Properties

Label 5225.2.a.bc.1.21
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $0$
Dimension $30$
Inner twists $2$

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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: \(0\)
Dimension: \(30\)
Twist minimal: no (minimal twist has level 1045)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.21
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)
 
\(f(q)\) \(=\) \(q+1.45151 q^{2} -0.0791235 q^{3} +0.106883 q^{4} -0.114849 q^{6} +2.96150 q^{7} -2.74788 q^{8} -2.99374 q^{9} +O(q^{10})\) \(q+1.45151 q^{2} -0.0791235 q^{3} +0.106883 q^{4} -0.114849 q^{6} +2.96150 q^{7} -2.74788 q^{8} -2.99374 q^{9} +1.00000 q^{11} -0.00845698 q^{12} +3.94825 q^{13} +4.29865 q^{14} -4.20234 q^{16} +4.79134 q^{17} -4.34544 q^{18} +1.00000 q^{19} -0.234324 q^{21} +1.45151 q^{22} -5.69554 q^{23} +0.217422 q^{24} +5.73093 q^{26} +0.474246 q^{27} +0.316535 q^{28} +2.19295 q^{29} -8.56978 q^{31} -0.603987 q^{32} -0.0791235 q^{33} +6.95468 q^{34} -0.319981 q^{36} +6.70306 q^{37} +1.45151 q^{38} -0.312399 q^{39} +5.78933 q^{41} -0.340124 q^{42} +11.8680 q^{43} +0.106883 q^{44} -8.26713 q^{46} +3.44076 q^{47} +0.332504 q^{48} +1.77048 q^{49} -0.379107 q^{51} +0.422002 q^{52} -2.73870 q^{53} +0.688373 q^{54} -8.13784 q^{56} -0.0791235 q^{57} +3.18309 q^{58} -1.87690 q^{59} -13.7308 q^{61} -12.4391 q^{62} -8.86596 q^{63} +7.52799 q^{64} -0.114849 q^{66} +10.1502 q^{67} +0.512114 q^{68} +0.450651 q^{69} -5.59507 q^{71} +8.22643 q^{72} -3.73181 q^{73} +9.72957 q^{74} +0.106883 q^{76} +2.96150 q^{77} -0.453451 q^{78} +16.6160 q^{79} +8.94369 q^{81} +8.40327 q^{82} +9.90171 q^{83} -0.0250453 q^{84} +17.2266 q^{86} -0.173514 q^{87} -2.74788 q^{88} -1.30858 q^{89} +11.6927 q^{91} -0.608757 q^{92} +0.678071 q^{93} +4.99431 q^{94} +0.0477895 q^{96} +9.85288 q^{97} +2.56986 q^{98} -2.99374 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q + 42 q^{4} + 12 q^{6} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 30 q + 42 q^{4} + 12 q^{6} + 40 q^{9} + 30 q^{11} - 4 q^{14} + 66 q^{16} + 30 q^{19} + 14 q^{21} + 22 q^{24} + 30 q^{29} + 26 q^{31} + 12 q^{34} + 78 q^{36} + 64 q^{39} + 22 q^{41} + 42 q^{44} + 28 q^{46} + 60 q^{49} + 64 q^{51} + 62 q^{54} - 32 q^{56} - 14 q^{59} + 78 q^{61} + 90 q^{64} + 12 q^{66} - 28 q^{69} + 20 q^{71} + 42 q^{74} + 42 q^{76} + 102 q^{79} + 42 q^{81} + 98 q^{84} - 52 q^{86} - 8 q^{89} + 56 q^{91} + 40 q^{94} - 74 q^{96} + 40 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.45151 1.02637 0.513187 0.858277i \(-0.328465\pi\)
0.513187 + 0.858277i \(0.328465\pi\)
\(3\) −0.0791235 −0.0456820 −0.0228410 0.999739i \(-0.507271\pi\)
−0.0228410 + 0.999739i \(0.507271\pi\)
\(4\) 0.106883 0.0534416
\(5\) 0 0
\(6\) −0.114849 −0.0468868
\(7\) 2.96150 1.11934 0.559671 0.828715i \(-0.310928\pi\)
0.559671 + 0.828715i \(0.310928\pi\)
\(8\) −2.74788 −0.971522
\(9\) −2.99374 −0.997913
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) −0.00845698 −0.00244132
\(13\) 3.94825 1.09505 0.547524 0.836790i \(-0.315571\pi\)
0.547524 + 0.836790i \(0.315571\pi\)
\(14\) 4.29865 1.14886
\(15\) 0 0
\(16\) −4.20234 −1.05059
\(17\) 4.79134 1.16207 0.581035 0.813879i \(-0.302648\pi\)
0.581035 + 0.813879i \(0.302648\pi\)
\(18\) −4.34544 −1.02423
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −0.234324 −0.0511337
\(22\) 1.45151 0.309463
\(23\) −5.69554 −1.18760 −0.593801 0.804612i \(-0.702373\pi\)
−0.593801 + 0.804612i \(0.702373\pi\)
\(24\) 0.217422 0.0443811
\(25\) 0 0
\(26\) 5.73093 1.12393
\(27\) 0.474246 0.0912686
\(28\) 0.316535 0.0598194
\(29\) 2.19295 0.407221 0.203611 0.979052i \(-0.434732\pi\)
0.203611 + 0.979052i \(0.434732\pi\)
\(30\) 0 0
\(31\) −8.56978 −1.53918 −0.769589 0.638539i \(-0.779539\pi\)
−0.769589 + 0.638539i \(0.779539\pi\)
\(32\) −0.603987 −0.106771
\(33\) −0.0791235 −0.0137736
\(34\) 6.95468 1.19272
\(35\) 0 0
\(36\) −0.319981 −0.0533301
\(37\) 6.70306 1.10198 0.550988 0.834513i \(-0.314251\pi\)
0.550988 + 0.834513i \(0.314251\pi\)
\(38\) 1.45151 0.235466
\(39\) −0.312399 −0.0500239
\(40\) 0 0
\(41\) 5.78933 0.904141 0.452071 0.891982i \(-0.350686\pi\)
0.452071 + 0.891982i \(0.350686\pi\)
\(42\) −0.340124 −0.0524823
\(43\) 11.8680 1.80986 0.904930 0.425560i \(-0.139923\pi\)
0.904930 + 0.425560i \(0.139923\pi\)
\(44\) 0.106883 0.0161133
\(45\) 0 0
\(46\) −8.26713 −1.21892
\(47\) 3.44076 0.501887 0.250944 0.968002i \(-0.419259\pi\)
0.250944 + 0.968002i \(0.419259\pi\)
\(48\) 0.332504 0.0479928
\(49\) 1.77048 0.252925
\(50\) 0 0
\(51\) −0.379107 −0.0530857
\(52\) 0.422002 0.0585211
\(53\) −2.73870 −0.376190 −0.188095 0.982151i \(-0.560231\pi\)
−0.188095 + 0.982151i \(0.560231\pi\)
\(54\) 0.688373 0.0936757
\(55\) 0 0
\(56\) −8.13784 −1.08746
\(57\) −0.0791235 −0.0104802
\(58\) 3.18309 0.417961
\(59\) −1.87690 −0.244351 −0.122176 0.992508i \(-0.538987\pi\)
