Properties

Label 3040.2.a.q.1.3
Level $3040$
Weight $2$
Character 3040.1
Self dual yes
Analytic conductor $24.275$
Analytic rank $0$
Dimension $4$
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] = [3040,2,Mod(1,3040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3040, 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("3040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3040.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: \(24.2745222145\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.78292.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 10x^{2} + 8x + 18 \) 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 1.3
Root \(-1.09502\) of defining polynomial
Character \(\chi\) \(=\) 3040.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.09502 q^{3} -1.00000 q^{5} -2.80094 q^{7} -1.80094 q^{9} +O(q^{10})\) \(q+1.09502 q^{3} -1.00000 q^{5} -2.80094 q^{7} -1.80094 q^{9} +2.45617 q^{11} +3.55118 q^{13} -1.09502 q^{15} -4.34477 q^{17} -1.00000 q^{19} -3.06707 q^{21} +1.19906 q^{23} +1.00000 q^{25} -5.25711 q^{27} +7.44714 q^{29} +2.91234 q^{31} +2.68954 q^{33} +2.80094 q^{35} +7.89596 q^{37} +3.88860 q^{39} -10.5142 q^{41} +2.95568 q^{43} +1.80094 q^{45} +13.1457 q^{47} +0.845261 q^{49} -4.75760 q^{51} +3.74122 q^{53} -2.45617 q^{55} -1.09502 q^{57} +12.3595 q^{59} -2.45617 q^{61} +5.04432 q^{63} -3.55118 q^{65} -9.59453 q^{67} +1.31299 q^{69} -7.60188 q^{71} +2.75760 q^{73} +1.09502 q^{75} -6.87958 q^{77} +9.79191 q^{79} -0.353800 q^{81} +6.95568 q^{83} +4.34477 q^{85} +8.15474 q^{87} +8.49951 q^{89} -9.94665 q^{91} +3.18905 q^{93} +1.00000 q^{95} +2.01540 q^{97} -4.42341 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} - 4 q^{5} + 5 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{3} - 4 q^{5} + 5 q^{7} + 9 q^{9} + 6 q^{11} + 5 q^{13} + q^{15} - 5 q^{17} - 4 q^{19} - 3 q^{21} + 21 q^{23} + 4 q^{25} - q^{27} - q^{29} + 4 q^{31} - 14 q^{33} - 5 q^{35} + 10 q^{37} + 7 q^{39} - 2 q^{41} - 6 q^{43} - 9 q^{45} + 24 q^{47} + 5 q^{49} - 13 q^{51} - 5 q^{53} - 6 q^{55} + q^{57} + 11 q^{59} - 6 q^{61} + 38 q^{63} - 5 q^{65} - 19 q^{67} - 7 q^{69} + 2 q^{71} + 5 q^{73} - q^{75} + 8 q^{77} - 4 q^{79} - 16 q^{81} + 10 q^{83} + 5 q^{85} + 31 q^{87} + 20 q^{89} + 5 q^{91} - 26 q^{93} + 4 q^{95} - 6 q^{97} + 14 q^{99}+O(q^{100}) \) 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
\(3\) 1.09502 0.632208 0.316104 0.948725i \(-0.397625\pi\)
0.316104 + 0.948725i \(0.397625\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.80094 −1.05866 −0.529328 0.848417i \(-0.677556\pi\)
−0.529328 + 0.848417i \(0.677556\pi\)
\(8\) 0 0
\(9\) −1.80094 −0.600313
\(10\) 0 0
\(11\) 2.45617 0.740562 0.370281 0.928920i \(-0.379261\pi\)
0.370281 + 0.928920i \(0.379261\pi\)
\(12\) 0 0
\(13\) 3.55118 0.984921 0.492461 0.870335i \(-0.336098\pi\)
0.492461 + 0.870335i \(0.336098\pi\)
\(14\) 0 0
\(15\) −1.09502 −0.282732
\(16\) 0 0
\(17\) −4.34477 −1.05376 −0.526881 0.849939i \(-0.676639\pi\)
−0.526881 + 0.849939i \(0.676639\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −3.06707 −0.669290
\(22\) 0 0
\(23\) 1.19906 0.250021 0.125011 0.992155i \(-0.460103\pi\)
0.125011 + 0.992155i \(0.460103\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.25711 −1.01173
\(28\) 0 0
\(29\) 7.44714 1.38290 0.691450 0.722425i \(-0.256972\pi\)
0.691450 + 0.722425i \(0.256972\pi\)
\(30\) 0 0
\(31\) 2.91234 0.523071 0.261535 0.965194i \(-0.415771\pi\)
0.261535 + 0.965194i \(0.415771\pi\)
\(32\) 0 0
\(33\) 2.68954 0.468189
\(34\) 0 0
\(35\) 2.80094 0.473445
\(36\) 0 0
\(37\) 7.89596 1.29809 0.649044 0.760751i \(-0.275169\pi\)
0.649044 + 0.760751i \(0.275169\pi\)
\(38\) 0 0
\(39\) 3.88860 0.622675
\(40\) 0 0
\(41\) −10.5142 −1.64204 −0.821022 0.570896i \(-0.806596\pi\)
−0.821022 + 0.570896i \(0.806596\pi\)
\(42\) 0 0
\(43\) 2.95568 0.450737 0.225368 0.974274i \(-0.427641\pi\)
0.225368 + 0.974274i \(0.427641\pi\)
\(44\) 0 0
\(45\) 1.80094 0.268468
\(46\) 0 0
\(47\) 13.1457 1.91750 0.958750 0.284252i \(-0.0917453\pi\)
0.958750 + 0.284252i \(0.0917453\pi\)
\(48\) 0 0
\(49\) 0.845261 0.120752
\(50\) 0 0
\(51\) −4.75760 −0.666197
\(52\) 0 0
\(53\) 3.74122 0.513896 0.256948 0.966425i \(-0.417283\pi\)
0.256948 + 0.966425i \(0.417283\pi\)
