Properties

Label 3800.2.d.k.3649.5
Level $3800$
Weight $2$
Character 3800.3649
Analytic conductor $30.343$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3800,2,Mod(3649,3800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3800.3649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.d (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(30.3431527681\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5161984.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 760)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3649.5
Root \(0.432320 + 0.432320i\) of defining polynomial
Character \(\chi\) \(=\) 3800.3649
Dual form 3800.2.d.k.3649.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.76156i q^{3} +4.62620i q^{7} -0.103084 q^{9} +O(q^{10})\) \(q+1.76156i q^{3} +4.62620i q^{7} -0.103084 q^{9} -5.52311 q^{11} +5.49084i q^{13} +6.62620i q^{17} +1.00000 q^{19} -8.14931 q^{21} +4.14931i q^{23} +5.10308i q^{27} +7.87859 q^{29} +1.25240 q^{31} -9.72928i q^{33} +0.387755i q^{37} -9.67243 q^{39} +6.77551 q^{41} -10.9817i q^{43} -1.72928i q^{47} -14.4017 q^{49} -11.6724 q^{51} +1.49084i q^{53} +1.76156i q^{57} -0.626198 q^{59} +15.0462 q^{61} -0.476886i q^{63} -5.22012i q^{67} -7.30925 q^{69} -11.0462 q^{71} -4.83237i q^{73} -25.5510i q^{77} +2.98168 q^{79} -9.29862 q^{81} -2.74760i q^{83} +13.8786i q^{87} -4.27072 q^{89} -25.4017 q^{91} +2.20617i q^{93} +13.7047i q^{97} +0.569343 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 8 q^{9} - 8 q^{11} + 6 q^{19} - 6 q^{21} - 6 q^{29} - 28 q^{31} + 10 q^{39} - 20 q^{41} - 8 q^{49} - 2 q^{51} + 14 q^{59} + 40 q^{61} - 66 q^{69} - 16 q^{71} - 28 q^{79} + 30 q^{81} - 36 q^{89} - 74 q^{91} - 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 1.76156i 1.01704i 0.861052 + 0.508518i \(0.169806\pi\)
−0.861052 + 0.508518i \(0.830194\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.62620i 1.74854i 0.485441 + 0.874269i \(0.338659\pi\)
−0.485441 + 0.874269i \(0.661341\pi\)
\(8\) 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
\(13\) 5.49084i 1.52288i 0.648233 + 0.761442i \(0.275508\pi\)
−0.648233 + 0.761442i \(0.724492\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.62620i 1.60709i 0.595245 + 0.803545i \(0.297055\pi\)
−0.595245 + 0.803545i \(0.702945\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −8.14931 −1.77833
\(22\) 0 0
\(23\) 4.14931i 0.865191i 0.901588 + 0.432596i \(0.142402\pi\)
−0.901588 + 0.432596i \(0.857598\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.10308i 0.982089i
\(28\) 0 0
\(29\) 7.87859 1.46302 0.731509 0.681832i \(-0.238816\pi\)
0.731509 + 0.681832i \(0.238816\pi\)
\(30\) 0 0
\(31\) 1.25240 0.224937 0.112468 0.993655i \(-0.464124\pi\)
0.112468 + 0.993655i \(0.464124\pi\)
\(32\) 0 0
\(33\) − 9.72928i − 1.69365i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.387755i 0.0637466i 0.999492 + 0.0318733i \(0.0101473\pi\)
−0.999492 + 0.0318733i \(0.989853\pi\)
\(38\) 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
\(43\) − 10.9817i − 1.67469i −0.546675 0.837345i \(-0.684107\pi\)
