Properties

Label 5225.2.a.t.1.14
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $0$
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: \(0\)
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: \( x^{15} - x^{14} - 21 x^{13} + 19 x^{12} + 170 x^{11} - 137 x^{10} - 669 x^{9} + 458 x^{8} + 1327 x^{7} + \cdots - 9 \) 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 1.14
Root \(-2.41798\) of defining polynomial
Character \(\chi\) \(=\) 5225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.41798 q^{2} -2.46128 q^{3} +3.84665 q^{4} -5.95133 q^{6} +1.10716 q^{7} +4.46516 q^{8} +3.05789 q^{9} +O(q^{10})\) \(q+2.41798 q^{2} -2.46128 q^{3} +3.84665 q^{4} -5.95133 q^{6} +1.10716 q^{7} +4.46516 q^{8} +3.05789 q^{9} -1.00000 q^{11} -9.46767 q^{12} -5.07159 q^{13} +2.67710 q^{14} +3.10340 q^{16} +2.37975 q^{17} +7.39392 q^{18} -1.00000 q^{19} -2.72503 q^{21} -2.41798 q^{22} +2.73783 q^{23} -10.9900 q^{24} -12.2630 q^{26} -0.142476 q^{27} +4.25886 q^{28} +9.42813 q^{29} -7.43254 q^{31} -1.42636 q^{32} +2.46128 q^{33} +5.75420 q^{34} +11.7626 q^{36} +7.39611 q^{37} -2.41798 q^{38} +12.4826 q^{39} +4.35090 q^{41} -6.58909 q^{42} +9.44261 q^{43} -3.84665 q^{44} +6.62002 q^{46} +12.5808 q^{47} -7.63833 q^{48} -5.77419 q^{49} -5.85723 q^{51} -19.5086 q^{52} +6.64927 q^{53} -0.344504 q^{54} +4.94366 q^{56} +2.46128 q^{57} +22.7971 q^{58} -3.55020 q^{59} -12.4902 q^{61} -17.9718 q^{62} +3.38558 q^{63} -9.65570 q^{64} +5.95133 q^{66} -1.62561 q^{67} +9.15406 q^{68} -6.73855 q^{69} +13.7547 q^{71} +13.6540 q^{72} -3.23967 q^{73} +17.8837 q^{74} -3.84665 q^{76} -1.10716 q^{77} +30.1827 q^{78} -0.514907 q^{79} -8.82299 q^{81} +10.5204 q^{82} +6.35171 q^{83} -10.4822 q^{84} +22.8321 q^{86} -23.2052 q^{87} -4.46516 q^{88} +14.6426 q^{89} -5.61508 q^{91} +10.5315 q^{92} +18.2935 q^{93} +30.4201 q^{94} +3.51066 q^{96} -8.43348 q^{97} -13.9619 q^{98} -3.05789 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - q^{2} + 4 q^{3} + 13 q^{4} - q^{6} + 11 q^{7} - 3 q^{8} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - q^{2} + 4 q^{3} + 13 q^{4} - q^{6} + 11 q^{7} - 3 q^{8} + 19 q^{9} - 15 q^{11} + 11 q^{12} + 3 q^{13} - 11 q^{14} + 13 q^{16} - 5 q^{17} + 12 q^{18} - 15 q^{19} + 10 q^{21} + q^{22} + 26 q^{23} - 11 q^{24} - 5 q^{26} + 19 q^{27} + 18 q^{28} + 7 q^{29} - 10 q^{31} - 12 q^{32} - 4 q^{33} + 17 q^{34} + 24 q^{36} + 31 q^{37} + q^{38} + 4 q^{39} + 2 q^{41} + 22 q^{42} + 26 q^{43} - 13 q^{44} - 23 q^{46} + 26 q^{47} + 46 q^{48} + 12 q^{49} + 12 q^{51} + 16 q^{52} + 21 q^{53} + 5 q^{54} - 10 q^{56} - 4 q^{57} + 34 q^{58} - 11 q^{59} + 20 q^{61} + 25 q^{62} + 27 q^{63} - 3 q^{64} + q^{66} + 41 q^{67} - 6 q^{68} + q^{69} + 25 q^{71} + 54 q^{72} - 6 q^{73} - 9 q^{74} - 13 q^{76} - 11 q^{77} + 28 q^{78} - 6 q^{79} + 43 q^{81} + 18 q^{82} - 20 q^{83} - 14 q^{84} + 35 q^{86} + 29 q^{87} + 3 q^{88} - 3 q^{89} + 30 q^{91} + 54 q^{92} + 2 q^{93} - 28 q^{94} - 61 q^{96} + 28 q^{97} - 2 q^{98} - 19 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.41798 1.70977 0.854886 0.518815i \(-0.173627\pi\)
0.854886 + 0.518815i \(0.173627\pi\)
\(3\) −2.46128 −1.42102 −0.710510 0.703687i \(-0.751536\pi\)
−0.710510 + 0.703687i \(0.751536\pi\)
\(4\) 3.84665 1.92332
\(5\) 0 0
\(6\) −5.95133 −2.42962
\(7\) 1.10716 0.418468 0.209234 0.977866i \(-0.432903\pi\)
0.209234 + 0.977866i \(0.432903\pi\)
\(8\) 4.46516 1.57867
\(9\) 3.05789 1.01930
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −9.46767 −2.73308
\(13\) −5.07159 −1.40661 −0.703303 0.710890i \(-0.748292\pi\)
−0.703303 + 0.710890i \(0.748292\pi\)
\(14\) 2.67710 0.715485
\(15\) 0 0
\(16\) 3.10340 0.775850
\(17\) 2.37975 0.577174 0.288587 0.957454i \(-0.406814\pi\)
0.288587 + 0.957454i \(0.406814\pi\)
\(18\) 7.39392 1.74276
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −2.72503 −0.594651
\(22\) −2.41798 −0.515516
\(23\) 2.73783 0.570876 0.285438 0.958397i \(-0.407861\pi\)
0.285438 + 0.958397i \(0.407861\pi\)
\(24\) −10.9900 −2.24333
\(25\) 0 0
\(26\) −12.2630 −2.40498
\(27\) −0.142476 −0.0274195
\(28\) 4.25886 0.804850
\(29\) 9.42813 1.75076 0.875380 0.483436i \(-0.160611\pi\)
0.875380 + 0.483436i \(0.160611\pi\)
\(30\) 0 0
\(31\) −7.43254 −1.33492 −0.667462 0.744644i \(-0.732619\pi\)
−0.667462 + 0.744644i \(0.732619\pi\)
\(32\) −1.42636 −0.252146
\(33\) 2.46128 0.428453
\(34\) 5.75420 0.986837
\(35\) 0 0
\(36\) 11.7626 1.96044
\(37\) 7.39611 1.21591 0.607956 0.793970i \(-0.291990\pi\)
0.607956 + 0.793970i \(0.291990\pi\)
\(38\) −2.41798 −0.392249
\(39\) 12.4826 1.99881
\(40\) 0 0
\(41\) 4.35090 0.679496 0.339748 0.940516i \(-0.389658\pi\)
0.339748 + 0.940516i \(0.389658\pi\)
\(42\) −6.58909 −1.01672
\(43\) 9.44261 1.43998 0.719992 0.693982i \(-0.244145\pi\)
0.719992 + 0.693982i \(0.244145\pi\)
\(44\) −3.84665 −0.579904
\(45\) 0 0
\(46\) 6.62002 0.976069