\(54\) 0 0
\(55\) −2.45617 −0.331190
\(56\) 0 0
\(57\) −1.09502 −0.145038
\(58\) 0 0
\(59\) 12.3595 1.60907 0.804533 0.593908i \(-0.202416\pi\)
0.804533 + 0.593908i \(0.202416\pi\)
\(60\) 0 0
\(61\) −2.45617 −0.314480 −0.157240 0.987560i \(-0.550260\pi\)
−0.157240 + 0.987560i \(0.550260\pi\)
\(62\) 0 0
\(63\) 5.04432 0.635525
\(64\) 0 0
\(65\) −3.55118 −0.440470
\(66\) 0 0
\(67\) −9.59453 −1.17216 −0.586079 0.810254i \(-0.699329\pi\)
−0.586079 + 0.810254i \(0.699329\pi\)
\(68\) 0 0
\(69\) 1.31299 0.158066
\(70\) 0 0
\(71\) −7.60188 −0.902177 −0.451089 0.892479i \(-0.648964\pi\)
−0.451089 + 0.892479i \(0.648964\pi\)
\(72\) 0 0
\(73\) 2.75760 0.322752 0.161376 0.986893i \(-0.448407\pi\)
0.161376 + 0.986893i \(0.448407\pi\)
\(74\) 0 0
\(75\) 1.09502 0.126442
\(76\) 0 0
\(77\) −6.87958 −0.784000
\(78\) 0 0
\(79\) 9.79191 1.10168 0.550838 0.834612i \(-0.314308\pi\)
0.550838 + 0.834612i \(0.314308\pi\)
\(80\) 0 0
\(81\) −0.353800 −0.0393111
\(82\) 0 0
\(83\) 6.95568 0.763485 0.381742 0.924269i \(-0.375324\pi\)
0.381742 + 0.924269i \(0.375324\pi\)
\(84\) 0 0
\(85\) 4.34477 0.471257
\(86\) 0 0
\(87\) 8.15474 0.874280
\(88\) 0 0
\(89\) 8.49951 0.900946 0.450473 0.892790i \(-0.351255\pi\)
0.450473 + 0.892790i \(0.351255\pi\)
\(90\) 0 0
\(91\) −9.94665 −1.04269
\(92\) 0 0
\(93\) 3.18905 0.330690
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 2.01540 0.204633 0.102316 0.994752i \(-0.467375\pi\)
0.102316 + 0.994752i \(0.467375\pi\)
\(98\) 0 0
\(99\) −4.42341 −0.444569
\(100\) 0 0
\(101\) 4.95568 0.493108 0.246554 0.969129i \(-0.420702\pi\)
0.246554 + 0.969129i \(0.420702\pi\)
\(102\) 0 0
\(103\) 3.83791 0.378160 0.189080 0.981962i \(-0.439449\pi\)
0.189080 + 0.981962i \(0.439449\pi\)
\(104\) 0 0
\(105\) 3.06707 0.299316
\(106\) 0 0
\(107\) 9.59453 0.927538 0.463769 0.885956i \(-0.346497\pi\)
0.463769 + 0.885956i \(0.346497\pi\)
\(108\) 0 0
\(109\) 0.534804 0.0512250 0.0256125 0.999672i \(-0.491846\pi\)
0.0256125 + 0.999672i \(0.491846\pi\)
\(110\) 0 0
\(111\) 8.64620 0.820661
\(112\) 0 0
\(113\) 13.6879 1.28765 0.643823 0.765174i \(-0.277347\pi\)
0.643823 + 0.765174i \(0.277347\pi\)
\(114\) 0 0
\(115\) −1.19906 −0.111813
\(116\) 0 0
\(117\) −6.39547 −0.591261
\(118\) 0 0
\(119\) 12.1694 1.11557
\(120\) 0 0
\(121\) −4.96724 −0.451567
\(122\) 0 0
\(123\) −11.5132 −1.03811
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 17.5739 1.55943 0.779717 0.626132i \(-0.215363\pi\)
0.779717 + 0.626132i \(0.215363\pi\)
\(128\) 0 0
\(129\) 3.23652 0.284959
\(130\) 0 0
\(131\) −4.32418 −0.377805 −0.188903 0.981996i \(-0.560493\pi\)
−0.188903 + 0.981996i \(0.560493\pi\)
\(132\) 0 0
\(133\) 2.80094 0.242872
\(134\) 0 0
\(135\) 5.25711 0.452460
\(136\) 0 0
\(137\) −5.63717 −0.481616 −0.240808 0.970573i \(-0.577412\pi\)
−0.240808 + 0.970573i \(0.577412\pi\)
\(138\) 0 0
\(139\) 20.7476 1.75979 0.879894 0.475170i \(-0.157614\pi\)
0.879894 + 0.475170i \(0.157614\pi\)
\(140\) 0 0
\(141\) 14.3948 1.21226
\(142\) 0 0
\(143\) 8.72230 0.729396
\(144\) 0 0
\(145\) −7.44714 −0.618451
\(146\) 0 0
\(147\) 0.925574 0.0763401
\(148\) 0 0
\(149\) −17.2799 −1.41562 −0.707811 0.706402i \(-0.750317\pi\)
−0.707811 + 0.706402i \(0.750317\pi\)
\(150\) 0 0
\(151\) 8.10139 0.659282 0.329641 0.944106i \(-0.393072\pi\)
0.329641 + 0.944106i \(0.393072\pi\)
\(152\) 0 0
\(153\) 7.82467 0.632587
\(154\) 0 0
\(155\) −2.91234 −0.233924
\(156\) 0 0
\(157\) −6.76013 −0.539517 −0.269759 0.962928i \(-0.586944\pi\)
−0.269759 + 0.962928i \(0.586944\pi\)
\(158\) 0 0
\(159\) 4.09669 0.324889
\(160\) 0 0
\(161\) −3.35850 −0.264687
\(162\) 0 0
\(163\) 3.86801 0.302966 0.151483 0.988460i \(-0.451595\pi\)
0.151483 + 0.988460i \(0.451595\pi\)
\(164\) 0 0
\(165\) −2.68954 −0.209381
\(166\) 0 0
\(167\) 23.4545 1.81496 0.907481 0.420092i \(-0.138002\pi\)
0.907481 + 0.420092i \(0.138002\pi\)
\(168\) 0 0
\(169\) −0.389093 −0.0299303
\(170\) 0 0
\(171\) 1.80094 0.137721
\(172\) 0 0
\(173\) −20.7902 −1.58065 −0.790326 0.612686i \(-0.790089\pi\)
−0.790326 + 0.612686i \(0.790089\pi\)
\(174\) 0 0
\(175\) −2.80094 −0.211731
\(176\) 0 0
\(177\) 13.5338 1.01726
\(178\) 0 0
\(179\) 7.27770 0.543961 0.271980 0.962303i \(-0.412321\pi\)
0.271980 + 0.962303i \(0.412321\pi\)
\(180\) 0 0
\(181\) 8.82369 0.655860 0.327930 0.944702i \(-0.393649\pi\)