−0.122176 + 0.992508i \(0.538987\pi\)
\(60\) 0 0
\(61\) −13.7308 −1.75804 −0.879021 0.476782i \(-0.841803\pi\)
−0.879021 + 0.476782i \(0.841803\pi\)
\(62\) −12.4391 −1.57977
\(63\) −8.86596 −1.11701
\(64\) 7.52799 0.940999
\(65\) 0 0
\(66\) −0.114849 −0.0141369
\(67\) 10.1502 1.24005 0.620025 0.784582i \(-0.287123\pi\)
0.620025 + 0.784582i \(0.287123\pi\)
\(68\) 0.512114 0.0621029
\(69\) 0.450651 0.0542520
\(70\) 0 0
\(71\) −5.59507 −0.664013 −0.332006 0.943277i \(-0.607726\pi\)
−0.332006 + 0.943277i \(0.607726\pi\)
\(72\) 8.22643 0.969495
\(73\) −3.73181 −0.436775 −0.218388 0.975862i \(-0.570080\pi\)
−0.218388 + 0.975862i \(0.570080\pi\)
\(74\) 9.72957 1.13104
\(75\) 0 0
\(76\) 0.106883 0.0122603
\(77\) 2.96150 0.337494
\(78\) −0.453451 −0.0513432
\(79\) 16.6160 1.86944 0.934722 0.355379i \(-0.115648\pi\)
0.934722 + 0.355379i \(0.115648\pi\)
\(80\) 0 0
\(81\) 8.94369 0.993744
\(82\) 8.40327 0.927986
\(83\) 9.90171 1.08685 0.543427 0.839456i \(-0.317126\pi\)
0.543427 + 0.839456i \(0.317126\pi\)
\(84\) −0.0250453 −0.00273267
\(85\) 0 0
\(86\) 17.2266 1.85759
\(87\) −0.173514 −0.0186027
\(88\) −2.74788 −0.292925
\(89\) −1.30858 −0.138709 −0.0693546 0.997592i \(-0.522094\pi\)
−0.0693546 + 0.997592i \(0.522094\pi\)
\(90\) 0 0
\(91\) 11.6927 1.22573
\(92\) −0.608757 −0.0634674
\(93\) 0.678071 0.0703127
\(94\) 4.99431 0.515123
\(95\) 0 0
\(96\) 0.0477895 0.00487750
\(97\) 9.85288 1.00041 0.500204 0.865908i \(-0.333258\pi\)
0.500204 + 0.865908i \(0.333258\pi\)
\(98\) 2.56986 0.259595
\(99\) −2.99374 −0.300882
\(100\) 0 0
\(101\) 18.1171 1.80272 0.901358 0.433076i \(-0.142572\pi\)
0.901358 + 0.433076i \(0.142572\pi\)
\(102\) −0.550279 −0.0544857
\(103\) 14.2177 1.40091 0.700455 0.713696i \(-0.252980\pi\)
0.700455 + 0.713696i \(0.252980\pi\)
\(104\) −10.8493 −1.06386
\(105\) 0 0
\(106\) −3.97526 −0.386111
\(107\) −1.93848 −0.187400 −0.0937001 0.995600i \(-0.529869\pi\)
−0.0937001 + 0.995600i \(0.529869\pi\)
\(108\) 0.0506889 0.00487754
\(109\) −10.8606 −1.04026 −0.520130 0.854087i \(-0.674117\pi\)
−0.520130 + 0.854087i \(0.674117\pi\)
\(110\) 0 0
\(111\) −0.530370 −0.0503405
\(112\) −12.4452 −1.17596
\(113\) −16.1744 −1.52156 −0.760782 0.649008i \(-0.775184\pi\)
−0.760782 + 0.649008i \(0.775184\pi\)
\(114\) −0.114849 −0.0107566
\(115\) 0 0
\(116\) 0.234390 0.0217626
\(117\) −11.8200 −1.09276
\(118\) −2.72434 −0.250796
\(119\) 14.1895 1.30075
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −19.9303 −1.80441
\(123\) −0.458072 −0.0413030
\(124\) −0.915966 −0.0822562
\(125\) 0 0
\(126\) −12.8690 −1.14646
\(127\) −13.0331 −1.15650 −0.578251 0.815859i \(-0.696264\pi\)
−0.578251 + 0.815859i \(0.696264\pi\)
\(128\) 12.1349 1.07259
\(129\) −0.939042 −0.0826780
\(130\) 0 0
\(131\) 13.8516 1.21022 0.605111 0.796141i \(-0.293129\pi\)
0.605111 + 0.796141i \(0.293129\pi\)
\(132\) −0.00845698 −0.000736085 0
\(133\) 2.96150 0.256795
\(134\) 14.7332 1.27275
\(135\) 0 0
\(136\) −13.1660 −1.12898
\(137\) 7.87545 0.672845 0.336422 0.941711i \(-0.390783\pi\)
0.336422 + 0.941711i \(0.390783\pi\)
\(138\) 0.654125 0.0556828
\(139\) −18.7282 −1.58850 −0.794251 0.607590i \(-0.792136\pi\)
−0.794251 + 0.607590i \(0.792136\pi\)
\(140\) 0 0
\(141\) −0.272245 −0.0229272
\(142\) −8.12131 −0.681525
\(143\) 3.94825 0.330169
\(144\) 12.5807 1.04839
\(145\) 0 0
\(146\) −5.41676 −0.448294
\(147\) −0.140086 −0.0115541
\(148\) 0.716445 0.0588914
\(149\) −5.28231 −0.432744 −0.216372 0.976311i \(-0.569422\pi\)
−0.216372 + 0.976311i \(0.569422\pi\)
\(150\) 0 0
\(151\) 15.4741 1.25926 0.629631 0.776895i \(-0.283206\pi\)
0.629631 + 0.776895i \(0.283206\pi\)
\(152\) −2.74788 −0.222882
\(153\) −14.3440 −1.15964
\(154\) 4.29865 0.346395
\(155\) 0 0
\(156\) −0.0333903 −0.00267336
\(157\) 19.8551 1.58461 0.792303 0.610128i \(-0.208882\pi\)
0.792303 + 0.610128i \(0.208882\pi\)
\(158\) 24.1183 1.91875
\(159\) 0.216696 0.0171851
\(160\) 0 0
\(161\) −16.8673 −1.32933
\(162\) 12.9819 1.01995
\(163\) 0.0217138 0.00170076 0.000850380 1.00000i \(-0.499729\pi\)
0.000850380 1.00000i \(0.499729\pi\)
\(164\) 0.618782 0.0483188
\(165\) 0 0
\(166\) 14.3724 1.11552
\(167\) −1.69054 −0.130818 −0.0654088 0.997859i \(-0.520835\pi\)
−0.0654088 + 0.997859i \(0.520835\pi\)
\(168\) 0.643895 0.0496775
\(169\) 2.58868 0.199129
\(170\) 0 0
\(171\) −2.99374 −0.228937
\(172\) 1.26850 0.0967219
\(173\) 24.1483 1.83596 0.917982 0.396623i \(-0.129818\pi\)
0.917982 + 0.396623i \(0.129818\pi\)
\(174\) −0.251858 −0.0190933
\(175\) 0 0
\(176\) −4.20234 −0.316763
\(177\) 0.148507 0.0111625
\(178\) −1.89942 −0.142367
\(179\) 4.79370 0.358298 0.179149 0.983822i \(-0.442666\pi\)
0.179149 + 0.983822i \(0.442666\pi\)
\(180\) 0 0
\(181\) 14.6891 1.09183 0.545917 0.837839i \(-0.316181\pi\)
0.545917 + 0.837839i \(0.316181\pi\)
\(182\) 16.9721 1.25806
\(183\) 1.08643 0.0803109
\(184\) 15.6506 1.15378
\(185\) 0 0
\(186\) 0.984228 0.0721671