\(47\) 12.5808 1.83510 0.917548 0.397626i \(-0.130166\pi\)
0.917548 + 0.397626i \(0.130166\pi\)
\(48\) −7.63833 −1.10250
\(49\) −5.77419 −0.824884
\(50\) 0 0
\(51\) −5.85723 −0.820176
\(52\) −19.5086 −2.70536
\(53\) 6.64927 0.913347 0.456673 0.889634i \(-0.349041\pi\)
0.456673 + 0.889634i \(0.349041\pi\)
\(54\) −0.344504 −0.0468811
\(55\) 0 0
\(56\) 4.94366 0.660625
\(57\) 2.46128 0.326004
\(58\) 22.7971 2.99340
\(59\) −3.55020 −0.462197 −0.231098 0.972930i \(-0.574232\pi\)
−0.231098 + 0.972930i \(0.574232\pi\)
\(60\) 0 0
\(61\) −12.4902 −1.59921 −0.799603 0.600529i \(-0.794957\pi\)
−0.799603 + 0.600529i \(0.794957\pi\)
\(62\) −17.9718 −2.28242
\(63\) 3.38558 0.426543
\(64\) −9.65570 −1.20696
\(65\) 0 0
\(66\) 5.95133 0.732558
\(67\) −1.62561 −0.198600 −0.0993002 0.995058i \(-0.531660\pi\)
−0.0993002 + 0.995058i \(0.531660\pi\)
\(68\) 9.15406 1.11009
\(69\) −6.73855 −0.811226
\(70\) 0 0
\(71\) 13.7547 1.63238 0.816192 0.577780i \(-0.196081\pi\)
0.816192 + 0.577780i \(0.196081\pi\)
\(72\) 13.6540 1.60914
\(73\) −3.23967 −0.379175 −0.189587 0.981864i \(-0.560715\pi\)
−0.189587 + 0.981864i \(0.560715\pi\)
\(74\) 17.8837 2.07893
\(75\) 0 0
\(76\) −3.84665 −0.441241
\(77\) −1.10716 −0.126173
\(78\) 30.1827 3.41752
\(79\) −0.514907 −0.0579316 −0.0289658 0.999580i \(-0.509221\pi\)
−0.0289658 + 0.999580i \(0.509221\pi\)
\(80\) 0 0
\(81\) −8.82299 −0.980332
\(82\) 10.5204 1.16178
\(83\) 6.35171 0.697191 0.348596 0.937273i \(-0.386659\pi\)
0.348596 + 0.937273i \(0.386659\pi\)
\(84\) −10.4822 −1.14371
\(85\) 0 0
\(86\) 22.8321 2.46205
\(87\) −23.2052 −2.48786
\(88\) −4.46516 −0.475988
\(89\) 14.6426 1.55212 0.776058 0.630661i \(-0.217216\pi\)
0.776058 + 0.630661i \(0.217216\pi\)
\(90\) 0 0
\(91\) −5.61508 −0.588620
\(92\) 10.5315 1.09798
\(93\) 18.2935 1.89695
\(94\) 30.4201 3.13760
\(95\) 0 0
\(96\) 3.51066 0.358305
\(97\) −8.43348 −0.856290 −0.428145 0.903710i \(-0.640833\pi\)
−0.428145 + 0.903710i \(0.640833\pi\)
\(98\) −13.9619 −1.41037
\(99\) −3.05789 −0.307329
\(100\) 0 0
\(101\) 11.5040 1.14469 0.572345 0.820013i \(-0.306034\pi\)
0.572345 + 0.820013i \(0.306034\pi\)
\(102\) −14.1627 −1.40231
\(103\) 15.2206 1.49973 0.749864 0.661592i \(-0.230119\pi\)
0.749864 + 0.661592i \(0.230119\pi\)
\(104\) −22.6455 −2.22057
\(105\) 0 0
\(106\) 16.0778 1.56162
\(107\) −0.307450 −0.0297223 −0.0148612 0.999890i \(-0.504731\pi\)
−0.0148612 + 0.999890i \(0.504731\pi\)
\(108\) −0.548054 −0.0527365
\(109\) 9.76849 0.935652 0.467826 0.883821i \(-0.345037\pi\)
0.467826 + 0.883821i \(0.345037\pi\)
\(110\) 0 0
\(111\) −18.2039 −1.72784
\(112\) 3.43597 0.324668
\(113\) −3.57447 −0.336258 −0.168129 0.985765i \(-0.553772\pi\)
−0.168129 + 0.985765i \(0.553772\pi\)
\(114\) 5.95133 0.557393
\(115\) 0 0
\(116\) 36.2667 3.36728
\(117\) −15.5084 −1.43375
\(118\) −8.58433 −0.790252
\(119\) 2.63477 0.241529
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −30.2011 −2.73428
\(123\) −10.7088 −0.965577
\(124\) −28.5904 −2.56749
\(125\) 0 0
\(126\) 8.18627 0.729291
\(127\) 8.91806 0.791350 0.395675 0.918391i \(-0.370511\pi\)
0.395675 + 0.918391i \(0.370511\pi\)
\(128\) −20.4946 −1.81149
\(129\) −23.2409 −2.04625
\(130\) 0 0
\(131\) 11.8092 1.03178 0.515888 0.856656i \(-0.327462\pi\)
0.515888 + 0.856656i \(0.327462\pi\)
\(132\) 9.46767 0.824055
\(133\) −1.10716 −0.0960032
\(134\) −3.93071 −0.339562
\(135\) 0 0
\(136\) 10.6260 0.911170
\(137\) 8.26599 0.706211 0.353106 0.935583i \(-0.385126\pi\)
0.353106 + 0.935583i \(0.385126\pi\)
\(138\) −16.2937 −1.38701
\(139\) −16.8238 −1.42697 −0.713486 0.700669i \(-0.752885\pi\)
−0.713486 + 0.700669i \(0.752885\pi\)
\(140\) 0 0
\(141\) −30.9648 −2.60771
\(142\) 33.2587 2.79101
\(143\) 5.07159 0.424108
\(144\) 9.48984 0.790820
\(145\) 0 0
\(146\) −7.83347 −0.648303
\(147\) 14.2119 1.17218
\(148\) 28.4502 2.33859
\(149\) −15.5030 −1.27005 −0.635027 0.772490i \(-0.719011\pi\)
−0.635027 + 0.772490i \(0.719011\pi\)
\(150\) 0 0
\(151\) −2.69818 −0.219575 −0.109788 0.993955i \(-0.535017\pi\)
−0.109788 + 0.993955i \(0.535017\pi\)
\(152\) −4.46516 −0.362173
\(153\) 7.27701 0.588311
\(154\) −2.67710 −0.215727
\(155\) 0 0
\(156\) 48.0161 3.84437
\(157\) 2.34539 0.187182 0.0935911 0.995611i \(-0.470165\pi\)
0.0935911 + 0.995611i \(0.470165\pi\)
\(158\) −1.24504 −0.0990499
\(159\) −16.3657 −1.29788
\(160\) 0 0
\(161\) 3.03122 0.238894
\(162\) −21.3338 −1.67615
\(163\) −6.14837 −0.481577 −0.240789 0.970578i \(-0.577406\pi\)
−0.240789 + 0.970578i \(0.577406\pi\)
\(164\) 16.7364 1.30689
\(165\) 0 0
\(166\) 15.3583 1.19204
\(167\) 8.21512 0.635705 0.317852 0.948140i \(-0.397038\pi\)
0.317852 + 0.948140i \(0.397038\pi\)
\(168\) −12.1677 −0.938760
\(169\) 12.7210 0.978542
\(170\) 0 0
\(171\) −3.05789 −0.233842
\(172\) 36.3224 2.76956
\(173\) −13.6117 −1.03488 −0.517440 0.855720i \(-0.673115\pi\)