0.327930 + 0.944702i \(0.393649\pi\)
\(182\) 0 0
\(183\) −2.68954 −0.198817
\(184\) 0 0
\(185\) −7.89596 −0.580522
\(186\) 0 0
\(187\) −10.6715 −0.780376
\(188\) 0 0
\(189\) 14.7248 1.07107
\(190\) 0 0
\(191\) 6.03529 0.436699 0.218349 0.975871i \(-0.429933\pi\)
0.218349 + 0.975871i \(0.429933\pi\)
\(192\) 0 0
\(193\) −4.42823 −0.318751 −0.159375 0.987218i \(-0.550948\pi\)
−0.159375 + 0.987218i \(0.550948\pi\)
\(194\) 0 0
\(195\) −3.88860 −0.278469
\(196\) 0 0
\(197\) −16.9957 −1.21089 −0.605446 0.795887i \(-0.707005\pi\)
−0.605446 + 0.795887i \(0.707005\pi\)
\(198\) 0 0
\(199\) −22.4609 −1.59221 −0.796104 0.605160i \(-0.793109\pi\)
−0.796104 + 0.605160i \(0.793109\pi\)
\(200\) 0 0
\(201\) −10.5062 −0.741048
\(202\) 0 0
\(203\) −20.8590 −1.46401
\(204\) 0 0
\(205\) 10.5142 0.734345
\(206\) 0 0
\(207\) −2.15944 −0.150091
\(208\) 0 0
\(209\) −2.45617 −0.169897
\(210\) 0 0
\(211\) 4.22533 0.290883 0.145442 0.989367i \(-0.453540\pi\)
0.145442 + 0.989367i \(0.453540\pi\)
\(212\) 0 0
\(213\) −8.32418 −0.570363
\(214\) 0 0
\(215\) −2.95568 −0.201576
\(216\) 0 0
\(217\) −8.15727 −0.553752
\(218\) 0 0
\(219\) 3.01961 0.204047
\(220\) 0 0
\(221\) −15.4291 −1.03787
\(222\) 0 0
\(223\) −13.8673 −0.928624 −0.464312 0.885672i \(-0.653698\pi\)
−0.464312 + 0.885672i \(0.653698\pi\)
\(224\) 0 0
\(225\) −1.80094 −0.120063
\(226\) 0 0
\(227\) 25.0211 1.66071 0.830354 0.557237i \(-0.188138\pi\)
0.830354 + 0.557237i \(0.188138\pi\)
\(228\) 0 0
\(229\) −19.4699 −1.28661 −0.643303 0.765611i \(-0.722437\pi\)
−0.643303 + 0.765611i \(0.722437\pi\)
\(230\) 0 0
\(231\) −7.53325 −0.495651
\(232\) 0 0
\(233\) −2.58815 −0.169556 −0.0847778 0.996400i \(-0.527018\pi\)
−0.0847778 + 0.996400i \(0.527018\pi\)
\(234\) 0 0
\(235\) −13.1457 −0.857532
\(236\) 0 0
\(237\) 10.7223 0.696488
\(238\) 0 0
\(239\) −10.0500 −0.650080 −0.325040 0.945700i \(-0.605378\pi\)
−0.325040 + 0.945700i \(0.605378\pi\)
\(240\) 0 0
\(241\) 11.4677 0.738701 0.369351 0.929290i \(-0.379580\pi\)
0.369351 + 0.929290i \(0.379580\pi\)
\(242\) 0 0
\(243\) 15.3839 0.986878
\(244\) 0 0
\(245\) −0.845261 −0.0540017
\(246\) 0 0
\(247\) −3.55118 −0.225956
\(248\) 0 0
\(249\) 7.61658 0.482681
\(250\) 0 0
\(251\) −17.6725 −1.11548 −0.557738 0.830017i \(-0.688331\pi\)
−0.557738 + 0.830017i \(0.688331\pi\)
\(252\) 0 0
\(253\) 2.94509 0.185156
\(254\) 0 0
\(255\) 4.75760 0.297932
\(256\) 0 0
\(257\) −23.6879 −1.47761 −0.738804 0.673920i \(-0.764609\pi\)
−0.738804 + 0.673920i \(0.764609\pi\)
\(258\) 0 0
\(259\) −22.1161 −1.37423
\(260\) 0 0
\(261\) −13.4118 −0.830172
\(262\) 0 0
\(263\) 10.4381 0.643642 0.321821 0.946801i \(-0.395705\pi\)
0.321821 + 0.946801i \(0.395705\pi\)
\(264\) 0 0
\(265\) −3.74122 −0.229821
\(266\) 0 0
\(267\) 9.30710 0.569585
\(268\) 0 0
\(269\) 21.0137 1.28123 0.640615 0.767862i \(-0.278680\pi\)
0.640615 + 0.767862i \(0.278680\pi\)
\(270\) 0 0
\(271\) 14.4028 0.874909 0.437454 0.899241i \(-0.355880\pi\)
0.437454 + 0.899241i \(0.355880\pi\)
\(272\) 0 0
\(273\) −10.8917 −0.659198
\(274\) 0 0
\(275\) 2.45617 0.148112
\(276\) 0 0
\(277\) −13.6905 −0.822584 −0.411292 0.911504i \(-0.634922\pi\)
−0.411292 + 0.911504i \(0.634922\pi\)
\(278\) 0 0
\(279\) −5.24494 −0.314006
\(280\) 0 0
\(281\) −3.31046 −0.197485 −0.0987426 0.995113i \(-0.531482\pi\)
−0.0987426 + 0.995113i \(0.531482\pi\)
\(282\) 0 0
\(283\) 4.62814 0.275115 0.137557 0.990494i \(-0.456075\pi\)
0.137557 + 0.990494i \(0.456075\pi\)
\(284\) 0 0
\(285\) 1.09502 0.0648632
\(286\) 0 0
\(287\) 29.4497 1.73836
\(288\) 0 0
\(289\) 1.87704 0.110414
\(290\) 0 0
\(291\) 2.20690 0.129371
\(292\) 0 0
\(293\) 27.7867 1.62332 0.811659 0.584132i \(-0.198565\pi\)
0.811659 + 0.584132i \(0.198565\pi\)
\(294\) 0 0
\(295\) −12.3595 −0.719596
\(296\) 0 0
\(297\) −12.9123 −0.749250
\(298\) 0 0
\(299\) 4.25809 0.246251
\(300\) 0 0
\(301\) −8.27868 −0.477175
\(302\) 0 0
\(303\) 5.42655 0.311747
\(304\) 0 0
\(305\) 2.45617 0.140640
\(306\) 0 0
\(307\) 20.1260 1.14865 0.574325 0.818627i \(-0.305265\pi\)
0.574325 + 0.818627i \(0.305265\pi\)
\(308\) 0 0
\(309\) 4.20257 0.239076
\(310\) 0 0
\(311\) −21.5611 −1.22262 −0.611308 0.791393i \(-0.709356\pi\)
−0.611308 + 0.791393i \(0.709356\pi\)
\(312\) 0 0
