Properties

Label 4008.2.a.f.1.3
Level $4008$
Weight $2$
Character 4008.1
Self dual yes
Analytic conductor $32.004$
Analytic rank $1$
Dimension $5$
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] = [4008,2,Mod(1,4008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4008, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4008.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4008.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: \(32.0040411301\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.284897.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 6x^{3} + 5x^{2} + 10x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.65065\) of defining polynomial
Character \(\chi\) \(=\) 4008.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -0.457884 q^{5} -2.37531 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -0.457884 q^{5} -2.37531 q^{7} +1.00000 q^{9} -1.86534 q^{11} +1.48384 q^{13} -0.457884 q^{15} -0.676771 q^{17} +4.18757 q^{19} -2.37531 q^{21} -1.60277 q^{23} -4.79034 q^{25} +1.00000 q^{27} +8.15542 q^{29} -1.46812 q^{31} -1.86534 q^{33} +1.08761 q^{35} -8.61076 q^{37} +1.48384 q^{39} -12.1631 q^{41} +1.80872 q^{43} -0.457884 q^{45} +4.53272 q^{47} -1.35793 q^{49} -0.676771 q^{51} -7.07284 q^{53} +0.854111 q^{55} +4.18757 q^{57} +15.0829 q^{59} +6.30014 q^{61} -2.37531 q^{63} -0.679427 q^{65} -10.7228 q^{67} -1.60277 q^{69} -1.23296 q^{71} -10.3160 q^{73} -4.79034 q^{75} +4.43076 q^{77} -6.40327 q^{79} +1.00000 q^{81} -10.9254 q^{83} +0.309883 q^{85} +8.15542 q^{87} -2.83435 q^{89} -3.52458 q^{91} -1.46812 q^{93} -1.91742 q^{95} -6.73223 q^{97} -1.86534 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{3} + q^{5} - 4 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 5 q^{3} + q^{5} - 4 q^{7} + 5 q^{9} - 3 q^{11} - 14 q^{13} + q^{15} - 13 q^{17} - 2 q^{19} - 4 q^{21} - 5 q^{23} + 2 q^{25} + 5 q^{27} + 13 q^{29} + 2 q^{31} - 3 q^{33} - 12 q^{35} - 5 q^{37} - 14 q^{39} - 20 q^{41} - 20 q^{43} + q^{45} + q^{47} - 9 q^{49} - 13 q^{51} - 3 q^{53} - 3 q^{55} - 2 q^{57} - q^{59} - 34 q^{61} - 4 q^{63} - 22 q^{65} - 16 q^{67} - 5 q^{69} - 5 q^{71} - 12 q^{73} + 2 q^{75} - 8 q^{77} - 20 q^{79} + 5 q^{81} - 15 q^{83} - 27 q^{85} + 13 q^{87} - 48 q^{89} + 7 q^{91} + 2 q^{93} - 5 q^{95} - 21 q^{97} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −0.457884 −0.204772 −0.102386 0.994745i \(-0.532648\pi\)
−0.102386 + 0.994745i \(0.532648\pi\)
\(6\) 0 0
\(7\) −2.37531 −0.897781 −0.448890 0.893587i \(-0.648181\pi\)
−0.448890 + 0.893587i \(0.648181\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.86534 −0.562423 −0.281211 0.959646i \(-0.590736\pi\)
−0.281211 + 0.959646i \(0.590736\pi\)
\(12\) 0 0
\(13\) 1.48384 0.411543 0.205772 0.978600i \(-0.434030\pi\)
0.205772 + 0.978600i \(0.434030\pi\)
\(14\) 0 0
\(15\) −0.457884 −0.118225
\(16\) 0 0
\(17\) −0.676771 −0.164141 −0.0820706 0.996627i \(-0.526153\pi\)
−0.0820706 + 0.996627i \(0.526153\pi\)
\(18\) 0 0
\(19\) 4.18757 0.960695 0.480347 0.877078i \(-0.340511\pi\)
0.480347 + 0.877078i \(0.340511\pi\)
\(20\) 0 0
\(21\) −2.37531 −0.518334
\(22\) 0 0
\(23\) −1.60277 −0.334201 −0.167100 0.985940i \(-0.553440\pi\)
−0.167100 + 0.985940i \(0.553440\pi\)
\(24\) 0 0
\(25\) −4.79034 −0.958068
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 8.15542 1.51442 0.757211 0.653170i \(-0.226561\pi\)
0.757211 + 0.653170i \(0.226561\pi\)
\(30\) 0 0
\(31\) −1.46812 −0.263681 −0.131841 0.991271i \(-0.542089\pi\)
−0.131841 + 0.991271i \(0.542089\pi\)
\(32\) 0 0
\(33\) −1.86534 −0.324715
\(34\) 0 0
\(35\) 1.08761 0.183840
\(36\) 0 0
\(37\) −8.61076 −1.41560 −0.707800 0.706413i \(-0.750312\pi\)
−0.707800 + 0.706413i \(0.750312\pi\)
\(38\) 0 0
\(39\) 1.48384 0.237605
\(40\) 0 0
\(41\) −12.1631 −1.89956 −0.949779 0.312921i \(-0.898692\pi\)
−0.949779 + 0.312921i \(0.898692\pi\)
\(42\) 0 0
\(43\) 1.80872 0.275828 0.137914 0.990444i \(-0.455960\pi\)
0.137914 + 0.990444i \(0.455960\pi\)
\(44\) 0 0
\(45\) −0.457884 −0.0682573
\(46\) 0 0
\(47\) 4.53272 0.661166 0.330583 0.943777i \(-0.392755\pi\)
0.330583 + 0.943777i \(0.392755\pi\)
\(48\) 0 0
\(49\) −1.35793 −0.193989
\(50\) 0 0
\(51\) −0.676771 −0.0947670
\(52\) 0 0
\(53\) −7.07284 −0.971529 −0.485764 0.874090i \(-0.661459\pi\)
