Properties

Label 760.2.a.h.1.3
Level $760$
Weight $2$
Character 760.1
Self dual yes
Analytic conductor $6.069$
Analytic rank $1$
Dimension $3$
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] = [760,2,Mod(1,760)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(760, 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("760.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 760.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: \(6.06863055362\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) 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 \(-1.76156\) of defining polynomial
Character \(\chi\) \(=\) 760.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.76156 q^{3} -1.00000 q^{5} -4.62620 q^{7} +0.103084 q^{9} +O(q^{10})\) \(q+1.76156 q^{3} -1.00000 q^{5} -4.62620 q^{7} +0.103084 q^{9} -5.52311 q^{11} +5.49084 q^{13} -1.76156 q^{15} -6.62620 q^{17} -1.00000 q^{19} -8.14931 q^{21} +4.14931 q^{23} +1.00000 q^{25} -5.10308 q^{27} -7.87859 q^{29} +1.25240 q^{31} -9.72928 q^{33} +4.62620 q^{35} -0.387755 q^{37} +9.67243 q^{39} +6.77551 q^{41} -10.9817 q^{43} -0.103084 q^{45} +1.72928 q^{47} +14.4017 q^{49} -11.6724 q^{51} +1.49084 q^{53} +5.52311 q^{55} -1.76156 q^{57} +0.626198 q^{59} +15.0462 q^{61} -0.476886 q^{63} -5.49084 q^{65} +5.22012 q^{67} +7.30925 q^{69} -11.0462 q^{71} -4.83237 q^{73} +1.76156 q^{75} +25.5510 q^{77} -2.98168 q^{79} -9.29862 q^{81} -2.74760 q^{83} +6.62620 q^{85} -13.8786 q^{87} +4.27072 q^{89} -25.4017 q^{91} +2.20617 q^{93} +1.00000 q^{95} -13.7047 q^{97} -0.569343 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{3} - 3 q^{5} - 5 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{3} - 3 q^{5} - 5 q^{7} + 4 q^{9} - 4 q^{11} + 5 q^{13} + q^{15} - 11 q^{17} - 3 q^{19} - 3 q^{21} - 9 q^{23} + 3 q^{25} - 19 q^{27} + 3 q^{29} - 14 q^{31} - 24 q^{33} + 5 q^{35} + 14 q^{37} - 5 q^{39} - 10 q^{41} - 10 q^{43} - 4 q^{45} + 4 q^{49} - q^{51} - 7 q^{53} + 4 q^{55} + q^{57} - 7 q^{59} + 20 q^{61} - 14 q^{63} - 5 q^{65} - q^{67} + 33 q^{69} - 8 q^{71} - 13 q^{73} - q^{75} + 16 q^{77} + 14 q^{79} + 15 q^{81} - 26 q^{83} + 11 q^{85} - 15 q^{87} + 18 q^{89} - 37 q^{91} + 14 q^{93} + 3 q^{95} - 6 q^{97} + 36 q^{99}+O(q^{100}) \) 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
\(3\) 1.76156 1.01704 0.508518 0.861052i \(-0.330194\pi\)
0.508518 + 0.861052i \(0.330194\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.62620 −1.74854 −0.874269 0.485441i \(-0.838659\pi\)
−0.874269 + 0.485441i \(0.838659\pi\)
\(8\) 0 0
\(9\) 0.103084 0.0343612
\(10\) 0 0
\(11\) −5.52311 −1.66528 −0.832641 0.553813i \(-0.813172\pi\)
−0.832641 + 0.553813i \(0.813172\pi\)
\(12\) 0 0
\(13\) 5.49084 1.52288 0.761442 0.648233i \(-0.224492\pi\)
0.761442 + 0.648233i \(0.224492\pi\)
\(14\) 0 0
\(15\) −1.76156 −0.454832
\(16\) 0 0
\(17\) −6.62620 −1.60709 −0.803545 0.595245i \(-0.797055\pi\)
−0.803545 + 0.595245i \(0.797055\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −8.14931 −1.77833
\(22\) 0 0
\(23\) 4.14931 0.865191 0.432596 0.901588i \(-0.357598\pi\)
0.432596 + 0.901588i \(0.357598\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.10308 −0.982089
\(28\) 0 0
\(29\) −7.87859 −1.46302 −0.731509 0.681832i \(-0.761184\pi\)
−0.731509 + 0.681832i \(0.761184\pi\)
\(30\) 0 0
\(31\) 1.25240 0.224937 0.112468 0.993655i \(-0.464124\pi\)
0.112468 + 0.993655i \(0.464124\pi\)
\(32\) 0 0
\(33\) −9.72928 −1.69365
\(34\) 0 0
\(35\) 4.62620 0.781970
\(36\) 0 0
\(37\) −0.387755 −0.0637466 −0.0318733 0.999492i \(-0.510147\pi\)
−0.0318733 + 0.999492i \(0.510147\pi\)
\(38\) 0 0
\(39\) 9.67243 1.54883
\(40\) 0 0
\(41\) 6.77551 1.05816 0.529078 0.848573i \(-0.322538\pi\)
0.529078 + 0.848573i \(0.322538\pi\)
\(42\) 0 0
\(43\) −10.9817 −1.67469 −0.837345 0.546675i \(-0.815893\pi\)
−0.837345 + 0.546675i \(0.815893\pi\)
\(44\) 0 0
\(45\) −0.103084 −0.0153668
\(46\) 0 0
\(47\) 1.72928 0.252242 0.126121 0.992015i \(-0.459747\pi\)
0.126121 + 0.992015i \(0.459747\pi\)
\(48\) 0 0
\(49\) 14.4017 2.05739
\(50\) 0 0
\(51\) −11.6724 −1.63447
\(52\) 0 0
\(53\) 1.49084 0.204782 0.102391 0.994744i \(-0.467351\pi\)
0.102391 + 0.994744i \(0.467351\pi\)
\(54\) 0 0
\(55\) 5.52311 0.744737
\(56\) 0 0
\(57\) −1.76156 −0.233324
\(58\) 0 0