\(313\) 8.02059 0.453351 0.226675 0.973970i \(-0.427214\pi\)
0.226675 + 0.973970i \(0.427214\pi\)
\(314\) 0 0
\(315\) −5.04432 −0.284215
\(316\) 0 0
\(317\) −29.0156 −1.62968 −0.814838 0.579689i \(-0.803174\pi\)
−0.814838 + 0.579689i \(0.803174\pi\)
\(318\) 0 0
\(319\) 18.2914 1.02412
\(320\) 0 0
\(321\) 10.5062 0.586397
\(322\) 0 0
\(323\) 4.34477 0.241750
\(324\) 0 0
\(325\) 3.55118 0.196984
\(326\) 0 0
\(327\) 0.585620 0.0323848
\(328\) 0 0
\(329\) −36.8203 −2.02997
\(330\) 0 0
\(331\) −18.5529 −1.01976 −0.509879 0.860246i \(-0.670310\pi\)
−0.509879 + 0.860246i \(0.670310\pi\)
\(332\) 0 0
\(333\) −14.2201 −0.779259
\(334\) 0 0
\(335\) 9.59453 0.524205
\(336\) 0 0
\(337\) 13.1144 0.714388 0.357194 0.934030i \(-0.383734\pi\)
0.357194 + 0.934030i \(0.383734\pi\)
\(338\) 0 0
\(339\) 14.9884 0.814060
\(340\) 0 0
\(341\) 7.15318 0.387367
\(342\) 0 0
\(343\) 17.2391 0.930821
\(344\) 0 0
\(345\) −1.31299 −0.0706891
\(346\) 0 0
\(347\) −23.1277 −1.24156 −0.620779 0.783986i \(-0.713183\pi\)
−0.620779 + 0.783986i \(0.713183\pi\)
\(348\) 0 0
\(349\) −16.9957 −0.909757 −0.454879 0.890553i \(-0.650317\pi\)
−0.454879 + 0.890553i \(0.650317\pi\)
\(350\) 0 0
\(351\) −18.6690 −0.996475
\(352\) 0 0
\(353\) 28.7542 1.53043 0.765217 0.643772i \(-0.222632\pi\)
0.765217 + 0.643772i \(0.222632\pi\)
\(354\) 0 0
\(355\) 7.60188 0.403466
\(356\) 0 0
\(357\) 13.3257 0.705273
\(358\) 0 0
\(359\) 1.97725 0.104355 0.0521776 0.998638i \(-0.483384\pi\)
0.0521776 + 0.998638i \(0.483384\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −5.43921 −0.285484
\(364\) 0 0
\(365\) −2.75760 −0.144339
\(366\) 0 0
\(367\) 7.90919 0.412857 0.206428 0.978462i \(-0.433816\pi\)
0.206428 + 0.978462i \(0.433816\pi\)
\(368\) 0 0
\(369\) 18.9355 0.985741
\(370\) 0 0
\(371\) −10.4789 −0.544038
\(372\) 0 0
\(373\) 8.96303 0.464088 0.232044 0.972705i \(-0.425459\pi\)
0.232044 + 0.972705i \(0.425459\pi\)
\(374\) 0 0
\(375\) −1.09502 −0.0565464
\(376\) 0 0
\(377\) 26.4462 1.36205
\(378\) 0 0
\(379\) −36.9751 −1.89928 −0.949641 0.313340i \(-0.898552\pi\)
−0.949641 + 0.313340i \(0.898552\pi\)
\(380\) 0 0
\(381\) 19.2437 0.985887
\(382\) 0 0
\(383\) 30.7469 1.57109 0.785546 0.618803i \(-0.212382\pi\)
0.785546 + 0.618803i \(0.212382\pi\)
\(384\) 0 0
\(385\) 6.87958 0.350616
\(386\) 0 0
\(387\) −5.32300 −0.270583
\(388\) 0 0
\(389\) 32.6862 1.65726 0.828628 0.559800i \(-0.189122\pi\)
0.828628 + 0.559800i \(0.189122\pi\)
\(390\) 0 0
\(391\) −5.20965 −0.263463
\(392\) 0 0
\(393\) −4.73505 −0.238852
\(394\) 0 0
\(395\) −9.79191 −0.492685
\(396\) 0 0
\(397\) −2.27868 −0.114363 −0.0571817 0.998364i \(-0.518211\pi\)
−0.0571817 + 0.998364i \(0.518211\pi\)
\(398\) 0 0
\(399\) 3.06707 0.153546
\(400\) 0 0
\(401\) 13.4971 0.674015 0.337007 0.941502i \(-0.390585\pi\)
0.337007 + 0.941502i \(0.390585\pi\)
\(402\) 0 0
\(403\) 10.3422 0.515184
\(404\) 0 0
\(405\) 0.353800 0.0175805
\(406\) 0 0
\(407\) 19.3938 0.961314
\(408\) 0 0
\(409\) 19.5691 0.967631 0.483815 0.875170i \(-0.339251\pi\)
0.483815 + 0.875170i \(0.339251\pi\)
\(410\) 0 0
\(411\) −6.17280 −0.304482
\(412\) 0 0
\(413\) −34.6181 −1.70345
\(414\) 0 0
\(415\) −6.95568 −0.341441
\(416\) 0 0
\(417\) 22.7189 1.11255
\(418\) 0 0
\(419\) −15.7507 −0.769473 −0.384737 0.923026i \(-0.625708\pi\)
−0.384737 + 0.923026i \(0.625708\pi\)
\(420\) 0 0
\(421\) 9.16846 0.446844 0.223422 0.974722i \(-0.428277\pi\)
0.223422 + 0.974722i \(0.428277\pi\)
\(422\) 0 0
\(423\) −23.6746 −1.15110
\(424\) 0 0
\(425\) −4.34477 −0.210752
\(426\) 0 0
\(427\) 6.87958 0.332926
\(428\) 0 0
\(429\) 9.55106 0.461130
\(430\) 0 0
\(431\) −10.2606 −0.494237 −0.247118 0.968985i \(-0.579484\pi\)
−0.247118 + 0.968985i \(0.579484\pi\)
\(432\) 0 0
\(433\) −26.1643 −1.25737 −0.628687 0.777659i \(-0.716407\pi\)
−0.628687 + 0.777659i \(0.716407\pi\)
\(434\) 0 0
\(435\) −8.15474 −0.390990
\(436\) 0 0
\(437\) −1.19906 −0.0573589
\(438\) 0 0
\(439\) 14.0853 0.672254 0.336127 0.941817i \(-0.390883\pi\)
0.336127 + 0.941817i \(0.390883\pi\)
\(440\) 0 0
\(441\) −1.52226 −0.0724887
\(442\) 0 0
\(443\) 17.1751 0.816014 0.408007 0.912979i \(-0.366224\pi\)
0.408007 + 0.912979i \(0.366224\pi\)
\(444\) 0 0
\(445\) −8.49951 −0.402915
\(446\) 0 0
\(447\) −18.9217 −0.894967
\(448\) 0 0
\(449\) −15.1024 −0.712725 −0.356362 0.934348i \(-0.615983\pi\)