−0.485764 + 0.874090i \(0.661459\pi\)
\(54\) 0 0
\(55\) 0.854111 0.115168
\(56\) 0 0
\(57\) 4.18757 0.554657
\(58\) 0 0
\(59\) 15.0829 1.96363 0.981814 0.189847i \(-0.0607993\pi\)
0.981814 + 0.189847i \(0.0607993\pi\)
\(60\) 0 0
\(61\) 6.30014 0.806650 0.403325 0.915057i \(-0.367854\pi\)
0.403325 + 0.915057i \(0.367854\pi\)
\(62\) 0 0
\(63\) −2.37531 −0.299260
\(64\) 0 0
\(65\) −0.679427 −0.0842725
\(66\) 0 0
\(67\) −10.7228 −1.31000 −0.655002 0.755628i \(-0.727332\pi\)
−0.655002 + 0.755628i \(0.727332\pi\)
\(68\) 0 0
\(69\) −1.60277 −0.192951
\(70\) 0 0
\(71\) −1.23296 −0.146325 −0.0731627 0.997320i \(-0.523309\pi\)
−0.0731627 + 0.997320i \(0.523309\pi\)
\(72\) 0 0
\(73\) −10.3160 −1.20740 −0.603700 0.797212i \(-0.706307\pi\)
−0.603700 + 0.797212i \(0.706307\pi\)
\(74\) 0 0
\(75\) −4.79034 −0.553141
\(76\) 0 0
\(77\) 4.43076 0.504932
\(78\) 0 0
\(79\) −6.40327 −0.720424 −0.360212 0.932871i \(-0.617296\pi\)
−0.360212 + 0.932871i \(0.617296\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −10.9254 −1.19922 −0.599610 0.800293i \(-0.704677\pi\)
−0.599610 + 0.800293i \(0.704677\pi\)
\(84\) 0 0
\(85\) 0.309883 0.0336115
\(86\) 0 0
\(87\) 8.15542 0.874352
\(88\) 0 0
\(89\) −2.83435 −0.300441 −0.150220 0.988653i \(-0.547998\pi\)
−0.150220 + 0.988653i \(0.547998\pi\)
\(90\) 0 0
\(91\) −3.52458 −0.369476
\(92\) 0 0
\(93\) −1.46812 −0.152236
\(94\) 0 0
\(95\) −1.91742 −0.196723
\(96\) 0 0
\(97\) −6.73223 −0.683554 −0.341777 0.939781i \(-0.611029\pi\)
−0.341777 + 0.939781i \(0.611029\pi\)
\(98\) 0 0
\(99\) −1.86534 −0.187474
\(100\) 0 0
\(101\) −7.45223 −0.741524 −0.370762 0.928728i \(-0.620904\pi\)
−0.370762 + 0.928728i \(0.620904\pi\)
\(102\) 0 0
\(103\) −6.11552 −0.602580 −0.301290 0.953533i \(-0.597417\pi\)
−0.301290 + 0.953533i \(0.597417\pi\)
\(104\) 0 0
\(105\) 1.08761 0.106140
\(106\) 0 0
\(107\) −13.0772 −1.26422 −0.632111 0.774878i \(-0.717811\pi\)
−0.632111 + 0.774878i \(0.717811\pi\)
\(108\) 0 0
\(109\) 7.00238 0.670706 0.335353 0.942092i \(-0.391144\pi\)
0.335353 + 0.942092i \(0.391144\pi\)
\(110\) 0 0
\(111\) −8.61076 −0.817297
\(112\) 0 0
\(113\) −9.06929 −0.853167 −0.426584 0.904448i \(-0.640283\pi\)
−0.426584 + 0.904448i \(0.640283\pi\)
\(114\) 0 0
\(115\) 0.733883 0.0684349
\(116\) 0 0
\(117\) 1.48384 0.137181
\(118\) 0 0
\(119\) 1.60754 0.147363
\(120\) 0 0
\(121\) −7.52049 −0.683681
\(122\) 0 0
\(123\) −12.1631 −1.09671
\(124\) 0 0
\(125\) 4.48284 0.400957
\(126\) 0 0
\(127\) 8.65569 0.768068 0.384034 0.923319i \(-0.374535\pi\)
0.384034 + 0.923319i \(0.374535\pi\)
\(128\) 0 0
\(129\) 1.80872 0.159249
\(130\) 0 0
\(131\) 8.05546 0.703809 0.351904 0.936036i \(-0.385534\pi\)
0.351904 + 0.936036i \(0.385534\pi\)
\(132\) 0 0
\(133\) −9.94676 −0.862493
\(134\) 0 0
\(135\) −0.457884 −0.0394084
\(136\) 0 0
\(137\) −5.88865 −0.503101 −0.251551 0.967844i \(-0.580940\pi\)
−0.251551 + 0.967844i \(0.580940\pi\)
\(138\) 0 0
\(139\) 9.61934 0.815901 0.407951 0.913004i \(-0.366244\pi\)
0.407951 + 0.913004i \(0.366244\pi\)
\(140\) 0 0
\(141\) 4.53272 0.381724
\(142\) 0 0
\(143\) −2.76788 −0.231461
\(144\) 0 0
\(145\) −3.73423 −0.310111
\(146\) 0 0
\(147\) −1.35793 −0.112000
\(148\) 0 0
\(149\) 10.3048 0.844206 0.422103 0.906548i \(-0.361292\pi\)
0.422103 + 0.906548i \(0.361292\pi\)
\(150\) 0 0
\(151\) −9.91415 −0.806802 −0.403401 0.915023i \(-0.632172\pi\)
−0.403401 + 0.915023i \(0.632172\pi\)
\(152\) 0 0
\(153\) −0.676771 −0.0547137
\(154\) 0 0
\(155\) 0.672226 0.0539945
\(156\) 0 0
\(157\) −15.1523 −1.20929 −0.604643 0.796497i \(-0.706684\pi\)
−0.604643 + 0.796497i \(0.706684\pi\)
\(158\) 0 0
\(159\) −7.07284 −0.560912
\(160\) 0 0
\(161\) 3.80707 0.300039
\(162\) 0 0
\(163\) 8.57960 0.672006 0.336003 0.941861i \(-0.390925\pi\)
0.336003 + 0.941861i \(0.390925\pi\)
\(164\) 0 0
\(165\) 0.854111 0.0664925
\(166\) 0 0
\(167\) 1.00000 0.0773823
\(168\) 0 0
\(169\) −10.7982 −0.830632
\(170\) 0 0
\(171\) 4.18757 0.320232
\(172\) 0 0
\(173\) 6.58645 0.500759 0.250379 0.968148i \(-0.419445\pi\)
0.250379 + 0.968148i \(0.419445\pi\)
\(174\) 0 0
\(175\) 11.3785 0.860136
\(176\) 0 0
\(177\) 15.0829 1.13370
\(178\) 0 0
\(179\) −17.1920 −1.28499 −0.642494 0.766290i \(-0.722100\pi\)
−0.642494 + 0.766290i \(0.722100\pi\)
\(180\) 0 0