0.546675 0.837345i \(-0.315893\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 1.72928i − 0.252242i −0.992015 0.126121i \(-0.959747\pi\)
0.992015 0.126121i \(-0.0402527\pi\)
\(48\) 0 0
\(49\) −14.4017 −2.05739
\(50\) 0 0
\(51\) −11.6724 −1.63447
\(52\) 0 0
\(53\) 1.49084i 0.204782i 0.994744 + 0.102391i \(0.0326494\pi\)
−0.994744 + 0.102391i \(0.967351\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.76156i 0.233324i
\(58\) 0 0
\(59\) −0.626198 −0.0815240 −0.0407620 0.999169i \(-0.512979\pi\)
−0.0407620 + 0.999169i \(0.512979\pi\)
\(60\) 0 0
\(61\) 15.0462 1.92647 0.963236 0.268656i \(-0.0865796\pi\)
0.963236 + 0.268656i \(0.0865796\pi\)
\(62\) 0 0
\(63\) − 0.476886i − 0.0600819i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 5.22012i − 0.637739i −0.947799 0.318870i \(-0.896697\pi\)
0.947799 0.318870i \(-0.103303\pi\)
\(68\) 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
\(73\) − 4.83237i − 0.565586i −0.959181 0.282793i \(-0.908739\pi\)
0.959181 0.282793i \(-0.0912609\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 25.5510i − 2.91181i
\(78\) 0 0
\(79\) 2.98168 0.335465 0.167732 0.985833i \(-0.446356\pi\)
0.167732 + 0.985833i \(0.446356\pi\)
\(80\) 0 0
\(81\) −9.29862 −1.03318
\(82\) 0 0
\(83\) − 2.74760i − 0.301589i −0.988565 0.150794i \(-0.951817\pi\)
0.988565 0.150794i \(-0.0481831\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 13.8786i 1.48794i
\(88\) 0 0
\(89\) −4.27072 −0.452695 −0.226348 0.974047i \(-0.572679\pi\)
−0.226348 + 0.974047i \(0.572679\pi\)
\(90\) 0 0
\(91\) −25.4017 −2.66282
\(92\) 0 0
\(93\) 2.20617i 0.228769i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 13.7047i 1.39150i 0.718283 + 0.695751i \(0.244928\pi\)
−0.718283 + 0.695751i \(0.755072\pi\)
\(98\) 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
\(103\) 0.658473i 0.0648813i 0.999474 + 0.0324407i \(0.0103280\pi\)
−0.999474 + 0.0324407i \(0.989672\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.22012i 0.504648i 0.967643 + 0.252324i \(0.0811949\pi\)
−0.967643 + 0.252324i \(0.918805\pi\)
\(108\) 0 0
\(109\) 11.8140 1.13158 0.565790 0.824549i \(-0.308571\pi\)
0.565790 + 0.824549i \(0.308571\pi\)
\(110\) 0 0
\(111\) −0.683053 −0.0648325
\(112\) 0 0
\(113\) − 0.0891304i − 0.00838468i −0.999991 0.00419234i \(-0.998666\pi\)
0.999991 0.00419234i \(-0.00133447\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 0.566016i − 0.0523282i
\(118\) 0 0
\(119\) −30.6541 −2.81006
\(120\) 0 0
\(121\) 19.5048 1.77316
\(122\) 0 0
\(123\) 11.9354i 1.07618i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 13.3694i − 1.18635i −0.805075 0.593173i \(-0.797875\pi\)
0.805075 0.593173i \(-0.202125\pi\)
\(128\) 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
\(133\) 4.62620i 0.401142i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 10.6262i 0.907857i 0.891038 + 0.453929i \(0.149978\pi\)
−0.891038 + 0.453929i \(0.850022\pi\)
\(138\) 0 0
\(139\) 12.5693 1.06612 0.533059 0.846078i \(-0.321042\pi\)
0.533059 + 0.846078i \(0.321042\pi\)
\(140\) 0 0
\(141\) 3.04623 0.256539
\(142\) 0 0