−0.356362 + 0.934348i \(0.615983\pi\)
\(450\) 0 0
\(451\) −25.8247 −1.21604
\(452\) 0 0
\(453\) 8.87115 0.416803
\(454\) 0 0
\(455\) 9.94665 0.466306
\(456\) 0 0
\(457\) −25.5960 −1.19733 −0.598665 0.801000i \(-0.704302\pi\)
−0.598665 + 0.801000i \(0.704302\pi\)
\(458\) 0 0
\(459\) 22.8409 1.06612
\(460\) 0 0
\(461\) 32.8531 1.53012 0.765061 0.643958i \(-0.222709\pi\)
0.765061 + 0.643958i \(0.222709\pi\)
\(462\) 0 0
\(463\) −26.8182 −1.24635 −0.623173 0.782084i \(-0.714157\pi\)
−0.623173 + 0.782084i \(0.714157\pi\)
\(464\) 0 0
\(465\) −3.18905 −0.147889
\(466\) 0 0
\(467\) −13.7074 −0.634302 −0.317151 0.948375i \(-0.602726\pi\)
−0.317151 + 0.948375i \(0.602726\pi\)
\(468\) 0 0
\(469\) 26.8737 1.24091
\(470\) 0 0
\(471\) −7.40245 −0.341087
\(472\) 0 0
\(473\) 7.25964 0.333799
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) −6.73770 −0.308498
\(478\) 0 0
\(479\) 15.3685 0.702205 0.351102 0.936337i \(-0.385807\pi\)
0.351102 + 0.936337i \(0.385807\pi\)
\(480\) 0 0
\(481\) 28.0400 1.27851
\(482\) 0 0
\(483\) −3.67761 −0.167337
\(484\) 0 0
\(485\) −2.01540 −0.0915147
\(486\) 0 0
\(487\) 11.6445 0.527664 0.263832 0.964569i \(-0.415014\pi\)
0.263832 + 0.964569i \(0.415014\pi\)
\(488\) 0 0
\(489\) 4.23554 0.191538
\(490\) 0 0
\(491\) −30.7421 −1.38737 −0.693685 0.720278i \(-0.744014\pi\)
−0.693685 + 0.720278i \(0.744014\pi\)
\(492\) 0 0
\(493\) −32.3561 −1.45725
\(494\) 0 0
\(495\) 4.42341 0.198817
\(496\) 0 0
\(497\) 21.2924 0.955095
\(498\) 0 0
\(499\) −4.43811 −0.198677 −0.0993386 0.995054i \(-0.531673\pi\)
−0.0993386 + 0.995054i \(0.531673\pi\)
\(500\) 0 0
\(501\) 25.6830 1.14743
\(502\) 0 0
\(503\) −23.2104 −1.03490 −0.517451 0.855713i \(-0.673119\pi\)
−0.517451 + 0.855713i \(0.673119\pi\)
\(504\) 0 0
\(505\) −4.95568 −0.220525
\(506\) 0 0
\(507\) −0.426064 −0.0189221
\(508\) 0 0
\(509\) 14.4403 0.640054 0.320027 0.947408i \(-0.396308\pi\)
0.320027 + 0.947408i \(0.396308\pi\)
\(510\) 0 0
\(511\) −7.72386 −0.341683
\(512\) 0 0
\(513\) 5.25711 0.232107
\(514\) 0 0
\(515\) −3.83791 −0.169118
\(516\) 0 0
\(517\) 32.2881 1.42003
\(518\) 0 0
\(519\) −22.7656 −0.999301
\(520\) 0 0
\(521\) 33.6293 1.47333 0.736664 0.676259i \(-0.236400\pi\)
0.736664 + 0.676259i \(0.236400\pi\)
\(522\) 0 0
\(523\) −18.8542 −0.824435 −0.412218 0.911085i \(-0.635246\pi\)
−0.412218 + 0.911085i \(0.635246\pi\)
\(524\) 0 0
\(525\) −3.06707 −0.133858
\(526\) 0 0
\(527\) −12.6534 −0.551192
\(528\) 0 0
\(529\) −21.5623 −0.937489
\(530\) 0 0
\(531\) −22.2587 −0.965944
\(532\) 0 0
\(533\) −37.3379 −1.61728
\(534\) 0 0
\(535\) −9.59453 −0.414808
\(536\) 0 0
\(537\) 7.96920 0.343896
\(538\) 0 0
\(539\) 2.07610 0.0894241
\(540\) 0 0
\(541\) −21.3651 −0.918560 −0.459280 0.888292i \(-0.651892\pi\)
−0.459280 + 0.888292i \(0.651892\pi\)
\(542\) 0 0
\(543\) 9.66209 0.414640
\(544\) 0 0
\(545\) −0.534804 −0.0229085
\(546\) 0 0
\(547\) −31.6118 −1.35162 −0.675811 0.737075i \(-0.736206\pi\)
−0.675811 + 0.737075i \(0.736206\pi\)
\(548\) 0 0
\(549\) 4.42341 0.188786
\(550\) 0 0
\(551\) −7.44714 −0.317259
\(552\) 0 0
\(553\) −27.4265 −1.16630
\(554\) 0 0
\(555\) −8.64620 −0.367011
\(556\) 0 0
\(557\) 11.4858 0.486668 0.243334 0.969943i \(-0.421759\pi\)
0.243334 + 0.969943i \(0.421759\pi\)
\(558\) 0 0
\(559\) 10.4962 0.443940
\(560\) 0 0
\(561\) −11.6855 −0.493360
\(562\) 0 0
\(563\) 17.4725 0.736380 0.368190 0.929751i \(-0.379978\pi\)
0.368190 + 0.929751i \(0.379978\pi\)
\(564\) 0 0
\(565\) −13.6879 −0.575853
\(566\) 0 0
\(567\) 0.990972 0.0416169
\(568\) 0 0
\(569\) −16.6748 −0.699046 −0.349523 0.936928i \(-0.613656\pi\)
−0.349523 + 0.936928i \(0.613656\pi\)
\(570\) 0 0
\(571\) −9.23435 −0.386446 −0.193223 0.981155i \(-0.561894\pi\)
−0.193223 + 0.981155i \(0.561894\pi\)
\(572\) 0 0
\(573\) 6.60874 0.276084
\(574\) 0 0
\(575\) 1.19906 0.0500043
\(576\) 0 0
\(577\) −30.4281 −1.26674 −0.633369 0.773850i \(-0.718329\pi\)
−0.633369 + 0.773850i \(0.718329\pi\)
\(578\) 0 0
\(579\) −4.84898 −0.201517
\(580\) 0 0
\(581\) −19.4824 −0.808268
\(582\) 0 0
\(583\) 9.18905 0.380572
\(584\) 0 0
\(585\) 6.39547 0.264420
\(586\) 0 0
\(587\) −15.6040 −0.644048 −0.322024 0.946732i \(-0.604363\pi\)
−0.322024 + 0.946732i \(0.604363\pi\)
\(588\) 0 0
\(589\) −2.91234 −0.120001