\(187\) 4.79134 0.350377
\(188\) 0.367760 0.0268217
\(189\) 1.40448 0.102161
\(190\) 0 0
\(191\) 25.2511 1.82711 0.913553 0.406719i \(-0.133327\pi\)
0.913553 + 0.406719i \(0.133327\pi\)
\(192\) −0.595641 −0.0429867
\(193\) −3.62179 −0.260702 −0.130351 0.991468i \(-0.541610\pi\)
−0.130351 + 0.991468i \(0.541610\pi\)
\(194\) 14.3016 1.02679
\(195\) 0 0
\(196\) 0.189234 0.0135167
\(197\) 7.51197 0.535206 0.267603 0.963529i \(-0.413768\pi\)
0.267603 + 0.963529i \(0.413768\pi\)
\(198\) −4.34544 −0.308817
\(199\) 3.89016 0.275766 0.137883 0.990448i \(-0.455970\pi\)
0.137883 + 0.990448i \(0.455970\pi\)
\(200\) 0 0
\(201\) −0.803123 −0.0566479
\(202\) 26.2971 1.85026
\(203\) 6.49443 0.455819
\(204\) −0.0405202 −0.00283698
\(205\) 0 0
\(206\) 20.6371 1.43786
\(207\) 17.0510 1.18512
\(208\) −16.5919 −1.15044
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 3.21646 0.221430 0.110715 0.993852i \(-0.464686\pi\)
0.110715 + 0.993852i \(0.464686\pi\)
\(212\) −0.292722 −0.0201042
\(213\) 0.442702 0.0303334
\(214\) −2.81373 −0.192343
\(215\) 0 0
\(216\) −1.30317 −0.0886695
\(217\) −25.3794 −1.72287
\(218\) −15.7643 −1.06770
\(219\) 0.295274 0.0199528
\(220\) 0 0
\(221\) 18.9174 1.27252
\(222\) −0.769838 −0.0516681
\(223\) 10.4914 0.702553 0.351277 0.936272i \(-0.385748\pi\)
0.351277 + 0.936272i \(0.385748\pi\)
\(224\) −1.78871 −0.119513
\(225\) 0 0
\(226\) −23.4774 −1.56169
\(227\) −0.798523 −0.0529999 −0.0264999 0.999649i \(-0.508436\pi\)
−0.0264999 + 0.999649i \(0.508436\pi\)
\(228\) −0.00845698 −0.000560077 0
\(229\) −16.3354 −1.07947 −0.539735 0.841835i \(-0.681476\pi\)
−0.539735 + 0.841835i \(0.681476\pi\)
\(230\) 0 0
\(231\) −0.234324 −0.0154174
\(232\) −6.02597 −0.395624
\(233\) −14.9185 −0.977340 −0.488670 0.872469i \(-0.662518\pi\)
−0.488670 + 0.872469i \(0.662518\pi\)
\(234\) −17.1569 −1.12158
\(235\) 0 0
\(236\) −0.200609 −0.0130585
\(237\) −1.31472 −0.0853999
\(238\) 20.5963 1.33506
\(239\) −23.5350 −1.52235 −0.761177 0.648544i \(-0.775378\pi\)
−0.761177 + 0.648544i \(0.775378\pi\)
\(240\) 0 0
\(241\) 24.7890 1.59680 0.798399 0.602129i \(-0.205681\pi\)
0.798399 + 0.602129i \(0.205681\pi\)
\(242\) 1.45151 0.0933066
\(243\) −2.13039 −0.136665
\(244\) −1.46759 −0.0939527
\(245\) 0 0
\(246\) −0.664896 −0.0423922
\(247\) 3.94825 0.251221
\(248\) 23.5487 1.49535
\(249\) −0.783458 −0.0496497
\(250\) 0 0
\(251\) −5.81152 −0.366820 −0.183410 0.983036i \(-0.558714\pi\)
−0.183410 + 0.983036i \(0.558714\pi\)
\(252\) −0.947622 −0.0596946
\(253\) −5.69554 −0.358075
\(254\) −18.9177 −1.18700
\(255\) 0 0
\(256\) 2.55800 0.159875
\(257\) −26.8322 −1.67375 −0.836873 0.547397i \(-0.815619\pi\)
−0.836873 + 0.547397i \(0.815619\pi\)
\(258\) −1.36303 −0.0848585
\(259\) 19.8511 1.23349
\(260\) 0 0
\(261\) −6.56513 −0.406371
\(262\) 20.1058 1.24214
\(263\) −24.5162 −1.51173 −0.755865 0.654727i \(-0.772784\pi\)
−0.755865 + 0.654727i \(0.772784\pi\)
\(264\) 0.217422 0.0133814
\(265\) 0 0
\(266\) 4.29865 0.263567
\(267\) 0.103539 0.00633651
\(268\) 1.08489 0.0662702
\(269\) 15.0720 0.918956 0.459478 0.888189i \(-0.348037\pi\)
0.459478 + 0.888189i \(0.348037\pi\)
\(270\) 0 0
\(271\) −18.3558 −1.11503 −0.557517 0.830165i \(-0.688246\pi\)
−0.557517 + 0.830165i \(0.688246\pi\)
\(272\) −20.1348 −1.22085
\(273\) −0.925171 −0.0559939
\(274\) 11.4313 0.690590
\(275\) 0 0
\(276\) 0.0481670 0.00289931
\(277\) −11.1338 −0.668964 −0.334482 0.942402i \(-0.608561\pi\)
−0.334482 + 0.942402i \(0.608561\pi\)
\(278\) −27.1841 −1.63040
\(279\) 25.6557 1.53597
\(280\) 0 0
\(281\) 3.26607 0.194837 0.0974187 0.995243i \(-0.468941\pi\)
0.0974187 + 0.995243i \(0.468941\pi\)
\(282\) −0.395167 −0.0235319
\(283\) 25.0622 1.48979 0.744897 0.667180i \(-0.232499\pi\)
0.744897 + 0.667180i \(0.232499\pi\)
\(284\) −0.598020 −0.0354859
\(285\) 0 0
\(286\) 5.73093 0.338877
\(287\) 17.1451 1.01204
\(288\) 1.80818 0.106548
\(289\) 5.95691 0.350407
\(290\) 0 0
\(291\) −0.779594 −0.0457006
\(292\) −0.398868 −0.0233420
\(293\) −21.9171 −1.28041 −0.640205 0.768204i \(-0.721151\pi\)
−0.640205 + 0.768204i \(0.721151\pi\)
\(294\) −0.203337 −0.0118588
\(295\) 0 0
\(296\) −18.4192 −1.07059
\(297\) 0.474246 0.0275185
\(298\) −7.66733 −0.444157
\(299\) −22.4874 −1.30048
\(300\) 0 0
\(301\) 35.1472 2.02585
\(302\) 22.4608 1.29247
\(303\) −1.43349 −0.0823516
\(304\) −4.20234 −0.241021
\(305\) 0 0
\(306\) −20.8205 −1.19023
\(307\) −16.0425 −0.915593 −0.457796 0.889057i \(-0.651361\pi\)
−0.457796 + 0.889057i \(0.651361\pi\)
\(308\) 0.316535 0.0180362
\(309\) −1.12495 −0.0639964
\(310\) 0 0
\(311\) 14.9546 0.847999 0.424000 0.905662i \(-0.360626\pi\)
0.424000 + 0.905662i \(0.360626\pi\)
\(312\) 0.858436 0.0485994
\(313\) 34.3903 1.94385 0.971927 0.235282i \(-0.0756014\pi\)
0.971927 + 0.235282i \(0.0756014\pi\)
\(314\) 28.8198 1.62640
\(315\) 0 0
\(316\) 1.77597 0.0999061
\(317\) −7.59079 −0.426341 −0.213171 0.977015i \(-0.568379\pi\)