\(143\) − 30.3265i − 2.53603i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 25.3694i − 2.09244i
\(148\) 0 0
\(149\) −7.04623 −0.577250 −0.288625 0.957442i \(-0.593198\pi\)
−0.288625 + 0.957442i \(0.593198\pi\)
\(150\) 0 0
\(151\) 4.47689 0.364324 0.182162 0.983269i \(-0.441691\pi\)
0.182162 + 0.983269i \(0.441691\pi\)
\(152\) 0 0
\(153\) − 0.683053i − 0.0552216i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 8.27072i − 0.660075i −0.943968 0.330038i \(-0.892939\pi\)
0.943968 0.330038i \(-0.107061\pi\)
\(158\) 0 0
\(159\) −2.62620 −0.208271
\(160\) 0 0
\(161\) −19.1955 −1.51282
\(162\) 0 0
\(163\) − 18.5693i − 1.45446i −0.686392 0.727232i \(-0.740807\pi\)
0.686392 0.727232i \(-0.259193\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 21.3694i 1.65362i 0.562484 + 0.826808i \(0.309846\pi\)
−0.562484 + 0.826808i \(0.690154\pi\)
\(168\) 0 0
\(169\) −17.1493 −1.31918
\(170\) 0 0
\(171\) −0.103084 −0.00788301
\(172\) 0 0
\(173\) − 12.3878i − 0.941824i −0.882180 0.470912i \(-0.843925\pi\)
0.882180 0.470912i \(-0.156075\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 1.10308i − 0.0829128i
\(178\) 0 0
\(179\) −12.7755 −0.954886 −0.477443 0.878663i \(-0.658436\pi\)
−0.477443 + 0.878663i \(0.658436\pi\)
\(180\) 0 0
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 0 0
\(183\) 26.5048i 1.95929i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 36.5972i − 2.67626i
\(188\) 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
\(193\) − 13.7047i − 0.986486i −0.869892 0.493243i \(-0.835811\pi\)
0.869892 0.493243i \(-0.164189\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.00000i 0.142494i 0.997459 + 0.0712470i \(0.0226979\pi\)
−0.997459 + 0.0712470i \(0.977302\pi\)
\(198\) 0 0
\(199\) 0.561647 0.0398141 0.0199071 0.999802i \(-0.493663\pi\)
0.0199071 + 0.999802i \(0.493663\pi\)
\(200\) 0 0
\(201\) 9.19554 0.648603
\(202\) 0 0
\(203\) 36.4479i 2.55814i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 0.427726i − 0.0297290i
\(208\) 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
\(213\) − 19.4586i − 1.33328i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 5.79383i 0.393311i
\(218\) 0 0
\(219\) 8.51249 0.575221
\(220\) 0 0
\(221\) −36.3834 −2.44741
\(222\) 0 0
\(223\) 12.0525i 0.807094i 0.914959 + 0.403547i \(0.132223\pi\)
−0.914959 + 0.403547i \(0.867777\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 3.07850i − 0.204327i −0.994768 0.102164i \(-0.967423\pi\)
0.994768 0.102164i \(-0.0325766\pi\)
\(228\) 0 0
\(229\) −17.7938 −1.17585 −0.587925 0.808916i \(-0.700055\pi\)
−0.587925 + 0.808916i \(0.700055\pi\)
\(230\) 0 0
\(231\) 45.0096 2.96141
\(232\) 0 0
\(233\) 8.50479i 0.557167i 0.960412 + 0.278584i \(0.0898650\pi\)
−0.960412 + 0.278584i \(0.910135\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 5.25240i 0.341180i
\(238\) 0 0
\(239\) 23.6079 1.52707 0.763533 0.645768i \(-0.223463\pi\)
0.763533 + 0.645768i \(0.223463\pi\)
\(240\) 0 0
\(241\) 9.11078 0.586877 0.293438 0.955978i \(-0.405200\pi\)
0.293438 + 0.955978i \(0.405200\pi\)
\(242\) 0 0
\(243\) − 1.07081i − 0.0686924i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 5.49084i 0.349374i