\(181\) −24.8573 −1.84763 −0.923815 0.382838i \(-0.874947\pi\)
−0.923815 + 0.382838i \(0.874947\pi\)
\(182\) 0 0
\(183\) 6.30014 0.465720
\(184\) 0 0
\(185\) 3.94273 0.289875
\(186\) 0 0
\(187\) 1.26241 0.0923167
\(188\) 0 0
\(189\) −2.37531 −0.172778
\(190\) 0 0
\(191\) −5.29302 −0.382990 −0.191495 0.981494i \(-0.561334\pi\)
−0.191495 + 0.981494i \(0.561334\pi\)
\(192\) 0 0
\(193\) 3.18857 0.229519 0.114759 0.993393i \(-0.463390\pi\)
0.114759 + 0.993393i \(0.463390\pi\)
\(194\) 0 0
\(195\) −0.679427 −0.0486548
\(196\) 0 0
\(197\) −0.992148 −0.0706876 −0.0353438 0.999375i \(-0.511253\pi\)
−0.0353438 + 0.999375i \(0.511253\pi\)
\(198\) 0 0
\(199\) 7.00374 0.496482 0.248241 0.968698i \(-0.420148\pi\)
0.248241 + 0.968698i \(0.420148\pi\)
\(200\) 0 0
\(201\) −10.7228 −0.756331
\(202\) 0 0
\(203\) −19.3716 −1.35962
\(204\) 0 0
\(205\) 5.56929 0.388976
\(206\) 0 0
\(207\) −1.60277 −0.111400
\(208\) 0 0
\(209\) −7.81126 −0.540316
\(210\) 0 0
\(211\) −10.7769 −0.741914 −0.370957 0.928650i \(-0.620970\pi\)
−0.370957 + 0.928650i \(0.620970\pi\)
\(212\) 0 0
\(213\) −1.23296 −0.0844810
\(214\) 0 0
\(215\) −0.828185 −0.0564818
\(216\) 0 0
\(217\) 3.48722 0.236728
\(218\) 0 0
\(219\) −10.3160 −0.697093
\(220\) 0 0
\(221\) −1.00422 −0.0675512
\(222\) 0 0
\(223\) −1.32658 −0.0888346 −0.0444173 0.999013i \(-0.514143\pi\)
−0.0444173 + 0.999013i \(0.514143\pi\)
\(224\) 0 0
\(225\) −4.79034 −0.319356
\(226\) 0 0
\(227\) −5.38477 −0.357400 −0.178700 0.983904i \(-0.557189\pi\)
−0.178700 + 0.983904i \(0.557189\pi\)
\(228\) 0 0
\(229\) −5.16299 −0.341180 −0.170590 0.985342i \(-0.554567\pi\)
−0.170590 + 0.985342i \(0.554567\pi\)
\(230\) 0 0
\(231\) 4.43076 0.291523
\(232\) 0 0
\(233\) −6.17331 −0.404427 −0.202213 0.979341i \(-0.564813\pi\)
−0.202213 + 0.979341i \(0.564813\pi\)
\(234\) 0 0
\(235\) −2.07546 −0.135388
\(236\) 0 0
\(237\) −6.40327 −0.415937
\(238\) 0 0
\(239\) 14.2571 0.922215 0.461108 0.887344i \(-0.347452\pi\)
0.461108 + 0.887344i \(0.347452\pi\)
\(240\) 0 0
\(241\) 13.2484 0.853407 0.426704 0.904392i \(-0.359675\pi\)
0.426704 + 0.904392i \(0.359675\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0.621772 0.0397236
\(246\) 0 0
\(247\) 6.21369 0.395368
\(248\) 0 0
\(249\) −10.9254 −0.692370
\(250\) 0 0
\(251\) −20.4505 −1.29082 −0.645411 0.763835i \(-0.723314\pi\)
−0.645411 + 0.763835i \(0.723314\pi\)
\(252\) 0 0
\(253\) 2.98972 0.187962
\(254\) 0 0
\(255\) 0.309883 0.0194056
\(256\) 0 0
\(257\) 29.6854 1.85172 0.925861 0.377865i \(-0.123342\pi\)
0.925861 + 0.377865i \(0.123342\pi\)
\(258\) 0 0
\(259\) 20.4532 1.27090
\(260\) 0 0
\(261\) 8.15542 0.504808
\(262\) 0 0
\(263\) −19.1828 −1.18286 −0.591431 0.806356i \(-0.701437\pi\)
−0.591431 + 0.806356i \(0.701437\pi\)
\(264\) 0 0
\(265\) 3.23854 0.198942
\(266\) 0 0
\(267\) −2.83435 −0.173460
\(268\) 0 0
\(269\) 30.3063 1.84781 0.923904 0.382624i \(-0.124979\pi\)
0.923904 + 0.382624i \(0.124979\pi\)
\(270\) 0 0
\(271\) 24.7437 1.50307 0.751537 0.659691i \(-0.229313\pi\)
0.751537 + 0.659691i \(0.229313\pi\)
\(272\) 0 0
\(273\) −3.52458 −0.213317
\(274\) 0 0
\(275\) 8.93564 0.538839
\(276\) 0 0
\(277\) 3.25833 0.195774 0.0978869 0.995198i \(-0.468792\pi\)
0.0978869 + 0.995198i \(0.468792\pi\)
\(278\) 0 0
\(279\) −1.46812 −0.0878938
\(280\) 0 0
\(281\) 16.6904 0.995668 0.497834 0.867272i \(-0.334129\pi\)
0.497834 + 0.867272i \(0.334129\pi\)
\(282\) 0 0
\(283\) −6.95153 −0.413225 −0.206613 0.978423i \(-0.566244\pi\)
−0.206613 + 0.978423i \(0.566244\pi\)
\(284\) 0 0
\(285\) −1.91742 −0.113578
\(286\) 0 0
\(287\) 28.8911 1.70539
\(288\) 0 0
\(289\) −16.5420 −0.973058
\(290\) 0 0
\(291\) −6.73223 −0.394650
\(292\) 0 0
\(293\) −8.01237 −0.468088 −0.234044 0.972226i \(-0.575196\pi\)
−0.234044 + 0.972226i \(0.575196\pi\)
\(294\) 0 0
\(295\) −6.90622 −0.402096
\(296\) 0 0
\(297\) −1.86534 −0.108238
\(298\) 0 0
\(299\) −2.37826 −0.137538
\(300\) 0 0
\(301\) −4.29627 −0.247633
\(302\) 0 0
\(303\) −7.45223 −0.428119
\(304\) 0 0
\(305\) −2.88473 −0.165179
\(306\) 0 0
\(307\) 27.0713 1.54504 0.772519 0.634991i \(-0.218996\pi\)
0.772519 + 0.634991i \(0.218996\pi\)
\(308\) 0 0
\(309\) −6.11552 −0.347900
\(310\) 0 0
\(311\) 24.6711 1.39897 0.699484 0.714648i \(-0.253413\pi\)
0.699484 + 0.714648i \(0.253413\pi\)
