Properties

Label 8012.2.a.a.1.6
Level $8012$
Weight $2$
Character 8012.1
Self dual yes
Analytic conductor $63.976$
Analytic rank $1$
Dimension $79$
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] = [8012,2,Mod(1,8012)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8012, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8012.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8012 = 2^{2} \cdot 2003 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8012.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: \(63.9761420994\)
Analytic rank: \(1\)
Dimension: \(79\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Character \(\chi\) \(=\) 8012.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.17871 q^{3} -1.27908 q^{5} +2.10162 q^{7} +7.10419 q^{9} +O(q^{10})\) \(q-3.17871 q^{3} -1.27908 q^{5} +2.10162 q^{7} +7.10419 q^{9} -3.72848 q^{11} +5.44031 q^{13} +4.06583 q^{15} +5.20119 q^{17} +2.34715 q^{19} -6.68045 q^{21} -5.59231 q^{23} -3.36395 q^{25} -13.0460 q^{27} +7.67788 q^{29} -6.93668 q^{31} +11.8518 q^{33} -2.68815 q^{35} -5.90921 q^{37} -17.2932 q^{39} -2.29198 q^{41} -10.7154 q^{43} -9.08683 q^{45} -5.18881 q^{47} -2.58318 q^{49} -16.5331 q^{51} +9.40890 q^{53} +4.76903 q^{55} -7.46092 q^{57} +14.3119 q^{59} +10.8624 q^{61} +14.9303 q^{63} -6.95860 q^{65} -0.890134 q^{67} +17.7763 q^{69} -14.1973 q^{71} -8.46780 q^{73} +10.6930 q^{75} -7.83587 q^{77} -3.25307 q^{79} +20.1569 q^{81} +14.1541 q^{83} -6.65274 q^{85} -24.4058 q^{87} +6.98160 q^{89} +11.4335 q^{91} +22.0497 q^{93} -3.00220 q^{95} -7.14684 q^{97} -26.4879 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 79 q - 19 q^{3} - 40 q^{7} + 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 79 q - 19 q^{3} - 40 q^{7} + 72 q^{9} - 14 q^{11} - 3 q^{13} - 19 q^{15} - 30 q^{17} - 49 q^{19} + 7 q^{21} - 40 q^{23} + 51 q^{25} - 70 q^{27} + 2 q^{29} - 48 q^{31} - 25 q^{33} - 34 q^{35} - 35 q^{39} - 20 q^{41} - 104 q^{43} + 12 q^{45} - 38 q^{47} + 51 q^{49} - 41 q^{51} - q^{53} - 112 q^{55} - 34 q^{57} - 24 q^{59} - 120 q^{63} - 21 q^{65} - 67 q^{67} + 15 q^{69} - 28 q^{71} - 88 q^{73} - 103 q^{75} + 4 q^{77} - 99 q^{79} + 47 q^{81} - 70 q^{83} + 7 q^{85} - 109 q^{87} - 50 q^{89} - 83 q^{91} - 7 q^{93} - 61 q^{95} - 93 q^{97} - 78 q^{99}+O(q^{100}) \) 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
\(3\) −3.17871 −1.83523 −0.917614 0.397473i \(-0.869887\pi\)
−0.917614 + 0.397473i \(0.869887\pi\)
\(4\) 0 0
\(5\) −1.27908 −0.572022 −0.286011 0.958226i \(-0.592329\pi\)
−0.286011 + 0.958226i \(0.592329\pi\)
\(6\) 0 0
\(7\) 2.10162 0.794339 0.397170 0.917745i \(-0.369992\pi\)
0.397170 + 0.917745i \(0.369992\pi\)
\(8\) 0 0
\(9\) 7.10419 2.36806
\(10\) 0 0
\(11\) −3.72848 −1.12418 −0.562090 0.827076i \(-0.690003\pi\)
−0.562090 + 0.827076i \(0.690003\pi\)
\(12\) 0 0
\(13\) 5.44031 1.50887 0.754435 0.656375i \(-0.227911\pi\)
0.754435 + 0.656375i \(0.227911\pi\)
\(14\) 0 0
\(15\) 4.06583 1.04979
\(16\) 0 0
\(17\) 5.20119 1.26147 0.630737 0.775997i \(-0.282753\pi\)
0.630737 + 0.775997i \(0.282753\pi\)
\(18\) 0 0
\(19\) 2.34715 0.538474 0.269237 0.963074i \(-0.413229\pi\)
0.269237 + 0.963074i \(0.413229\pi\)
\(20\) 0 0
\(21\) −6.68045 −1.45779
\(22\) 0 0
\(23\) −5.59231 −1.16608 −0.583039 0.812444i \(-0.698137\pi\)
−0.583039 + 0.812444i \(0.698137\pi\)
\(24\) 0 0
\(25\) −3.36395 −0.672790
\(26\) 0 0
\(27\) −13.0460 −2.51071
\(28\) 0 0
\(29\) 7.67788 1.42575 0.712874 0.701292i \(-0.247393\pi\)
0.712874 + 0.701292i \(0.247393\pi\)
\(30\) 0 0
\(31\) −6.93668 −1.24587 −0.622933 0.782276i \(-0.714059\pi\)
−0.622933 + 0.782276i \(0.714059\pi\)
\(32\) 0 0
\(33\) 11.8518 2.06313
\(34\) 0 0
\(35\) −2.68815 −0.454380
\(36\) 0 0
\(37\) −5.90921 −0.971468 −0.485734 0.874107i \(-0.661448\pi\)
−0.485734 + 0.874107i \(0.661448\pi\)
\(38\) 0 0
\(39\) −17.2932 −2.76912
\(40\) 0 0
\(41\) −2.29198 −0.357947 −0.178973 0.983854i \(-0.557278\pi\)
−0.178973 + 0.983854i \(0.557278\pi\)
\(42\) 0 0
\(43\) −10.7154 −1.63409 −0.817044 0.576575i \(-0.804389\pi\)
−0.817044 + 0.576575i \(0.804389\pi\)
\(44\) 0 0
\(45\) −9.08683 −1.35458
\(46\) 0 0
\(47\) −5.18881 −0.756866 −0.378433 0.925629i \(-0.623537\pi\)
−0.378433 + 0.925629i \(0.623537\pi\)
\(48\) 0 0
\(49\) −2.58318 −0.369025