\(59\) 0.626198 0.0815240 0.0407620 0.999169i \(-0.487021\pi\)
0.0407620 + 0.999169i \(0.487021\pi\)
\(60\) 0 0
\(61\) 15.0462 1.92647 0.963236 0.268656i \(-0.0865796\pi\)
0.963236 + 0.268656i \(0.0865796\pi\)
\(62\) 0 0
\(63\) −0.476886 −0.0600819
\(64\) 0 0
\(65\) −5.49084 −0.681055
\(66\) 0 0
\(67\) 5.22012 0.637739 0.318870 0.947799i \(-0.396697\pi\)
0.318870 + 0.947799i \(0.396697\pi\)
\(68\) 0 0
\(69\) 7.30925 0.879930
\(70\) 0 0
\(71\) −11.0462 −1.31095 −0.655473 0.755219i \(-0.727531\pi\)
−0.655473 + 0.755219i \(0.727531\pi\)
\(72\) 0 0
\(73\) −4.83237 −0.565586 −0.282793 0.959181i \(-0.591261\pi\)
−0.282793 + 0.959181i \(0.591261\pi\)
\(74\) 0 0
\(75\) 1.76156 0.203407
\(76\) 0 0
\(77\) 25.5510 2.91181
\(78\) 0 0
\(79\) −2.98168 −0.335465 −0.167732 0.985833i \(-0.553644\pi\)
−0.167732 + 0.985833i \(0.553644\pi\)
\(80\) 0 0
\(81\) −9.29862 −1.03318
\(82\) 0 0
\(83\) −2.74760 −0.301589 −0.150794 0.988565i \(-0.548183\pi\)
−0.150794 + 0.988565i \(0.548183\pi\)
\(84\) 0 0
\(85\) 6.62620 0.718712
\(86\) 0 0
\(87\) −13.8786 −1.48794
\(88\) 0 0
\(89\) 4.27072 0.452695 0.226348 0.974047i \(-0.427321\pi\)
0.226348 + 0.974047i \(0.427321\pi\)
\(90\) 0 0
\(91\) −25.4017 −2.66282
\(92\) 0 0
\(93\) 2.20617 0.228769
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −13.7047 −1.39150 −0.695751 0.718283i \(-0.744928\pi\)
−0.695751 + 0.718283i \(0.744928\pi\)
\(98\) 0 0
\(99\) −0.569343 −0.0572211
\(100\) 0 0
\(101\) 7.04623 0.701126 0.350563 0.936539i \(-0.385990\pi\)
0.350563 + 0.936539i \(0.385990\pi\)
\(102\) 0 0
\(103\) 0.658473 0.0648813 0.0324407 0.999474i \(-0.489672\pi\)
0.0324407 + 0.999474i \(0.489672\pi\)
\(104\) 0 0
\(105\) 8.14931 0.795291
\(106\) 0 0
\(107\) −5.22012 −0.504648 −0.252324 0.967643i \(-0.581195\pi\)
−0.252324 + 0.967643i \(0.581195\pi\)
\(108\) 0 0
\(109\) −11.8140 −1.13158 −0.565790 0.824549i \(-0.691429\pi\)
−0.565790 + 0.824549i \(0.691429\pi\)
\(110\) 0 0
\(111\) −0.683053 −0.0648325
\(112\) 0 0
\(113\) −0.0891304 −0.00838468 −0.00419234 0.999991i \(-0.501334\pi\)
−0.00419234 + 0.999991i \(0.501334\pi\)
\(114\) 0 0
\(115\) −4.14931 −0.386925
\(116\) 0 0
\(117\) 0.566016 0.0523282
\(118\) 0 0
\(119\) 30.6541 2.81006
\(120\) 0 0
\(121\) 19.5048 1.77316
\(122\) 0 0
\(123\) 11.9354 1.07618
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 13.3694 1.18635 0.593173 0.805075i \(-0.297875\pi\)
0.593173 + 0.805075i \(0.297875\pi\)
\(128\) 0 0
\(129\) −19.3449 −1.70322
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) 4.62620 0.401142
\(134\) 0 0
\(135\) 5.10308 0.439204
\(136\) 0 0
\(137\) −10.6262 −0.907857 −0.453929 0.891038i \(-0.649978\pi\)
−0.453929 + 0.891038i \(0.649978\pi\)
\(138\) 0 0
\(139\) −12.5693 −1.06612 −0.533059 0.846078i \(-0.678958\pi\)
−0.533059 + 0.846078i \(0.678958\pi\)
\(140\) 0 0
\(141\) 3.04623 0.256539
\(142\) 0 0
\(143\) −30.3265 −2.53603
\(144\) 0 0
\(145\) 7.87859 0.654282
\(146\) 0 0
\(147\) 25.3694 2.09244
\(148\) 0 0
\(149\) 7.04623 0.577250 0.288625 0.957442i \(-0.406802\pi\)
0.288625 + 0.957442i \(0.406802\pi\)
\(150\) 0 0
\(151\) 4.47689 0.364324 0.182162 0.983269i \(-0.441691\pi\)
0.182162 + 0.983269i \(0.441691\pi\)
\(152\) 0 0
\(153\) −0.683053 −0.0552216
\(154\) 0 0
\(155\) −1.25240 −0.100595
\(156\) 0 0
\(157\) 8.27072 0.660075 0.330038 0.943968i \(-0.392939\pi\)
0.330038 + 0.943968i \(0.392939\pi\)
\(158\) 0 0
\(159\) 2.62620 0.208271
\(160\) 0 0
\(161\) −19.1955 −1.51282
\(162\) 0 0
\(163\) −18.5693 −1.45446 −0.727232 0.686392i \(-0.759193\pi\)
−0.727232 + 0.686392i \(0.759193\pi\)
\(164\) 0 0
\(165\) 9.72928 0.757424
\(166\) 0 0
\(167\) −21.3694 −1.65362 −0.826808 0.562484i \(-0.809846\pi\)
−0.826808 + 0.562484i \(0.809846\pi\)
\(168\) 0 0
\(169\) 17.1493 1.31918
\(170\) 0 0
\(171\) −0.103084 −0.00788301
\(172\) 0 0
\(173\) −12.3878 −0.941824 −0.470912 0.882180i \(-0.656075\pi\)
−0.470912 + 0.882180i \(0.656075\pi\)
\(174\) 0 0
\(175\) −4.62620 −0.349708
\(176\) 0 0
\(177\) 1.10308 0.0829128
\(178\) 0 0
\(179\) 12.7755 0.954886 0.477443 0.878663i \(-0.341564\pi\)
0.477443 + 0.878663i \(0.341564\pi\)
\(180\) 0 0
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 0 0
\(183\) 26.5048 1.95929
\(184\) 0 0
\(185\) 0.387755 0.0285083
\(186\) 0 0
\(187\) 36.5972 2.67626
