.6492 1.45416
\(902\) −0.122610 −0.00408247
\(903\) −42.2112 −1.40470
\(904\) 2.56888 0.0854397
\(905\) −17.8694 −0.593999
\(906\) 50.0295 1.66212
\(907\) 32.5497 1.08079 0.540397 0.841410i \(-0.318274\pi\)
0.540397 + 0.841410i \(0.318274\pi\)
\(908\) −2.07233 −0.0687727
\(909\) −55.6468 −1.84569
\(910\) 0 0
\(911\) 12.1736 0.403331 0.201665 0.979454i \(-0.435365\pi\)
0.201665 + 0.979454i \(0.435365\pi\)
\(912\) −13.5892 −0.449985
\(913\) −0.0955146 −0.00316107
\(914\) −60.5645 −2.00330
\(915\) −43.5745 −1.44053
\(916\) −0.681813 −0.0225277
\(917\) 75.7932 2.50291
\(918\) 36.5930 1.20775
\(919\) 33.3627 1.10053 0.550266 0.834989i \(-0.314526\pi\)
0.550266 + 0.834989i \(0.314526\pi\)
\(920\) 56.7179 1.86993
\(921\) 61.7528 2.03482
\(922\) −13.4131 −0.441738
\(923\) 0 0
\(924\) −0.0563833 −0.00185487
\(925\) 12.5005 0.411014
\(926\) −14.8628 −0.488422
\(927\) 60.9706 2.00254
\(928\) −2.94047 −0.0965258
\(929\) −4.29919 −0.141052 −0.0705259 0.997510i \(-0.522468\pi\)
−0.0705259 + 0.997510i \(0.522468\pi\)
\(930\) −10.5227 −0.345052
\(931\) −21.1133 −0.691961
\(932\) −1.44618 −0.0473711
\(933\) −47.8091 −1.56520
\(934\) −0.953126 −0.0311872
\(935\) 0.726087 0.0237456
\(936\) 0 0
\(937\) 48.4050 1.58132 0.790662 0.612253i \(-0.209737\pi\)
0.790662 + 0.612253i \(0.209737\pi\)
\(938\) 10.8169 0.353184
\(939\) 80.9006 2.64009
\(940\) 0.0371105 0.00121041
\(941\) −31.2320 −1.01813 −0.509067 0.860727i \(-0.670010\pi\)
−0.509067 + 0.860727i \(0.670010\pi\)
\(942\) 62.6698 2.04189
\(943\) 12.1804 0.396649
\(944\) 12.0718 0.392903
\(945\) −65.4175 −2.12803
\(946\) −0.237992 −0.00773780
\(947\) 27.4625 0.892410 0.446205 0.894931i \(-0.352775\pi\)
0.446205 + 0.894931i \(0.352775\pi\)
\(948\) −0.229016 −0.00743809
\(949\) 0 0
\(950\) 3.13380 0.101674
\(951\) −17.3743 −0.563401
\(952\) 70.4044 2.28182
\(953\) 39.5670 1.28170 0.640850 0.767666i \(-0.278582\pi\)
0.640850 + 0.767666i \(0.278582\pi\)
\(954\) −59.2950 −1.91975
\(955\) −36.1409 −1.16949
\(956\) −1.25338 −0.0405371
\(957\) 1.07071 0.0346112
\(958\) 5.23409 0.169106
\(959\) −57.3997 −1.85353
\(960\) −56.1322 −1.81166
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) −14.1581 −0.456239
\(964\) 1.08137 0.0348285
\(965\) 23.3530 0.751760
\(966\) 156.917 5.04871
\(967\) 0.618915 0.0199030 0.00995149 0.999950i \(-0.496832\pi\)
0.00995149 + 0.999950i \(0.496832\pi\)
\(968\) −30.5022 −0.980379
\(969\) 16.6645 0.535341
\(970\) −45.7479 −1.46888
\(971\) 9.35978 0.300370 0.150185 0.988658i \(-0.452013\pi\)
0.150185 + 0.988658i \(0.452013\pi\)
\(972\) 1.19640 0.0383746
\(973\) −33.2170 −1.06489
\(974\) 44.8221 1.43619
\(975\) 0 0
\(976\) −24.7051 −0.790790
\(977\) −21.1557 −0.676830 −0.338415 0.940997i \(-0.609891\pi\)
−0.338415 + 0.940997i \(0.609891\pi\)
\(978\) −10.1481 −0.324502
\(979\) 0.128862 0.00411846
\(980\) 3.48163 0.111217
\(981\) −95.3979 −3.04582
\(982\) 25.8661 0.825421
\(983\) −18.5493 −0.591630 −0.295815 0.955245i \(-0.595591\pi\)
−0.295815 + 0.955245i \(0.595591\pi\)
\(984\) −12.0718 −0.384836
\(985\) 21.3008 0.678701
\(986\) 51.3933 1.63670
\(987\) −2.67092 −0.0850164
\(988\) 0 0
\(989\) 23.6428 0.751797
\(990\) −0.986350 −0.0313483
\(991\) −32.6306 −1.03654 −0.518272 0.855216i \(-0.673425\pi\)
−0.518272 + 0.855216i \(0.673425\pi\)
\(992\) −0.418588 −0.0132902
\(993\) −14.7507 −0.468100
\(994\) 10.3647 0.328748
\(995\) −32.7039 −1.03678
\(996\) 0.361493 0.0114543
\(997\) −12.4748 −0.395082 −0.197541 0.980295i \(-0.563296\pi\)
−0.197541 + 0.980295i \(0.563296\pi\)
\(998\) −11.6998 −0.370351
\(999\) −33.7667 −1.06833
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 5239.2.a.l.1.4 16
13.5 odd 4 403.2.c.b.311.23 yes 32
13.8 odd 4 403.2.c.b.311.10 32
13.12 even 2 5239.2.a.k.1.13 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
403.2.c.b.311.10 32 13.8 odd 4
403.2.c.b.311.23 yes 32 13.5 odd 4
5239.2.a.k.1.13 16 13.12 even 2
5239.2.a.l.1.4 16 1.1 even 1 trivial