\(50\) 0 0
\(51\) −16.5331 −2.31509
\(52\) 0 0
\(53\) 9.40890 1.29241 0.646206 0.763163i \(-0.276355\pi\)
0.646206 + 0.763163i \(0.276355\pi\)
\(54\) 0 0
\(55\) 4.76903 0.643056
\(56\) 0 0
\(57\) −7.46092 −0.988223
\(58\) 0 0
\(59\) 14.3119 1.86325 0.931627 0.363415i \(-0.118389\pi\)
0.931627 + 0.363415i \(0.118389\pi\)
\(60\) 0 0
\(61\) 10.8624 1.39078 0.695391 0.718631i \(-0.255231\pi\)
0.695391 + 0.718631i \(0.255231\pi\)
\(62\) 0 0
\(63\) 14.9303 1.88104
\(64\) 0 0
\(65\) −6.95860 −0.863108
\(66\) 0 0
\(67\) −0.890134 −0.108747 −0.0543736 0.998521i \(-0.517316\pi\)
−0.0543736 + 0.998521i \(0.517316\pi\)
\(68\) 0 0
\(69\) 17.7763 2.14002
\(70\) 0 0
\(71\) −14.1973 −1.68491 −0.842456 0.538764i \(-0.818891\pi\)
−0.842456 + 0.538764i \(0.818891\pi\)
\(72\) 0 0
\(73\) −8.46780 −0.991081 −0.495540 0.868585i \(-0.665030\pi\)
−0.495540 + 0.868585i \(0.665030\pi\)
\(74\) 0 0
\(75\) 10.6930 1.23472
\(76\) 0 0
\(77\) −7.83587 −0.892980
\(78\) 0 0
\(79\) −3.25307 −0.365999 −0.183000 0.983113i \(-0.558581\pi\)
−0.183000 + 0.983113i \(0.558581\pi\)
\(80\) 0 0
\(81\) 20.1569 2.23966
\(82\) 0 0
\(83\) 14.1541 1.55361 0.776805 0.629741i \(-0.216839\pi\)
0.776805 + 0.629741i \(0.216839\pi\)
\(84\) 0 0
\(85\) −6.65274 −0.721591
\(86\) 0 0
\(87\) −24.4058 −2.61657
\(88\) 0 0
\(89\) 6.98160 0.740048 0.370024 0.929022i \(-0.379349\pi\)
0.370024 + 0.929022i \(0.379349\pi\)
\(90\) 0 0
\(91\) 11.4335 1.19855
\(92\) 0 0
\(93\) 22.0497 2.28645
\(94\) 0 0
\(95\) −3.00220 −0.308019
\(96\) 0 0
\(97\) −7.14684 −0.725652 −0.362826 0.931857i \(-0.618188\pi\)
−0.362826 + 0.931857i \(0.618188\pi\)
\(98\) 0 0
\(99\) −26.4879 −2.66213
\(100\) 0 0
\(101\) −2.99616 −0.298129 −0.149065 0.988827i \(-0.547626\pi\)
−0.149065 + 0.988827i \(0.547626\pi\)
\(102\) 0 0
\(103\) 13.0645 1.28728 0.643642 0.765327i \(-0.277423\pi\)
0.643642 + 0.765327i \(0.277423\pi\)
\(104\) 0 0
\(105\) 8.54484 0.833891
\(106\) 0 0
\(107\) −10.5063 −1.01568 −0.507840 0.861451i \(-0.669556\pi\)
−0.507840 + 0.861451i \(0.669556\pi\)
\(108\) 0 0
\(109\) −9.88964 −0.947256 −0.473628 0.880725i \(-0.657056\pi\)
−0.473628 + 0.880725i \(0.657056\pi\)
\(110\) 0 0
\(111\) 18.7836 1.78286
\(112\) 0 0
\(113\) −6.21987 −0.585116 −0.292558 0.956248i \(-0.594506\pi\)
−0.292558 + 0.956248i \(0.594506\pi\)
\(114\) 0 0
\(115\) 7.15302 0.667023
\(116\) 0 0
\(117\) 38.6490 3.57310
\(118\) 0 0
\(119\) 10.9309 1.00204
\(120\) 0 0
\(121\) 2.90160 0.263782
\(122\) 0 0
\(123\) 7.28553 0.656914
\(124\) 0 0
\(125\) 10.6982 0.956874
\(126\) 0 0
\(127\) 20.8050 1.84614 0.923072 0.384626i \(-0.125670\pi\)
0.923072 + 0.384626i \(0.125670\pi\)
\(128\) 0 0
\(129\) 34.0612 2.99892
\(130\) 0 0
\(131\) −3.14247 −0.274559 −0.137279 0.990532i \(-0.543836\pi\)
−0.137279 + 0.990532i \(0.543836\pi\)
\(132\) 0 0
\(133\) 4.93283 0.427731
\(134\) 0 0
\(135\) 16.6869 1.43618
\(136\) 0 0
\(137\) 3.61305 0.308684 0.154342 0.988018i \(-0.450674\pi\)
0.154342 + 0.988018i \(0.450674\pi\)
\(138\) 0 0
\(139\) 6.95234 0.589690 0.294845 0.955545i \(-0.404732\pi\)
0.294845 + 0.955545i \(0.404732\pi\)
\(140\) 0 0
\(141\) 16.4937 1.38902
\(142\) 0 0
\(143\) −20.2841 −1.69624
\(144\) 0 0
\(145\) −9.82064 −0.815559
\(146\) 0 0
\(147\) 8.21117 0.677246
\(148\) 0 0
\(149\) −13.9782 −1.14514 −0.572568 0.819857i \(-0.694053\pi\)
−0.572568 + 0.819857i \(0.694053\pi\)
\(150\) 0 0
\(151\) −4.90853 −0.399451 −0.199725 0.979852i \(-0.564005\pi\)
−0.199725 + 0.979852i \(0.564005\pi\)
\(152\) 0 0
\(153\) 36.9502 2.98725
\(154\) 0 0
\(155\) 8.87258 0.712663
\(156\) 0 0
\(157\) −1.52372 −0.121606 −0.0608029 0.998150i \(-0.519366\pi\)
−0.0608029 + 0.998150i \(0.519366\pi\)
\(158\) 0 0
\(159\) −29.9082 −2.37187
\(160\) 0 0
\(161\) −11.7529 −0.926261
\(162\) 0 0
\(163\) 5.13735 0.402388 0.201194 0.979551i \(-0.435518\pi\)
0.201194 + 0.979551i \(0.435518\pi\)
\(164\) 0 0
\(165\) −15.1594 −1.18016
\(166\) 0 0
\(167\) 4.78075 0.369945 0.184973 0.982744i \(-0.440780\pi\)
0.184973 + 0.982744i \(0.440780\pi\)
\(168\) 0 0
\(169\) 16.5970 1.27669
\(170\) 0 0
\(171\) 16.6746 1.27514
\(172\) 0 0
\(173\) −13.4275 −1.02088 −0.510438 0.859915i \(-0.670517\pi\)
−0.510438 + 0.859915i \(0.670517\pi\)
\(174\) 0 0
\(175\) −7.06976 −0.534424
\(176\) 0 0