\(188\) 0 0
\(189\) 23.6079 1.71722
\(190\) 0 0
\(191\) −19.1955 −1.38894 −0.694470 0.719521i \(-0.744361\pi\)
−0.694470 + 0.719521i \(0.744361\pi\)
\(192\) 0 0
\(193\) −13.7047 −0.986486 −0.493243 0.869892i \(-0.664189\pi\)
−0.493243 + 0.869892i \(0.664189\pi\)
\(194\) 0 0
\(195\) −9.67243 −0.692657
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) −0.561647 −0.0398141 −0.0199071 0.999802i \(-0.506337\pi\)
−0.0199071 + 0.999802i \(0.506337\pi\)
\(200\) 0 0
\(201\) 9.19554 0.648603
\(202\) 0 0
\(203\) 36.4479 2.55814
\(204\) 0 0
\(205\) −6.77551 −0.473222
\(206\) 0 0
\(207\) 0.427726 0.0297290
\(208\) 0 0
\(209\) 5.52311 0.382042
\(210\) 0 0
\(211\) 15.6724 1.07893 0.539467 0.842007i \(-0.318626\pi\)
0.539467 + 0.842007i \(0.318626\pi\)
\(212\) 0 0
\(213\) −19.4586 −1.33328
\(214\) 0 0
\(215\) 10.9817 0.748944
\(216\) 0 0
\(217\) −5.79383 −0.393311
\(218\) 0 0
\(219\) −8.51249 −0.575221
\(220\) 0 0
\(221\) −36.3834 −2.44741
\(222\) 0 0
\(223\) 12.0525 0.807094 0.403547 0.914959i \(-0.367777\pi\)
0.403547 + 0.914959i \(0.367777\pi\)
\(224\) 0 0
\(225\) 0.103084 0.00687225
\(226\) 0 0
\(227\) 3.07850 0.204327 0.102164 0.994768i \(-0.467423\pi\)
0.102164 + 0.994768i \(0.467423\pi\)
\(228\) 0 0
\(229\) 17.7938 1.17585 0.587925 0.808916i \(-0.299945\pi\)
0.587925 + 0.808916i \(0.299945\pi\)
\(230\) 0 0
\(231\) 45.0096 2.96141
\(232\) 0 0
\(233\) 8.50479 0.557167 0.278584 0.960412i \(-0.410135\pi\)
0.278584 + 0.960412i \(0.410135\pi\)
\(234\) 0 0
\(235\) −1.72928 −0.112806
\(236\) 0 0
\(237\) −5.25240 −0.341180
\(238\) 0 0
\(239\) −23.6079 −1.52707 −0.763533 0.645768i \(-0.776537\pi\)
−0.763533 + 0.645768i \(0.776537\pi\)
\(240\) 0 0
\(241\) 9.11078 0.586877 0.293438 0.955978i \(-0.405200\pi\)
0.293438 + 0.955978i \(0.405200\pi\)
\(242\) 0 0
\(243\) −1.07081 −0.0686924
\(244\) 0 0
\(245\) −14.4017 −0.920091
\(246\) 0 0
\(247\) −5.49084 −0.349374
\(248\) 0 0
\(249\) −4.84006 −0.306726
\(250\) 0 0
\(251\) 13.5510 0.855333 0.427666 0.903937i \(-0.359336\pi\)
0.427666 + 0.903937i \(0.359336\pi\)
\(252\) 0 0
\(253\) −22.9171 −1.44079
\(254\) 0 0
\(255\) 11.6724 0.730956
\(256\) 0 0
\(257\) 10.6585 0.664857 0.332429 0.943128i \(-0.392132\pi\)
0.332429 + 0.943128i \(0.392132\pi\)
\(258\) 0 0
\(259\) 1.79383 0.111463
\(260\) 0 0
\(261\) −0.812155 −0.0502711
\(262\) 0 0
\(263\) −17.9634 −1.10767 −0.553834 0.832627i \(-0.686836\pi\)
−0.553834 + 0.832627i \(0.686836\pi\)
\(264\) 0 0
\(265\) −1.49084 −0.0915815
\(266\) 0 0
\(267\) 7.52311 0.460407
\(268\) 0 0
\(269\) 15.5510 0.948162 0.474081 0.880481i \(-0.342780\pi\)
0.474081 + 0.880481i \(0.342780\pi\)
\(270\) 0 0
\(271\) 8.29093 0.503638 0.251819 0.967774i \(-0.418971\pi\)
0.251819 + 0.967774i \(0.418971\pi\)
\(272\) 0 0
\(273\) −44.7466 −2.70819
\(274\) 0 0
\(275\) −5.52311 −0.333056
\(276\) 0 0
\(277\) −28.5693 −1.71657 −0.858283 0.513177i \(-0.828468\pi\)
−0.858283 + 0.513177i \(0.828468\pi\)
\(278\) 0 0
\(279\) 0.129102 0.00772911
\(280\) 0 0
\(281\) −19.3169 −1.15235 −0.576176 0.817325i \(-0.695456\pi\)
−0.576176 + 0.817325i \(0.695456\pi\)
\(282\) 0 0
\(283\) 16.2986 0.968853 0.484426 0.874832i \(-0.339028\pi\)
0.484426 + 0.874832i \(0.339028\pi\)
\(284\) 0 0
\(285\) 1.76156 0.104346
\(286\) 0 0
\(287\) −31.3449 −1.85023
\(288\) 0 0
\(289\) 26.9065 1.58274
\(290\) 0 0
\(291\) −24.1416 −1.41521
\(292\) 0 0
\(293\) −26.3955 −1.54204 −0.771019 0.636812i \(-0.780253\pi\)
−0.771019 + 0.636812i \(0.780253\pi\)
\(294\) 0 0
\(295\) −0.626198 −0.0364587
\(296\) 0 0
\(297\) 28.1849 1.63545
\(298\) 0 0
\(299\) 22.7832 1.31759
\(300\) 0 0
\(301\) 50.8034 2.92826
\(302\) 0 0
\(303\) 12.4123 0.713070
\(304\) 0 0
\(305\) −15.0462 −0.861545
\(306\) 0 0
\(307\) −30.2095 −1.72415 −0.862073 0.506783i \(-0.830834\pi\)
−0.862073 + 0.506783i \(0.830834\pi\)
\(308\) 0 0
\(309\) 1.15994 0.0659866
\(310\) 0 0
\(311\) −4.12141 −0.233703 −0.116852 0.993149i \(-0.537280\pi\)
−0.116852 + 0.993149i \(0.537280\pi\)
\(312\) 0 0
\(313\) −15.1031 −0.853677 −0.426838 0.904328i \(-0.640373\pi\)
−0.426838 + 0.904328i \(0.640373\pi\)
\(314\) 0 0
\(315\) 0.476886 0.0268695
\(316\) 0 0
\(317\) −13.3126 −0.747709 −0.373854 0.927487i \(-0.621964\pi\)
−0.373854 + 0.927487i \(0.621964\pi\)
\(318\) 0 0
\(319\) 43.5144 2.43634
\(320\) 0 0