\(590\) 0 0
\(591\) −18.6105 −0.765535
\(592\) 0 0
\(593\) 13.9639 0.573428 0.286714 0.958016i \(-0.407437\pi\)
0.286714 + 0.958016i \(0.407437\pi\)
\(594\) 0 0
\(595\) −12.1694 −0.498898
\(596\) 0 0
\(597\) −24.5950 −1.00661
\(598\) 0 0
\(599\) −23.7033 −0.968489 −0.484245 0.874933i \(-0.660906\pi\)
−0.484245 + 0.874933i \(0.660906\pi\)
\(600\) 0 0
\(601\) −46.5648 −1.89942 −0.949709 0.313135i \(-0.898621\pi\)
−0.949709 + 0.313135i \(0.898621\pi\)
\(602\) 0 0
\(603\) 17.2792 0.703662
\(604\) 0 0
\(605\) 4.96724 0.201947
\(606\) 0 0
\(607\) 32.7957 1.33114 0.665569 0.746336i \(-0.268189\pi\)
0.665569 + 0.746336i \(0.268189\pi\)
\(608\) 0 0
\(609\) −22.8409 −0.925561
\(610\) 0 0
\(611\) 46.6828 1.88859
\(612\) 0 0
\(613\) 2.98432 0.120536 0.0602678 0.998182i \(-0.480805\pi\)
0.0602678 + 0.998182i \(0.480805\pi\)
\(614\) 0 0
\(615\) 11.5132 0.464258
\(616\) 0 0
\(617\) 28.0833 1.13059 0.565296 0.824888i \(-0.308762\pi\)
0.565296 + 0.824888i \(0.308762\pi\)
\(618\) 0 0
\(619\) 31.1551 1.25223 0.626115 0.779731i \(-0.284644\pi\)
0.626115 + 0.779731i \(0.284644\pi\)
\(620\) 0 0
\(621\) −6.30359 −0.252954
\(622\) 0 0
\(623\) −23.8066 −0.953792
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −2.68954 −0.107410
\(628\) 0 0
\(629\) −34.3061 −1.36787
\(630\) 0 0
\(631\) −25.3665 −1.00983 −0.504913 0.863170i \(-0.668475\pi\)
−0.504913 + 0.863170i \(0.668475\pi\)
\(632\) 0 0
\(633\) 4.62680 0.183899
\(634\) 0 0
\(635\) −17.5739 −0.697400
\(636\) 0 0
\(637\) 3.00168 0.118931
\(638\) 0 0
\(639\) 13.6905 0.541589
\(640\) 0 0
\(641\) 30.4942 1.20445 0.602224 0.798327i \(-0.294281\pi\)
0.602224 + 0.798327i \(0.294281\pi\)
\(642\) 0 0
\(643\) −23.6599 −0.933056 −0.466528 0.884506i \(-0.654495\pi\)
−0.466528 + 0.884506i \(0.654495\pi\)
\(644\) 0 0
\(645\) −3.23652 −0.127438
\(646\) 0 0
\(647\) 48.9298 1.92363 0.961814 0.273704i \(-0.0882488\pi\)
0.961814 + 0.273704i \(0.0882488\pi\)
\(648\) 0 0
\(649\) 30.3569 1.19161
\(650\) 0 0
\(651\) −8.93235 −0.350086
\(652\) 0 0
\(653\) 9.98530 0.390755 0.195377 0.980728i \(-0.437407\pi\)
0.195377 + 0.980728i \(0.437407\pi\)
\(654\) 0 0
\(655\) 4.32418 0.168960
\(656\) 0 0
\(657\) −4.96626 −0.193752
\(658\) 0 0
\(659\) 28.2889 1.10198 0.550989 0.834512i \(-0.314251\pi\)
0.550989 + 0.834512i \(0.314251\pi\)
\(660\) 0 0
\(661\) −32.3722 −1.25913 −0.629567 0.776946i \(-0.716768\pi\)
−0.629567 + 0.776946i \(0.716768\pi\)
\(662\) 0 0
\(663\) −16.8951 −0.656151
\(664\) 0 0
\(665\) −2.80094 −0.108616
\(666\) 0 0
\(667\) 8.92957 0.345754
\(668\) 0 0
\(669\) −15.1849 −0.587084
\(670\) 0 0
\(671\) −6.03276 −0.232892
\(672\) 0 0
\(673\) 6.15993 0.237448 0.118724 0.992927i \(-0.462120\pi\)
0.118724 + 0.992927i \(0.462120\pi\)
\(674\) 0 0
\(675\) −5.25711 −0.202346
\(676\) 0 0
\(677\) 28.1962 1.08367 0.541834 0.840486i \(-0.317730\pi\)
0.541834 + 0.840486i \(0.317730\pi\)
\(678\) 0 0
\(679\) −5.64502 −0.216636
\(680\) 0 0
\(681\) 27.3985 1.04991
\(682\) 0 0
\(683\) 6.85163 0.262170 0.131085 0.991371i \(-0.458154\pi\)
0.131085 + 0.991371i \(0.458154\pi\)
\(684\) 0 0
\(685\) 5.63717 0.215385
\(686\) 0 0
\(687\) −21.3198 −0.813403
\(688\) 0 0
\(689\) 13.2857 0.506147
\(690\) 0 0
\(691\) −14.2334 −0.541463 −0.270732 0.962655i \(-0.587266\pi\)
−0.270732 + 0.962655i \(0.587266\pi\)
\(692\) 0 0
\(693\) 12.3897 0.470646
\(694\) 0 0
\(695\) −20.7476 −0.787001
\(696\) 0 0
\(697\) 45.6819 1.73032
\(698\) 0 0
\(699\) −2.83407 −0.107194
\(700\) 0 0
\(701\) −38.1561 −1.44114 −0.720568 0.693385i \(-0.756119\pi\)
−0.720568 + 0.693385i \(0.756119\pi\)
\(702\) 0 0
\(703\) −7.89596 −0.297802
\(704\) 0 0
\(705\) −14.3948 −0.542138
\(706\) 0 0
\(707\) −13.8806 −0.522032
\(708\) 0 0
\(709\) 24.3767 0.915487 0.457743 0.889084i \(-0.348658\pi\)
0.457743 + 0.889084i \(0.348658\pi\)
\(710\) 0 0
\(711\) −17.6346 −0.661351
\(712\) 0 0
\(713\) 3.49207 0.130779
\(714\) 0 0
\(715\) −8.72230 −0.326196
\(716\) 0 0
\(717\) −11.0049 −0.410986
\(718\) 0 0
\(719\) −2.24554 −0.0837447 −0.0418723 0.999123i \(-0.513332\pi\)
−0.0418723 + 0.999123i \(0.513332\pi\)
\(720\) 0 0
\(721\) −10.7498 −0.400342
\(722\) 0 0
\(723\) 12.5574 0.467013
\(724\) 0 0
\(725\) 7.44714 0.276580
\(726\) 0 0
\(727\) 4.61091 0.171009 0.0855045 0.996338i \(-0.472750\pi\)