\(248\) 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
\(253\) − 22.9171i − 1.44079i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 10.6585i − 0.664857i −0.943128 0.332429i \(-0.892132\pi\)
0.943128 0.332429i \(-0.107868\pi\)
\(258\) 0 0
\(259\) −1.79383 −0.111463
\(260\) 0 0
\(261\) −0.812155 −0.0502711
\(262\) 0 0
\(263\) − 17.9634i − 1.10767i −0.832627 0.553834i \(-0.813164\pi\)
0.832627 0.553834i \(-0.186836\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 7.52311i − 0.460407i
\(268\) 0 0
\(269\) −15.5510 −0.948162 −0.474081 0.880481i \(-0.657220\pi\)
−0.474081 + 0.880481i \(0.657220\pi\)
\(270\) 0 0
\(271\) 8.29093 0.503638 0.251819 0.967774i \(-0.418971\pi\)
0.251819 + 0.967774i \(0.418971\pi\)
\(272\) 0 0
\(273\) − 44.7466i − 2.70819i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 28.5693i 1.71657i 0.513177 + 0.858283i \(0.328468\pi\)
−0.513177 + 0.858283i \(0.671532\pi\)
\(278\) 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
\(283\) 16.2986i 0.968853i 0.874832 + 0.484426i \(0.160972\pi\)
−0.874832 + 0.484426i \(0.839028\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 31.3449i 1.85023i
\(288\) 0 0
\(289\) −26.9065 −1.58274
\(290\) 0 0
\(291\) −24.1416 −1.41521
\(292\) 0 0
\(293\) − 26.3955i − 1.54204i −0.636812 0.771019i \(-0.719747\pi\)
0.636812 0.771019i \(-0.280253\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 28.1849i − 1.63545i
\(298\) 0 0
\(299\) −22.7832 −1.31759
\(300\) 0 0
\(301\) 50.8034 2.92826
\(302\) 0 0
\(303\) 12.4123i 0.713070i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 30.2095i 1.72415i 0.506783 + 0.862073i \(0.330834\pi\)
−0.506783 + 0.862073i \(0.669166\pi\)
\(308\) 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
\(313\) − 15.1031i − 0.853677i −0.904328 0.426838i \(-0.859627\pi\)
0.904328 0.426838i \(-0.140373\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 13.3126i 0.747709i 0.927487 + 0.373854i \(0.121964\pi\)
−0.927487 + 0.373854i \(0.878036\pi\)
\(318\) 0 0
\(319\) −43.5144 −2.43634
\(320\) 0 0
\(321\) −9.19554 −0.513245
\(322\) 0 0
\(323\) 6.62620i 0.368692i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 20.8111i 1.15086i
\(328\) 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
\(333\) − 0.0399712i − 0.00219041i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 1.16327i − 0.0633671i −0.999498 0.0316836i \(-0.989913\pi\)
0.999498 0.0316836i \(-0.0100869\pi\)
\(338\) 0 0
\(339\) 0.157008 0.00852752
\(340\) 0 0
\(341\) −6.91713 −0.374583
\(342\) 0 0
\(343\) − 34.2418i − 1.84888i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 3.70138i − 0.198700i −0.995053 0.0993501i \(-0.968324\pi\)
0.995053 0.0993501i \(-0.0316764\pi\)
\(348\) 0 0
\(349\) 2.54144 0.136040 0.0680200 0.997684i \(-0.478332\pi\)
0.0680200 + 0.997684i \(0.478332\pi\)
\(350\) 0 0
\(351\) −28.0202 −1.49561
\(352\) 0 0
\(353\) − 21.3738i − 1.13761i −0.822471 0.568806i \(-0.807405\pi\)
0.822471 0.568806i \(-0.192595\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 53.9990i − 2.85793i
\(358\) 0 0