\(321\) −9.19554 −0.513245
\(322\) 0 0
\(323\) 6.62620 0.368692
\(324\) 0 0
\(325\) 5.49084 0.304577
\(326\) 0 0
\(327\) −20.8111 −1.15086
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) 16.9248 0.930272 0.465136 0.885239i \(-0.346005\pi\)
0.465136 + 0.885239i \(0.346005\pi\)
\(332\) 0 0
\(333\) −0.0399712 −0.00219041
\(334\) 0 0
\(335\) −5.22012 −0.285206
\(336\) 0 0
\(337\) 1.16327 0.0633671 0.0316836 0.999498i \(-0.489913\pi\)
0.0316836 + 0.999498i \(0.489913\pi\)
\(338\) 0 0
\(339\) −0.157008 −0.00852752
\(340\) 0 0
\(341\) −6.91713 −0.374583
\(342\) 0 0
\(343\) −34.2418 −1.84888
\(344\) 0 0
\(345\) −7.30925 −0.393517
\(346\) 0 0
\(347\) 3.70138 0.198700 0.0993501 0.995053i \(-0.468324\pi\)
0.0993501 + 0.995053i \(0.468324\pi\)
\(348\) 0 0
\(349\) −2.54144 −0.136040 −0.0680200 0.997684i \(-0.521668\pi\)
−0.0680200 + 0.997684i \(0.521668\pi\)
\(350\) 0 0
\(351\) −28.0202 −1.49561
\(352\) 0 0
\(353\) −21.3738 −1.13761 −0.568806 0.822471i \(-0.692595\pi\)
−0.568806 + 0.822471i \(0.692595\pi\)
\(354\) 0 0
\(355\) 11.0462 0.586273
\(356\) 0 0
\(357\) 53.9990 2.85793
\(358\) 0 0
\(359\) −4.42003 −0.233280 −0.116640 0.993174i \(-0.537212\pi\)
−0.116640 + 0.993174i \(0.537212\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 34.3588 1.80337
\(364\) 0 0
\(365\) 4.83237 0.252938
\(366\) 0 0
\(367\) 3.28030 0.171230 0.0856152 0.996328i \(-0.472714\pi\)
0.0856152 + 0.996328i \(0.472714\pi\)
\(368\) 0 0
\(369\) 0.698445 0.0363596
\(370\) 0 0
\(371\) −6.89692 −0.358070
\(372\) 0 0
\(373\) −14.8078 −0.766718 −0.383359 0.923599i \(-0.625233\pi\)
−0.383359 + 0.923599i \(0.625233\pi\)
\(374\) 0 0
\(375\) −1.76156 −0.0909664
\(376\) 0 0
\(377\) −43.2601 −2.22801
\(378\) 0 0
\(379\) −19.7370 −1.01382 −0.506910 0.861999i \(-0.669212\pi\)
−0.506910 + 0.861999i \(0.669212\pi\)
\(380\) 0 0
\(381\) 23.5510 1.20656
\(382\) 0 0
\(383\) −10.2095 −0.521681 −0.260840 0.965382i \(-0.584000\pi\)
−0.260840 + 0.965382i \(0.584000\pi\)
\(384\) 0 0
\(385\) −25.5510 −1.30220
\(386\) 0 0
\(387\) −1.13203 −0.0575444
\(388\) 0 0
\(389\) 10.7110 0.543067 0.271534 0.962429i \(-0.412469\pi\)
0.271534 + 0.962429i \(0.412469\pi\)
\(390\) 0 0
\(391\) −27.4942 −1.39044
\(392\) 0 0
\(393\) −7.04623 −0.355435
\(394\) 0 0
\(395\) 2.98168 0.150024
\(396\) 0 0
\(397\) 20.5693 1.03235 0.516173 0.856484i \(-0.327356\pi\)
0.516173 + 0.856484i \(0.327356\pi\)
\(398\) 0 0
\(399\) 8.14931 0.407976
\(400\) 0 0
\(401\) −23.9634 −1.19667 −0.598336 0.801245i \(-0.704171\pi\)
−0.598336 + 0.801245i \(0.704171\pi\)
\(402\) 0 0
\(403\) 6.87671 0.342553
\(404\) 0 0
\(405\) 9.29862 0.462052
\(406\) 0 0
\(407\) 2.14162 0.106156
\(408\) 0 0
\(409\) −4.50479 −0.222748 −0.111374 0.993779i \(-0.535525\pi\)
−0.111374 + 0.993779i \(0.535525\pi\)
\(410\) 0 0
\(411\) −18.7187 −0.923323
\(412\) 0 0
\(413\) −2.89692 −0.142548
\(414\) 0 0
\(415\) 2.74760 0.134875
\(416\) 0 0
\(417\) −22.1416 −1.08428
\(418\) 0 0
\(419\) −31.7572 −1.55144 −0.775720 0.631077i \(-0.782613\pi\)
−0.775720 + 0.631077i \(0.782613\pi\)
\(420\) 0 0
\(421\) −21.4296 −1.04442 −0.522208 0.852818i \(-0.674891\pi\)
−0.522208 + 0.852818i \(0.674891\pi\)
\(422\) 0 0
\(423\) 0.178261 0.00866734
\(424\) 0 0
\(425\) −6.62620 −0.321418
\(426\) 0 0
\(427\) −69.6068 −3.36851
\(428\) 0 0
\(429\) −53.4219 −2.57923
\(430\) 0 0
\(431\) −10.6831 −0.514585 −0.257292 0.966334i \(-0.582830\pi\)
−0.257292 + 0.966334i \(0.582830\pi\)
\(432\) 0 0
\(433\) −24.6864 −1.18635 −0.593176 0.805073i \(-0.702126\pi\)
−0.593176 + 0.805073i \(0.702126\pi\)
\(434\) 0 0
\(435\) 13.8786 0.665428
\(436\) 0 0
\(437\) −4.14931 −0.198489
\(438\) 0 0
\(439\) −14.9817 −0.715036 −0.357518 0.933906i \(-0.616377\pi\)
−0.357518 + 0.933906i \(0.616377\pi\)
\(440\) 0 0
\(441\) 1.48458 0.0706944
\(442\) 0 0
\(443\) −4.54144 −0.215770 −0.107885 0.994163i \(-0.534408\pi\)
−0.107885 + 0.994163i \(0.534408\pi\)
\(444\) 0 0
\(445\) −4.27072 −0.202451
\(446\) 0 0
\(447\) 12.4123 0.587083
\(448\) 0 0
\(449\) −16.0925 −0.759450 −0.379725 0.925099i \(-0.623981\pi\)
−0.379725 + 0.925099i \(0.623981\pi\)
\(450\) 0 0
\(451\) −37.4219 −1.76213
\(452\) 0 0
\(453\) 7.88629 0.370530
\(454\) 0 0
\(455\) 25.4017 1.19085
\(456\) 0 0
\(457\) −4.65410 −0.217710 −0.108855 0.994058i \(-0.534718\pi\)