\(177\) −45.4935 −3.41950
\(178\) 0 0
\(179\) 4.34796 0.324982 0.162491 0.986710i \(-0.448047\pi\)
0.162491 + 0.986710i \(0.448047\pi\)
\(180\) 0 0
\(181\) −5.74834 −0.427271 −0.213635 0.976913i \(-0.568530\pi\)
−0.213635 + 0.976913i \(0.568530\pi\)
\(182\) 0 0
\(183\) −34.5283 −2.55240
\(184\) 0 0
\(185\) 7.55836 0.555701
\(186\) 0 0
\(187\) −19.3926 −1.41812
\(188\) 0 0
\(189\) −27.4178 −1.99435
\(190\) 0 0
\(191\) 2.61082 0.188913 0.0944563 0.995529i \(-0.469889\pi\)
0.0944563 + 0.995529i \(0.469889\pi\)
\(192\) 0 0
\(193\) −9.78415 −0.704278 −0.352139 0.935948i \(-0.614546\pi\)
−0.352139 + 0.935948i \(0.614546\pi\)
\(194\) 0 0
\(195\) 22.1193 1.58400
\(196\) 0 0
\(197\) 11.1065 0.791307 0.395653 0.918400i \(-0.370518\pi\)
0.395653 + 0.918400i \(0.370518\pi\)
\(198\) 0 0
\(199\) −25.8972 −1.83580 −0.917901 0.396810i \(-0.870117\pi\)
−0.917901 + 0.396810i \(0.870117\pi\)
\(200\) 0 0
\(201\) 2.82948 0.199576
\(202\) 0 0
\(203\) 16.1360 1.13253
\(204\) 0 0
\(205\) 2.93162 0.204754
\(206\) 0 0
\(207\) −39.7288 −2.76135
\(208\) 0 0
\(209\) −8.75133 −0.605342
\(210\) 0 0
\(211\) 4.67120 0.321579 0.160789 0.986989i \(-0.448596\pi\)
0.160789 + 0.986989i \(0.448596\pi\)
\(212\) 0 0
\(213\) 45.1292 3.09220
\(214\) 0 0
\(215\) 13.7059 0.934735
\(216\) 0 0
\(217\) −14.5783 −0.989639
\(218\) 0 0
\(219\) 26.9167 1.81886
\(220\) 0 0
\(221\) 28.2961 1.90340
\(222\) 0 0
\(223\) 17.0194 1.13970 0.569851 0.821748i \(-0.307001\pi\)
0.569851 + 0.821748i \(0.307001\pi\)
\(224\) 0 0
\(225\) −23.8981 −1.59321
\(226\) 0 0
\(227\) 0.00692156 0.000459400 0 0.000229700 1.00000i \(-0.499927\pi\)
0.000229700 1.00000i \(0.499927\pi\)
\(228\) 0 0
\(229\) −19.8935 −1.31460 −0.657299 0.753630i \(-0.728301\pi\)
−0.657299 + 0.753630i \(0.728301\pi\)
\(230\) 0 0
\(231\) 24.9080 1.63882
\(232\) 0 0
\(233\) 25.9958 1.70304 0.851521 0.524320i \(-0.175681\pi\)
0.851521 + 0.524320i \(0.175681\pi\)
\(234\) 0 0
\(235\) 6.63691 0.432944
\(236\) 0 0
\(237\) 10.3406 0.671692
\(238\) 0 0
\(239\) 22.1381 1.43199 0.715997 0.698103i \(-0.245972\pi\)
0.715997 + 0.698103i \(0.245972\pi\)
\(240\) 0 0
\(241\) −10.2920 −0.662969 −0.331484 0.943461i \(-0.607549\pi\)
−0.331484 + 0.943461i \(0.607549\pi\)
\(242\) 0 0
\(243\) −24.9349 −1.59957
\(244\) 0 0
\(245\) 3.30409 0.211091
\(246\) 0 0
\(247\) 12.7692 0.812487
\(248\) 0 0
\(249\) −44.9916 −2.85123
\(250\) 0 0
\(251\) −21.6564 −1.36694 −0.683470 0.729978i \(-0.739530\pi\)
−0.683470 + 0.729978i \(0.739530\pi\)
\(252\) 0 0
\(253\) 20.8509 1.31088
\(254\) 0 0
\(255\) 21.1471 1.32428
\(256\) 0 0
\(257\) −17.4986 −1.09154 −0.545768 0.837936i \(-0.683762\pi\)
−0.545768 + 0.837936i \(0.683762\pi\)
\(258\) 0 0
\(259\) −12.4189 −0.771675
\(260\) 0 0
\(261\) 54.5451 3.37626
\(262\) 0 0
\(263\) 10.7109 0.660463 0.330231 0.943900i \(-0.392873\pi\)
0.330231 + 0.943900i \(0.392873\pi\)
\(264\) 0 0
\(265\) −12.0347 −0.739289
\(266\) 0 0
\(267\) −22.1925 −1.35816
\(268\) 0 0
\(269\) 4.48215 0.273281 0.136641 0.990621i \(-0.456369\pi\)
0.136641 + 0.990621i \(0.456369\pi\)
\(270\) 0 0
\(271\) −15.1009 −0.917316 −0.458658 0.888613i \(-0.651670\pi\)
−0.458658 + 0.888613i \(0.651670\pi\)
\(272\) 0 0
\(273\) −36.3437 −2.19962
\(274\) 0 0
\(275\) 12.5424 0.756338
\(276\) 0 0
\(277\) −24.9920 −1.50162 −0.750811 0.660517i \(-0.770337\pi\)
−0.750811 + 0.660517i \(0.770337\pi\)
\(278\) 0 0
\(279\) −49.2795 −2.95029
\(280\) 0 0
\(281\) 18.9284 1.12917 0.564586 0.825374i \(-0.309036\pi\)
0.564586 + 0.825374i \(0.309036\pi\)
\(282\) 0 0
\(283\) 28.0710 1.66865 0.834325 0.551274i \(-0.185858\pi\)
0.834325 + 0.551274i \(0.185858\pi\)
\(284\) 0 0
\(285\) 9.54312 0.565286
\(286\) 0 0
\(287\) −4.81687 −0.284331
\(288\) 0 0
\(289\) 10.0524 0.591317
\(290\) 0 0
\(291\) 22.7177 1.33174
\(292\) 0 0
\(293\) −7.30729 −0.426896 −0.213448 0.976954i \(-0.568469\pi\)
−0.213448 + 0.976954i \(0.568469\pi\)
\(294\) 0 0
\(295\) −18.3061 −1.06582
\(296\) 0 0
\(297\) 48.6419 2.82249
\(298\) 0 0
\(299\) −30.4239 −1.75946
\(300\) 0 0
\(301\) −22.5198 −1.29802
\(302\) 0 0
\(303\) 9.52392 0.547135
\(304\) 0 0
\(305\) −13.8938 −0.795559
\(306\) 0 0
\(307\) −21.3771 −1.22006 −0.610029 0.792379i \(-0.708842\pi\)
−0.610029 + 0.792379i \(0.708842\pi\)
\(308\) 0 0
\(309\) −41.5283 −2.36246