\(312\) 0 0
\(313\) −3.74346 −0.211593 −0.105797 0.994388i \(-0.533739\pi\)
−0.105797 + 0.994388i \(0.533739\pi\)
\(314\) 0 0
\(315\) 1.08761 0.0612801
\(316\) 0 0
\(317\) 34.0139 1.91041 0.955205 0.295944i \(-0.0956343\pi\)
0.955205 + 0.295944i \(0.0956343\pi\)
\(318\) 0 0
\(319\) −15.2127 −0.851746
\(320\) 0 0
\(321\) −13.0772 −0.729899
\(322\) 0 0
\(323\) −2.83403 −0.157690
\(324\) 0 0
\(325\) −7.10811 −0.394287
\(326\) 0 0
\(327\) 7.00238 0.387232
\(328\) 0 0
\(329\) −10.7666 −0.593582
\(330\) 0 0
\(331\) −22.0014 −1.20931 −0.604653 0.796489i \(-0.706688\pi\)
−0.604653 + 0.796489i \(0.706688\pi\)
\(332\) 0 0
\(333\) −8.61076 −0.471867
\(334\) 0 0
\(335\) 4.90981 0.268252
\(336\) 0 0
\(337\) −34.0326 −1.85387 −0.926936 0.375218i \(-0.877568\pi\)
−0.926936 + 0.375218i \(0.877568\pi\)
\(338\) 0 0
\(339\) −9.06929 −0.492576
\(340\) 0 0
\(341\) 2.73854 0.148300
\(342\) 0 0
\(343\) 19.8526 1.07194
\(344\) 0 0
\(345\) 0.733883 0.0395109
\(346\) 0 0
\(347\) 9.32656 0.500676 0.250338 0.968159i \(-0.419458\pi\)
0.250338 + 0.968159i \(0.419458\pi\)
\(348\) 0 0
\(349\) −12.5114 −0.669719 −0.334860 0.942268i \(-0.608689\pi\)
−0.334860 + 0.942268i \(0.608689\pi\)
\(350\) 0 0
\(351\) 1.48384 0.0792016
\(352\) 0 0
\(353\) −18.8349 −1.00248 −0.501242 0.865307i \(-0.667123\pi\)
−0.501242 + 0.865307i \(0.667123\pi\)
\(354\) 0 0
\(355\) 0.564552 0.0299633
\(356\) 0 0
\(357\) 1.60754 0.0850800
\(358\) 0 0
\(359\) 22.0405 1.16325 0.581626 0.813456i \(-0.302417\pi\)
0.581626 + 0.813456i \(0.302417\pi\)
\(360\) 0 0
\(361\) −1.46424 −0.0770655
\(362\) 0 0
\(363\) −7.52049 −0.394723
\(364\) 0 0
\(365\) 4.72354 0.247242
\(366\) 0 0
\(367\) −24.4521 −1.27639 −0.638195 0.769874i \(-0.720319\pi\)
−0.638195 + 0.769874i \(0.720319\pi\)
\(368\) 0 0
\(369\) −12.1631 −0.633186
\(370\) 0 0
\(371\) 16.8001 0.872220
\(372\) 0 0
\(373\) −25.9516 −1.34372 −0.671862 0.740676i \(-0.734505\pi\)
−0.671862 + 0.740676i \(0.734505\pi\)
\(374\) 0 0
\(375\) 4.48284 0.231493
\(376\) 0 0
\(377\) 12.1013 0.623251
\(378\) 0 0
\(379\) 17.6988 0.909128 0.454564 0.890714i \(-0.349795\pi\)
0.454564 + 0.890714i \(0.349795\pi\)
\(380\) 0 0
\(381\) 8.65569 0.443444
\(382\) 0 0
\(383\) 29.3960 1.50206 0.751032 0.660266i \(-0.229557\pi\)
0.751032 + 0.660266i \(0.229557\pi\)
\(384\) 0 0
\(385\) −2.02877 −0.103396
\(386\) 0 0
\(387\) 1.80872 0.0919426
\(388\) 0 0
\(389\) 1.25391 0.0635759 0.0317880 0.999495i \(-0.489880\pi\)
0.0317880 + 0.999495i \(0.489880\pi\)
\(390\) 0 0
\(391\) 1.08471 0.0548561
\(392\) 0 0
\(393\) 8.05546 0.406344
\(394\) 0 0
\(395\) 2.93195 0.147523
\(396\) 0 0
\(397\) −14.3131 −0.718354 −0.359177 0.933269i \(-0.616943\pi\)
−0.359177 + 0.933269i \(0.616943\pi\)
\(398\) 0 0
\(399\) −9.94676 −0.497961
\(400\) 0 0
\(401\) −26.3545 −1.31608 −0.658040 0.752983i \(-0.728614\pi\)
−0.658040 + 0.752983i \(0.728614\pi\)
\(402\) 0 0
\(403\) −2.17845 −0.108516
\(404\) 0 0
\(405\) −0.457884 −0.0227524
\(406\) 0 0
\(407\) 16.0620 0.796165
\(408\) 0 0
\(409\) −24.1880 −1.19602 −0.598009 0.801489i \(-0.704041\pi\)
−0.598009 + 0.801489i \(0.704041\pi\)
\(410\) 0 0
\(411\) −5.88865 −0.290466
\(412\) 0 0
\(413\) −35.8265 −1.76291
\(414\) 0 0
\(415\) 5.00257 0.245566
\(416\) 0 0
\(417\) 9.61934 0.471061
\(418\) 0 0
\(419\) 36.5880 1.78744 0.893720 0.448625i \(-0.148086\pi\)
0.893720 + 0.448625i \(0.148086\pi\)
\(420\) 0 0
\(421\) 7.82504 0.381369 0.190685 0.981651i \(-0.438929\pi\)
0.190685 + 0.981651i \(0.438929\pi\)
\(422\) 0 0
\(423\) 4.53272 0.220389
\(424\) 0 0
\(425\) 3.24197 0.157258
\(426\) 0 0
\(427\) −14.9648 −0.724195
\(428\) 0 0
\(429\) −2.76788 −0.133634
\(430\) 0 0
\(431\) 28.1822 1.35749 0.678745 0.734374i \(-0.262524\pi\)
0.678745 + 0.734374i \(0.262524\pi\)
\(432\) 0 0
\(433\) 21.7667 1.04604 0.523021 0.852320i \(-0.324805\pi\)
0.523021 + 0.852320i \(0.324805\pi\)
\(434\) 0 0
\(435\) −3.73423 −0.179043
\(436\) 0 0
\(437\) −6.71172 −0.321065
\(438\) 0 0
\(439\) −38.7660 −1.85020 −0.925099 0.379726i \(-0.876018\pi\)
−0.925099 + 0.379726i \(0.876018\pi\)
\(440\) 0 0
\(441\) −1.35793 −0.0646631
\(442\) 0 0
\(443\) −4.14294 −0.196837 −0.0984185 0.995145i \(-0.531378\pi\)
−0.0984185 + 0.995145i \(0.531378\pi\)
\(444\) 0 0
\(445\) 1.29780 0.0615218
\(446\) 0 0