−0.517440 + 0.855720i \(0.673115\pi\)
\(174\) −56.1099 −4.25368
\(175\) 0 0
\(176\) −3.10340 −0.233928
\(177\) 8.73803 0.656791
\(178\) 35.4057 2.65377
\(179\) −7.09103 −0.530009 −0.265004 0.964247i \(-0.585373\pi\)
−0.265004 + 0.964247i \(0.585373\pi\)
\(180\) 0 0
\(181\) −7.79652 −0.579511 −0.289755 0.957101i \(-0.593574\pi\)
−0.289755 + 0.957101i \(0.593574\pi\)
\(182\) −13.5772 −1.00641
\(183\) 30.7419 2.27250
\(184\) 12.2248 0.901227
\(185\) 0 0
\(186\) 44.2335 3.24336
\(187\) −2.37975 −0.174025
\(188\) 48.3938 3.52948
\(189\) −0.157744 −0.0114742
\(190\) 0 0
\(191\) −5.07779 −0.367416 −0.183708 0.982981i \(-0.558810\pi\)
−0.183708 + 0.982981i \(0.558810\pi\)
\(192\) 23.7654 1.71512
\(193\) 7.70684 0.554751 0.277375 0.960762i \(-0.410535\pi\)
0.277375 + 0.960762i \(0.410535\pi\)
\(194\) −20.3920 −1.46406
\(195\) 0 0
\(196\) −22.2113 −1.58652
\(197\) −1.27143 −0.0905858 −0.0452929 0.998974i \(-0.514422\pi\)
−0.0452929 + 0.998974i \(0.514422\pi\)
\(198\) −7.39392 −0.525463
\(199\) −7.28878 −0.516688 −0.258344 0.966053i \(-0.583177\pi\)
−0.258344 + 0.966053i \(0.583177\pi\)
\(200\) 0 0
\(201\) 4.00109 0.282215
\(202\) 27.8165 1.95716
\(203\) 10.4385 0.732637
\(204\) −22.5307 −1.57746
\(205\) 0 0
\(206\) 36.8031 2.56419
\(207\) 8.37196 0.581892
\(208\) −15.7392 −1.09132
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 9.45015 0.650575 0.325287 0.945615i \(-0.394539\pi\)
0.325287 + 0.945615i \(0.394539\pi\)
\(212\) 25.5774 1.75666
\(213\) −33.8542 −2.31965
\(214\) −0.743409 −0.0508184
\(215\) 0 0
\(216\) −0.636177 −0.0432864
\(217\) −8.22903 −0.558623
\(218\) 23.6201 1.59975
\(219\) 7.97373 0.538815
\(220\) 0 0
\(221\) −12.0691 −0.811857
\(222\) −44.0167 −2.95421
\(223\) −7.06703 −0.473244 −0.236622 0.971602i \(-0.576040\pi\)
−0.236622 + 0.971602i \(0.576040\pi\)
\(224\) −1.57921 −0.105515
\(225\) 0 0
\(226\) −8.64301 −0.574924
\(227\) 18.6126 1.23536 0.617679 0.786430i \(-0.288073\pi\)
0.617679 + 0.786430i \(0.288073\pi\)
\(228\) 9.46767 0.627011
\(229\) 14.9571 0.988390 0.494195 0.869351i \(-0.335463\pi\)
0.494195 + 0.869351i \(0.335463\pi\)
\(230\) 0 0
\(231\) 2.72503 0.179294
\(232\) 42.0981 2.76388
\(233\) 16.2468 1.06436 0.532181 0.846631i \(-0.321373\pi\)
0.532181 + 0.846631i \(0.321373\pi\)
\(234\) −37.4990 −2.45138
\(235\) 0 0
\(236\) −13.6564 −0.888954
\(237\) 1.26733 0.0823219
\(238\) 6.37083 0.412960
\(239\) 27.4232 1.77386 0.886931 0.461903i \(-0.152833\pi\)
0.886931 + 0.461903i \(0.152833\pi\)
\(240\) 0 0
\(241\) 23.8678 1.53746 0.768731 0.639572i \(-0.220888\pi\)
0.768731 + 0.639572i \(0.220888\pi\)
\(242\) 2.41798 0.155434
\(243\) 22.1432 1.42049
\(244\) −48.0454 −3.07579
\(245\) 0 0
\(246\) −25.8936 −1.65092
\(247\) 5.07159 0.322698
\(248\) −33.1875 −2.10741
\(249\) −15.6333 −0.990722
\(250\) 0 0
\(251\) −5.64308 −0.356188 −0.178094 0.984013i \(-0.556993\pi\)
−0.178094 + 0.984013i \(0.556993\pi\)
\(252\) 13.0231 0.820380
\(253\) −2.73783 −0.172126
\(254\) 21.5637 1.35303
\(255\) 0 0
\(256\) −30.2443 −1.89027
\(257\) 27.8036 1.73434 0.867170 0.498012i \(-0.165936\pi\)
0.867170 + 0.498012i \(0.165936\pi\)
\(258\) −56.1961 −3.49862
\(259\) 8.18869 0.508821
\(260\) 0 0
\(261\) 28.8301 1.78454
\(262\) 28.5545 1.76410
\(263\) 17.2368 1.06287 0.531435 0.847099i \(-0.321653\pi\)
0.531435 + 0.847099i \(0.321653\pi\)
\(264\) 10.9900 0.676388
\(265\) 0 0
\(266\) −2.67710 −0.164144
\(267\) −36.0396 −2.20559
\(268\) −6.25316 −0.381973
\(269\) 25.2689 1.54067 0.770337 0.637637i \(-0.220088\pi\)
0.770337 + 0.637637i \(0.220088\pi\)
\(270\) 0 0
\(271\) −31.7545 −1.92895 −0.964474 0.264179i \(-0.914899\pi\)
−0.964474 + 0.264179i \(0.914899\pi\)
\(272\) 7.38532 0.447801
\(273\) 13.8203 0.836440
\(274\) 19.9870 1.20746
\(275\) 0 0
\(276\) −25.9208 −1.56025
\(277\) −14.9451 −0.897965 −0.448982 0.893541i \(-0.648213\pi\)
−0.448982 + 0.893541i \(0.648213\pi\)
\(278\) −40.6796 −2.43980
\(279\) −22.7279 −1.36068
\(280\) 0 0
\(281\) −4.01194 −0.239332 −0.119666 0.992814i \(-0.538182\pi\)
−0.119666 + 0.992814i \(0.538182\pi\)
\(282\) −74.8724 −4.45858
\(283\) 1.98533 0.118015 0.0590077 0.998258i \(-0.481206\pi\)
0.0590077 + 0.998258i \(0.481206\pi\)
\(284\) 52.9095 3.13960
\(285\) 0 0
\(286\) 12.2630 0.725128
\(287\) 4.81715 0.284348
\(288\) −4.36163 −0.257012
\(289\) −11.3368 −0.666870
\(290\) 0 0
\(291\) 20.7571 1.21680
\(292\) −12.4619 −0.729276
\(293\) −24.8700 −1.45292 −0.726459 0.687210i \(-0.758835\pi\)
−0.726459 + 0.687210i \(0.758835\pi\)
\(294\) 34.3641 2.00416
\(295\) 0 0
\(296\) 33.0248 1.91953
\(297\) 0.142476 0.00826728
\(298\) −37.4860 −2.17150
\(299\) −13.8851 −0.802998
\(300\) 0 0
\(301\) 10.4545 0.602588
\(302\) −6.52416 −0.375423
\(303\) −28.3145 −1.62663
\(304\) −3.10340 −0.177992
\(305\) 0 0
\(306\) 17.5957 1.00588
\(307\) 23.8855 1.36322 0.681610 0.731716i \(-0.261280\pi\)