\(310\) 0 0
\(311\) 15.6329 0.886458 0.443229 0.896408i \(-0.353833\pi\)
0.443229 + 0.896408i \(0.353833\pi\)
\(312\) 0 0
\(313\) −20.7482 −1.17276 −0.586378 0.810038i \(-0.699447\pi\)
−0.586378 + 0.810038i \(0.699447\pi\)
\(314\) 0 0
\(315\) −19.0971 −1.07600
\(316\) 0 0
\(317\) −1.88566 −0.105909 −0.0529545 0.998597i \(-0.516864\pi\)
−0.0529545 + 0.998597i \(0.516864\pi\)
\(318\) 0 0
\(319\) −28.6269 −1.60280
\(320\) 0 0
\(321\) 33.3964 1.86401
\(322\) 0 0
\(323\) 12.2080 0.679271
\(324\) 0 0
\(325\) −18.3009 −1.01515
\(326\) 0 0
\(327\) 31.4363 1.73843
\(328\) 0 0
\(329\) −10.9049 −0.601208
\(330\) 0 0
\(331\) −2.64651 −0.145465 −0.0727327 0.997351i \(-0.523172\pi\)
−0.0727327 + 0.997351i \(0.523172\pi\)
\(332\) 0 0
\(333\) −41.9801 −2.30050
\(334\) 0 0
\(335\) 1.13855 0.0622059
\(336\) 0 0
\(337\) −11.3187 −0.616571 −0.308286 0.951294i \(-0.599755\pi\)
−0.308286 + 0.951294i \(0.599755\pi\)
\(338\) 0 0
\(339\) 19.7711 1.07382
\(340\) 0 0
\(341\) 25.8633 1.40058
\(342\) 0 0
\(343\) −20.1402 −1.08747
\(344\) 0 0
\(345\) −22.7374 −1.22414
\(346\) 0 0
\(347\) −28.4058 −1.52490 −0.762451 0.647045i \(-0.776004\pi\)
−0.762451 + 0.647045i \(0.776004\pi\)
\(348\) 0 0
\(349\) 10.1102 0.541189 0.270594 0.962693i \(-0.412780\pi\)
0.270594 + 0.962693i \(0.412780\pi\)
\(350\) 0 0
\(351\) −70.9743 −3.78833
\(352\) 0 0
\(353\) 4.77887 0.254354 0.127177 0.991880i \(-0.459408\pi\)
0.127177 + 0.991880i \(0.459408\pi\)
\(354\) 0 0
\(355\) 18.1595 0.963808
\(356\) 0 0
\(357\) −34.7463 −1.83897
\(358\) 0 0
\(359\) 8.26001 0.435947 0.217973 0.975955i \(-0.430055\pi\)
0.217973 + 0.975955i \(0.430055\pi\)
\(360\) 0 0
\(361\) −13.4909 −0.710046
\(362\) 0 0
\(363\) −9.22333 −0.484100
\(364\) 0 0
\(365\) 10.8310 0.566920
\(366\) 0 0
\(367\) −1.69811 −0.0886408 −0.0443204 0.999017i \(-0.514112\pi\)
−0.0443204 + 0.999017i \(0.514112\pi\)
\(368\) 0 0
\(369\) −16.2826 −0.847640
\(370\) 0 0
\(371\) 19.7740 1.02661
\(372\) 0 0
\(373\) 15.4445 0.799688 0.399844 0.916583i \(-0.369064\pi\)
0.399844 + 0.916583i \(0.369064\pi\)
\(374\) 0 0
\(375\) −34.0064 −1.75608
\(376\) 0 0
\(377\) 41.7701 2.15127
\(378\) 0 0
\(379\) 3.25410 0.167152 0.0835759 0.996501i \(-0.473366\pi\)
0.0835759 + 0.996501i \(0.473366\pi\)
\(380\) 0 0
\(381\) −66.1330 −3.38810
\(382\) 0 0
\(383\) −36.7738 −1.87905 −0.939525 0.342479i \(-0.888733\pi\)
−0.939525 + 0.342479i \(0.888733\pi\)
\(384\) 0 0
\(385\) 10.0227 0.510805
\(386\) 0 0
\(387\) −76.1244 −3.86962
\(388\) 0 0
\(389\) 31.9193 1.61837 0.809187 0.587551i \(-0.199908\pi\)
0.809187 + 0.587551i \(0.199908\pi\)
\(390\) 0 0
\(391\) −29.0867 −1.47098
\(392\) 0 0
\(393\) 9.98899 0.503878
\(394\) 0 0
\(395\) 4.16094 0.209360
\(396\) 0 0
\(397\) 34.1598 1.71443 0.857215 0.514959i \(-0.172193\pi\)
0.857215 + 0.514959i \(0.172193\pi\)
\(398\) 0 0
\(399\) −15.6800 −0.784984
\(400\) 0 0
\(401\) 11.6354 0.581045 0.290523 0.956868i \(-0.406171\pi\)
0.290523 + 0.956868i \(0.406171\pi\)
\(402\) 0 0
\(403\) −37.7377 −1.87985
\(404\) 0 0
\(405\) −25.7823 −1.28113
\(406\) 0 0
\(407\) 22.0324 1.09211
\(408\) 0 0
\(409\) −16.0899 −0.795593 −0.397796 0.917474i \(-0.630225\pi\)
−0.397796 + 0.917474i \(0.630225\pi\)
\(410\) 0 0
\(411\) −11.4848 −0.566505
\(412\) 0 0
\(413\) 30.0783 1.48006
\(414\) 0 0
\(415\) −18.1042 −0.888700
\(416\) 0 0
\(417\) −22.0995 −1.08222
\(418\) 0 0
\(419\) 36.1402 1.76556 0.882782 0.469783i \(-0.155668\pi\)
0.882782 + 0.469783i \(0.155668\pi\)
\(420\) 0 0
\(421\) −32.6735 −1.59241 −0.796205 0.605027i \(-0.793162\pi\)
−0.796205 + 0.605027i \(0.793162\pi\)
\(422\) 0 0
\(423\) −36.8623 −1.79231
\(424\) 0 0
\(425\) −17.4966 −0.848707
\(426\) 0 0
\(427\) 22.8286 1.10475
\(428\) 0 0
\(429\) 64.4773 3.11299
\(430\) 0 0
\(431\) −1.50339 −0.0724160 −0.0362080 0.999344i \(-0.511528\pi\)
−0.0362080 + 0.999344i \(0.511528\pi\)
\(432\) 0 0
\(433\) 13.9950 0.672555 0.336277 0.941763i \(-0.390832\pi\)
0.336277 + 0.941763i \(0.390832\pi\)
\(434\) 0 0
\(435\) 31.2169 1.49674
\(436\) 0 0
\(437\) −13.1260 −0.627903
\(438\) 0 0
\(439\) −4.35894 −0.208041 −0.104020 0.994575i \(-0.533171\pi\)
−0.104020 + 0.994575i \(0.533171\pi\)
\(440\) 0 0
\(441\) −18.3514 −0.873875
\(442\) 0 0
\(443\) −24.0318 −1.14178 −0.570892 0.821025i \(-0.693403\pi\)