\(447\) 10.3048 0.487403
\(448\) 0 0
\(449\) 23.8195 1.12411 0.562055 0.827100i \(-0.310011\pi\)
0.562055 + 0.827100i \(0.310011\pi\)
\(450\) 0 0
\(451\) 22.6884 1.06835
\(452\) 0 0
\(453\) −9.91415 −0.465807
\(454\) 0 0
\(455\) 1.61385 0.0756583
\(456\) 0 0
\(457\) −23.7137 −1.10928 −0.554640 0.832090i \(-0.687144\pi\)
−0.554640 + 0.832090i \(0.687144\pi\)
\(458\) 0 0
\(459\) −0.676771 −0.0315890
\(460\) 0 0
\(461\) −11.4104 −0.531437 −0.265719 0.964051i \(-0.585609\pi\)
−0.265719 + 0.964051i \(0.585609\pi\)
\(462\) 0 0
\(463\) 31.1257 1.44653 0.723267 0.690568i \(-0.242640\pi\)
0.723267 + 0.690568i \(0.242640\pi\)
\(464\) 0 0
\(465\) 0.672226 0.0311738
\(466\) 0 0
\(467\) 15.0522 0.696532 0.348266 0.937396i \(-0.386771\pi\)
0.348266 + 0.937396i \(0.386771\pi\)
\(468\) 0 0
\(469\) 25.4700 1.17610
\(470\) 0 0
\(471\) −15.1523 −0.698181
\(472\) 0 0
\(473\) −3.37389 −0.155132
\(474\) 0 0
\(475\) −20.0599 −0.920411
\(476\) 0 0
\(477\) −7.07284 −0.323843
\(478\) 0 0
\(479\) 9.31427 0.425580 0.212790 0.977098i \(-0.431745\pi\)
0.212790 + 0.977098i \(0.431745\pi\)
\(480\) 0 0
\(481\) −12.7770 −0.582581
\(482\) 0 0
\(483\) 3.80707 0.173228
\(484\) 0 0
\(485\) 3.08258 0.139973
\(486\) 0 0
\(487\) −24.4701 −1.10884 −0.554422 0.832235i \(-0.687061\pi\)
−0.554422 + 0.832235i \(0.687061\pi\)
\(488\) 0 0
\(489\) 8.57960 0.387983
\(490\) 0 0
\(491\) −3.82613 −0.172671 −0.0863354 0.996266i \(-0.527516\pi\)
−0.0863354 + 0.996266i \(0.527516\pi\)
\(492\) 0 0
\(493\) −5.51935 −0.248579
\(494\) 0 0
\(495\) 0.854111 0.0383894
\(496\) 0 0
\(497\) 2.92865 0.131368
\(498\) 0 0
\(499\) −7.14511 −0.319859 −0.159929 0.987128i \(-0.551127\pi\)
−0.159929 + 0.987128i \(0.551127\pi\)
\(500\) 0 0
\(501\) 1.00000 0.0446767
\(502\) 0 0
\(503\) 1.70327 0.0759451 0.0379726 0.999279i \(-0.487910\pi\)
0.0379726 + 0.999279i \(0.487910\pi\)
\(504\) 0 0
\(505\) 3.41225 0.151843
\(506\) 0 0
\(507\) −10.7982 −0.479566
\(508\) 0 0
\(509\) −10.7859 −0.478074 −0.239037 0.971010i \(-0.576832\pi\)
−0.239037 + 0.971010i \(0.576832\pi\)
\(510\) 0 0
\(511\) 24.5037 1.08398
\(512\) 0 0
\(513\) 4.18757 0.184886
\(514\) 0 0
\(515\) 2.80020 0.123391
\(516\) 0 0
\(517\) −8.45509 −0.371855
\(518\) 0 0
\(519\) 6.58645 0.289113
\(520\) 0 0
\(521\) 18.6019 0.814962 0.407481 0.913214i \(-0.366407\pi\)
0.407481 + 0.913214i \(0.366407\pi\)
\(522\) 0 0
\(523\) 34.9400 1.52782 0.763910 0.645323i \(-0.223277\pi\)
0.763910 + 0.645323i \(0.223277\pi\)
\(524\) 0 0
\(525\) 11.3785 0.496600
\(526\) 0 0
\(527\) 0.993579 0.0432810
\(528\) 0 0
\(529\) −20.4311 −0.888310
\(530\) 0 0
\(531\) 15.0829 0.654542
\(532\) 0 0
\(533\) −18.0481 −0.781751
\(534\) 0 0
\(535\) 5.98785 0.258877
\(536\) 0 0
\(537\) −17.1920 −0.741889
\(538\) 0 0
\(539\) 2.53300 0.109104
\(540\) 0 0
\(541\) 5.89556 0.253470 0.126735 0.991937i \(-0.459550\pi\)
0.126735 + 0.991937i \(0.459550\pi\)
\(542\) 0 0
\(543\) −24.8573 −1.06673
\(544\) 0 0
\(545\) −3.20628 −0.137342
\(546\) 0 0
\(547\) 32.6442 1.39577 0.697883 0.716212i \(-0.254126\pi\)
0.697883 + 0.716212i \(0.254126\pi\)
\(548\) 0 0
\(549\) 6.30014 0.268883
\(550\) 0 0
\(551\) 34.1514 1.45490
\(552\) 0 0
\(553\) 15.2097 0.646783
\(554\) 0 0
\(555\) 3.94273 0.167359
\(556\) 0 0
\(557\) −0.252911 −0.0107162 −0.00535810 0.999986i \(-0.501706\pi\)
−0.00535810 + 0.999986i \(0.501706\pi\)
\(558\) 0 0
\(559\) 2.68386 0.113515
\(560\) 0 0
\(561\) 1.26241 0.0532991
\(562\) 0 0
\(563\) −43.2541 −1.82294 −0.911471 0.411364i \(-0.865052\pi\)
−0.911471 + 0.411364i \(0.865052\pi\)
\(564\) 0 0
\(565\) 4.15268 0.174705
\(566\) 0 0
\(567\) −2.37531 −0.0997534
\(568\) 0 0
\(569\) 0.698883 0.0292987 0.0146494 0.999893i \(-0.495337\pi\)
0.0146494 + 0.999893i \(0.495337\pi\)
\(570\) 0 0
\(571\) −3.30453 −0.138290 −0.0691452 0.997607i \(-0.522027\pi\)
−0.0691452 + 0.997607i \(0.522027\pi\)
\(572\) 0 0
\(573\) −5.29302 −0.221119
\(574\) 0 0
\(575\) 7.67782 0.320187
\(576\) 0 0
\(577\) −4.99066 −0.207764 −0.103882 0.994590i \(-0.533126\pi\)
−0.103882 + 0.994590i \(0.533126\pi\)
\(578\) 0 0
\(579\) 3.18857 0.132513
\(580\) 0 0
\(581\) 25.9512 1.07664
\(582\) 0 0
\(583\) 13.1933 0.546410
\(584\) 0 0
\(585\) −0.679427 −0.0280908
\(586\) 0 0
\(587\) −20.3173 −0.838586 −0.419293 0.907851i \(-0.637722\pi\)