−0.108855 + 0.994058i \(0.534718\pi\)
\(458\) 0 0
\(459\) 33.8140 1.57830
\(460\) 0 0
\(461\) 19.0096 0.885365 0.442682 0.896679i \(-0.354027\pi\)
0.442682 + 0.896679i \(0.354027\pi\)
\(462\) 0 0
\(463\) −10.6339 −0.494199 −0.247099 0.968990i \(-0.579477\pi\)
−0.247099 + 0.968990i \(0.579477\pi\)
\(464\) 0 0
\(465\) −2.20617 −0.102309
\(466\) 0 0
\(467\) 32.8805 1.52153 0.760764 0.649029i \(-0.224825\pi\)
0.760764 + 0.649029i \(0.224825\pi\)
\(468\) 0 0
\(469\) −24.1493 −1.11511
\(470\) 0 0
\(471\) 14.5693 0.671320
\(472\) 0 0
\(473\) 60.6531 2.78883
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 0.153681 0.00703658
\(478\) 0 0
\(479\) 12.4402 0.568409 0.284205 0.958764i \(-0.408271\pi\)
0.284205 + 0.958764i \(0.408271\pi\)
\(480\) 0 0
\(481\) −2.12910 −0.0970787
\(482\) 0 0
\(483\) −33.8140 −1.53859
\(484\) 0 0
\(485\) 13.7047 0.622298
\(486\) 0 0
\(487\) 40.9571 1.85594 0.927972 0.372651i \(-0.121551\pi\)
0.927972 + 0.372651i \(0.121551\pi\)
\(488\) 0 0
\(489\) −32.7110 −1.47924
\(490\) 0 0
\(491\) 43.3449 1.95613 0.978063 0.208310i \(-0.0667962\pi\)
0.978063 + 0.208310i \(0.0667962\pi\)
\(492\) 0 0
\(493\) 52.2051 2.35120
\(494\) 0 0
\(495\) 0.569343 0.0255901
\(496\) 0 0
\(497\) 51.1020 2.29224
\(498\) 0 0
\(499\) 25.4094 1.13748 0.568741 0.822517i \(-0.307431\pi\)
0.568741 + 0.822517i \(0.307431\pi\)
\(500\) 0 0
\(501\) −37.6435 −1.68179
\(502\) 0 0
\(503\) 34.1127 1.52101 0.760504 0.649333i \(-0.224952\pi\)
0.760504 + 0.649333i \(0.224952\pi\)
\(504\) 0 0
\(505\) −7.04623 −0.313553
\(506\) 0 0
\(507\) 30.2095 1.34165
\(508\) 0 0
\(509\) −22.3632 −0.991230 −0.495615 0.868542i \(-0.665057\pi\)
−0.495615 + 0.868542i \(0.665057\pi\)
\(510\) 0 0
\(511\) 22.3555 0.988948
\(512\) 0 0
\(513\) 5.10308 0.225307
\(514\) 0 0
\(515\) −0.658473 −0.0290158
\(516\) 0 0
\(517\) −9.55102 −0.420053
\(518\) 0 0
\(519\) −21.8217 −0.957868
\(520\) 0 0
\(521\) 2.29862 0.100705 0.0503523 0.998732i \(-0.483966\pi\)
0.0503523 + 0.998732i \(0.483966\pi\)
\(522\) 0 0
\(523\) 27.3771 1.19712 0.598559 0.801079i \(-0.295740\pi\)
0.598559 + 0.801079i \(0.295740\pi\)
\(524\) 0 0
\(525\) −8.14931 −0.355665
\(526\) 0 0
\(527\) −8.29862 −0.361494
\(528\) 0 0
\(529\) −5.78321 −0.251444
\(530\) 0 0
\(531\) 0.0645508 0.00280127
\(532\) 0 0
\(533\) 37.2032 1.61145
\(534\) 0 0
\(535\) 5.22012 0.225685
\(536\) 0 0
\(537\) 22.5048 0.971153
\(538\) 0 0
\(539\) −79.5423 −3.42613
\(540\) 0 0
\(541\) 15.4586 0.664616 0.332308 0.943171i \(-0.392173\pi\)
0.332308 + 0.943171i \(0.392173\pi\)
\(542\) 0 0
\(543\) 38.7543 1.66310
\(544\) 0 0
\(545\) 11.8140 0.506058
\(546\) 0 0
\(547\) −13.9754 −0.597546 −0.298773 0.954324i \(-0.596577\pi\)
−0.298773 + 0.954324i \(0.596577\pi\)
\(548\) 0 0
\(549\) 1.55102 0.0661960
\(550\) 0 0
\(551\) 7.87859 0.335639
\(552\) 0 0
\(553\) 13.7938 0.586573
\(554\) 0 0
\(555\) 0.683053 0.0289940
\(556\) 0 0
\(557\) 30.6618 1.29918 0.649591 0.760284i \(-0.274940\pi\)
0.649591 + 0.760284i \(0.274940\pi\)
\(558\) 0 0
\(559\) −60.2986 −2.55036
\(560\) 0 0
\(561\) 64.4681 2.72185
\(562\) 0 0
\(563\) −2.02458 −0.0853259 −0.0426629 0.999090i \(-0.513584\pi\)
−0.0426629 + 0.999090i \(0.513584\pi\)
\(564\) 0 0
\(565\) 0.0891304 0.00374974
\(566\) 0 0
\(567\) 43.0173 1.80656
\(568\) 0 0
\(569\) −28.2620 −1.18480 −0.592402 0.805643i \(-0.701820\pi\)
−0.592402 + 0.805643i \(0.701820\pi\)
\(570\) 0 0
\(571\) 10.6464 0.445538 0.222769 0.974871i \(-0.428490\pi\)
0.222769 + 0.974871i \(0.428490\pi\)
\(572\) 0 0
\(573\) −33.8140 −1.41260
\(574\) 0 0
\(575\) 4.14931 0.173038
\(576\) 0 0
\(577\) −9.30925 −0.387549 −0.193775 0.981046i \(-0.562073\pi\)
−0.193775 + 0.981046i \(0.562073\pi\)
\(578\) 0 0
\(579\) −24.1416 −1.00329
\(580\) 0 0
\(581\) 12.7110 0.527339
\(582\) 0 0
\(583\) −8.23407 −0.341021
\(584\) 0 0
\(585\) −0.566016 −0.0234019
\(586\) 0 0
\(587\) −42.3911 −1.74967 −0.874834 0.484424i \(-0.839029\pi\)
−0.874834 + 0.484424i \(0.839029\pi\)
\(588\) 0 0
\(589\) −1.25240 −0.0516041
\(590\) 0 0
\(591\) −3.52311 −0.144922
\(592\) 0 0
\(593\) 28.5048 1.17055 0.585276 0.810834i \(-0.300986\pi\)
0.585276 + 0.810834i \(0.300986\pi\)
\(594\) 0 0
\(595\) −30.6541 −1.25670
\(596\) 0 0
\(597\) −0.989374 −0.0404924
\(598\) 0 0
\(599\) 25.6156 1.04662 0.523312 0.852141i \(-0.324696\pi\)