0.681610 + 0.731716i \(0.261280\pi\)
\(308\) −4.25886 −0.242671
\(309\) −37.4621 −2.13114
\(310\) 0 0
\(311\) −12.3074 −0.697887 −0.348943 0.937144i \(-0.613459\pi\)
−0.348943 + 0.937144i \(0.613459\pi\)
\(312\) 55.7368 3.15548
\(313\) −4.39566 −0.248457 −0.124229 0.992254i \(-0.539646\pi\)
−0.124229 + 0.992254i \(0.539646\pi\)
\(314\) 5.67111 0.320039
\(315\) 0 0
\(316\) −1.98067 −0.111421
\(317\) −19.9798 −1.12218 −0.561088 0.827756i \(-0.689617\pi\)
−0.561088 + 0.827756i \(0.689617\pi\)
\(318\) −39.5720 −2.21909
\(319\) −9.42813 −0.527874
\(320\) 0 0
\(321\) 0.756720 0.0422360
\(322\) 7.32944 0.408454
\(323\) −2.37975 −0.132413
\(324\) −33.9389 −1.88550
\(325\) 0 0
\(326\) −14.8667 −0.823388
\(327\) −24.0430 −1.32958
\(328\) 19.4275 1.07270
\(329\) 13.9290 0.767929
\(330\) 0 0
\(331\) 1.04497 0.0574369 0.0287184 0.999588i \(-0.490857\pi\)
0.0287184 + 0.999588i \(0.490857\pi\)
\(332\) 24.4328 1.34092
\(333\) 22.6165 1.23937
\(334\) 19.8640 1.08691
\(335\) 0 0
\(336\) −8.45687 −0.461360
\(337\) −6.81770 −0.371384 −0.185692 0.982608i \(-0.559453\pi\)
−0.185692 + 0.982608i \(0.559453\pi\)
\(338\) 30.7593 1.67308
\(339\) 8.79776 0.477829
\(340\) 0 0
\(341\) 7.43254 0.402495
\(342\) −7.39392 −0.399818
\(343\) −14.1431 −0.763656
\(344\) 42.1628 2.27327
\(345\) 0 0
\(346\) −32.9129 −1.76941
\(347\) −15.0138 −0.805981 −0.402990 0.915204i \(-0.632029\pi\)
−0.402990 + 0.915204i \(0.632029\pi\)
\(348\) −89.2624 −4.78497
\(349\) −23.6909 −1.26814 −0.634072 0.773274i \(-0.718618\pi\)
−0.634072 + 0.773274i \(0.718618\pi\)
\(350\) 0 0
\(351\) 0.722579 0.0385684
\(352\) 1.42636 0.0760250
\(353\) 22.4138 1.19297 0.596484 0.802625i \(-0.296564\pi\)
0.596484 + 0.802625i \(0.296564\pi\)
\(354\) 21.1284 1.12296
\(355\) 0 0
\(356\) 56.3251 2.98522
\(357\) −6.48490 −0.343217
\(358\) −17.1460 −0.906194
\(359\) 7.97079 0.420682 0.210341 0.977628i \(-0.432543\pi\)
0.210341 + 0.977628i \(0.432543\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −18.8519 −0.990832
\(363\) −2.46128 −0.129184
\(364\) −21.5992 −1.13211
\(365\) 0 0
\(366\) 74.3333 3.88546
\(367\) 25.5390 1.33313 0.666563 0.745449i \(-0.267765\pi\)
0.666563 + 0.745449i \(0.267765\pi\)
\(368\) 8.49657 0.442914
\(369\) 13.3046 0.692608
\(370\) 0 0
\(371\) 7.36182 0.382207
\(372\) 70.3688 3.64845
\(373\) 25.1942 1.30450 0.652252 0.758002i \(-0.273824\pi\)
0.652252 + 0.758002i \(0.273824\pi\)
\(374\) −5.75420 −0.297543
\(375\) 0 0
\(376\) 56.1752 2.89702
\(377\) −47.8156 −2.46263
\(378\) −0.381422 −0.0196182
\(379\) 30.8946 1.58695 0.793476 0.608602i \(-0.208269\pi\)
0.793476 + 0.608602i \(0.208269\pi\)
\(380\) 0 0
\(381\) −21.9498 −1.12452
\(382\) −12.2780 −0.628197
\(383\) −3.14686 −0.160797 −0.0803984 0.996763i \(-0.525619\pi\)
−0.0803984 + 0.996763i \(0.525619\pi\)
\(384\) 50.4430 2.57416
\(385\) 0 0
\(386\) 18.6350 0.948497
\(387\) 28.8744 1.46777
\(388\) −32.4406 −1.64692
\(389\) −10.6962 −0.542321 −0.271160 0.962534i \(-0.587407\pi\)
−0.271160 + 0.962534i \(0.587407\pi\)
\(390\) 0 0
\(391\) 6.51534 0.329495
\(392\) −25.7827 −1.30222
\(393\) −29.0657 −1.46617
\(394\) −3.07430 −0.154881
\(395\) 0 0
\(396\) −11.7626 −0.591093
\(397\) −31.3710 −1.57446 −0.787232 0.616656i \(-0.788487\pi\)
−0.787232 + 0.616656i \(0.788487\pi\)
\(398\) −17.6242 −0.883419
\(399\) 2.72503 0.136422
\(400\) 0 0
\(401\) −1.32625 −0.0662299 −0.0331150 0.999452i \(-0.510543\pi\)
−0.0331150 + 0.999452i \(0.510543\pi\)
\(402\) 9.67456 0.482523
\(403\) 37.6948 1.87771
\(404\) 44.2518 2.20161
\(405\) 0 0
\(406\) 25.2401 1.25264
\(407\) −7.39611 −0.366611
\(408\) −26.1535 −1.29479
\(409\) −32.2345 −1.59389 −0.796946 0.604051i \(-0.793552\pi\)
−0.796946 + 0.604051i \(0.793552\pi\)
\(410\) 0 0
\(411\) −20.3449 −1.00354
\(412\) 58.5482 2.88446
\(413\) −3.93065 −0.193415
\(414\) 20.2433 0.994903
\(415\) 0 0
\(416\) 7.23389 0.354671
\(417\) 41.4080 2.02776
\(418\) 2.41798 0.118267
\(419\) −3.07983 −0.150460 −0.0752299 0.997166i \(-0.523969\pi\)
−0.0752299 + 0.997166i \(0.523969\pi\)
\(420\) 0 0
\(421\) −12.8233 −0.624971 −0.312486 0.949923i \(-0.601162\pi\)
−0.312486 + 0.949923i \(0.601162\pi\)
\(422\) 22.8503 1.11234
\(423\) 38.4706 1.87050
\(424\) 29.6901 1.44188
\(425\) 0 0
\(426\) −81.8589 −3.96607
\(427\) −13.8287 −0.669217
\(428\) −1.18265 −0.0571656
\(429\) −12.4826 −0.602665
\(430\) 0 0
\(431\) −10.3542 −0.498743 −0.249372 0.968408i \(-0.580224\pi\)
−0.249372 + 0.968408i \(0.580224\pi\)
\(432\) −0.442159 −0.0212734
\(433\) −28.8062 −1.38434 −0.692168 0.721737i \(-0.743344\pi\)
−0.692168 + 0.721737i \(0.743344\pi\)
\(434\) −19.8977 −0.955118
\(435\) 0 0
\(436\) 37.5759 1.79956
\(437\) −2.73783 −0.130968
\(438\) 19.2803 0.921251
\(439\) 0.245125 0.0116992 0.00584960 0.999983i \(-0.498138\pi\)
0.00584960 + 0.999983i \(0.498138\pi\)
\(440\) 0 0
\(441\) −17.6568 −0.840801
\(442\) −29.1830 −1.38809