−0.419293 + 0.907851i \(0.637722\pi\)
\(588\) 0 0
\(589\) −6.14784 −0.253317
\(590\) 0 0
\(591\) −0.992148 −0.0408115
\(592\) 0 0
\(593\) 38.9314 1.59872 0.799361 0.600851i \(-0.205172\pi\)
0.799361 + 0.600851i \(0.205172\pi\)
\(594\) 0 0
\(595\) −0.736066 −0.0301758
\(596\) 0 0
\(597\) 7.00374 0.286644
\(598\) 0 0
\(599\) −31.7669 −1.29796 −0.648980 0.760805i \(-0.724804\pi\)
−0.648980 + 0.760805i \(0.724804\pi\)
\(600\) 0 0
\(601\) −18.8357 −0.768323 −0.384162 0.923266i \(-0.625509\pi\)
−0.384162 + 0.923266i \(0.625509\pi\)
\(602\) 0 0
\(603\) −10.7228 −0.436668
\(604\) 0 0
\(605\) 3.44351 0.139999
\(606\) 0 0
\(607\) 21.0801 0.855614 0.427807 0.903870i \(-0.359286\pi\)
0.427807 + 0.903870i \(0.359286\pi\)
\(608\) 0 0
\(609\) −19.3716 −0.784977
\(610\) 0 0
\(611\) 6.72584 0.272098
\(612\) 0 0
\(613\) 45.5832 1.84109 0.920543 0.390641i \(-0.127747\pi\)
0.920543 + 0.390641i \(0.127747\pi\)
\(614\) 0 0
\(615\) 5.56929 0.224575
\(616\) 0 0
\(617\) −17.9426 −0.722341 −0.361170 0.932500i \(-0.617623\pi\)
−0.361170 + 0.932500i \(0.617623\pi\)
\(618\) 0 0
\(619\) −22.6408 −0.910011 −0.455005 0.890489i \(-0.650363\pi\)
−0.455005 + 0.890489i \(0.650363\pi\)
\(620\) 0 0
\(621\) −1.60277 −0.0643170
\(622\) 0 0
\(623\) 6.73245 0.269730
\(624\) 0 0
\(625\) 21.8991 0.875964
\(626\) 0 0
\(627\) −7.81126 −0.311952
\(628\) 0 0
\(629\) 5.82751 0.232358
\(630\) 0 0
\(631\) −20.2768 −0.807207 −0.403604 0.914934i \(-0.632242\pi\)
−0.403604 + 0.914934i \(0.632242\pi\)
\(632\) 0 0
\(633\) −10.7769 −0.428344
\(634\) 0 0
\(635\) −3.96330 −0.157279
\(636\) 0 0
\(637\) −2.01495 −0.0798351
\(638\) 0 0
\(639\) −1.23296 −0.0487751
\(640\) 0 0
\(641\) 14.6088 0.577012 0.288506 0.957478i \(-0.406841\pi\)
0.288506 + 0.957478i \(0.406841\pi\)
\(642\) 0 0
\(643\) −21.8745 −0.862648 −0.431324 0.902197i \(-0.641953\pi\)
−0.431324 + 0.902197i \(0.641953\pi\)
\(644\) 0 0
\(645\) −0.828185 −0.0326098
\(646\) 0 0
\(647\) −43.4270 −1.70729 −0.853646 0.520854i \(-0.825614\pi\)
−0.853646 + 0.520854i \(0.825614\pi\)
\(648\) 0 0
\(649\) −28.1348 −1.10439
\(650\) 0 0
\(651\) 3.48722 0.136675
\(652\) 0 0
\(653\) 12.5298 0.490328 0.245164 0.969482i \(-0.421158\pi\)
0.245164 + 0.969482i \(0.421158\pi\)
\(654\) 0 0
\(655\) −3.68846 −0.144120
\(656\) 0 0
\(657\) −10.3160 −0.402467
\(658\) 0 0
\(659\) −8.77130 −0.341682 −0.170841 0.985299i \(-0.554648\pi\)
−0.170841 + 0.985299i \(0.554648\pi\)
\(660\) 0 0
\(661\) −3.37077 −0.131108 −0.0655538 0.997849i \(-0.520881\pi\)
−0.0655538 + 0.997849i \(0.520881\pi\)
\(662\) 0 0
\(663\) −1.00422 −0.0390007
\(664\) 0 0
\(665\) 4.55446 0.176614
\(666\) 0 0
\(667\) −13.0713 −0.506121
\(668\) 0 0
\(669\) −1.32658 −0.0512887
\(670\) 0 0
\(671\) −11.7519 −0.453678
\(672\) 0 0
\(673\) −41.0801 −1.58352 −0.791761 0.610831i \(-0.790836\pi\)
−0.791761 + 0.610831i \(0.790836\pi\)
\(674\) 0 0
\(675\) −4.79034 −0.184380
\(676\) 0 0
\(677\) −18.5374 −0.712448 −0.356224 0.934401i \(-0.615936\pi\)
−0.356224 + 0.934401i \(0.615936\pi\)
\(678\) 0 0
\(679\) 15.9911 0.613682
\(680\) 0 0
\(681\) −5.38477 −0.206345
\(682\) 0 0
\(683\) 35.2309 1.34807 0.674037 0.738697i \(-0.264559\pi\)
0.674037 + 0.738697i \(0.264559\pi\)
\(684\) 0 0
\(685\) 2.69632 0.103021
\(686\) 0 0
\(687\) −5.16299 −0.196980
\(688\) 0 0
\(689\) −10.4950 −0.399826
\(690\) 0 0
\(691\) 13.7748 0.524018 0.262009 0.965065i \(-0.415615\pi\)
0.262009 + 0.965065i \(0.415615\pi\)
\(692\) 0 0
\(693\) 4.43076 0.168311
\(694\) 0 0
\(695\) −4.40454 −0.167074
\(696\) 0 0
\(697\) 8.23164 0.311796
\(698\) 0 0
\(699\) −6.17331 −0.233496
\(700\) 0 0
\(701\) −34.3881 −1.29882 −0.649411 0.760438i \(-0.724985\pi\)
−0.649411 + 0.760438i \(0.724985\pi\)
\(702\) 0 0
\(703\) −36.0582 −1.35996
\(704\) 0 0
\(705\) −2.07546 −0.0781664
\(706\) 0 0
\(707\) 17.7013 0.665726
\(708\) 0 0
\(709\) 31.5151 1.18357 0.591787 0.806094i \(-0.298423\pi\)
0.591787 + 0.806094i \(0.298423\pi\)
\(710\) 0 0
\(711\) −6.40327 −0.240141
\(712\) 0 0
\(713\) 2.35305 0.0881225
\(714\) 0 0
\(715\) 1.26737 0.0473968
\(716\) 0 0
\(717\) 14.2571 0.532441
\(718\) 0 0
\(719\) 33.6542 1.25509 0.627546 0.778580i \(-0.284060\pi\)
0.627546 + 0.778580i \(0.284060\pi\)
\(720\) 0 0
\(721\) 14.5262 0.540985
\(722\) 0 0
\(723\) 13.2484 0.492715
\(724\) 0 0