0.523312 + 0.852141i \(0.324696\pi\)
\(600\) 0 0
\(601\) −29.2803 −1.19437 −0.597184 0.802104i \(-0.703714\pi\)
−0.597184 + 0.802104i \(0.703714\pi\)
\(602\) 0 0
\(603\) 0.538109 0.0219135
\(604\) 0 0
\(605\) −19.5048 −0.792983
\(606\) 0 0
\(607\) −20.0525 −0.813905 −0.406953 0.913449i \(-0.633409\pi\)
−0.406953 + 0.913449i \(0.633409\pi\)
\(608\) 0 0
\(609\) 64.2051 2.60172
\(610\) 0 0
\(611\) 9.49521 0.384135
\(612\) 0 0
\(613\) −7.89881 −0.319030 −0.159515 0.987196i \(-0.550993\pi\)
−0.159515 + 0.987196i \(0.550993\pi\)
\(614\) 0 0
\(615\) −11.9354 −0.481284
\(616\) 0 0
\(617\) −41.4094 −1.66708 −0.833540 0.552459i \(-0.813689\pi\)
−0.833540 + 0.552459i \(0.813689\pi\)
\(618\) 0 0
\(619\) 26.9450 1.08301 0.541506 0.840697i \(-0.317854\pi\)
0.541506 + 0.840697i \(0.317854\pi\)
\(620\) 0 0
\(621\) −21.1743 −0.849695
\(622\) 0 0
\(623\) −19.7572 −0.791555
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 9.72928 0.388550
\(628\) 0 0
\(629\) 2.56934 0.102446
\(630\) 0 0
\(631\) −15.4865 −0.616507 −0.308253 0.951304i \(-0.599744\pi\)
−0.308253 + 0.951304i \(0.599744\pi\)
\(632\) 0 0
\(633\) 27.6079 1.09731
\(634\) 0 0
\(635\) −13.3694 −0.530550
\(636\) 0 0
\(637\) 79.0775 3.13316
\(638\) 0 0
\(639\) −1.13869 −0.0450457
\(640\) 0 0
\(641\) 9.93545 0.392427 0.196213 0.980561i \(-0.437135\pi\)
0.196213 + 0.980561i \(0.437135\pi\)
\(642\) 0 0
\(643\) 5.25240 0.207134 0.103567 0.994622i \(-0.466974\pi\)
0.103567 + 0.994622i \(0.466974\pi\)
\(644\) 0 0
\(645\) 19.3449 0.761703
\(646\) 0 0
\(647\) 1.64452 0.0646528 0.0323264 0.999477i \(-0.489708\pi\)
0.0323264 + 0.999477i \(0.489708\pi\)
\(648\) 0 0
\(649\) −3.45856 −0.135760
\(650\) 0 0
\(651\) −10.2062 −0.400011
\(652\) 0 0
\(653\) −30.9450 −1.21097 −0.605486 0.795856i \(-0.707021\pi\)
−0.605486 + 0.795856i \(0.707021\pi\)
\(654\) 0 0
\(655\) 4.00000 0.156293
\(656\) 0 0
\(657\) −0.498138 −0.0194342
\(658\) 0 0
\(659\) 17.3372 0.675360 0.337680 0.941261i \(-0.390358\pi\)
0.337680 + 0.941261i \(0.390358\pi\)
\(660\) 0 0
\(661\) 8.83237 0.343539 0.171770 0.985137i \(-0.445052\pi\)
0.171770 + 0.985137i \(0.445052\pi\)
\(662\) 0 0
\(663\) −64.0914 −2.48910
\(664\) 0 0
\(665\) −4.62620 −0.179396
\(666\) 0 0
\(667\) −32.6907 −1.26579
\(668\) 0 0
\(669\) 21.2311 0.820843
\(670\) 0 0
\(671\) −83.1020 −3.20812
\(672\) 0 0
\(673\) 29.6402 1.14254 0.571272 0.820761i \(-0.306450\pi\)
0.571272 + 0.820761i \(0.306450\pi\)
\(674\) 0 0
\(675\) −5.10308 −0.196418
\(676\) 0 0
\(677\) 23.2847 0.894903 0.447451 0.894308i \(-0.352332\pi\)
0.447451 + 0.894308i \(0.352332\pi\)
\(678\) 0 0
\(679\) 63.4007 2.43309
\(680\) 0 0
\(681\) 5.42296 0.207808
\(682\) 0 0
\(683\) 27.3415 1.04619 0.523097 0.852273i \(-0.324776\pi\)
0.523097 + 0.852273i \(0.324776\pi\)
\(684\) 0 0
\(685\) 10.6262 0.406006
\(686\) 0 0
\(687\) 31.3449 1.19588
\(688\) 0 0
\(689\) 8.18596 0.311860
\(690\) 0 0
\(691\) 38.8313 1.47721 0.738607 0.674137i \(-0.235484\pi\)
0.738607 + 0.674137i \(0.235484\pi\)
\(692\) 0 0
\(693\) 2.63389 0.100053
\(694\) 0 0
\(695\) 12.5693 0.476782
\(696\) 0 0
\(697\) −44.8959 −1.70055
\(698\) 0 0
\(699\) 14.9817 0.566659
\(700\) 0 0
\(701\) −19.5144 −0.737048 −0.368524 0.929618i \(-0.620137\pi\)
−0.368524 + 0.929618i \(0.620137\pi\)
\(702\) 0 0
\(703\) 0.387755 0.0146245
\(704\) 0 0
\(705\) −3.04623 −0.114728
\(706\) 0 0
\(707\) −32.5972 −1.22595
\(708\) 0 0
\(709\) −11.1387 −0.418322 −0.209161 0.977881i \(-0.567073\pi\)
−0.209161 + 0.977881i \(0.567073\pi\)
\(710\) 0 0
\(711\) −0.307362 −0.0115270
\(712\) 0 0
\(713\) 5.19658 0.194614
\(714\) 0 0
\(715\) 30.3265 1.13415
\(716\) 0 0
\(717\) −41.5866 −1.55308
\(718\) 0 0
\(719\) −16.1214 −0.601227 −0.300613 0.953746i \(-0.597191\pi\)
−0.300613 + 0.953746i \(0.597191\pi\)
\(720\) 0 0
\(721\) −3.04623 −0.113447
\(722\) 0 0
\(723\) 16.0492 0.596875
\(724\) 0 0
\(725\) −7.87859 −0.292604
\(726\) 0 0
\(727\) −8.21386 −0.304635 −0.152318 0.988332i \(-0.548674\pi\)
−0.152318 + 0.988332i \(0.548674\pi\)
\(728\) 0 0
\(729\) 26.0096 0.963318
\(730\) 0 0
\(731\) 72.7668 2.69138
\(732\) 0 0
\(733\) 28.6743 1.05911 0.529555 0.848276i \(-0.322359\pi\)
0.529555 + 0.848276i \(0.322359\pi\)
\(734\) 0 0
\(735\) −25.3694 −0.935766
\(736\) 0 0
\(737\) −28.8313 −1.06202