\(443\) −2.98982 −0.142051 −0.0710254 0.997475i \(-0.522627\pi\)
−0.0710254 + 0.997475i \(0.522627\pi\)
\(444\) −70.0239 −3.32319
\(445\) 0 0
\(446\) −17.0880 −0.809139
\(447\) 38.1571 1.80477
\(448\) −10.6904 −0.505076
\(449\) 12.4206 0.586163 0.293082 0.956087i \(-0.405319\pi\)
0.293082 + 0.956087i \(0.405319\pi\)
\(450\) 0 0
\(451\) −4.35090 −0.204876
\(452\) −13.7497 −0.646732
\(453\) 6.64098 0.312020
\(454\) 45.0049 2.11218
\(455\) 0 0
\(456\) 10.9900 0.514654
\(457\) −14.2153 −0.664965 −0.332482 0.943109i \(-0.607886\pi\)
−0.332482 + 0.943109i \(0.607886\pi\)
\(458\) 36.1659 1.68992
\(459\) −0.339057 −0.0158258
\(460\) 0 0
\(461\) −18.6876 −0.870366 −0.435183 0.900342i \(-0.643316\pi\)
−0.435183 + 0.900342i \(0.643316\pi\)
\(462\) 6.58909 0.306552
\(463\) 28.4132 1.32047 0.660237 0.751057i \(-0.270456\pi\)
0.660237 + 0.751057i \(0.270456\pi\)
\(464\) 29.2592 1.35833
\(465\) 0 0
\(466\) 39.2844 1.81982
\(467\) 37.3978 1.73056 0.865281 0.501287i \(-0.167140\pi\)
0.865281 + 0.501287i \(0.167140\pi\)
\(468\) −59.6552 −2.75756
\(469\) −1.79982 −0.0831079
\(470\) 0 0
\(471\) −5.77265 −0.265990
\(472\) −15.8522 −0.729658
\(473\) −9.44261 −0.434172
\(474\) 3.06438 0.140752
\(475\) 0 0
\(476\) 10.1350 0.464539
\(477\) 20.3327 0.930971
\(478\) 66.3089 3.03290
\(479\) 0.441317 0.0201643 0.0100821 0.999949i \(-0.496791\pi\)
0.0100821 + 0.999949i \(0.496791\pi\)
\(480\) 0 0
\(481\) −37.5100 −1.71031
\(482\) 57.7121 2.62871
\(483\) −7.46067 −0.339472
\(484\) 3.84665 0.174848
\(485\) 0 0
\(486\) 53.5420 2.42872
\(487\) −15.0236 −0.680783 −0.340392 0.940284i \(-0.610560\pi\)
−0.340392 + 0.940284i \(0.610560\pi\)
\(488\) −55.7708 −2.52463
\(489\) 15.1328 0.684330
\(490\) 0 0
\(491\) −13.6853 −0.617611 −0.308805 0.951125i \(-0.599929\pi\)
−0.308805 + 0.951125i \(0.599929\pi\)
\(492\) −41.1929 −1.85712
\(493\) 22.4366 1.01049
\(494\) 12.2630 0.551740
\(495\) 0 0
\(496\) −23.0661 −1.03570
\(497\) 15.2287 0.683101
\(498\) −37.8011 −1.69391
\(499\) 2.77902 0.124406 0.0622031 0.998064i \(-0.480187\pi\)
0.0622031 + 0.998064i \(0.480187\pi\)
\(500\) 0 0
\(501\) −20.2197 −0.903349
\(502\) −13.6449 −0.609001
\(503\) −32.2630 −1.43853 −0.719267 0.694733i \(-0.755522\pi\)
−0.719267 + 0.694733i \(0.755522\pi\)
\(504\) 15.1172 0.673372
\(505\) 0 0
\(506\) −6.62002 −0.294296
\(507\) −31.3100 −1.39053
\(508\) 34.3046 1.52202
\(509\) 19.9232 0.883079 0.441539 0.897242i \(-0.354433\pi\)
0.441539 + 0.897242i \(0.354433\pi\)
\(510\) 0 0
\(511\) −3.58684 −0.158673
\(512\) −32.1409 −1.42044
\(513\) 0.142476 0.00629046
\(514\) 67.2286 2.96533
\(515\) 0 0
\(516\) −89.3995 −3.93559
\(517\) −12.5808 −0.553302
\(518\) 19.8001 0.869968
\(519\) 33.5022 1.47058
\(520\) 0 0
\(521\) −37.7880 −1.65552 −0.827762 0.561080i \(-0.810386\pi\)
−0.827762 + 0.561080i \(0.810386\pi\)
\(522\) 69.7108 3.05116
\(523\) −26.3858 −1.15377 −0.576885 0.816825i \(-0.695732\pi\)
−0.576885 + 0.816825i \(0.695732\pi\)
\(524\) 45.4258 1.98444
\(525\) 0 0
\(526\) 41.6784 1.81727
\(527\) −17.6876 −0.770483
\(528\) 7.63833 0.332416
\(529\) −15.5043 −0.674100
\(530\) 0 0
\(531\) −10.8561 −0.471115
\(532\) −4.25886 −0.184645
\(533\) −22.0660 −0.955784
\(534\) −87.1432 −3.77105
\(535\) 0 0
\(536\) −7.25863 −0.313525
\(537\) 17.4530 0.753152
\(538\) 61.0999 2.63420
\(539\) 5.77419 0.248712
\(540\) 0 0
\(541\) 20.9351 0.900071 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(542\) −76.7818 −3.29806
\(543\) 19.1894 0.823496
\(544\) −3.39437 −0.145532
\(545\) 0 0
\(546\) 33.4172 1.43012
\(547\) −7.46047 −0.318987 −0.159493 0.987199i \(-0.550986\pi\)
−0.159493 + 0.987199i \(0.550986\pi\)
\(548\) 31.7964 1.35827
\(549\) −38.1936 −1.63006
\(550\) 0 0
\(551\) −9.42813 −0.401652
\(552\) −30.0887 −1.28066
\(553\) −0.570086 −0.0242425
\(554\) −36.1371 −1.53532
\(555\) 0 0
\(556\) −64.7151 −2.74453
\(557\) −43.7709 −1.85463 −0.927317 0.374278i \(-0.877891\pi\)
−0.927317 + 0.374278i \(0.877891\pi\)
\(558\) −54.9556 −2.32646
\(559\) −47.8891 −2.02549
\(560\) 0 0
\(561\) 5.85723 0.247292
\(562\) −9.70081 −0.409204
\(563\) 8.39563 0.353833 0.176917 0.984226i \(-0.443388\pi\)
0.176917 + 0.984226i \(0.443388\pi\)
\(564\) −119.111 −5.01546
\(565\) 0 0
\(566\) 4.80049 0.201779
\(567\) −9.76848 −0.410238
\(568\) 61.4171 2.57700
\(569\) −10.4987 −0.440126 −0.220063 0.975486i \(-0.570626\pi\)
−0.220063 + 0.975486i \(0.570626\pi\)
\(570\) 0 0
\(571\) −8.24895 −0.345208 −0.172604 0.984991i \(-0.555218\pi\)
−0.172604 + 0.984991i \(0.555218\pi\)
\(572\) 19.5086 0.815697
\(573\) 12.4978 0.522105
\(574\) 11.6478 0.486170
\(575\) 0 0
\(576\) −29.5261 −1.23025
\(577\) −11.4272 −0.475720 −0.237860 0.971299i \(-0.576446\pi\)
−0.237860 + 0.971299i \(0.576446\pi\)
\(578\) −27.4122 −1.14020
\(579\) −18.9687 −0.788311
\(580\) 0 0
\(581\) 7.03238 0.291752
\(582\) 50.1904 2.08046
\(583\) −6.64927 −0.275384