\(359\) 4.42003 0.233280 0.116640 0.993174i \(-0.462788\pi\)
0.116640 + 0.993174i \(0.462788\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 34.3588i 1.80337i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 3.28030i − 0.171230i −0.996328 0.0856152i \(-0.972714\pi\)
0.996328 0.0856152i \(-0.0272856\pi\)
\(368\) 0 0
\(369\) −0.698445 −0.0363596
\(370\) 0 0
\(371\) −6.89692 −0.358070
\(372\) 0 0
\(373\) − 14.8078i − 0.766718i −0.923599 0.383359i \(-0.874767\pi\)
0.923599 0.383359i \(-0.125233\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 43.2601i 2.22801i
\(378\) 0 0
\(379\) 19.7370 1.01382 0.506910 0.861999i \(-0.330788\pi\)
0.506910 + 0.861999i \(0.330788\pi\)
\(380\) 0 0
\(381\) 23.5510 1.20656
\(382\) 0 0
\(383\) − 10.2095i − 0.521681i −0.965382 0.260840i \(-0.916000\pi\)
0.965382 0.260840i \(-0.0839996\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.13203i 0.0575444i
\(388\) 0 0
\(389\) −10.7110 −0.543067 −0.271534 0.962429i \(-0.587531\pi\)
−0.271534 + 0.962429i \(0.587531\pi\)
\(390\) 0 0
\(391\) −27.4942 −1.39044
\(392\) 0 0
\(393\) − 7.04623i − 0.355435i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 20.5693i − 1.03235i −0.856484 0.516173i \(-0.827356\pi\)
0.856484 0.516173i \(-0.172644\pi\)
\(398\) 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
\(403\) 6.87671i 0.342553i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 2.14162i − 0.106156i
\(408\) 0 0
\(409\) 4.50479 0.222748 0.111374 0.993779i \(-0.464475\pi\)
0.111374 + 0.993779i \(0.464475\pi\)
\(410\) 0 0
\(411\) −18.7187 −0.923323
\(412\) 0 0
\(413\) − 2.89692i − 0.142548i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 22.1416i 1.08428i
\(418\) 0 0
\(419\) 31.7572 1.55144 0.775720 0.631077i \(-0.217387\pi\)
0.775720 + 0.631077i \(0.217387\pi\)
\(420\) 0 0
\(421\) −21.4296 −1.04442 −0.522208 0.852818i \(-0.674891\pi\)
−0.522208 + 0.852818i \(0.674891\pi\)
\(422\) 0 0
\(423\) 0.178261i 0.00866734i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 69.6068i 3.36851i
\(428\) 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
\(433\) − 24.6864i − 1.18635i −0.805073 0.593176i \(-0.797874\pi\)
0.805073 0.593176i \(-0.202126\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.14931i 0.198489i
\(438\) 0 0
\(439\) 14.9817 0.715036 0.357518 0.933906i \(-0.383623\pi\)
0.357518 + 0.933906i \(0.383623\pi\)
\(440\) 0 0
\(441\) 1.48458 0.0706944
\(442\) 0 0
\(443\) − 4.54144i − 0.215770i −0.994163 0.107885i \(-0.965592\pi\)
0.994163 0.107885i \(-0.0344079\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 12.4123i − 0.587083i
\(448\) 0 0
\(449\) 16.0925 0.759450 0.379725 0.925099i \(-0.376019\pi\)
0.379725 + 0.925099i \(0.376019\pi\)
\(450\) 0 0
\(451\) −37.4219 −1.76213
\(452\) 0 0
\(453\) 7.88629i 0.370530i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 4.65410i 0.217710i 0.994058 + 0.108855i \(0.0347184\pi\)
−0.994058 + 0.108855i \(0.965282\pi\)
\(458\) 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
\(463\) − 10.6339i − 0.494199i −0.968990 0.247099i \(-0.920523\pi\)