−0.570892 + 0.821025i \(0.693403\pi\)
\(444\) 0 0
\(445\) −8.93003 −0.423324
\(446\) 0 0
\(447\) 44.4325 2.10158
\(448\) 0 0
\(449\) −23.6604 −1.11660 −0.558302 0.829638i \(-0.688547\pi\)
−0.558302 + 0.829638i \(0.688547\pi\)
\(450\) 0 0
\(451\) 8.54560 0.402397
\(452\) 0 0
\(453\) 15.6028 0.733083
\(454\) 0 0
\(455\) −14.6243 −0.685600
\(456\) 0 0
\(457\) −31.7010 −1.48291 −0.741455 0.671003i \(-0.765864\pi\)
−0.741455 + 0.671003i \(0.765864\pi\)
\(458\) 0 0
\(459\) −67.8548 −3.16719
\(460\) 0 0
\(461\) 33.6680 1.56808 0.784038 0.620712i \(-0.213157\pi\)
0.784038 + 0.620712i \(0.213157\pi\)
\(462\) 0 0
\(463\) −12.7432 −0.592228 −0.296114 0.955153i \(-0.595691\pi\)
−0.296114 + 0.955153i \(0.595691\pi\)
\(464\) 0 0
\(465\) −28.2033 −1.30790
\(466\) 0 0
\(467\) −12.0257 −0.556485 −0.278242 0.960511i \(-0.589752\pi\)
−0.278242 + 0.960511i \(0.589752\pi\)
\(468\) 0 0
\(469\) −1.87073 −0.0863822
\(470\) 0 0
\(471\) 4.84345 0.223174
\(472\) 0 0
\(473\) 39.9523 1.83701
\(474\) 0 0
\(475\) −7.89571 −0.362280
\(476\) 0 0
\(477\) 66.8426 3.06051
\(478\) 0 0
\(479\) −32.2215 −1.47224 −0.736118 0.676853i \(-0.763343\pi\)
−0.736118 + 0.676853i \(0.763343\pi\)
\(480\) 0 0
\(481\) −32.1479 −1.46582
\(482\) 0 0
\(483\) 37.3592 1.69990
\(484\) 0 0
\(485\) 9.14139 0.415089
\(486\) 0 0
\(487\) −11.6158 −0.526361 −0.263181 0.964747i \(-0.584772\pi\)
−0.263181 + 0.964747i \(0.584772\pi\)
\(488\) 0 0
\(489\) −16.3301 −0.738474
\(490\) 0 0
\(491\) −4.50475 −0.203297 −0.101648 0.994820i \(-0.532412\pi\)
−0.101648 + 0.994820i \(0.532412\pi\)
\(492\) 0 0
\(493\) 39.9341 1.79854
\(494\) 0 0
\(495\) 33.8801 1.52280
\(496\) 0 0
\(497\) −29.8374 −1.33839
\(498\) 0 0
\(499\) 17.7092 0.792774 0.396387 0.918084i \(-0.370264\pi\)
0.396387 + 0.918084i \(0.370264\pi\)
\(500\) 0 0
\(501\) −15.1966 −0.678934
\(502\) 0 0
\(503\) −33.3299 −1.48611 −0.743053 0.669232i \(-0.766623\pi\)
−0.743053 + 0.669232i \(0.766623\pi\)
\(504\) 0 0
\(505\) 3.83233 0.170537
\(506\) 0 0
\(507\) −52.7569 −2.34302
\(508\) 0 0
\(509\) 3.96218 0.175620 0.0878102 0.996137i \(-0.472013\pi\)
0.0878102 + 0.996137i \(0.472013\pi\)
\(510\) 0 0
\(511\) −17.7961 −0.787254
\(512\) 0 0
\(513\) −30.6210 −1.35195
\(514\) 0 0
\(515\) −16.7106 −0.736356
\(516\) 0 0
\(517\) 19.3464 0.850854
\(518\) 0 0
\(519\) 42.6822 1.87354
\(520\) 0 0
\(521\) −29.8476 −1.30765 −0.653824 0.756647i \(-0.726836\pi\)
−0.653824 + 0.756647i \(0.726836\pi\)
\(522\) 0 0
\(523\) −4.37508 −0.191309 −0.0956544 0.995415i \(-0.530494\pi\)
−0.0956544 + 0.995415i \(0.530494\pi\)
\(524\) 0 0
\(525\) 22.4727 0.980789
\(526\) 0 0
\(527\) −36.0790 −1.57163
\(528\) 0 0
\(529\) 8.27397 0.359738
\(530\) 0 0
\(531\) 101.675 4.41230
\(532\) 0 0
\(533\) −12.4691 −0.540095
\(534\) 0 0
\(535\) 13.4384 0.580992
\(536\) 0 0
\(537\) −13.8209 −0.596416
\(538\) 0 0
\(539\) 9.63134 0.414851
\(540\) 0 0
\(541\) 15.6271 0.671862 0.335931 0.941887i \(-0.390949\pi\)
0.335931 + 0.941887i \(0.390949\pi\)
\(542\) 0 0
\(543\) 18.2723 0.784139
\(544\) 0 0
\(545\) 12.6497 0.541852
\(546\) 0 0
\(547\) −27.3422 −1.16907 −0.584534 0.811369i \(-0.698723\pi\)
−0.584534 + 0.811369i \(0.698723\pi\)
\(548\) 0 0
\(549\) 77.1682 3.29346
\(550\) 0 0
\(551\) 18.0212 0.767728
\(552\) 0 0
\(553\) −6.83674 −0.290728
\(554\) 0 0
\(555\) −24.0258 −1.01984
\(556\) 0 0
\(557\) −5.75092 −0.243674 −0.121837 0.992550i \(-0.538879\pi\)
−0.121837 + 0.992550i \(0.538879\pi\)
\(558\) 0 0
\(559\) −58.2953 −2.46563
\(560\) 0 0
\(561\) 61.6433 2.60258
\(562\) 0 0
\(563\) 2.20075 0.0927507 0.0463753 0.998924i \(-0.485233\pi\)
0.0463753 + 0.998924i \(0.485233\pi\)
\(564\) 0 0
\(565\) 7.95572 0.334699
\(566\) 0 0
\(567\) 42.3622 1.77905
\(568\) 0 0
\(569\) −3.99283 −0.167388 −0.0836942 0.996491i \(-0.526672\pi\)
−0.0836942 + 0.996491i \(0.526672\pi\)
\(570\) 0 0
\(571\) 4.73008 0.197948 0.0989738 0.995090i \(-0.468444\pi\)
0.0989738 + 0.995090i \(0.468444\pi\)
\(572\) 0 0
\(573\) −8.29905 −0.346698
\(574\) 0 0
\(575\) 18.8123 0.784526
\(576\) 0 0
\(577\) 38.0254 1.58302 0.791508 0.611158i \(-0.209296\pi\)
0.791508 + 0.611158i \(0.209296\pi\)
\(578\) 0 0
\(579\) 31.1010 1.29251
\(580\) 0 0
\(581\) 29.7465 1.23409
\(582\) 0 0
\(583\) −35.0809 −1.45290
\(584\) 0 0
\(585\) −49.4352 −2.04389