−0.213171 + 0.977015i \(0.568379\pi\)
\(318\) 0.314536 0.0176383
\(319\) 2.19295 0.122782
\(320\) 0 0
\(321\) 0.153380 0.00856081
\(322\) −24.4831 −1.36439
\(323\) 4.79134 0.266597
\(324\) 0.955931 0.0531073
\(325\) 0 0
\(326\) 0.0315179 0.00174561
\(327\) 0.859332 0.0475212
\(328\) −15.9084 −0.878393
\(329\) 10.1898 0.561783
\(330\) 0 0
\(331\) −28.0896 −1.54395 −0.771973 0.635655i \(-0.780730\pi\)
−0.771973 + 0.635655i \(0.780730\pi\)
\(332\) 1.05833 0.0580832
\(333\) −20.0672 −1.09968
\(334\) −2.45383 −0.134268
\(335\) 0 0
\(336\) 0.984711 0.0537204
\(337\) −20.8306 −1.13472 −0.567358 0.823472i \(-0.692034\pi\)
−0.567358 + 0.823472i \(0.692034\pi\)
\(338\) 3.75750 0.204381
\(339\) 1.27978 0.0695081
\(340\) 0 0
\(341\) −8.56978 −0.464080
\(342\) −4.34544 −0.234975
\(343\) −15.4872 −0.836232
\(344\) −32.6120 −1.75832
\(345\) 0 0
\(346\) 35.0516 1.88438
\(347\) 14.9351 0.801760 0.400880 0.916130i \(-0.368704\pi\)
0.400880 + 0.916130i \(0.368704\pi\)
\(348\) −0.0185458 −0.000994157 0
\(349\) 36.0679 1.93067 0.965335 0.261013i \(-0.0840567\pi\)
0.965335 + 0.261013i \(0.0840567\pi\)
\(350\) 0 0
\(351\) 1.87244 0.0999435
\(352\) −0.603987 −0.0321926
\(353\) −28.0400 −1.49242 −0.746208 0.665713i \(-0.768128\pi\)
−0.746208 + 0.665713i \(0.768128\pi\)
\(354\) 0.215559 0.0114568
\(355\) 0 0
\(356\) −0.139865 −0.00741284
\(357\) −1.12273 −0.0594210
\(358\) 6.95811 0.367747
\(359\) 1.28132 0.0676256 0.0338128 0.999428i \(-0.489235\pi\)
0.0338128 + 0.999428i \(0.489235\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 21.3214 1.12063
\(363\) −0.0791235 −0.00415291
\(364\) 1.24976 0.0655051
\(365\) 0 0
\(366\) 1.57696 0.0824289
\(367\) −2.01037 −0.104940 −0.0524702 0.998622i \(-0.516709\pi\)
−0.0524702 + 0.998622i \(0.516709\pi\)
\(368\) 23.9346 1.24768
\(369\) −17.3317 −0.902254
\(370\) 0 0
\(371\) −8.11067 −0.421085
\(372\) 0.0724745 0.00375763
\(373\) 2.59241 0.134230 0.0671150 0.997745i \(-0.478621\pi\)
0.0671150 + 0.997745i \(0.478621\pi\)
\(374\) 6.95468 0.359618
\(375\) 0 0
\(376\) −9.45480 −0.487594
\(377\) 8.65833 0.445926
\(378\) 2.03862 0.104855
\(379\) 14.3543 0.737330 0.368665 0.929562i \(-0.379815\pi\)
0.368665 + 0.929562i \(0.379815\pi\)
\(380\) 0 0
\(381\) 1.03123 0.0528313
\(382\) 36.6523 1.87529
\(383\) 8.17879 0.417917 0.208958 0.977925i \(-0.432993\pi\)
0.208958 + 0.977925i \(0.432993\pi\)
\(384\) −0.960159 −0.0489979
\(385\) 0 0
\(386\) −5.25706 −0.267577
\(387\) −35.5298 −1.80608
\(388\) 1.05311 0.0534634
\(389\) −19.7560 −1.00167 −0.500833 0.865544i \(-0.666973\pi\)
−0.500833 + 0.865544i \(0.666973\pi\)
\(390\) 0 0
\(391\) −27.2892 −1.38008
\(392\) −4.86505 −0.245722
\(393\) −1.09599 −0.0552854
\(394\) 10.9037 0.549321
\(395\) 0 0
\(396\) −0.319981 −0.0160796
\(397\) 12.6658 0.635678 0.317839 0.948145i \(-0.397043\pi\)
0.317839 + 0.948145i \(0.397043\pi\)
\(398\) 5.64661 0.283039
\(399\) −0.234324 −0.0117309
\(400\) 0 0
\(401\) −36.6271 −1.82907 −0.914534 0.404509i \(-0.867442\pi\)
−0.914534 + 0.404509i \(0.867442\pi\)
\(402\) −1.16574 −0.0581419
\(403\) −33.8357 −1.68547
\(404\) 1.93641 0.0963400
\(405\) 0 0
\(406\) 9.42673 0.467841
\(407\) 6.70306 0.332258
\(408\) 1.04174 0.0515739
\(409\) −8.99494 −0.444771 −0.222386 0.974959i \(-0.571384\pi\)
−0.222386 + 0.974959i \(0.571384\pi\)
\(410\) 0 0
\(411\) −0.623133 −0.0307369
\(412\) 1.51963 0.0748669
\(413\) −5.55843 −0.273513
\(414\) 24.7496 1.21638
\(415\) 0 0
\(416\) −2.38469 −0.116919
\(417\) 1.48184 0.0725659
\(418\) 1.45151 0.0709957
\(419\) 27.2758 1.33251 0.666254 0.745725i \(-0.267897\pi\)
0.666254 + 0.745725i \(0.267897\pi\)
\(420\) 0 0
\(421\) 33.0585 1.61117 0.805587 0.592477i \(-0.201850\pi\)
0.805587 + 0.592477i \(0.201850\pi\)
\(422\) 4.66873 0.227270
\(423\) −10.3008 −0.500840
\(424\) 7.52563 0.365477
\(425\) 0 0
\(426\) 0.642586 0.0311334
\(427\) −40.6636 −1.96785
\(428\) −0.207191 −0.0100150
\(429\) −0.312399 −0.0150828
\(430\) 0 0
\(431\) −13.7567 −0.662635 −0.331317 0.943519i \(-0.607493\pi\)
−0.331317 + 0.943519i \(0.607493\pi\)
\(432\) −1.99294 −0.0958855
\(433\) −16.8098 −0.807828 −0.403914 0.914797i \(-0.632350\pi\)
−0.403914 + 0.914797i \(0.632350\pi\)
\(434\) −36.8385 −1.76830
\(435\) 0 0
\(436\) −1.16082 −0.0555932
\(437\) −5.69554 −0.272454
\(438\) 0.428593 0.0204790
\(439\) −6.24019 −0.297828 −0.148914 0.988850i \(-0.547578\pi\)
−0.148914 + 0.988850i \(0.547578\pi\)
\(440\) 0 0
\(441\) −5.30034 −0.252397
\(442\) 27.4588 1.30608
\(443\) −24.8505 −1.18068 −0.590341 0.807154i \(-0.701007\pi\)
−0.590341 + 0.807154i \(0.701007\pi\)
\(444\) −0.0566877 −0.00269028
\(445\) 0 0
\(446\) 15.2283 0.721081
\(447\) 0.417955 0.0197686
\(448\) 22.2941 1.05330
\(449\) −0.426567 −0.0201309 −0.0100655 0.999949i \(-0.503204\pi\)
−0.0100655 + 0.999949i \(0.503204\pi\)
\(450\) 0 0
\(451\) 5.78933 0.272609
\(452\) −1.72878 −0.0813148