0.0855045 + 0.996338i \(0.472750\pi\)
\(728\) 0 0
\(729\) 17.9070 0.663223
\(730\) 0 0
\(731\) −12.8417 −0.474969
\(732\) 0 0
\(733\) 35.2820 1.30317 0.651586 0.758575i \(-0.274104\pi\)
0.651586 + 0.758575i \(0.274104\pi\)
\(734\) 0 0
\(735\) −0.925574 −0.0341403
\(736\) 0 0
\(737\) −23.5658 −0.868056
\(738\) 0 0
\(739\) 51.5065 1.89470 0.947349 0.320202i \(-0.103751\pi\)
0.947349 + 0.320202i \(0.103751\pi\)
\(740\) 0 0
\(741\) −3.88860 −0.142851
\(742\) 0 0
\(743\) 12.1587 0.446061 0.223030 0.974811i \(-0.428405\pi\)
0.223030 + 0.974811i \(0.428405\pi\)
\(744\) 0 0
\(745\) 17.2799 0.633085
\(746\) 0 0
\(747\) −12.5268 −0.458330
\(748\) 0 0
\(749\) −26.8737 −0.981943
\(750\) 0 0
\(751\) 21.7739 0.794539 0.397270 0.917702i \(-0.369958\pi\)
0.397270 + 0.917702i \(0.369958\pi\)
\(752\) 0 0
\(753\) −19.3516 −0.705213
\(754\) 0 0
\(755\) −8.10139 −0.294840
\(756\) 0 0
\(757\) 28.3473 1.03030 0.515150 0.857100i \(-0.327736\pi\)
0.515150 + 0.857100i \(0.327736\pi\)
\(758\) 0 0
\(759\) 3.22493 0.117057
\(760\) 0 0
\(761\) −54.1535 −1.96306 −0.981532 0.191297i \(-0.938731\pi\)
−0.981532 + 0.191297i \(0.938731\pi\)
\(762\) 0 0
\(763\) −1.49795 −0.0542296
\(764\) 0 0
\(765\) −7.82467 −0.282902
\(766\) 0 0
\(767\) 43.8908 1.58480
\(768\) 0 0
\(769\) −44.7795 −1.61479 −0.807396 0.590011i \(-0.799124\pi\)
−0.807396 + 0.590011i \(0.799124\pi\)
\(770\) 0 0
\(771\) −25.9386 −0.934156
\(772\) 0 0
\(773\) 17.5331 0.630623 0.315311 0.948988i \(-0.397891\pi\)
0.315311 + 0.948988i \(0.397891\pi\)
\(774\) 0 0
\(775\) 2.91234 0.104614
\(776\) 0 0
\(777\) −24.2175 −0.868797
\(778\) 0 0
\(779\) 10.5142 0.376711
\(780\) 0 0
\(781\) −18.6715 −0.668118
\(782\) 0 0
\(783\) −39.1504 −1.39912
\(784\) 0 0
\(785\) 6.76013 0.241279
\(786\) 0 0
\(787\) −6.34763 −0.226269 −0.113134 0.993580i \(-0.536089\pi\)
−0.113134 + 0.993580i \(0.536089\pi\)
\(788\) 0 0
\(789\) 11.4299 0.406915
\(790\) 0 0
\(791\) −38.3389 −1.36317
\(792\) 0 0
\(793\) −8.72230 −0.309738
\(794\) 0 0
\(795\) −4.09669 −0.145295
\(796\) 0 0
\(797\) 3.56393 0.126241 0.0631204 0.998006i \(-0.479895\pi\)
0.0631204 + 0.998006i \(0.479895\pi\)
\(798\) 0 0
\(799\) −57.1151 −2.02059
\(800\) 0 0
\(801\) −15.3071 −0.540850
\(802\) 0 0
\(803\) 6.77312 0.239018
\(804\) 0 0
\(805\) 3.35850 0.118371
\(806\) 0 0
\(807\) 23.0104 0.810003
\(808\) 0 0
\(809\) 0.330069 0.0116046 0.00580230 0.999983i \(-0.498153\pi\)
0.00580230 + 0.999983i \(0.498153\pi\)
\(810\) 0 0
\(811\) −29.6827 −1.04230 −0.521150 0.853465i \(-0.674497\pi\)
−0.521150 + 0.853465i \(0.674497\pi\)
\(812\) 0 0
\(813\) 15.7713 0.553124
\(814\) 0 0
\(815\) −3.86801 −0.135491
\(816\) 0 0
\(817\) −2.95568 −0.103406
\(818\) 0 0
\(819\) 17.9133 0.625942
\(820\) 0 0
\(821\) −20.4179 −0.712589 −0.356295 0.934374i \(-0.615960\pi\)
−0.356295 + 0.934374i \(0.615960\pi\)
\(822\) 0 0
\(823\) 8.14922 0.284064 0.142032 0.989862i \(-0.454636\pi\)
0.142032 + 0.989862i \(0.454636\pi\)
\(824\) 0 0
\(825\) 2.68954 0.0936379
\(826\) 0 0
\(827\) 25.7993 0.897128 0.448564 0.893751i \(-0.351936\pi\)
0.448564 + 0.893751i \(0.351936\pi\)
\(828\) 0 0
\(829\) −3.39968 −0.118076 −0.0590378 0.998256i \(-0.518803\pi\)
−0.0590378 + 0.998256i \(0.518803\pi\)
\(830\) 0 0
\(831\) −14.9913 −0.520044
\(832\) 0 0
\(833\) −3.67247 −0.127243
\(834\) 0 0
\(835\) −23.4545 −0.811676
\(836\) 0 0
\(837\) −15.3105 −0.529207
\(838\) 0 0
\(839\) −12.8384 −0.443231 −0.221615 0.975134i \(-0.571133\pi\)
−0.221615 + 0.975134i \(0.571133\pi\)
\(840\) 0 0
\(841\) 26.4599 0.912410
\(842\) 0 0
\(843\) −3.62500 −0.124852
\(844\) 0 0
\(845\) 0.389093 0.0133852
\(846\) 0 0
\(847\) 13.9129 0.478054
\(848\) 0 0
\(849\) 5.06789 0.173930
\(850\) 0 0
\(851\) 9.46773 0.324550
\(852\) 0 0
\(853\) 1.29436 0.0443179 0.0221590 0.999754i \(-0.492946\pi\)
0.0221590 + 0.999754i \(0.492946\pi\)
\(854\) 0 0
\(855\) −1.80094 −0.0615908
\(856\) 0 0
\(857\) −1.20977 −0.0413248 −0.0206624 0.999787i \(-0.506578\pi\)
−0.0206624 + 0.999787i \(0.506578\pi\)
\(858\) 0 0
\(859\) −54.0261 −1.84335 −0.921673 0.387969i \(-0.873177\pi\)
−0.921673 + 0.387969i \(0.873177\pi\)
\(860\) 0 0
\(861\) 32.2479 1.09900
\(862\) 0 0
\(863\) 27.9654 0.951952 0.475976 0.879458i \(-0.342095\pi\)
0.475976 + 0.879458i \(0.342095\pi\)
\(864\) 0 0
\(865\) 20.7902 0.706889