\(586\) 0 0
\(587\) 16.0470 0.662329 0.331164 0.943573i \(-0.392559\pi\)
0.331164 + 0.943573i \(0.392559\pi\)
\(588\) 0 0
\(589\) −16.2815 −0.670866
\(590\) 0 0
\(591\) −35.3044 −1.45223
\(592\) 0 0
\(593\) −18.3836 −0.754924 −0.377462 0.926025i \(-0.623203\pi\)
−0.377462 + 0.926025i \(0.623203\pi\)
\(594\) 0 0
\(595\) −13.9816 −0.573188
\(596\) 0 0
\(597\) 82.3195 3.36911
\(598\) 0 0
\(599\) −21.5833 −0.881868 −0.440934 0.897539i \(-0.645353\pi\)
−0.440934 + 0.897539i \(0.645353\pi\)
\(600\) 0 0
\(601\) 8.04840 0.328301 0.164150 0.986435i \(-0.447512\pi\)
0.164150 + 0.986435i \(0.447512\pi\)
\(602\) 0 0
\(603\) −6.32368 −0.257520
\(604\) 0 0
\(605\) −3.71138 −0.150889
\(606\) 0 0
\(607\) −35.6486 −1.44693 −0.723466 0.690360i \(-0.757452\pi\)
−0.723466 + 0.690360i \(0.757452\pi\)
\(608\) 0 0
\(609\) −51.2917 −2.07845
\(610\) 0 0
\(611\) −28.2287 −1.14201
\(612\) 0 0
\(613\) −13.7657 −0.555992 −0.277996 0.960582i \(-0.589670\pi\)
−0.277996 + 0.960582i \(0.589670\pi\)
\(614\) 0 0
\(615\) −9.31878 −0.375769
\(616\) 0 0
\(617\) 10.7078 0.431078 0.215539 0.976495i \(-0.430849\pi\)
0.215539 + 0.976495i \(0.430849\pi\)
\(618\) 0 0
\(619\) 7.38688 0.296904 0.148452 0.988920i \(-0.452571\pi\)
0.148452 + 0.988920i \(0.452571\pi\)
\(620\) 0 0
\(621\) 72.9574 2.92768
\(622\) 0 0
\(623\) 14.6727 0.587849
\(624\) 0 0
\(625\) 3.13593 0.125437
\(626\) 0 0
\(627\) 27.8179 1.11094
\(628\) 0 0
\(629\) −30.7349 −1.22548
\(630\) 0 0
\(631\) 15.7947 0.628779 0.314390 0.949294i \(-0.398200\pi\)
0.314390 + 0.949294i \(0.398200\pi\)
\(632\) 0 0
\(633\) −14.8484 −0.590171
\(634\) 0 0
\(635\) −26.6113 −1.05604
\(636\) 0 0
\(637\) −14.0533 −0.556811
\(638\) 0 0
\(639\) −100.860 −3.98998
\(640\) 0 0
\(641\) 0.387491 0.0153050 0.00765249 0.999971i \(-0.497564\pi\)
0.00765249 + 0.999971i \(0.497564\pi\)
\(642\) 0 0
\(643\) 14.2249 0.560977 0.280488 0.959857i \(-0.409504\pi\)
0.280488 + 0.959857i \(0.409504\pi\)
\(644\) 0 0
\(645\) −43.5671 −1.71545
\(646\) 0 0
\(647\) −2.91597 −0.114639 −0.0573193 0.998356i \(-0.518255\pi\)
−0.0573193 + 0.998356i \(0.518255\pi\)
\(648\) 0 0
\(649\) −53.3618 −2.09463
\(650\) 0 0
\(651\) 46.3402 1.81621
\(652\) 0 0
\(653\) 6.13227 0.239974 0.119987 0.992775i \(-0.461715\pi\)
0.119987 + 0.992775i \(0.461715\pi\)
\(654\) 0 0
\(655\) 4.01947 0.157054
\(656\) 0 0
\(657\) −60.1568 −2.34694
\(658\) 0 0
\(659\) 2.28241 0.0889102 0.0444551 0.999011i \(-0.485845\pi\)
0.0444551 + 0.999011i \(0.485845\pi\)
\(660\) 0 0
\(661\) −34.2139 −1.33077 −0.665384 0.746502i \(-0.731732\pi\)
−0.665384 + 0.746502i \(0.731732\pi\)
\(662\) 0 0
\(663\) −89.9450 −3.49317
\(664\) 0 0
\(665\) −6.30949 −0.244672
\(666\) 0 0
\(667\) −42.9371 −1.66253
\(668\) 0 0
\(669\) −54.0996 −2.09161
\(670\) 0 0
\(671\) −40.5001 −1.56349
\(672\) 0 0
\(673\) −30.4778 −1.17483 −0.587416 0.809285i \(-0.699855\pi\)
−0.587416 + 0.809285i \(0.699855\pi\)
\(674\) 0 0
\(675\) 43.8862 1.68918
\(676\) 0 0
\(677\) −36.1054 −1.38764 −0.693821 0.720147i \(-0.744074\pi\)
−0.693821 + 0.720147i \(0.744074\pi\)
\(678\) 0 0
\(679\) −15.0200 −0.576413
\(680\) 0 0
\(681\) −0.0220016 −0.000843104 0
\(682\) 0 0
\(683\) 1.14902 0.0439659 0.0219830 0.999758i \(-0.493002\pi\)
0.0219830 + 0.999758i \(0.493002\pi\)
\(684\) 0 0
\(685\) −4.62138 −0.176574
\(686\) 0 0
\(687\) 63.2355 2.41259
\(688\) 0 0
\(689\) 51.1873 1.95008
\(690\) 0 0
\(691\) 26.2295 0.997817 0.498909 0.866655i \(-0.333734\pi\)
0.498909 + 0.866655i \(0.333734\pi\)
\(692\) 0 0
\(693\) −55.6675 −2.11463
\(694\) 0 0
\(695\) −8.89261 −0.337316
\(696\) 0 0
\(697\) −11.9210 −0.451540
\(698\) 0 0
\(699\) −82.6331 −3.12547
\(700\) 0 0
\(701\) 30.0227 1.13394 0.566972 0.823737i \(-0.308115\pi\)
0.566972 + 0.823737i \(0.308115\pi\)
\(702\) 0 0
\(703\) −13.8698 −0.523110
\(704\) 0 0
\(705\) −21.0968 −0.794551
\(706\) 0 0
\(707\) −6.29680 −0.236816
\(708\) 0 0
\(709\) −16.4762 −0.618777 −0.309388 0.950936i \(-0.600124\pi\)
−0.309388 + 0.950936i \(0.600124\pi\)
\(710\) 0 0
\(711\) −23.1104 −0.866709
\(712\) 0 0
\(713\) 38.7921 1.45278
\(714\) 0 0
\(715\) 25.9450 0.970289
\(716\) 0 0
\(717\) −70.3705 −2.62804
\(718\) 0 0
\(719\) 45.7056 1.70453 0.852265 0.523110i \(-0.175228\pi\)
0.852265 + 0.523110i \(0.175228\pi\)
\(720\) 0 0