\(453\) −1.22436 −0.0575255
\(454\) −1.15907 −0.0543976
\(455\) 0 0
\(456\) 0.217422 0.0101817
\(457\) −17.3831 −0.813146 −0.406573 0.913618i \(-0.633276\pi\)
−0.406573 + 0.913618i \(0.633276\pi\)
\(458\) −23.7109 −1.10794
\(459\) 2.27227 0.106061
\(460\) 0 0
\(461\) −24.8094 −1.15549 −0.577745 0.816218i \(-0.696067\pi\)
−0.577745 + 0.816218i \(0.696067\pi\)
\(462\) −0.340124 −0.0158240
\(463\) 7.55986 0.351337 0.175668 0.984449i \(-0.443791\pi\)
0.175668 + 0.984449i \(0.443791\pi\)
\(464\) −9.21554 −0.427821
\(465\) 0 0
\(466\) −21.6543 −1.00312
\(467\) −11.6238 −0.537884 −0.268942 0.963156i \(-0.586674\pi\)
−0.268942 + 0.963156i \(0.586674\pi\)
\(468\) −1.26336 −0.0583990
\(469\) 30.0599 1.38804
\(470\) 0 0
\(471\) −1.57100 −0.0723879
\(472\) 5.15749 0.237393
\(473\) 11.8680 0.545694
\(474\) −1.90832 −0.0876522
\(475\) 0 0
\(476\) 1.51662 0.0695143
\(477\) 8.19897 0.375405
\(478\) −34.1614 −1.56250
\(479\) 25.9454 1.18548 0.592739 0.805395i \(-0.298047\pi\)
0.592739 + 0.805395i \(0.298047\pi\)
\(480\) 0 0
\(481\) 26.4654 1.20672
\(482\) 35.9815 1.63891
\(483\) 1.33460 0.0607265
\(484\) 0.106883 0.00485833
\(485\) 0 0
\(486\) −3.09229 −0.140269
\(487\) −8.22122 −0.372539 −0.186270 0.982499i \(-0.559640\pi\)
−0.186270 + 0.982499i \(0.559640\pi\)
\(488\) 37.7305 1.70798
\(489\) −0.00171808 −7.76941e−5 0
\(490\) 0 0
\(491\) 8.68356 0.391884 0.195942 0.980616i \(-0.437224\pi\)
0.195942 + 0.980616i \(0.437224\pi\)
\(492\) −0.0489602 −0.00220730
\(493\) 10.5072 0.473219
\(494\) 5.73093 0.257847
\(495\) 0 0
\(496\) 36.0132 1.61704
\(497\) −16.5698 −0.743257
\(498\) −1.13720 −0.0509591
\(499\) −22.3175 −0.999068 −0.499534 0.866294i \(-0.666495\pi\)
−0.499534 + 0.866294i \(0.666495\pi\)
\(500\) 0 0
\(501\) 0.133761 0.00597601
\(502\) −8.43549 −0.376494
\(503\) −18.6767 −0.832754 −0.416377 0.909192i \(-0.636700\pi\)
−0.416377 + 0.909192i \(0.636700\pi\)
\(504\) 24.3626 1.08520
\(505\) 0 0
\(506\) −8.26713 −0.367519
\(507\) −0.204826 −0.00909662
\(508\) −1.39302 −0.0618053
\(509\) −27.7521 −1.23009 −0.615045 0.788492i \(-0.710862\pi\)
−0.615045 + 0.788492i \(0.710862\pi\)
\(510\) 0 0
\(511\) −11.0518 −0.488901
\(512\) −20.5569 −0.908495
\(513\) 0.474246 0.0209385
\(514\) −38.9472 −1.71789
\(515\) 0 0
\(516\) −0.100368 −0.00441845
\(517\) 3.44076 0.151325
\(518\) 28.8141 1.26602
\(519\) −1.91070 −0.0838704
\(520\) 0 0
\(521\) −10.3834 −0.454905 −0.227452 0.973789i \(-0.573040\pi\)
−0.227452 + 0.973789i \(0.573040\pi\)
\(522\) −9.52936 −0.417089
\(523\) 22.1998 0.970729 0.485365 0.874312i \(-0.338687\pi\)
0.485365 + 0.874312i \(0.338687\pi\)
\(524\) 1.48051 0.0646763
\(525\) 0 0
\(526\) −35.5855 −1.55160
\(527\) −41.0607 −1.78863
\(528\) 0.332504 0.0144704
\(529\) 9.43914 0.410398
\(530\) 0 0
\(531\) 5.61895 0.243842
\(532\) 0.316535 0.0137235
\(533\) 22.8577 0.990078
\(534\) 0.150289 0.00650362
\(535\) 0 0
\(536\) −27.8916 −1.20474
\(537\) −0.379294 −0.0163678
\(538\) 21.8772 0.943191
\(539\) 1.77048 0.0762598
\(540\) 0 0
\(541\) 37.8116 1.62565 0.812824 0.582510i \(-0.197929\pi\)
0.812824 + 0.582510i \(0.197929\pi\)
\(542\) −26.6436 −1.14444
\(543\) −1.16226 −0.0498772
\(544\) −2.89390 −0.124075
\(545\) 0 0
\(546\) −1.34290 −0.0574706
\(547\) 11.2780 0.482212 0.241106 0.970499i \(-0.422490\pi\)
0.241106 + 0.970499i \(0.422490\pi\)
\(548\) 0.841753 0.0359579
\(549\) 41.1063 1.75437
\(550\) 0 0
\(551\) 2.19295 0.0934229
\(552\) −1.23833 −0.0527070
\(553\) 49.2082 2.09255
\(554\) −16.1608 −0.686606
\(555\) 0 0
\(556\) −2.00173 −0.0848921
\(557\) −7.20468 −0.305272 −0.152636 0.988282i \(-0.548776\pi\)
−0.152636 + 0.988282i \(0.548776\pi\)
\(558\) 37.2395 1.57647
\(559\) 46.8580 1.98188
\(560\) 0 0
\(561\) −0.379107 −0.0160059
\(562\) 4.74073 0.199976
\(563\) 1.81101 0.0763248 0.0381624 0.999272i \(-0.487850\pi\)
0.0381624 + 0.999272i \(0.487850\pi\)
\(564\) −0.0290985 −0.00122527
\(565\) 0 0
\(566\) 36.3781 1.52908
\(567\) 26.4867 1.11234
\(568\) 15.3746 0.645103
\(569\) 12.4854 0.523415 0.261707 0.965147i \(-0.415714\pi\)
0.261707 + 0.965147i \(0.415714\pi\)
\(570\) 0 0
\(571\) −37.5830 −1.57280 −0.786400 0.617718i \(-0.788057\pi\)
−0.786400 + 0.617718i \(0.788057\pi\)
\(572\) 0.422002 0.0176448
\(573\) −1.99796 −0.0834659
\(574\) 24.8863 1.03873
\(575\) 0 0
\(576\) −22.5368 −0.939035
\(577\) −7.68994 −0.320136 −0.160068 0.987106i \(-0.551171\pi\)
−0.160068 + 0.987106i \(0.551171\pi\)
\(578\) 8.64652 0.359648
\(579\) 0.286568 0.0119094
\(580\) 0 0
\(581\) 29.3239 1.21656
\(582\) −1.13159 −0.0469059
\(583\) −2.73870 −0.113426
\(584\) 10.2546 0.424337
\(585\) 0 0
\(586\) −31.8129 −1.31418
\(587\) −8.55695 −0.353183 −0.176592 0.984284i \(-0.556507\pi\)
−0.176592 + 0.984284i \(0.556507\pi\)
\(588\) −0.0149729 −0.000617471 0
\(589\) −8.56978 −0.353112
\(590\) 0 0
\(591\) −0.594374 −0.0244493
\(592\) −28.1686 −1.15772
\(593\) −17.3812 −0.713761 −0.356881 0.934150i \(-0.616160\pi\)