\(584\) −14.4657 −0.598593
\(585\) 0 0
\(586\) −60.1351 −2.48416
\(587\) 4.73721 0.195526 0.0977628 0.995210i \(-0.468831\pi\)
0.0977628 + 0.995210i \(0.468831\pi\)
\(588\) 54.6681 2.25447
\(589\) 7.43254 0.306252
\(590\) 0 0
\(591\) 3.12935 0.128724
\(592\) 22.9531 0.943366
\(593\) −9.39982 −0.386004 −0.193002 0.981198i \(-0.561822\pi\)
−0.193002 + 0.981198i \(0.561822\pi\)
\(594\) 0.344504 0.0141352
\(595\) 0 0
\(596\) −59.6345 −2.44272
\(597\) 17.9397 0.734223
\(598\) −33.5740 −1.37294
\(599\) −4.63794 −0.189501 −0.0947505 0.995501i \(-0.530205\pi\)
−0.0947505 + 0.995501i \(0.530205\pi\)
\(600\) 0 0
\(601\) 33.9459 1.38468 0.692341 0.721571i \(-0.256580\pi\)
0.692341 + 0.721571i \(0.256580\pi\)
\(602\) 25.2788 1.03029
\(603\) −4.97094 −0.202432
\(604\) −10.3790 −0.422314
\(605\) 0 0
\(606\) −68.4640 −2.78116
\(607\) 13.0888 0.531259 0.265630 0.964075i \(-0.414420\pi\)
0.265630 + 0.964075i \(0.414420\pi\)
\(608\) 1.42636 0.0578464
\(609\) −25.6920 −1.04109
\(610\) 0 0
\(611\) −63.8046 −2.58126
\(612\) 27.9921 1.13151
\(613\) −15.1892 −0.613485 −0.306742 0.951793i \(-0.599239\pi\)
−0.306742 + 0.951793i \(0.599239\pi\)
\(614\) 57.7548 2.33080
\(615\) 0 0
\(616\) −4.94366 −0.199186
\(617\) 24.6488 0.992323 0.496161 0.868230i \(-0.334742\pi\)
0.496161 + 0.868230i \(0.334742\pi\)
\(618\) −90.5827 −3.64377
\(619\) −10.1170 −0.406635 −0.203318 0.979113i \(-0.565172\pi\)
−0.203318 + 0.979113i \(0.565172\pi\)
\(620\) 0 0
\(621\) −0.390074 −0.0156531
\(622\) −29.7590 −1.19323
\(623\) 16.2118 0.649511
\(624\) 38.7385 1.55078
\(625\) 0 0
\(626\) −10.6286 −0.424805
\(627\) −2.46128 −0.0982940
\(628\) 9.02187 0.360012
\(629\) 17.6009 0.701794
\(630\) 0 0
\(631\) −28.5411 −1.13621 −0.568103 0.822958i \(-0.692322\pi\)
−0.568103 + 0.822958i \(0.692322\pi\)
\(632\) −2.29915 −0.0914551
\(633\) −23.2594 −0.924479
\(634\) −48.3108 −1.91867
\(635\) 0 0
\(636\) −62.9530 −2.49625
\(637\) 29.2843 1.16029
\(638\) −22.7971 −0.902544
\(639\) 42.0604 1.66388
\(640\) 0 0
\(641\) 1.13231 0.0447235 0.0223618 0.999750i \(-0.492881\pi\)
0.0223618 + 0.999750i \(0.492881\pi\)
\(642\) 1.82974 0.0722139
\(643\) −44.0466 −1.73703 −0.868514 0.495666i \(-0.834924\pi\)
−0.868514 + 0.495666i \(0.834924\pi\)
\(644\) 11.6600 0.459470
\(645\) 0 0
\(646\) −5.75420 −0.226396
\(647\) −34.6226 −1.36116 −0.680578 0.732675i \(-0.738271\pi\)
−0.680578 + 0.732675i \(0.738271\pi\)
\(648\) −39.3961 −1.54762
\(649\) 3.55020 0.139358
\(650\) 0 0
\(651\) 20.2539 0.793814
\(652\) −23.6506 −0.926229
\(653\) 33.3308 1.30433 0.652167 0.758076i \(-0.273860\pi\)
0.652167 + 0.758076i \(0.273860\pi\)
\(654\) −58.1355 −2.27328
\(655\) 0 0
\(656\) 13.5026 0.527187
\(657\) −9.90655 −0.386491
\(658\) 33.6800 1.31298
\(659\) −13.1051 −0.510503 −0.255251 0.966875i \(-0.582158\pi\)
−0.255251 + 0.966875i \(0.582158\pi\)
\(660\) 0 0
\(661\) 0.773684 0.0300928 0.0150464 0.999887i \(-0.495210\pi\)
0.0150464 + 0.999887i \(0.495210\pi\)
\(662\) 2.52673 0.0982041
\(663\) 29.7055 1.15366
\(664\) 28.3614 1.10064
\(665\) 0 0
\(666\) 54.6862 2.11905
\(667\) 25.8126 0.999467
\(668\) 31.6007 1.22267
\(669\) 17.3939 0.672488
\(670\) 0 0
\(671\) 12.4902 0.482179
\(672\) 3.88687 0.149939
\(673\) 48.5562 1.87171 0.935853 0.352392i \(-0.114632\pi\)
0.935853 + 0.352392i \(0.114632\pi\)
\(674\) −16.4851 −0.634982
\(675\) 0 0
\(676\) 48.9334 1.88205
\(677\) −7.59408 −0.291864 −0.145932 0.989295i \(-0.546618\pi\)
−0.145932 + 0.989295i \(0.546618\pi\)
\(678\) 21.2728 0.816978
\(679\) −9.33723 −0.358330
\(680\) 0 0
\(681\) −45.8107 −1.75547
\(682\) 17.9718 0.688174
\(683\) −31.4250 −1.20244 −0.601222 0.799082i \(-0.705319\pi\)
−0.601222 + 0.799082i \(0.705319\pi\)
\(684\) −11.7626 −0.449755
\(685\) 0 0
\(686\) −34.1978 −1.30568
\(687\) −36.8135 −1.40452
\(688\) 29.3042 1.11721
\(689\) −33.7224 −1.28472
\(690\) 0 0
\(691\) −28.1541 −1.07103 −0.535517 0.844524i \(-0.679883\pi\)
−0.535517 + 0.844524i \(0.679883\pi\)
\(692\) −52.3595 −1.99041
\(693\) −3.38558 −0.128607
\(694\) −36.3030 −1.37804
\(695\) 0 0
\(696\) −103.615 −3.92752
\(697\) 10.3541 0.392188
\(698\) −57.2841 −2.16824
\(699\) −39.9878 −1.51248
\(700\) 0 0
\(701\) −40.9965 −1.54842 −0.774208 0.632931i \(-0.781852\pi\)
−0.774208 + 0.632931i \(0.781852\pi\)
\(702\) 1.74718 0.0659432
\(703\) −7.39611 −0.278950
\(704\) 9.65570 0.363913
\(705\) 0 0
\(706\) 54.1963 2.03971
\(707\) 12.7368 0.479016
\(708\) 33.6121 1.26322
\(709\) −44.3343 −1.66501 −0.832505 0.554018i \(-0.813094\pi\)
−0.832505 + 0.554018i \(0.813094\pi\)
\(710\) 0 0
\(711\) −1.57453 −0.0590494
\(712\) 65.3818 2.45029
\(713\) −20.3490 −0.762076
\(714\) −15.6804 −0.586824
\(715\) 0 0
\(716\) −27.2767 −1.01938
\(717\) −67.4962 −2.52069
\(718\) 19.2732 0.719271
\(719\) 28.6087 1.06693 0.533463 0.845823i \(-0.320890\pi\)
0.533463 + 0.845823i \(0.320890\pi\)
\(720\) 0 0