\(721\) 27.4567 1.02254
\(722\) 0 0
\(723\) 32.7154 1.21670
\(724\) 0 0
\(725\) −25.8280 −0.959229
\(726\) 0 0
\(727\) −38.8430 −1.44061 −0.720303 0.693659i \(-0.755997\pi\)
−0.720303 + 0.693659i \(0.755997\pi\)
\(728\) 0 0
\(729\) 18.7901 0.695928
\(730\) 0 0
\(731\) −55.7330 −2.06136
\(732\) 0 0
\(733\) −43.8160 −1.61838 −0.809191 0.587546i \(-0.800094\pi\)
−0.809191 + 0.587546i \(0.800094\pi\)
\(734\) 0 0
\(735\) −10.5028 −0.387400
\(736\) 0 0
\(737\) 3.31885 0.122252
\(738\) 0 0
\(739\) −12.2001 −0.448789 −0.224394 0.974498i \(-0.572040\pi\)
−0.224394 + 0.974498i \(0.572040\pi\)
\(740\) 0 0
\(741\) −40.5897 −1.49110
\(742\) 0 0
\(743\) 3.02770 0.111075 0.0555377 0.998457i \(-0.482313\pi\)
0.0555377 + 0.998457i \(0.482313\pi\)
\(744\) 0 0
\(745\) 17.8792 0.655043
\(746\) 0 0
\(747\) 100.553 3.67904
\(748\) 0 0
\(749\) −22.0802 −0.806795
\(750\) 0 0
\(751\) −10.5312 −0.384287 −0.192144 0.981367i \(-0.561544\pi\)
−0.192144 + 0.981367i \(0.561544\pi\)
\(752\) 0 0
\(753\) 68.8394 2.50865
\(754\) 0 0
\(755\) 6.27841 0.228495
\(756\) 0 0
\(757\) −38.0982 −1.38470 −0.692351 0.721561i \(-0.743425\pi\)
−0.692351 + 0.721561i \(0.743425\pi\)
\(758\) 0 0
\(759\) −66.2788 −2.40577
\(760\) 0 0
\(761\) −33.8117 −1.22567 −0.612837 0.790210i \(-0.709972\pi\)
−0.612837 + 0.790210i \(0.709972\pi\)
\(762\) 0 0
\(763\) −20.7843 −0.752442
\(764\) 0 0
\(765\) −47.2623 −1.70877
\(766\) 0 0
\(767\) 77.8613 2.81141
\(768\) 0 0
\(769\) −32.4037 −1.16851 −0.584254 0.811571i \(-0.698613\pi\)
−0.584254 + 0.811571i \(0.698613\pi\)
\(770\) 0 0
\(771\) 55.6231 2.00322
\(772\) 0 0
\(773\) 30.2161 1.08680 0.543399 0.839475i \(-0.317137\pi\)
0.543399 + 0.839475i \(0.317137\pi\)
\(774\) 0 0
\(775\) 23.3347 0.838206
\(776\) 0 0
\(777\) 39.4762 1.41620
\(778\) 0 0
\(779\) −5.37962 −0.192745
\(780\) 0 0
\(781\) 52.9345 1.89415
\(782\) 0 0
\(783\) −100.166 −3.57963
\(784\) 0 0
\(785\) 1.94896 0.0695612
\(786\) 0 0
\(787\) −12.2739 −0.437516 −0.218758 0.975779i \(-0.570201\pi\)
−0.218758 + 0.975779i \(0.570201\pi\)
\(788\) 0 0
\(789\) −34.0468 −1.21210
\(790\) 0 0
\(791\) −13.0718 −0.464781
\(792\) 0 0
\(793\) 59.0946 2.09851
\(794\) 0 0
\(795\) 38.2549 1.35676
\(796\) 0 0
\(797\) −0.132785 −0.00470349 −0.00235174 0.999997i \(-0.500749\pi\)
−0.00235174 + 0.999997i \(0.500749\pi\)
\(798\) 0 0
\(799\) −26.9880 −0.954767
\(800\) 0 0
\(801\) 49.5986 1.75248
\(802\) 0 0
\(803\) 31.5721 1.11415
\(804\) 0 0
\(805\) 15.0330 0.529842
\(806\) 0 0
\(807\) −14.2474 −0.501534
\(808\) 0 0
\(809\) 0.141139 0.00496218 0.00248109 0.999997i \(-0.499210\pi\)
0.00248109 + 0.999997i \(0.499210\pi\)
\(810\) 0 0
\(811\) 24.2468 0.851419 0.425709 0.904860i \(-0.360025\pi\)
0.425709 + 0.904860i \(0.360025\pi\)
\(812\) 0 0
\(813\) 48.0015 1.68348
\(814\) 0 0
\(815\) −6.57108 −0.230175
\(816\) 0 0
\(817\) −25.1508 −0.879914
\(818\) 0 0
\(819\) 81.2256 2.83825
\(820\) 0 0
\(821\) −33.3285 −1.16317 −0.581587 0.813484i \(-0.697568\pi\)
−0.581587 + 0.813484i \(0.697568\pi\)
\(822\) 0 0
\(823\) 3.20364 0.111672 0.0558359 0.998440i \(-0.482218\pi\)
0.0558359 + 0.998440i \(0.482218\pi\)
\(824\) 0 0
\(825\) −39.8688 −1.38805
\(826\) 0 0
\(827\) 21.1257 0.734612 0.367306 0.930100i \(-0.380280\pi\)
0.367306 + 0.930100i \(0.380280\pi\)
\(828\) 0 0
\(829\) −45.8570 −1.59268 −0.796340 0.604850i \(-0.793233\pi\)
−0.796340 + 0.604850i \(0.793233\pi\)
\(830\) 0 0
\(831\) 79.4422 2.75582
\(832\) 0 0
\(833\) −13.4356 −0.465516
\(834\) 0 0
\(835\) −6.11496 −0.211617
\(836\) 0 0
\(837\) 90.4961 3.12800
\(838\) 0 0
\(839\) 41.8162 1.44366 0.721829 0.692072i \(-0.243302\pi\)
0.721829 + 0.692072i \(0.243302\pi\)
\(840\) 0 0
\(841\) 29.9499 1.03276
\(842\) 0 0
\(843\) −60.1677 −2.07229
\(844\) 0 0
\(845\) −21.2288 −0.730295
\(846\) 0 0
\(847\) 6.09807 0.209532
\(848\) 0 0
\(849\) −89.2296 −3.06235
\(850\) 0 0
\(851\) 33.0461 1.13281
\(852\) 0 0
\(853\) −25.8933 −0.886568 −0.443284 0.896381i \(-0.646187\pi\)
−0.443284 + 0.896381i \(0.646187\pi\)
\(854\) 0 0
\(855\) −21.3282 −0.729409
\(856\) 0 0
\(857\) 5.23515 0.178829 0.0894147 0.995994i \(-0.471500\pi\)
0.0894147 + 0.995994i \(0.471500\pi\)
\(858\) 0 0
\(859\) 16.0132 0.546363 0.273182 0.961962i \(-0.411924\pi\)
0.273182 + 0.961962i \(0.411924\pi\)