\(738\) 0 0
\(739\) −11.7014 −0.430442 −0.215221 0.976565i \(-0.569047\pi\)
−0.215221 + 0.976565i \(0.569047\pi\)
\(740\) 0 0
\(741\) −9.67243 −0.355325
\(742\) 0 0
\(743\) 1.91087 0.0701030 0.0350515 0.999386i \(-0.488840\pi\)
0.0350515 + 0.999386i \(0.488840\pi\)
\(744\) 0 0
\(745\) −7.04623 −0.258154
\(746\) 0 0
\(747\) −0.283233 −0.0103630
\(748\) 0 0
\(749\) 24.1493 0.882397
\(750\) 0 0
\(751\) 9.55976 0.348841 0.174420 0.984671i \(-0.444195\pi\)
0.174420 + 0.984671i \(0.444195\pi\)
\(752\) 0 0
\(753\) 23.8709 0.869904
\(754\) 0 0
\(755\) −4.47689 −0.162931
\(756\) 0 0
\(757\) −33.6435 −1.22279 −0.611397 0.791324i \(-0.709392\pi\)
−0.611397 + 0.791324i \(0.709392\pi\)
\(758\) 0 0
\(759\) −40.3698 −1.46533
\(760\) 0 0
\(761\) −22.4113 −0.812409 −0.406204 0.913782i \(-0.633148\pi\)
−0.406204 + 0.913782i \(0.633148\pi\)
\(762\) 0 0
\(763\) 54.6541 1.97861
\(764\) 0 0
\(765\) 0.683053 0.0246958
\(766\) 0 0
\(767\) 3.43835 0.124152
\(768\) 0 0
\(769\) 4.98937 0.179921 0.0899607 0.995945i \(-0.471326\pi\)
0.0899607 + 0.995945i \(0.471326\pi\)
\(770\) 0 0
\(771\) 18.7755 0.676183
\(772\) 0 0
\(773\) 16.6508 0.598887 0.299443 0.954114i \(-0.403199\pi\)
0.299443 + 0.954114i \(0.403199\pi\)
\(774\) 0 0
\(775\) 1.25240 0.0449874
\(776\) 0 0
\(777\) 3.15994 0.113362
\(778\) 0 0
\(779\) −6.77551 −0.242758
\(780\) 0 0
\(781\) 61.0096 2.18309
\(782\) 0 0
\(783\) 40.2051 1.43681
\(784\) 0 0
\(785\) −8.27072 −0.295195
\(786\) 0 0
\(787\) −43.8453 −1.56292 −0.781458 0.623958i \(-0.785524\pi\)
−0.781458 + 0.623958i \(0.785524\pi\)
\(788\) 0 0
\(789\) −31.6435 −1.12654
\(790\) 0 0
\(791\) 0.412335 0.0146609
\(792\) 0 0
\(793\) 82.6164 2.93380
\(794\) 0 0
\(795\) −2.62620 −0.0931416
\(796\) 0 0
\(797\) 49.0052 1.73585 0.867927 0.496692i \(-0.165452\pi\)
0.867927 + 0.496692i \(0.165452\pi\)
\(798\) 0 0
\(799\) −11.4586 −0.405375
\(800\) 0 0
\(801\) 0.440241 0.0155552
\(802\) 0 0
\(803\) 26.6897 0.941859
\(804\) 0 0
\(805\) 19.1955 0.676554
\(806\) 0 0
\(807\) 27.3940 0.964315
\(808\) 0 0
\(809\) 9.19554 0.323298 0.161649 0.986848i \(-0.448319\pi\)
0.161649 + 0.986848i \(0.448319\pi\)
\(810\) 0 0
\(811\) −18.9527 −0.665520 −0.332760 0.943011i \(-0.607980\pi\)
−0.332760 + 0.943011i \(0.607980\pi\)
\(812\) 0 0
\(813\) 14.6049 0.512218
\(814\) 0 0
\(815\) 18.5693 0.650456
\(816\) 0 0
\(817\) 10.9817 0.384200
\(818\) 0 0
\(819\) −2.61850 −0.0914979
\(820\) 0 0
\(821\) −45.7572 −1.59694 −0.798468 0.602037i \(-0.794356\pi\)
−0.798468 + 0.602037i \(0.794356\pi\)
\(822\) 0 0
\(823\) −11.3738 −0.396466 −0.198233 0.980155i \(-0.563520\pi\)
−0.198233 + 0.980155i \(0.563520\pi\)
\(824\) 0 0
\(825\) −9.72928 −0.338730
\(826\) 0 0
\(827\) −11.8386 −0.411669 −0.205835 0.978587i \(-0.565991\pi\)
−0.205835 + 0.978587i \(0.565991\pi\)
\(828\) 0 0
\(829\) −26.9894 −0.937380 −0.468690 0.883363i \(-0.655274\pi\)
−0.468690 + 0.883363i \(0.655274\pi\)
\(830\) 0 0
\(831\) −50.3265 −1.74581
\(832\) 0 0
\(833\) −95.4286 −3.30640
\(834\) 0 0
\(835\) 21.3694 0.739520
\(836\) 0 0
\(837\) −6.39108 −0.220908
\(838\) 0 0
\(839\) 2.81215 0.0970864 0.0485432 0.998821i \(-0.484542\pi\)
0.0485432 + 0.998821i \(0.484542\pi\)
\(840\) 0 0
\(841\) 33.0722 1.14042
\(842\) 0 0
\(843\) −34.0279 −1.17198
\(844\) 0 0
\(845\) −17.1493 −0.589954
\(846\) 0 0
\(847\) −90.2330 −3.10044
\(848\) 0 0
\(849\) 28.7110 0.985358
\(850\) 0 0
\(851\) −1.60892 −0.0551530
\(852\) 0 0
\(853\) −39.6068 −1.35611 −0.678056 0.735010i \(-0.737177\pi\)
−0.678056 + 0.735010i \(0.737177\pi\)
\(854\) 0 0
\(855\) 0.103084 0.00352539
\(856\) 0 0
\(857\) 17.8742 0.610572 0.305286 0.952261i \(-0.401248\pi\)
0.305286 + 0.952261i \(0.401248\pi\)
\(858\) 0 0
\(859\) 4.12910 0.140883 0.0704416 0.997516i \(-0.477559\pi\)
0.0704416 + 0.997516i \(0.477559\pi\)
\(860\) 0 0
\(861\) −55.2158 −1.88175
\(862\) 0 0
\(863\) 14.7509 0.502128 0.251064 0.967971i \(-0.419220\pi\)
0.251064 + 0.967971i \(0.419220\pi\)
\(864\) 0 0
\(865\) 12.3878 0.421196
\(866\) 0 0
\(867\) 47.3973 1.60970
\(868\) 0 0
\(869\) 16.4681 0.558644
\(870\) 0 0
\(871\) 28.6628 0.971203
\(872\) 0 0
\(873\) −1.41273 −0.0478137
\(874\) 0 0
\(875\) 4.62620 0.156394
\(876\) 0 0
\(877\) 16.8357 0.568501 0.284250 0.958750i \(-0.408255\pi\)
0.284250 + 0.958750i \(0.408255\pi\)