\(725\) −39.0672 −1.45092
\(726\) 0 0
\(727\) 39.7782 1.47529 0.737645 0.675188i \(-0.235938\pi\)
0.737645 + 0.675188i \(0.235938\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −1.22409 −0.0452747
\(732\) 0 0
\(733\) −35.1183 −1.29712 −0.648562 0.761162i \(-0.724629\pi\)
−0.648562 + 0.761162i \(0.724629\pi\)
\(734\) 0 0
\(735\) 0.621772 0.0229344
\(736\) 0 0
\(737\) 20.0018 0.736775
\(738\) 0 0
\(739\) 43.7065 1.60777 0.803885 0.594785i \(-0.202763\pi\)
0.803885 + 0.594785i \(0.202763\pi\)
\(740\) 0 0
\(741\) 6.21369 0.228266
\(742\) 0 0
\(743\) −26.8588 −0.985353 −0.492676 0.870213i \(-0.663981\pi\)
−0.492676 + 0.870213i \(0.663981\pi\)
\(744\) 0 0
\(745\) −4.71842 −0.172870
\(746\) 0 0
\(747\) −10.9254 −0.399740
\(748\) 0 0
\(749\) 31.0624 1.13499
\(750\) 0 0
\(751\) 19.2922 0.703983 0.351992 0.936003i \(-0.385505\pi\)
0.351992 + 0.936003i \(0.385505\pi\)
\(752\) 0 0
\(753\) −20.4505 −0.745257
\(754\) 0 0
\(755\) 4.53953 0.165210
\(756\) 0 0
\(757\) −15.9118 −0.578323 −0.289161 0.957280i \(-0.593376\pi\)
−0.289161 + 0.957280i \(0.593376\pi\)
\(758\) 0 0
\(759\) 2.98972 0.108520
\(760\) 0 0
\(761\) 18.2200 0.660473 0.330237 0.943898i \(-0.392871\pi\)
0.330237 + 0.943898i \(0.392871\pi\)
\(762\) 0 0
\(763\) −16.6328 −0.602147
\(764\) 0 0
\(765\) 0.309883 0.0112038
\(766\) 0 0
\(767\) 22.3806 0.808118
\(768\) 0 0
\(769\) −19.9699 −0.720134 −0.360067 0.932926i \(-0.617246\pi\)
−0.360067 + 0.932926i \(0.617246\pi\)
\(770\) 0 0
\(771\) 29.6854 1.06909
\(772\) 0 0
\(773\) 40.7753 1.46659 0.733294 0.679912i \(-0.237982\pi\)
0.733294 + 0.679912i \(0.237982\pi\)
\(774\) 0 0
\(775\) 7.03278 0.252625
\(776\) 0 0
\(777\) 20.4532 0.733754
\(778\) 0 0
\(779\) −50.9339 −1.82490
\(780\) 0 0
\(781\) 2.29989 0.0822967
\(782\) 0 0
\(783\) 8.15542 0.291451
\(784\) 0 0
\(785\) 6.93799 0.247628
\(786\) 0 0
\(787\) 0.273881 0.00976282 0.00488141 0.999988i \(-0.498446\pi\)
0.00488141 + 0.999988i \(0.498446\pi\)
\(788\) 0 0
\(789\) −19.1828 −0.682925
\(790\) 0 0
\(791\) 21.5423 0.765957
\(792\) 0 0
\(793\) 9.34841 0.331972
\(794\) 0 0
\(795\) 3.23854 0.114859
\(796\) 0 0
\(797\) 36.2528 1.28414 0.642070 0.766646i \(-0.278076\pi\)
0.642070 + 0.766646i \(0.278076\pi\)
\(798\) 0 0
\(799\) −3.06762 −0.108525
\(800\) 0 0
\(801\) −2.83435 −0.100147
\(802\) 0 0
\(803\) 19.2429 0.679069
\(804\) 0 0
\(805\) −1.74320 −0.0614396
\(806\) 0 0
\(807\) 30.3063 1.06683
\(808\) 0 0
\(809\) −44.0893 −1.55010 −0.775049 0.631902i \(-0.782275\pi\)
−0.775049 + 0.631902i \(0.782275\pi\)
\(810\) 0 0
\(811\) 34.4312 1.20904 0.604522 0.796589i \(-0.293364\pi\)
0.604522 + 0.796589i \(0.293364\pi\)
\(812\) 0 0
\(813\) 24.7437 0.867800
\(814\) 0 0
\(815\) −3.92846 −0.137608
\(816\) 0 0
\(817\) 7.57416 0.264986
\(818\) 0 0
\(819\) −3.52458 −0.123159
\(820\) 0 0
\(821\) 28.6715 1.00064 0.500321 0.865840i \(-0.333215\pi\)
0.500321 + 0.865840i \(0.333215\pi\)
\(822\) 0 0
\(823\) 53.2814 1.85727 0.928636 0.370991i \(-0.120982\pi\)
0.928636 + 0.370991i \(0.120982\pi\)
\(824\) 0 0
\(825\) 8.93564 0.311099
\(826\) 0 0
\(827\) 19.0653 0.662964 0.331482 0.943462i \(-0.392451\pi\)
0.331482 + 0.943462i \(0.392451\pi\)
\(828\) 0 0
\(829\) −49.5445 −1.72075 −0.860375 0.509662i \(-0.829771\pi\)
−0.860375 + 0.509662i \(0.829771\pi\)
\(830\) 0 0
\(831\) 3.25833 0.113030
\(832\) 0 0
\(833\) 0.919005 0.0318417
\(834\) 0 0
\(835\) −0.457884 −0.0158457
\(836\) 0 0
\(837\) −1.46812 −0.0507455
\(838\) 0 0
\(839\) 19.2378 0.664162 0.332081 0.943251i \(-0.392249\pi\)
0.332081 + 0.943251i \(0.392249\pi\)
\(840\) 0 0
\(841\) 37.5108 1.29348
\(842\) 0 0
\(843\) 16.6904 0.574849
\(844\) 0 0
\(845\) 4.94433 0.170090
\(846\) 0 0
\(847\) 17.8635 0.613796
\(848\) 0 0
\(849\) −6.95153 −0.238576
\(850\) 0 0
\(851\) 13.8011 0.473095
\(852\) 0 0
\(853\) 12.1982 0.417657 0.208829 0.977952i \(-0.433035\pi\)
0.208829 + 0.977952i \(0.433035\pi\)
\(854\) 0 0
\(855\) −1.91742 −0.0655744
\(856\) 0 0
\(857\) 26.7473 0.913671 0.456836 0.889551i \(-0.348983\pi\)
0.456836 + 0.889551i \(0.348983\pi\)
\(858\) 0 0
\(859\) −32.5008 −1.10891 −0.554457 0.832213i \(-0.687074\pi\)
−0.554457 + 0.832213i \(0.687074\pi\)
\(860\) 0 0
\(861\) 28.8911 0.984606
\(862\) 0 0
\(863\) 30.9910 1.05495 0.527473 0.849572i \(-0.323140\pi\)