\(866\) 0 0
\(867\) 2.05539 0.0698047
\(868\) 0 0
\(869\) 24.0506 0.815860
\(870\) 0 0
\(871\) −34.0719 −1.15448
\(872\) 0 0
\(873\) −3.62962 −0.122844
\(874\) 0 0
\(875\) 2.80094 0.0946890
\(876\) 0 0
\(877\) 6.75829 0.228211 0.114106 0.993469i \(-0.463600\pi\)
0.114106 + 0.993469i \(0.463600\pi\)
\(878\) 0 0
\(879\) 30.4269 1.02627
\(880\) 0 0
\(881\) 49.1665 1.65646 0.828230 0.560388i \(-0.189348\pi\)
0.828230 + 0.560388i \(0.189348\pi\)
\(882\) 0 0
\(883\) −16.8001 −0.565369 −0.282685 0.959213i \(-0.591225\pi\)
−0.282685 + 0.959213i \(0.591225\pi\)
\(884\) 0 0
\(885\) −13.5338 −0.454935
\(886\) 0 0
\(887\) 9.42509 0.316463 0.158232 0.987402i \(-0.449421\pi\)
0.158232 + 0.987402i \(0.449421\pi\)
\(888\) 0 0
\(889\) −49.2235 −1.65090
\(890\) 0 0
\(891\) −0.868992 −0.0291123
\(892\) 0 0
\(893\) −13.1457 −0.439904
\(894\) 0 0
\(895\) −7.27770 −0.243267
\(896\) 0 0
\(897\) 4.66267 0.155682
\(898\) 0 0
\(899\) 21.6886 0.723354
\(900\) 0 0
\(901\) −16.2547 −0.541524
\(902\) 0 0
\(903\) −9.06529 −0.301674
\(904\) 0 0
\(905\) −8.82369 −0.293309
\(906\) 0 0
\(907\) −23.4045 −0.777133 −0.388567 0.921421i \(-0.627030\pi\)
−0.388567 + 0.921421i \(0.627030\pi\)
\(908\) 0 0
\(909\) −8.92488 −0.296019
\(910\) 0 0
\(911\) −15.2596 −0.505574 −0.252787 0.967522i \(-0.581347\pi\)
−0.252787 + 0.967522i \(0.581347\pi\)
\(912\) 0 0
\(913\) 17.0843 0.565408
\(914\) 0 0
\(915\) 2.68954 0.0889136
\(916\) 0 0
\(917\) 12.1118 0.399966
\(918\) 0 0
\(919\) −57.2588 −1.88879 −0.944397 0.328807i \(-0.893353\pi\)
−0.944397 + 0.328807i \(0.893353\pi\)
\(920\) 0 0
\(921\) 22.0383 0.726186
\(922\) 0 0
\(923\) −26.9957 −0.888573
\(924\) 0 0
\(925\) 7.89596 0.259617
\(926\) 0 0
\(927\) −6.91184 −0.227015
\(928\) 0 0
\(929\) 2.63112 0.0863244 0.0431622 0.999068i \(-0.486257\pi\)
0.0431622 + 0.999068i \(0.486257\pi\)
\(930\) 0 0
\(931\) −0.845261 −0.0277023
\(932\) 0 0
\(933\) −23.6097 −0.772948
\(934\) 0 0
\(935\) 10.6715 0.348995
\(936\) 0 0
\(937\) 56.8145 1.85605 0.928024 0.372521i \(-0.121507\pi\)
0.928024 + 0.372521i \(0.121507\pi\)
\(938\) 0 0
\(939\) 8.78268 0.286612
\(940\) 0 0
\(941\) 26.8590 0.875578 0.437789 0.899078i \(-0.355762\pi\)
0.437789 + 0.899078i \(0.355762\pi\)
\(942\) 0 0
\(943\) −12.6072 −0.410546
\(944\) 0 0
\(945\) −14.7248 −0.478999
\(946\) 0 0
\(947\) 3.09318 0.100515 0.0502574 0.998736i \(-0.483996\pi\)
0.0502574 + 0.998736i \(0.483996\pi\)
\(948\) 0 0
\(949\) 9.79273 0.317885
\(950\) 0 0
\(951\) −31.7725 −1.03029
\(952\) 0 0
\(953\) 27.3225 0.885063 0.442531 0.896753i \(-0.354081\pi\)
0.442531 + 0.896753i \(0.354081\pi\)
\(954\) 0 0
\(955\) −6.03529 −0.195298
\(956\) 0 0
\(957\) 20.0294 0.647459
\(958\) 0 0
\(959\) 15.7894 0.509866
\(960\) 0 0
\(961\) −22.5183 −0.726397
\(962\) 0 0
\(963\) −17.2792 −0.556813
\(964\) 0 0
\(965\) 4.42823 0.142550
\(966\) 0 0
\(967\) 2.08452 0.0670338 0.0335169 0.999438i \(-0.489329\pi\)
0.0335169 + 0.999438i \(0.489329\pi\)
\(968\) 0 0
\(969\) 4.75760 0.152836
\(970\) 0 0
\(971\) 51.0629 1.63869 0.819343 0.573303i \(-0.194338\pi\)
0.819343 + 0.573303i \(0.194338\pi\)
\(972\) 0 0
\(973\) −58.1127 −1.86301
\(974\) 0 0
\(975\) 3.88860 0.124535
\(976\) 0 0
\(977\) 0.710255 0.0227231 0.0113615 0.999935i \(-0.496383\pi\)
0.0113615 + 0.999935i \(0.496383\pi\)
\(978\) 0 0
\(979\) 20.8762 0.667207
\(980\) 0 0
\(981\) −0.963150 −0.0307510
\(982\) 0 0
\(983\) −3.84730 −0.122710 −0.0613549 0.998116i \(-0.519542\pi\)
−0.0613549 + 0.998116i \(0.519542\pi\)
\(984\) 0 0
\(985\) 16.9957 0.541527
\(986\) 0 0
\(987\) −40.3189 −1.28336
\(988\) 0 0
\(989\) 3.54404 0.112694
\(990\) 0 0
\(991\) −11.3413 −0.360267 −0.180133 0.983642i \(-0.557653\pi\)
−0.180133 + 0.983642i \(0.557653\pi\)
\(992\) 0 0
\(993\) −20.3157 −0.644699
\(994\) 0 0
\(995\) 22.4609 0.712057
\(996\) 0 0
\(997\) −23.5313 −0.745243 −0.372622 0.927983i \(-0.621541\pi\)
−0.372622 + 0.927983i \(0.621541\pi\)
\(998\) 0 0
\(999\) −41.5099 −1.31331
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 3040.2.a.q.1.3 4
4.3 odd 2 3040.2.a.s.1.2 yes 4
8.3 odd 2 6080.2.a.ce.1.3 4
8.5 even 2 6080.2.a.cg.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.q.1.3 4 1.1 even 1 trivial
3040.2.a.s.1.2 yes 4 4.3 odd 2
6080.2.a.ce.1.3 4 8.3 odd 2
6080.2.a.cg.1.2 4 8.5 even 2