\(878\) 0 0
\(879\) −46.4971 −1.56831
\(880\) 0 0
\(881\) 45.4373 1.53082 0.765411 0.643542i \(-0.222536\pi\)
0.765411 + 0.643542i \(0.222536\pi\)
\(882\) 0 0
\(883\) 38.0837 1.28162 0.640810 0.767700i \(-0.278599\pi\)
0.640810 + 0.767700i \(0.278599\pi\)
\(884\) 0 0
\(885\) −1.10308 −0.0370798
\(886\) 0 0
\(887\) 21.3049 0.715348 0.357674 0.933847i \(-0.383570\pi\)
0.357674 + 0.933847i \(0.383570\pi\)
\(888\) 0 0
\(889\) −61.8496 −2.07437
\(890\) 0 0
\(891\) 51.3574 1.72054
\(892\) 0 0
\(893\) −1.72928 −0.0578682
\(894\) 0 0
\(895\) −12.7755 −0.427038
\(896\) 0 0
\(897\) 40.1339 1.34003
\(898\) 0 0
\(899\) −9.86712 −0.329087
\(900\) 0 0
\(901\) −9.87859 −0.329104
\(902\) 0 0
\(903\) 89.4931 2.97814
\(904\) 0 0
\(905\) −22.0000 −0.731305
\(906\) 0 0
\(907\) 12.4600 0.413728 0.206864 0.978370i \(-0.433674\pi\)
0.206864 + 0.978370i \(0.433674\pi\)
\(908\) 0 0
\(909\) 0.726351 0.0240916
\(910\) 0 0
\(911\) 35.2803 1.16889 0.584444 0.811434i \(-0.301313\pi\)
0.584444 + 0.811434i \(0.301313\pi\)
\(912\) 0 0
\(913\) 15.1753 0.502230
\(914\) 0 0
\(915\) −26.5048 −0.876221
\(916\) 0 0
\(917\) 18.5048 0.611082
\(918\) 0 0
\(919\) −0.392124 −0.0129350 −0.00646749 0.999979i \(-0.502059\pi\)
−0.00646749 + 0.999979i \(0.502059\pi\)
\(920\) 0 0
\(921\) −53.2158 −1.75352
\(922\) 0 0
\(923\) −60.6531 −1.99642
\(924\) 0 0
\(925\) −0.387755 −0.0127493
\(926\) 0 0
\(927\) 0.0678779 0.00222940
\(928\) 0 0
\(929\) −13.4942 −0.442729 −0.221365 0.975191i \(-0.571051\pi\)
−0.221365 + 0.975191i \(0.571051\pi\)
\(930\) 0 0
\(931\) −14.4017 −0.471997
\(932\) 0 0
\(933\) −7.26009 −0.237685
\(934\) 0 0
\(935\) −36.5972 −1.19686
\(936\) 0 0
\(937\) −53.4296 −1.74547 −0.872735 0.488195i \(-0.837656\pi\)
−0.872735 + 0.488195i \(0.837656\pi\)
\(938\) 0 0
\(939\) −26.6049 −0.868220
\(940\) 0 0
\(941\) −0.0568550 −0.00185342 −0.000926710 1.00000i \(-0.500295\pi\)
−0.000926710 1.00000i \(0.500295\pi\)
\(942\) 0 0
\(943\) 28.1137 0.915508
\(944\) 0 0
\(945\) −23.6079 −0.767964
\(946\) 0 0
\(947\) 2.03664 0.0661820 0.0330910 0.999452i \(-0.489465\pi\)
0.0330910 + 0.999452i \(0.489465\pi\)
\(948\) 0 0
\(949\) −26.5337 −0.861322
\(950\) 0 0
\(951\) −23.4509 −0.760446
\(952\) 0 0
\(953\) 17.5352 0.568020 0.284010 0.958821i \(-0.408335\pi\)
0.284010 + 0.958821i \(0.408335\pi\)
\(954\) 0 0
\(955\) 19.1955 0.621153
\(956\) 0 0
\(957\) 76.6531 2.47784
\(958\) 0 0
\(959\) 49.1589 1.58742
\(960\) 0 0
\(961\) −29.4315 −0.949403
\(962\) 0 0
\(963\) −0.538109 −0.0173403
\(964\) 0 0
\(965\) 13.7047 0.441170
\(966\) 0 0
\(967\) −14.9904 −0.482059 −0.241030 0.970518i \(-0.577485\pi\)
−0.241030 + 0.970518i \(0.577485\pi\)
\(968\) 0 0
\(969\) 11.6724 0.374972
\(970\) 0 0
\(971\) −20.8805 −0.670087 −0.335043 0.942203i \(-0.608751\pi\)
−0.335043 + 0.942203i \(0.608751\pi\)
\(972\) 0 0
\(973\) 58.1483 1.86415
\(974\) 0 0
\(975\) 9.67243 0.309766
\(976\) 0 0
\(977\) −24.0246 −0.768614 −0.384307 0.923205i \(-0.625560\pi\)
−0.384307 + 0.923205i \(0.625560\pi\)
\(978\) 0 0
\(979\) −23.5877 −0.753865
\(980\) 0 0
\(981\) −1.21784 −0.0388825
\(982\) 0 0
\(983\) −30.9205 −0.986209 −0.493105 0.869970i \(-0.664138\pi\)
−0.493105 + 0.869970i \(0.664138\pi\)
\(984\) 0 0
\(985\) 2.00000 0.0637253
\(986\) 0 0
\(987\) −14.0925 −0.448568
\(988\) 0 0
\(989\) −45.5664 −1.44893
\(990\) 0 0
\(991\) 2.56934 0.0816179 0.0408089 0.999167i \(-0.487007\pi\)
0.0408089 + 0.999167i \(0.487007\pi\)
\(992\) 0 0
\(993\) 29.8140 0.946120
\(994\) 0 0
\(995\) 0.561647 0.0178054
\(996\) 0 0
\(997\) 35.1878 1.11441 0.557205 0.830375i \(-0.311874\pi\)
0.557205 + 0.830375i \(0.311874\pi\)
\(998\) 0 0
\(999\) 1.97875 0.0626048
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 760.2.a.h.1.3 3
3.2 odd 2 6840.2.a.bj.1.1 3
4.3 odd 2 1520.2.a.r.1.1 3
5.2 odd 4 3800.2.d.k.3649.2 6
5.3 odd 4 3800.2.d.k.3649.5 6
5.4 even 2 3800.2.a.y.1.1 3
8.3 odd 2 6080.2.a.bs.1.3 3
8.5 even 2 6080.2.a.bw.1.1 3
20.19 odd 2 7600.2.a.bo.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.h.1.3 3 1.1 even 1 trivial
1520.2.a.r.1.1 3 4.3 odd 2
3800.2.a.y.1.1 3 5.4 even 2
3800.2.d.k.3649.2 6 5.2 odd 4
3800.2.d.k.3649.5 6 5.3 odd 4
6080.2.a.bs.1.3 3 8.3 odd 2
6080.2.a.bw.1.1 3 8.5 even 2
6840.2.a.bj.1.1 3 3.2 odd 2
7600.2.a.bo.1.3 3 20.19 odd 2