−0.356881 + 0.934150i \(0.616160\pi\)
\(594\) 0.688373 0.0282443
\(595\) 0 0
\(596\) −0.564591 −0.0231265
\(597\) −0.307803 −0.0125976
\(598\) −32.6407 −1.33478
\(599\) −15.2377 −0.622594 −0.311297 0.950313i \(-0.600763\pi\)
−0.311297 + 0.950313i \(0.600763\pi\)
\(600\) 0 0
\(601\) 7.32582 0.298826 0.149413 0.988775i \(-0.452262\pi\)
0.149413 + 0.988775i \(0.452262\pi\)
\(602\) 51.0166 2.07928
\(603\) −30.3872 −1.23746
\(604\) 1.65392 0.0672970
\(605\) 0 0
\(606\) −2.08072 −0.0845235
\(607\) −46.8213 −1.90042 −0.950208 0.311615i \(-0.899130\pi\)
−0.950208 + 0.311615i \(0.899130\pi\)
\(608\) −0.603987 −0.0244949
\(609\) −0.513862 −0.0208227
\(610\) 0 0
\(611\) 13.5850 0.549590
\(612\) −1.53313 −0.0619733
\(613\) 0.581511 0.0234870 0.0117435 0.999931i \(-0.496262\pi\)
0.0117435 + 0.999931i \(0.496262\pi\)
\(614\) −23.2858 −0.939740
\(615\) 0 0
\(616\) −8.13784 −0.327883
\(617\) −31.3211 −1.26094 −0.630470 0.776214i \(-0.717138\pi\)
−0.630470 + 0.776214i \(0.717138\pi\)
\(618\) −1.63288 −0.0656841
\(619\) 4.35588 0.175078 0.0875388 0.996161i \(-0.472100\pi\)
0.0875388 + 0.996161i \(0.472100\pi\)
\(620\) 0 0
\(621\) −2.70108 −0.108391
\(622\) 21.7068 0.870364
\(623\) −3.87536 −0.155263
\(624\) 1.31281 0.0525544
\(625\) 0 0
\(626\) 49.9179 1.99512
\(627\) −0.0791235 −0.00315989
\(628\) 2.12217 0.0846839
\(629\) 32.1166 1.28057
\(630\) 0 0
\(631\) 9.65769 0.384467 0.192233 0.981349i \(-0.438427\pi\)
0.192233 + 0.981349i \(0.438427\pi\)
\(632\) −45.6587 −1.81621
\(633\) −0.254498 −0.0101154
\(634\) −11.0181 −0.437585
\(635\) 0 0
\(636\) 0.0231612 0.000918400 0
\(637\) 6.99028 0.276965
\(638\) 3.18309 0.126020
\(639\) 16.7502 0.662627
\(640\) 0 0
\(641\) 33.4025 1.31932 0.659661 0.751564i \(-0.270700\pi\)
0.659661 + 0.751564i \(0.270700\pi\)
\(642\) 0.222632 0.00878659
\(643\) −23.3959 −0.922646 −0.461323 0.887232i \(-0.652625\pi\)
−0.461323 + 0.887232i \(0.652625\pi\)
\(644\) −1.80283 −0.0710416
\(645\) 0 0
\(646\) 6.95468 0.273628
\(647\) 25.8707 1.01708 0.508540 0.861038i \(-0.330185\pi\)
0.508540 + 0.861038i \(0.330185\pi\)
\(648\) −24.5762 −0.965444
\(649\) −1.87690 −0.0736747
\(650\) 0 0
\(651\) 2.00811 0.0787039
\(652\) 0.00232085 9.08914e−5 0
\(653\) −9.99941 −0.391307 −0.195654 0.980673i \(-0.562683\pi\)
−0.195654 + 0.980673i \(0.562683\pi\)
\(654\) 1.24733 0.0487744
\(655\) 0 0
\(656\) −24.3287 −0.949878
\(657\) 11.1721 0.435864
\(658\) 14.7906 0.576599
\(659\) 15.8834 0.618729 0.309365 0.950944i \(-0.399884\pi\)
0.309365 + 0.950944i \(0.399884\pi\)
\(660\) 0 0
\(661\) 20.4289 0.794590 0.397295 0.917691i \(-0.369949\pi\)
0.397295 + 0.917691i \(0.369949\pi\)
\(662\) −40.7724 −1.58466
\(663\) −1.49681 −0.0581313
\(664\) −27.2087 −1.05590
\(665\) 0 0
\(666\) −29.1278 −1.12868
\(667\) −12.4900 −0.483616
\(668\) −0.180690 −0.00699111
\(669\) −0.830113 −0.0320940
\(670\) 0 0
\(671\) −13.7308 −0.530070
\(672\) 0.141529 0.00545959
\(673\) −8.43312 −0.325073 −0.162536 0.986703i \(-0.551968\pi\)
−0.162536 + 0.986703i \(0.551968\pi\)
\(674\) −30.2358 −1.16464
\(675\) 0 0
\(676\) 0.276687 0.0106418
\(677\) 0.570325 0.0219194 0.0109597 0.999940i \(-0.496511\pi\)
0.0109597 + 0.999940i \(0.496511\pi\)
\(678\) 1.85761 0.0713412
\(679\) 29.1793 1.11980
\(680\) 0 0
\(681\) 0.0631820 0.00242114
\(682\) −12.4391 −0.476319
\(683\) 15.1949 0.581415 0.290708 0.956812i \(-0.406109\pi\)
0.290708 + 0.956812i \(0.406109\pi\)
\(684\) −0.319981 −0.0122348
\(685\) 0 0
\(686\) −22.4799 −0.858286
\(687\) 1.29251 0.0493123
\(688\) −49.8736 −1.90141
\(689\) −10.8131 −0.411946
\(690\) 0 0
\(691\) 3.49109 0.132807 0.0664036 0.997793i \(-0.478848\pi\)
0.0664036 + 0.997793i \(0.478848\pi\)
\(692\) 2.58105 0.0981169
\(693\) −8.86596 −0.336790
\(694\) 21.6785 0.822905
\(695\) 0 0
\(696\) 0.476796 0.0180729
\(697\) 27.7386 1.05068
\(698\) 52.3529 1.98159
\(699\) 1.18040 0.0446468
\(700\) 0 0
\(701\) 2.17137 0.0820116 0.0410058 0.999159i \(-0.486944\pi\)
0.0410058 + 0.999159i \(0.486944\pi\)
\(702\) 2.71787 0.102579
\(703\) 6.70306 0.252811
\(704\) 7.52799 0.283722
\(705\) 0 0
\(706\) −40.7003 −1.53178
\(707\) 53.6537 2.01785
\(708\) 0.0158729 0.000596540 0
\(709\) −3.77213 −0.141665 −0.0708326 0.997488i \(-0.522566\pi\)
−0.0708326 + 0.997488i \(0.522566\pi\)
\(710\) 0 0
\(711\) −49.7439 −1.86554
\(712\) 3.59582 0.134759
\(713\) 48.8095 1.82793
\(714\) −1.62965 −0.0609881
\(715\) 0 0
\(716\) 0.512366 0.0191480
\(717\) 1.86217 0.0695442
\(718\) 1.85985 0.0694091
\(719\) −13.6203 −0.507952 −0.253976 0.967211i \(-0.581738\pi\)
−0.253976 + 0.967211i \(0.581738\pi\)
\(720\) 0 0
\(721\) 42.1057 1.56810
\(722\) 1.45151 0.0540196
\(723\) −1.96139 −0.0729449
\(724\) 1.57002 0.0583494
\(725\) 0 0
\(726\) −0.114849 −0.00426243
\(727\) −1.60527 −0.0595361 −0.0297681 0.999557i \(-0.509477\pi\)
−0.0297681 + 0.999557i \(0.509477\pi\)
\(728\) −32.1302 −1.19083