\(860\) 0 0
\(861\) 15.3114 0.521812
\(862\) 0 0
\(863\) 18.1259 0.617014 0.308507 0.951222i \(-0.400171\pi\)
0.308507 + 0.951222i \(0.400171\pi\)
\(864\) 0 0
\(865\) 17.1749 0.583964
\(866\) 0 0
\(867\) −31.9536 −1.08520
\(868\) 0 0
\(869\) 12.1290 0.411449
\(870\) 0 0
\(871\) −4.84261 −0.164085
\(872\) 0 0
\(873\) −50.7725 −1.71839
\(874\) 0 0
\(875\) 22.4835 0.760082
\(876\) 0 0
\(877\) −19.5933 −0.661620 −0.330810 0.943697i \(-0.607322\pi\)
−0.330810 + 0.943697i \(0.607322\pi\)
\(878\) 0 0
\(879\) 23.2277 0.783452
\(880\) 0 0
\(881\) −1.23899 −0.0417428 −0.0208714 0.999782i \(-0.506644\pi\)
−0.0208714 + 0.999782i \(0.506644\pi\)
\(882\) 0 0
\(883\) 25.4579 0.856727 0.428364 0.903606i \(-0.359090\pi\)
0.428364 + 0.903606i \(0.359090\pi\)
\(884\) 0 0
\(885\) 58.1898 1.95603
\(886\) 0 0
\(887\) −42.1798 −1.41626 −0.708130 0.706083i \(-0.750461\pi\)
−0.708130 + 0.706083i \(0.750461\pi\)
\(888\) 0 0
\(889\) 43.7243 1.46646
\(890\) 0 0
\(891\) −75.1547 −2.51778
\(892\) 0 0
\(893\) −12.1789 −0.407553
\(894\) 0 0
\(895\) −5.56139 −0.185897
\(896\) 0 0
\(897\) 96.7087 3.22901
\(898\) 0 0
\(899\) −53.2591 −1.77629
\(900\) 0 0
\(901\) 48.9375 1.63034
\(902\) 0 0
\(903\) 71.5839 2.38216
\(904\) 0 0
\(905\) 7.35259 0.244408
\(906\) 0 0
\(907\) −22.3862 −0.743322 −0.371661 0.928369i \(-0.621212\pi\)
−0.371661 + 0.928369i \(0.621212\pi\)
\(908\) 0 0
\(909\) −21.2853 −0.705989
\(910\) 0 0
\(911\) −24.8677 −0.823904 −0.411952 0.911206i \(-0.635153\pi\)
−0.411952 + 0.911206i \(0.635153\pi\)
\(912\) 0 0
\(913\) −52.7732 −1.74654
\(914\) 0 0
\(915\) 44.1645 1.46003
\(916\) 0 0
\(917\) −6.60429 −0.218093
\(918\) 0 0
\(919\) 58.7609 1.93834 0.969172 0.246386i \(-0.0792432\pi\)
0.969172 + 0.246386i \(0.0792432\pi\)
\(920\) 0 0
\(921\) 67.9517 2.23909
\(922\) 0 0
\(923\) −77.2378 −2.54231
\(924\) 0 0
\(925\) 19.8783 0.653594
\(926\) 0 0
\(927\) 92.8127 3.04837
\(928\) 0 0
\(929\) 43.5440 1.42863 0.714317 0.699822i \(-0.246738\pi\)
0.714317 + 0.699822i \(0.246738\pi\)
\(930\) 0 0
\(931\) −6.06312 −0.198711
\(932\) 0 0
\(933\) −49.6923 −1.62685
\(934\) 0 0
\(935\) 24.8047 0.811199
\(936\) 0 0
\(937\) −1.25779 −0.0410903 −0.0205451 0.999789i \(-0.506540\pi\)
−0.0205451 + 0.999789i \(0.506540\pi\)
\(938\) 0 0
\(939\) 65.9524 2.15227
\(940\) 0 0
\(941\) −42.1912 −1.37539 −0.687697 0.725998i \(-0.741378\pi\)
−0.687697 + 0.725998i \(0.741378\pi\)
\(942\) 0 0
\(943\) 12.8175 0.417394
\(944\) 0 0
\(945\) 35.0696 1.14081
\(946\) 0 0
\(947\) 29.4161 0.955894 0.477947 0.878389i \(-0.341381\pi\)
0.477947 + 0.878389i \(0.341381\pi\)
\(948\) 0 0
\(949\) −46.0674 −1.49541
\(950\) 0 0
\(951\) 5.99396 0.194367
\(952\) 0 0
\(953\) −9.24426 −0.299451 −0.149726 0.988728i \(-0.547839\pi\)
−0.149726 + 0.988728i \(0.547839\pi\)
\(954\) 0 0
\(955\) −3.33946 −0.108062
\(956\) 0 0
\(957\) 90.9965 2.94150
\(958\) 0 0
\(959\) 7.59327 0.245199
\(960\) 0 0
\(961\) 17.1176 0.552180
\(962\) 0 0
\(963\) −74.6386 −2.40519
\(964\) 0 0
\(965\) 12.5147 0.402863
\(966\) 0 0
\(967\) 10.5302 0.338627 0.169313 0.985562i \(-0.445845\pi\)
0.169313 + 0.985562i \(0.445845\pi\)
\(968\) 0 0
\(969\) −38.8057 −1.24662
\(970\) 0 0
\(971\) −46.1256 −1.48024 −0.740120 0.672475i \(-0.765231\pi\)
−0.740120 + 0.672475i \(0.765231\pi\)
\(972\) 0 0
\(973\) 14.6112 0.468414
\(974\) 0 0
\(975\) 58.1733 1.86304
\(976\) 0 0
\(977\) −26.5947 −0.850841 −0.425420 0.904996i \(-0.639874\pi\)
−0.425420 + 0.904996i \(0.639874\pi\)
\(978\) 0 0
\(979\) −26.0308 −0.831947
\(980\) 0 0
\(981\) −70.2579 −2.24316
\(982\) 0 0
\(983\) −22.9146 −0.730861 −0.365430 0.930839i \(-0.619078\pi\)
−0.365430 + 0.930839i \(0.619078\pi\)
\(984\) 0 0
\(985\) −14.2061 −0.452645
\(986\) 0 0
\(987\) 34.6636 1.10335
\(988\) 0 0
\(989\) 59.9241 1.90547
\(990\) 0 0
\(991\) 40.8112 1.29641 0.648205 0.761466i \(-0.275520\pi\)
0.648205 + 0.761466i \(0.275520\pi\)
\(992\) 0 0
\(993\) 8.41249 0.266962
\(994\) 0 0
\(995\) 33.1246 1.05012
\(996\) 0 0
\(997\) −10.4399 −0.330636 −0.165318 0.986240i \(-0.552865\pi\)
−0.165318 + 0.986240i \(0.552865\pi\)
\(998\) 0 0
\(999\) 77.0916 2.43907
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 8012.2.a.a.1.6 79
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8012.2.a.a.1.6 79 1.1 even 1 trivial