0.968990 0.247099i \(-0.0794774\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 32.8805i − 1.52153i −0.649029 0.760764i \(-0.724825\pi\)
0.649029 0.760764i \(-0.275175\pi\)
\(468\) 0 0
\(469\) 24.1493 1.11511
\(470\) 0 0
\(471\) 14.5693 0.671320
\(472\) 0 0
\(473\) 60.6531i 2.78883i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 0.153681i − 0.00703658i
\(478\) 0 0
\(479\) −12.4402 −0.568409 −0.284205 0.958764i \(-0.591729\pi\)
−0.284205 + 0.958764i \(0.591729\pi\)
\(480\) 0 0
\(481\) −2.12910 −0.0970787
\(482\) 0 0
\(483\) − 33.8140i − 1.53859i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 40.9571i − 1.85594i −0.372651 0.927972i \(-0.621551\pi\)
0.372651 0.927972i \(-0.378449\pi\)
\(488\) 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
\(493\) 52.2051i 2.35120i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 51.1020i − 2.29224i
\(498\) 0 0
\(499\) −25.4094 −1.13748 −0.568741 0.822517i \(-0.692569\pi\)
−0.568741 + 0.822517i \(0.692569\pi\)
\(500\) 0 0
\(501\) −37.6435 −1.68179
\(502\) 0 0
\(503\) 34.1127i 1.52101i 0.649333 + 0.760504i \(0.275048\pi\)
−0.649333 + 0.760504i \(0.724952\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 30.2095i − 1.34165i
\(508\) 0 0
\(509\) 22.3632 0.991230 0.495615 0.868542i \(-0.334943\pi\)
0.495615 + 0.868542i \(0.334943\pi\)
\(510\) 0 0
\(511\) 22.3555 0.988948
\(512\) 0 0
\(513\) 5.10308i 0.225307i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 9.55102i 0.420053i
\(518\) 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
\(523\) 27.3771i 1.19712i 0.801079 + 0.598559i \(0.204260\pi\)
−0.801079 + 0.598559i \(0.795740\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.29862i 0.361494i
\(528\) 0 0
\(529\) 5.78321 0.251444
\(530\) 0 0
\(531\) 0.0645508 0.00280127
\(532\) 0 0
\(533\) 37.2032i 1.61145i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 22.5048i − 0.971153i
\(538\) 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
\(543\) 38.7543i 1.66310i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 13.9754i 0.597546i 0.954324 + 0.298773i \(0.0965773\pi\)
−0.954324 + 0.298773i \(0.903423\pi\)
\(548\) 0 0
\(549\) −1.55102 −0.0661960
\(550\) 0 0
\(551\) 7.87859 0.335639
\(552\) 0 0
\(553\) 13.7938i 0.586573i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 30.6618i − 1.29918i −0.760284 0.649591i \(-0.774940\pi\)
0.760284 0.649591i \(-0.225060\pi\)
\(558\) 0 0
\(559\) 60.2986 2.55036
\(560\) 0 0
\(561\) 64.4681 2.72185
\(562\) 0 0
\(563\) − 2.02458i − 0.0853259i −0.999090 0.0426629i \(-0.986416\pi\)
0.999090 0.0426629i \(-0.0135842\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 43.0173i − 1.80656i
\(568\) 0 0
\(569\) 28.2620 1.18480 0.592402 0.805643i \(-0.298180\pi\)
0.592402 + 0.805643i \(0.298180\pi\)
\(570\) 0 0
\(571\) 10.6464 0.445538 0.222769 0.974871i \(-0.428490\pi\)
0.222769 + 0.974871i \(0.428490\pi\)
\(572\) 0 0
\(573\) − 33.8140i − 1.41260i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 9.30925i 0.387549i 0.981046 + 0.193775i \(0.0620730\pi\)
−0.981046 + 0.193775i \(0.937927\pi\)
\(578\) 0 0
\(579\) 24.1416 1.00329