\(729\) −26.6625 −0.987501
\(730\) 0 0
\(731\) 56.8638 2.10318
\(732\) 0.116121 0.00429194
\(733\) −21.6583 −0.799968 −0.399984 0.916522i \(-0.630984\pi\)
−0.399984 + 0.916522i \(0.630984\pi\)
\(734\) −2.91807 −0.107708
\(735\) 0 0
\(736\) 3.44003 0.126801
\(737\) 10.1502 0.373889
\(738\) −25.1572 −0.926050
\(739\) −6.77955 −0.249390 −0.124695 0.992195i \(-0.539795\pi\)
−0.124695 + 0.992195i \(0.539795\pi\)
\(740\) 0 0
\(741\) −0.312399 −0.0114763
\(742\) −11.7727 −0.432190
\(743\) −26.6272 −0.976857 −0.488429 0.872604i \(-0.662430\pi\)
−0.488429 + 0.872604i \(0.662430\pi\)
\(744\) −1.86326 −0.0683104
\(745\) 0 0
\(746\) 3.76291 0.137770
\(747\) −29.6431 −1.08459
\(748\) 0.512114 0.0187247
\(749\) −5.74082 −0.209765
\(750\) 0 0
\(751\) 32.6441 1.19120 0.595600 0.803281i \(-0.296914\pi\)
0.595600 + 0.803281i \(0.296914\pi\)
\(752\) −14.4593 −0.527275
\(753\) 0.459828 0.0167571
\(754\) 12.5677 0.457687
\(755\) 0 0
\(756\) 0.150115 0.00545964
\(757\) 0.751499 0.0273137 0.0136569 0.999907i \(-0.495653\pi\)
0.0136569 + 0.999907i \(0.495653\pi\)
\(758\) 20.8354 0.756775
\(759\) 0.450651 0.0163576
\(760\) 0 0
\(761\) −17.5654 −0.636746 −0.318373 0.947966i \(-0.603136\pi\)
−0.318373 + 0.947966i \(0.603136\pi\)
\(762\) 1.49683 0.0542246
\(763\) −32.1638 −1.16441
\(764\) 2.69892 0.0976435
\(765\) 0 0
\(766\) 11.8716 0.428938
\(767\) −7.41047 −0.267576
\(768\) −0.202398 −0.00730341
\(769\) −35.7457 −1.28902 −0.644512 0.764595i \(-0.722939\pi\)
−0.644512 + 0.764595i \(0.722939\pi\)
\(770\) 0 0
\(771\) 2.12306 0.0764600
\(772\) −0.387108 −0.0139323
\(773\) 14.7295 0.529783 0.264891 0.964278i \(-0.414664\pi\)
0.264891 + 0.964278i \(0.414664\pi\)
\(774\) −51.5720 −1.85372
\(775\) 0 0
\(776\) −27.0745 −0.971919
\(777\) −1.57069 −0.0563482
\(778\) −28.6760 −1.02808
\(779\) 5.78933 0.207424
\(780\) 0 0
\(781\) −5.59507 −0.200207
\(782\) −39.6106 −1.41647
\(783\) 1.04000 0.0371665
\(784\) −7.44014 −0.265719
\(785\) 0 0
\(786\) −1.59084 −0.0567434
\(787\) 40.2297 1.43403 0.717016 0.697056i \(-0.245507\pi\)
0.717016 + 0.697056i \(0.245507\pi\)
\(788\) 0.802904 0.0286023
\(789\) 1.93980 0.0690589
\(790\) 0 0
\(791\) −47.9006 −1.70315
\(792\) 8.22643 0.292314
\(793\) −54.2125 −1.92514
\(794\) 18.3845 0.652442
\(795\) 0 0
\(796\) 0.415793 0.0147374
\(797\) −4.53756 −0.160729 −0.0803643 0.996766i \(-0.525608\pi\)
−0.0803643 + 0.996766i \(0.525608\pi\)
\(798\) −0.340124 −0.0120403
\(799\) 16.4859 0.583228
\(800\) 0 0
\(801\) 3.91755 0.138420
\(802\) −53.1646 −1.87731
\(803\) −3.73181 −0.131693
\(804\) −0.0858404 −0.00302736
\(805\) 0 0
\(806\) −49.1128 −1.72992
\(807\) −1.19255 −0.0419797
\(808\) −49.7835 −1.75138
\(809\) 1.86885 0.0657052 0.0328526 0.999460i \(-0.489541\pi\)
0.0328526 + 0.999460i \(0.489541\pi\)
\(810\) 0 0
\(811\) 26.5983 0.933994 0.466997 0.884259i \(-0.345336\pi\)
0.466997 + 0.884259i \(0.345336\pi\)
\(812\) 0.694145 0.0243597
\(813\) 1.45237 0.0509370
\(814\) 9.72957 0.341021
\(815\) 0 0
\(816\) 1.59314 0.0557710
\(817\) 11.8680 0.415211
\(818\) −13.0563 −0.456501
\(819\) −35.0050 −1.22317
\(820\) 0 0
\(821\) 11.4661 0.400171 0.200086 0.979778i \(-0.435878\pi\)
0.200086 + 0.979778i \(0.435878\pi\)
\(822\) −0.904484 −0.0315475
\(823\) −13.8917 −0.484233 −0.242117 0.970247i \(-0.577842\pi\)
−0.242117 + 0.970247i \(0.577842\pi\)
\(824\) −39.0685 −1.36102
\(825\) 0 0
\(826\) −8.06813 −0.280726
\(827\) −53.0943 −1.84627 −0.923134 0.384478i \(-0.874382\pi\)
−0.923134 + 0.384478i \(0.874382\pi\)
\(828\) 1.82246 0.0633349
\(829\) −0.910926 −0.0316378 −0.0158189 0.999875i \(-0.505036\pi\)
−0.0158189 + 0.999875i \(0.505036\pi\)
\(830\) 0 0
\(831\) 0.880944 0.0305596
\(832\) 29.7224 1.03044
\(833\) 8.48294 0.293917
\(834\) 2.15090 0.0744797
\(835\) 0 0
\(836\) 0.106883 0.00369663
\(837\) −4.06418 −0.140479
\(838\) 39.5911 1.36765
\(839\) −45.6602 −1.57636 −0.788182 0.615442i \(-0.788977\pi\)
−0.788182 + 0.615442i \(0.788977\pi\)
\(840\) 0 0
\(841\) −24.1910 −0.834171
\(842\) 47.9848 1.65367
\(843\) −0.258423 −0.00890056
\(844\) 0.343786 0.0118336
\(845\) 0 0
\(846\) −14.9517 −0.514048
\(847\) 2.96150 0.101758
\(848\) 11.5090 0.395220
\(849\) −1.98301 −0.0680567
\(850\) 0 0
\(851\) −38.1775 −1.30871
\(852\) 0.0473174 0.00162107
\(853\) 15.5961 0.534001 0.267000 0.963696i \(-0.413967\pi\)
0.267000 + 0.963696i \(0.413967\pi\)
\(854\) −59.0237 −2.01975
\(855\) 0 0
\(856\) 5.32672 0.182063
\(857\) 3.21516 0.109828 0.0549138 0.998491i \(-0.482512\pi\)
0.0549138 + 0.998491i \(0.482512\pi\)
\(858\) −0.453451 −0.0154806
\(859\) 20.1995 0.689198 0.344599 0.938750i \(-0.388015\pi\)
0.344599 + 0.938750i \(0.388015\pi\)
\(860\) 0 0
\(861\) −1.35658 −0.0462321
\(862\) −19.9679 −0.680111
\(863\) 3.58013 0.121869 0.0609345 0.998142i \(-0.480592\pi\)
0.0609345 + 0.998142i \(0.480592\pi\)
\(864\) −0.286438 −0.00974482
\(865\) 0 0
\(866\) −24.3996 −0.829133