0.527473 + 0.849572i \(0.323140\pi\)
\(864\) 0 0
\(865\) −3.01583 −0.102541
\(866\) 0 0
\(867\) −16.5420 −0.561795
\(868\) 0 0
\(869\) 11.9443 0.405183
\(870\) 0 0
\(871\) −15.9110 −0.539123
\(872\) 0 0
\(873\) −6.73223 −0.227851
\(874\) 0 0
\(875\) −10.6481 −0.359972
\(876\) 0 0
\(877\) −11.1380 −0.376103 −0.188052 0.982159i \(-0.560217\pi\)
−0.188052 + 0.982159i \(0.560217\pi\)
\(878\) 0 0
\(879\) −8.01237 −0.270251
\(880\) 0 0
\(881\) −20.1387 −0.678491 −0.339245 0.940698i \(-0.610172\pi\)
−0.339245 + 0.940698i \(0.610172\pi\)
\(882\) 0 0
\(883\) −54.0866 −1.82016 −0.910079 0.414435i \(-0.863979\pi\)
−0.910079 + 0.414435i \(0.863979\pi\)
\(884\) 0 0
\(885\) −6.90622 −0.232150
\(886\) 0 0
\(887\) 9.52921 0.319960 0.159980 0.987120i \(-0.448857\pi\)
0.159980 + 0.987120i \(0.448857\pi\)
\(888\) 0 0
\(889\) −20.5599 −0.689557
\(890\) 0 0
\(891\) −1.86534 −0.0624914
\(892\) 0 0
\(893\) 18.9811 0.635179
\(894\) 0 0
\(895\) 7.87193 0.263130
\(896\) 0 0
\(897\) −2.37826 −0.0794077
\(898\) 0 0
\(899\) −11.9731 −0.399325
\(900\) 0 0
\(901\) 4.78669 0.159468
\(902\) 0 0
\(903\) −4.29627 −0.142971
\(904\) 0 0
\(905\) 11.3818 0.378343
\(906\) 0 0
\(907\) 6.16421 0.204679 0.102340 0.994750i \(-0.467367\pi\)
0.102340 + 0.994750i \(0.467367\pi\)
\(908\) 0 0
\(909\) −7.45223 −0.247175
\(910\) 0 0
\(911\) −34.4827 −1.14246 −0.571231 0.820790i \(-0.693534\pi\)
−0.571231 + 0.820790i \(0.693534\pi\)
\(912\) 0 0
\(913\) 20.3797 0.674468
\(914\) 0 0
\(915\) −2.88473 −0.0953663
\(916\) 0 0
\(917\) −19.1342 −0.631866
\(918\) 0 0
\(919\) 6.45550 0.212947 0.106474 0.994316i \(-0.466044\pi\)
0.106474 + 0.994316i \(0.466044\pi\)
\(920\) 0 0
\(921\) 27.0713 0.892029
\(922\) 0 0
\(923\) −1.82952 −0.0602192
\(924\) 0 0
\(925\) 41.2485 1.35624
\(926\) 0 0
\(927\) −6.11552 −0.200860
\(928\) 0 0
\(929\) −2.22817 −0.0731037 −0.0365519 0.999332i \(-0.511637\pi\)
−0.0365519 + 0.999332i \(0.511637\pi\)
\(930\) 0 0
\(931\) −5.68641 −0.186365
\(932\) 0 0
\(933\) 24.6711 0.807694
\(934\) 0 0
\(935\) −0.578038 −0.0189039
\(936\) 0 0
\(937\) 36.3491 1.18747 0.593737 0.804659i \(-0.297652\pi\)
0.593737 + 0.804659i \(0.297652\pi\)
\(938\) 0 0
\(939\) −3.74346 −0.122163
\(940\) 0 0
\(941\) −20.8809 −0.680696 −0.340348 0.940299i \(-0.610545\pi\)
−0.340348 + 0.940299i \(0.610545\pi\)
\(942\) 0 0
\(943\) 19.4947 0.634834
\(944\) 0 0
\(945\) 1.08761 0.0353801
\(946\) 0 0
\(947\) −25.8019 −0.838448 −0.419224 0.907883i \(-0.637698\pi\)
−0.419224 + 0.907883i \(0.637698\pi\)
\(948\) 0 0
\(949\) −15.3073 −0.496897
\(950\) 0 0
\(951\) 34.0139 1.10298
\(952\) 0 0
\(953\) −14.8439 −0.480841 −0.240421 0.970669i \(-0.577285\pi\)
−0.240421 + 0.970669i \(0.577285\pi\)
\(954\) 0 0
\(955\) 2.42359 0.0784255
\(956\) 0 0
\(957\) −15.2127 −0.491756
\(958\) 0 0
\(959\) 13.9873 0.451675
\(960\) 0 0
\(961\) −28.8446 −0.930472
\(962\) 0 0
\(963\) −13.0772 −0.421408
\(964\) 0 0
\(965\) −1.46000 −0.0469989
\(966\) 0 0
\(967\) −34.4866 −1.10901 −0.554507 0.832179i \(-0.687093\pi\)
−0.554507 + 0.832179i \(0.687093\pi\)
\(968\) 0 0
\(969\) −2.83403 −0.0910421
\(970\) 0 0
\(971\) 19.1329 0.614004 0.307002 0.951709i \(-0.400674\pi\)
0.307002 + 0.951709i \(0.400674\pi\)
\(972\) 0 0
\(973\) −22.8489 −0.732501
\(974\) 0 0
\(975\) −7.10811 −0.227642
\(976\) 0 0
\(977\) −19.0244 −0.608646 −0.304323 0.952569i \(-0.598430\pi\)
−0.304323 + 0.952569i \(0.598430\pi\)
\(978\) 0 0
\(979\) 5.28704 0.168975
\(980\) 0 0
\(981\) 7.00238 0.223569
\(982\) 0 0
\(983\) 57.6794 1.83969 0.919843 0.392286i \(-0.128316\pi\)
0.919843 + 0.392286i \(0.128316\pi\)
\(984\) 0 0
\(985\) 0.454288 0.0144748
\(986\) 0 0
\(987\) −10.7666 −0.342705
\(988\) 0 0
\(989\) −2.89897 −0.0921818
\(990\) 0 0
\(991\) −4.02410 −0.127830 −0.0639149 0.997955i \(-0.520359\pi\)
−0.0639149 + 0.997955i \(0.520359\pi\)
\(992\) 0 0
\(993\) −22.0014 −0.698194
\(994\) 0 0
\(995\) −3.20690 −0.101665
\(996\) 0 0
\(997\) −12.0269 −0.380895 −0.190447 0.981697i \(-0.560994\pi\)
−0.190447 + 0.981697i \(0.560994\pi\)
\(998\) 0 0
\(999\) −8.61076 −0.272432
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 4008.2.a.f.1.3 5
4.3 odd 2 8016.2.a.s.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4008.2.a.f.1.3 5 1.1 even 1 trivial
8016.2.a.s.1.3 5 4.3 odd 2