\(580\) 0 0
\(581\) 12.7110 0.527339
\(582\) 0 0
\(583\) − 8.23407i − 0.341021i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 42.3911i 1.74967i 0.484424 + 0.874834i \(0.339029\pi\)
−0.484424 + 0.874834i \(0.660971\pi\)
\(588\) 0 0
\(589\) 1.25240 0.0516041
\(590\) 0 0
\(591\) −3.52311 −0.144922
\(592\) 0 0
\(593\) 28.5048i 1.17055i 0.810834 + 0.585276i \(0.199014\pi\)
−0.810834 + 0.585276i \(0.800986\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0.989374i 0.0404924i
\(598\) 0 0
\(599\) −25.6156 −1.04662 −0.523312 0.852141i \(-0.675304\pi\)
−0.523312 + 0.852141i \(0.675304\pi\)
\(600\) 0 0
\(601\) −29.2803 −1.19437 −0.597184 0.802104i \(-0.703714\pi\)
−0.597184 + 0.802104i \(0.703714\pi\)
\(602\) 0 0
\(603\) 0.538109i 0.0219135i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 20.0525i 0.813905i 0.913449 + 0.406953i \(0.133409\pi\)
−0.913449 + 0.406953i \(0.866591\pi\)
\(608\) 0 0
\(609\) −64.2051 −2.60172
\(610\) 0 0
\(611\) 9.49521 0.384135
\(612\) 0 0
\(613\) − 7.89881i − 0.319030i −0.987196 0.159515i \(-0.949007\pi\)
0.987196 0.159515i \(-0.0509930\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 41.4094i 1.66708i 0.552459 + 0.833540i \(0.313689\pi\)
−0.552459 + 0.833540i \(0.686311\pi\)
\(618\) 0 0
\(619\) −26.9450 −1.08301 −0.541506 0.840697i \(-0.682146\pi\)
−0.541506 + 0.840697i \(0.682146\pi\)
\(620\) 0 0
\(621\) −21.1743 −0.849695
\(622\) 0 0
\(623\) − 19.7572i − 0.791555i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 9.72928i − 0.388550i
\(628\) 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
\(633\) 27.6079i 1.09731i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 79.0775i − 3.13316i
\(638\) 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
\(643\) 5.25240i 0.207134i 0.994622 + 0.103567i \(0.0330257\pi\)
−0.994622 + 0.103567i \(0.966974\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 1.64452i − 0.0646528i −0.999477 0.0323264i \(-0.989708\pi\)
0.999477 0.0323264i \(-0.0102916\pi\)
\(648\) 0 0
\(649\) 3.45856 0.135760
\(650\) 0 0
\(651\) −10.2062 −0.400011
\(652\) 0 0
\(653\) − 30.9450i − 1.21097i −0.795856 0.605486i \(-0.792979\pi\)
0.795856 0.605486i \(-0.207021\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0.498138i 0.0194342i
\(658\) 0 0
\(659\) −17.3372 −0.675360 −0.337680 0.941261i \(-0.609642\pi\)
−0.337680 + 0.941261i \(0.609642\pi\)
\(660\) 0 0
\(661\) 8.83237 0.343539 0.171770 0.985137i \(-0.445052\pi\)
0.171770 + 0.985137i \(0.445052\pi\)
\(662\) 0 0
\(663\) − 64.0914i − 2.48910i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 32.6907i 1.26579i
\(668\) 0 0
\(669\) −21.2311 −0.820843
\(670\) 0 0
\(671\) −83.1020 −3.20812
\(672\) 0 0
\(673\) 29.6402i 1.14254i 0.820761 + 0.571272i \(0.193550\pi\)
−0.820761 + 0.571272i \(0.806450\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 23.2847i − 0.894903i −0.894308 0.447451i \(-0.852332\pi\)
0.894308 0.447451i \(-0.147668\pi\)
\(678\) 0 0
\(679\) −63.4007 −2.43309
\(680\) 0 0
\(681\) 5.42296 0.207808
\(682\) 0 0
\(683\) 27.3415i 1.04619i 0.852273 + 0.523097i \(0.175224\pi\)