\(867\) −0.471332 −0.0160073
\(868\) −2.71263 −0.0920728
\(869\) 16.6160 0.563659
\(870\) 0 0
\(871\) 40.0757 1.35791
\(872\) 29.8437 1.01064
\(873\) −29.4970 −0.998321
\(874\) −8.26713 −0.279640
\(875\) 0 0
\(876\) 0.0315598 0.00106631
\(877\) −1.12880 −0.0381169 −0.0190585 0.999818i \(-0.506067\pi\)
−0.0190585 + 0.999818i \(0.506067\pi\)
\(878\) −9.05771 −0.305683
\(879\) 1.73416 0.0584917
\(880\) 0 0
\(881\) −9.42205 −0.317437 −0.158719 0.987324i \(-0.550736\pi\)
−0.158719 + 0.987324i \(0.550736\pi\)
\(882\) −7.69350 −0.259054
\(883\) 33.9568 1.14274 0.571368 0.820694i \(-0.306413\pi\)
0.571368 + 0.820694i \(0.306413\pi\)
\(884\) 2.02195 0.0680056
\(885\) 0 0
\(886\) −36.0708 −1.21182
\(887\) −23.9547 −0.804321 −0.402161 0.915569i \(-0.631741\pi\)
−0.402161 + 0.915569i \(0.631741\pi\)
\(888\) 1.45739 0.0489069
\(889\) −38.5975 −1.29452
\(890\) 0 0
\(891\) 8.94369 0.299625
\(892\) 1.12135 0.0375456
\(893\) 3.44076 0.115141
\(894\) 0.606666 0.0202900
\(895\) 0 0
\(896\) 35.9376 1.20059
\(897\) 1.77928 0.0594085
\(898\) −0.619167 −0.0206619
\(899\) −18.7931 −0.626786
\(900\) 0 0
\(901\) −13.1221 −0.437159
\(902\) 8.40327 0.279798
\(903\) −2.78097 −0.0925449
\(904\) 44.4454 1.47823
\(905\) 0 0
\(906\) −1.77717 −0.0590427
\(907\) 20.1340 0.668538 0.334269 0.942478i \(-0.391511\pi\)
0.334269 + 0.942478i \(0.391511\pi\)
\(908\) −0.0853488 −0.00283240
\(909\) −54.2378 −1.79895
\(910\) 0 0
\(911\) −51.7375 −1.71414 −0.857070 0.515201i \(-0.827717\pi\)
−0.857070 + 0.515201i \(0.827717\pi\)
\(912\) 0.332504 0.0110103
\(913\) 9.90171 0.327699
\(914\) −25.2317 −0.834591
\(915\) 0 0
\(916\) −1.74598 −0.0576886
\(917\) 41.0216 1.35465
\(918\) 3.29823 0.108858
\(919\) 19.5840 0.646017 0.323009 0.946396i \(-0.395306\pi\)
0.323009 + 0.946396i \(0.395306\pi\)
\(920\) 0 0
\(921\) 1.26934 0.0418261
\(922\) −36.0111 −1.18596
\(923\) −22.0907 −0.727126
\(924\) −0.0250453 −0.000823931 0
\(925\) 0 0
\(926\) 10.9732 0.360602
\(927\) −42.5640 −1.39799
\(928\) −1.32451 −0.0434793
\(929\) −26.0616 −0.855055 −0.427527 0.904002i \(-0.640615\pi\)
−0.427527 + 0.904002i \(0.640615\pi\)
\(930\) 0 0
\(931\) 1.77048 0.0580250
\(932\) −1.59453 −0.0522307
\(933\) −1.18326 −0.0387383
\(934\) −16.8720 −0.552069
\(935\) 0 0
\(936\) 32.4800 1.06164
\(937\) 37.3846 1.22130 0.610650 0.791900i \(-0.290908\pi\)
0.610650 + 0.791900i \(0.290908\pi\)
\(938\) 43.6323 1.42465
\(939\) −2.72108 −0.0887991
\(940\) 0 0
\(941\) −33.0380 −1.07701 −0.538504 0.842623i \(-0.681010\pi\)
−0.538504 + 0.842623i \(0.681010\pi\)
\(942\) −2.28033 −0.0742970
\(943\) −32.9733 −1.07376
\(944\) 7.88737 0.256712
\(945\) 0 0
\(946\) 17.2266 0.560085
\(947\) 47.6863 1.54960 0.774799 0.632208i \(-0.217851\pi\)
0.774799 + 0.632208i \(0.217851\pi\)
\(948\) −0.140521 −0.00456391
\(949\) −14.7341 −0.478290
\(950\) 0 0
\(951\) 0.600610 0.0194761
\(952\) −38.9911 −1.26371
\(953\) 22.0635 0.714708 0.357354 0.933969i \(-0.383679\pi\)
0.357354 + 0.933969i \(0.383679\pi\)
\(954\) 11.9009 0.385306
\(955\) 0 0
\(956\) −2.51550 −0.0813571
\(957\) −0.173514 −0.00560892
\(958\) 37.6601 1.21674
\(959\) 23.3231 0.753143
\(960\) 0 0
\(961\) 42.4412 1.36907
\(962\) 38.4148 1.23854
\(963\) 5.80331 0.187009
\(964\) 2.64953 0.0853355
\(965\) 0 0
\(966\) 1.93719 0.0623280
\(967\) −53.0016 −1.70442 −0.852209 0.523201i \(-0.824738\pi\)
−0.852209 + 0.523201i \(0.824738\pi\)
\(968\) −2.74788 −0.0883202
\(969\) −0.379107 −0.0121787
\(970\) 0 0
\(971\) −10.0046 −0.321062 −0.160531 0.987031i \(-0.551321\pi\)
−0.160531 + 0.987031i \(0.551321\pi\)
\(972\) −0.227703 −0.00730359
\(973\) −55.4634 −1.77808
\(974\) −11.9332 −0.382364
\(975\) 0 0
\(976\) 57.7013 1.84697
\(977\) 0.508472 0.0162675 0.00813373 0.999967i \(-0.497411\pi\)
0.00813373 + 0.999967i \(0.497411\pi\)
\(978\) −0.00249381 −7.97431e−5 0
\(979\) −1.30858 −0.0418224
\(980\) 0 0
\(981\) 32.5139 1.03809
\(982\) 12.6043 0.402219
\(983\) 22.0873 0.704476 0.352238 0.935910i \(-0.385421\pi\)
0.352238 + 0.935910i \(0.385421\pi\)
\(984\) 1.25873 0.0401267
\(985\) 0 0
\(986\) 15.2513 0.485700
\(987\) −0.806254 −0.0256634
\(988\) 0.422002 0.0134257
\(989\) −67.5949 −2.14939
\(990\) 0 0
\(991\) −48.5368 −1.54182 −0.770911 0.636943i \(-0.780199\pi\)
−0.770911 + 0.636943i \(0.780199\pi\)
\(992\) 5.17603 0.164339
\(993\) 2.22255 0.0705305
\(994\) −24.0512 −0.762859
\(995\) 0 0
\(996\) −0.0837386 −0.00265336
\(997\) 20.4836 0.648723 0.324362 0.945933i \(-0.394851\pi\)
0.324362 + 0.945933i \(0.394851\pi\)
\(998\) −32.3941 −1.02542
\(999\) 3.17890 0.100576
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.bc.1.21 30
5.2 odd 4 1045.2.b.e.419.21 yes 30
5.3 odd 4 1045.2.b.e.419.10 30
5.4 even 2 inner 5225.2.a.bc.1.10 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.b.e.419.10 30 5.3 odd 4
1045.2.b.e.419.21 yes 30 5.2 odd 4
5225.2.a.bc.1.10 30 5.4 even 2 inner
5225.2.a.bc.1.21 30 1.1 even 1 trivial