\(721\) 16.8517 0.627588
\(722\) 2.41798 0.0899880
\(723\) −58.7454 −2.18476
\(724\) −29.9905 −1.11459
\(725\) 0 0
\(726\) −5.95133 −0.220875
\(727\) 18.7083 0.693851 0.346926 0.937893i \(-0.387226\pi\)
0.346926 + 0.937893i \(0.387226\pi\)
\(728\) −25.0722 −0.929239
\(729\) −28.0317 −1.03821
\(730\) 0 0
\(731\) 22.4711 0.831122
\(732\) 118.253 4.37076
\(733\) 30.8129 1.13810 0.569050 0.822303i \(-0.307311\pi\)
0.569050 + 0.822303i \(0.307311\pi\)
\(734\) 61.7529 2.27934
\(735\) 0 0
\(736\) −3.90511 −0.143944
\(737\) 1.62561 0.0598803
\(738\) 32.1702 1.18420
\(739\) 1.09461 0.0402658 0.0201329 0.999797i \(-0.493591\pi\)
0.0201329 + 0.999797i \(0.493591\pi\)
\(740\) 0 0
\(741\) −12.4826 −0.458560
\(742\) 17.8008 0.653486
\(743\) 13.3441 0.489549 0.244774 0.969580i \(-0.421286\pi\)
0.244774 + 0.969580i \(0.421286\pi\)
\(744\) 81.6836 2.99467
\(745\) 0 0
\(746\) 60.9191 2.23041
\(747\) 19.4228 0.710644
\(748\) −9.15406 −0.334706
\(749\) −0.340397 −0.0124378
\(750\) 0 0
\(751\) 38.5663 1.40730 0.703652 0.710545i \(-0.251551\pi\)
0.703652 + 0.710545i \(0.251551\pi\)
\(752\) 39.0432 1.42376
\(753\) 13.8892 0.506150
\(754\) −115.617 −4.21054
\(755\) 0 0
\(756\) −0.606785 −0.0220685
\(757\) −6.30329 −0.229097 −0.114549 0.993418i \(-0.536542\pi\)
−0.114549 + 0.993418i \(0.536542\pi\)
\(758\) 74.7028 2.71333
\(759\) 6.73855 0.244594
\(760\) 0 0
\(761\) −8.58504 −0.311207 −0.155604 0.987820i \(-0.549732\pi\)
−0.155604 + 0.987820i \(0.549732\pi\)
\(762\) −53.0743 −1.92268
\(763\) 10.8153 0.391541
\(764\) −19.5325 −0.706659
\(765\) 0 0
\(766\) −7.60905 −0.274926
\(767\) 18.0052 0.650129
\(768\) 74.4396 2.68611
\(769\) 48.7219 1.75696 0.878479 0.477781i \(-0.158559\pi\)
0.878479 + 0.477781i \(0.158559\pi\)
\(770\) 0 0
\(771\) −68.4324 −2.46453
\(772\) 29.6455 1.06696
\(773\) −53.9495 −1.94043 −0.970214 0.242249i \(-0.922115\pi\)
−0.970214 + 0.242249i \(0.922115\pi\)
\(774\) 69.8179 2.50955
\(775\) 0 0
\(776\) −37.6569 −1.35180
\(777\) −20.1546 −0.723044
\(778\) −25.8633 −0.927245
\(779\) −4.35090 −0.155887
\(780\) 0 0
\(781\) −13.7547 −0.492183
\(782\) 15.7540 0.563362
\(783\) −1.34328 −0.0480049
\(784\) −17.9196 −0.639987
\(785\) 0 0
\(786\) −70.2805 −2.50682
\(787\) 14.5730 0.519470 0.259735 0.965680i \(-0.416365\pi\)
0.259735 + 0.965680i \(0.416365\pi\)
\(788\) −4.89075 −0.174226
\(789\) −42.4247 −1.51036
\(790\) 0 0
\(791\) −3.95752 −0.140713
\(792\) −13.6540 −0.485172
\(793\) 63.3452 2.24945
\(794\) −75.8546 −2.69198
\(795\) 0 0
\(796\) −28.0374 −0.993758
\(797\) 25.6413 0.908261 0.454130 0.890935i \(-0.349950\pi\)
0.454130 + 0.890935i \(0.349950\pi\)
\(798\) 6.58909 0.233251
\(799\) 29.9391 1.05917
\(800\) 0 0
\(801\) 44.7755 1.58207
\(802\) −3.20686 −0.113238
\(803\) 3.23967 0.114325
\(804\) 15.3908 0.542791
\(805\) 0 0
\(806\) 91.1454 3.21046
\(807\) −62.1938 −2.18933
\(808\) 51.3672 1.80709
\(809\) −4.91560 −0.172823 −0.0864117 0.996260i \(-0.527540\pi\)
−0.0864117 + 0.996260i \(0.527540\pi\)
\(810\) 0 0
\(811\) −10.5485 −0.370407 −0.185203 0.982700i \(-0.559294\pi\)
−0.185203 + 0.982700i \(0.559294\pi\)
\(812\) 40.1531 1.40910
\(813\) 78.1566 2.74107
\(814\) −17.8837 −0.626822
\(815\) 0 0
\(816\) −18.1773 −0.636333
\(817\) −9.44261 −0.330355
\(818\) −77.9424 −2.72519
\(819\) −17.1703 −0.599978
\(820\) 0 0
\(821\) −34.7786 −1.21378 −0.606891 0.794785i \(-0.707584\pi\)
−0.606891 + 0.794785i \(0.707584\pi\)
\(822\) −49.1936 −1.71583
\(823\) −44.4038 −1.54782 −0.773909 0.633297i \(-0.781701\pi\)
−0.773909 + 0.633297i \(0.781701\pi\)
\(824\) 67.9624 2.36758
\(825\) 0 0
\(826\) −9.50425 −0.330695
\(827\) 26.7888 0.931537 0.465769 0.884907i \(-0.345778\pi\)
0.465769 + 0.884907i \(0.345778\pi\)
\(828\) 32.2040 1.11917
\(829\) −49.3143 −1.71276 −0.856379 0.516348i \(-0.827291\pi\)
−0.856379 + 0.516348i \(0.827291\pi\)
\(830\) 0 0
\(831\) 36.7841 1.27603
\(832\) 48.9698 1.69772
\(833\) −13.7411 −0.476102
\(834\) 100.124 3.46700
\(835\) 0 0
\(836\) 3.84665 0.133039
\(837\) 1.05896 0.0366029
\(838\) −7.44699 −0.257252
\(839\) 5.30801 0.183253 0.0916264 0.995793i \(-0.470793\pi\)
0.0916264 + 0.995793i \(0.470793\pi\)
\(840\) 0 0
\(841\) 59.8896 2.06516
\(842\) −31.0066 −1.06856
\(843\) 9.87450 0.340096
\(844\) 36.3514 1.25127
\(845\) 0 0
\(846\) 93.0213 3.19814
\(847\) 1.10716 0.0380426
\(848\) 20.6353 0.708620
\(849\) −4.88644 −0.167702
\(850\) 0 0
\(851\) 20.2493 0.694136
\(852\) −130.225 −4.46144
\(853\) 45.1035 1.54431 0.772157 0.635432i \(-0.219178\pi\)
0.772157 + 0.635432i \(0.219178\pi\)
\(854\) −33.4375 −1.14421
\(855\) 0 0
\(856\) −1.37281 −0.0469218
\(857\) −30.5012 −1.04190 −0.520951 0.853587i \(-0.674423\pi\)
−0.520951 + 0.853587i \(0.674423\pi\)
\(858\) −30.1827 −1.03042
\(859\) 19.6545 0.670604 0.335302 0.942111i \(-0.391162\pi\)
0.335302 + 0.942111i \(0.391162\pi\)
\(860\) 0 0
\(861\) −11.8563 −0.404063
\(862\) −25.0362 −0.852738
\(863\) 32.7845 1.11600 0.557999 0.829842i \(-0.311569\pi\)
0.557999 + 0.829842i \(0.311569\pi\)
\(864\) 0.203221 0.00691372
\(865\) 0 0
\(866\) −69.6528 −2.36690
\(867\) 27.9030 0.947635
\(868\) −31.6542 −1.07441
\(869\) 0.514907 0.0174670
\(870\) 0 0
\(871\) 8.24445 0.279353
\(872\) 43.6179 1.47709
\(873\) −25.7886 −0.872813
\(874\) −6.62002 −0.223926
\(875\) 0 0
\(876\) 30.6721 1.03631
\(877\) 49.2755 1.66392 0.831958 0.554839i \(-0.187220\pi\)
0.831958 + 0.554839i \(0.187220\pi\)
\(878\) 0.592709 0.0200030
\(879\) 61.2118 2.06462
\(880\) 0 0
\(881\) 10.3782 0.349652 0.174826 0.984599i \(-0.444064\pi\)
0.174826 + 0.984599i \(0.444064\pi\)
\(882\) −42.6939 −1.43758
\(883\) −2.17485 −0.0731897 −0.0365948 0.999330i \(-0.511651\pi\)
−0.0365948 + 0.999330i \(0.511651\pi\)
\(884\) −46.4257 −1.56146
\(885\) 0 0
\(886\) −7.22934 −0.242874
\(887\) 26.3608 0.885110 0.442555 0.896741i \(-0.354072\pi\)
0.442555 + 0.896741i \(0.354072\pi\)
\(888\) −81.2833 −2.72769
\(889\) 9.87375 0.331155
\(890\) 0 0
\(891\) 8.82299 0.295581
\(892\) −27.1844 −0.910200
\(893\) −12.5808 −0.421000
\(894\) 92.2633 3.08575
\(895\) 0 0
\(896\) −22.6909 −0.758049
\(897\) 34.1752 1.14108
\(898\) 30.0327 1.00221
\(899\) −70.0749 −2.33713
\(900\) 0 0
\(901\) 15.8236 0.527160
\(902\) −10.5204 −0.350291
\(903\) −25.7314 −0.856289
\(904\) −15.9606 −0.530841
\(905\) 0 0
\(906\) 16.0578 0.533484
\(907\) 45.8623 1.52283 0.761416 0.648263i \(-0.224504\pi\)
0.761416 + 0.648263i \(0.224504\pi\)
\(908\) 71.5959 2.37599
\(909\) 35.1779 1.16678
\(910\) 0 0
\(911\) 18.8420 0.624262 0.312131 0.950039i \(-0.398957\pi\)
0.312131 + 0.950039i \(0.398957\pi\)
\(912\) 7.63833 0.252930
\(913\) −6.35171 −0.210211
\(914\) −34.3724 −1.13694
\(915\) 0 0
\(916\) 57.5345 1.90099
\(917\) 13.0747 0.431765
\(918\) −0.819834 −0.0270585
\(919\) 6.86431 0.226433 0.113216 0.993570i \(-0.463885\pi\)
0.113216 + 0.993570i \(0.463885\pi\)
\(920\) 0 0
\(921\) −58.7889 −1.93716
\(922\) −45.1862 −1.48813
\(923\) −69.7583 −2.29612
\(924\) 10.4822 0.344841
\(925\) 0 0
\(926\) 68.7027 2.25771
\(927\) 46.5428 1.52867
\(928\) −13.4479 −0.441448
\(929\) 47.0098 1.54234 0.771171 0.636628i \(-0.219671\pi\)
0.771171 + 0.636628i \(0.219671\pi\)
\(930\) 0 0
\(931\) 5.77419 0.189241
\(932\) 62.4956 2.04711
\(933\) 30.2918 0.991710
\(934\) 90.4272 2.95887
\(935\) 0 0
\(936\) −69.2473 −2.26342
\(937\) −48.8157 −1.59474 −0.797369 0.603492i \(-0.793776\pi\)
−0.797369 + 0.603492i \(0.793776\pi\)
\(938\) −4.35193 −0.142096
\(939\) 10.8189 0.353062
\(940\) 0 0
\(941\) −41.5998 −1.35612 −0.678058 0.735009i \(-0.737178\pi\)
−0.678058 + 0.735009i \(0.737178\pi\)
\(942\) −13.9582 −0.454782
\(943\) 11.9120 0.387908
\(944\) −11.0177 −0.358595
\(945\) 0 0
\(946\) −22.8321 −0.742335
\(947\) 41.6052 1.35199 0.675994 0.736907i \(-0.263714\pi\)
0.675994 + 0.736907i \(0.263714\pi\)
\(948\) 4.87497 0.158332
\(949\) 16.4303 0.533350
\(950\) 0 0
\(951\) 49.1758 1.59463
\(952\) 11.7647 0.381296
\(953\) 1.18377 0.0383460 0.0191730 0.999816i \(-0.493897\pi\)
0.0191730 + 0.999816i \(0.493897\pi\)
\(954\) 49.1642 1.59175
\(955\) 0 0
\(956\) 105.487 3.41171
\(957\) 23.2052 0.750119
\(958\) 1.06710 0.0344764
\(959\) 9.15180 0.295527
\(960\) 0 0
\(961\) 24.2426 0.782020
\(962\) −90.6987 −2.92424
\(963\) −0.940147 −0.0302958
\(964\) 91.8111 2.95704
\(965\) 0 0
\(966\) −18.0398 −0.580421
\(967\) −26.2645 −0.844611 −0.422305 0.906454i \(-0.638779\pi\)
−0.422305 + 0.906454i \(0.638779\pi\)
\(968\) 4.46516 0.143516
\(969\) 5.85723 0.188161
\(970\) 0 0
\(971\) 19.5161 0.626300 0.313150 0.949704i \(-0.398616\pi\)
0.313150 + 0.949704i \(0.398616\pi\)
\(972\) 85.1773 2.73206
\(973\) −18.6266 −0.597143
\(974\) −36.3268 −1.16399
\(975\) 0 0
\(976\) −38.7621 −1.24074
\(977\) −34.3264 −1.09820 −0.549100 0.835757i \(-0.685029\pi\)
−0.549100 + 0.835757i \(0.685029\pi\)
\(978\) 36.5910 1.17005
\(979\) −14.6426 −0.467981
\(980\) 0 0
\(981\) 29.8710 0.953706
\(982\) −33.0909 −1.05597
\(983\) 35.4328 1.13013 0.565065 0.825047i \(-0.308851\pi\)
0.565065 + 0.825047i \(0.308851\pi\)
\(984\) −47.8164 −1.52433
\(985\) 0 0
\(986\) 54.2513 1.72771
\(987\) −34.2831 −1.09124
\(988\) 19.5086 0.620652
\(989\) 25.8522 0.822053
\(990\) 0 0
\(991\) 8.16624 0.259409 0.129705 0.991553i \(-0.458597\pi\)
0.129705 + 0.991553i \(0.458597\pi\)
\(992\) 10.6014 0.336596
\(993\) −2.57197 −0.0816189
\(994\) 36.8228 1.16795
\(995\) 0 0
\(996\) −60.1359 −1.90548
\(997\) −43.0463 −1.36329 −0.681646 0.731682i \(-0.738735\pi\)
−0.681646 + 0.731682i \(0.738735\pi\)
\(998\) 6.71963 0.212706
\(999\) −1.05377 −0.0333397
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.t.1.14 15
5.4 even 2 5225.2.a.w.1.2 yes 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5225.2.a.t.1.14 15 1.1 even 1 trivial
5225.2.a.w.1.2 yes 15 5.4 even 2