−0.852273 + 0.523097i \(0.824776\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 31.3449i − 1.19588i
\(688\) 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
\(693\) 2.63389i 0.100053i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 44.8959i 1.70055i
\(698\) 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
\(703\) 0.387755i 0.0146245i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 32.5972i 1.22595i
\(708\) 0 0
\(709\) 11.1387 0.418322 0.209161 0.977881i \(-0.432927\pi\)
0.209161 + 0.977881i \(0.432927\pi\)
\(710\) 0 0
\(711\) −0.307362 −0.0115270
\(712\) 0 0
\(713\) 5.19658i 0.194614i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 41.5866i 1.55308i
\(718\) 0 0
\(719\) 16.1214 0.601227 0.300613 0.953746i \(-0.402809\pi\)
0.300613 + 0.953746i \(0.402809\pi\)
\(720\) 0 0
\(721\) −3.04623 −0.113447
\(722\) 0 0
\(723\) 16.0492i 0.596875i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 8.21386i 0.304635i 0.988332 + 0.152318i \(0.0486737\pi\)
−0.988332 + 0.152318i \(0.951326\pi\)
\(728\) 0 0
\(729\) −26.0096 −0.963318
\(730\) 0 0
\(731\) 72.7668 2.69138
\(732\) 0 0
\(733\) 28.6743i 1.05911i 0.848276 + 0.529555i \(0.177641\pi\)
−0.848276 + 0.529555i \(0.822359\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 28.8313i 1.06202i
\(738\) 0 0
\(739\) 11.7014 0.430442 0.215221 0.976565i \(-0.430953\pi\)
0.215221 + 0.976565i \(0.430953\pi\)
\(740\) 0 0
\(741\) −9.67243 −0.355325
\(742\) 0 0
\(743\) 1.91087i 0.0701030i 0.999386 + 0.0350515i \(0.0111595\pi\)
−0.999386 + 0.0350515i \(0.988840\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0.283233i 0.0103630i
\(748\) 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
\(753\) 23.8709i 0.869904i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 33.6435i 1.22279i 0.791324 + 0.611397i \(0.209392\pi\)
−0.791324 + 0.611397i \(0.790608\pi\)
\(758\) 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
\(763\) 54.6541i 1.97861i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 3.43835i − 0.124152i
\(768\) 0 0
\(769\) −4.98937 −0.179921 −0.0899607 0.995945i \(-0.528674\pi\)
−0.0899607 + 0.995945i \(0.528674\pi\)
\(770\) 0 0
\(771\) 18.7755 0.676183
\(772\) 0 0
\(773\) 16.6508i 0.598887i 0.954114 + 0.299443i \(0.0968010\pi\)
−0.954114 + 0.299443i \(0.903199\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 3.15994i − 0.113362i
\(778\) 0 0
\(779\) 6.77551 0.242758
\(780\) 0 0
\(781\) 61.0096 2.18309
\(782\) 0 0
\(783\) 40.2051i 1.43681i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 43.8453i 1.56292i 0.623958 + 0.781458i \(0.285524\pi\)
−0.623958 + 0.781458i \(0.714476\pi\)
\(788\) 0 0
\(789\) 31.6435 1.12654
\(790\) 0 0
\(791\) 0.412335 0.0146609
\(792\) 0 0
\(793\) 82.6164i 2.93380i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 49.0052i − 1.73585i −0.496692 0.867927i \(-0.665452\pi\)
0.496692 0.867927i \(-0.334548\pi\)
\(798\) 0 0
\(799\) 11.4586 0.405375
\(800\) 0 0
\(801\) 0.440241 0.0155552
\(802\) 0 0
\(803\) 26.6897i 0.941859i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 27.3940i − 0.964315i
\(808\) 0 0
\(809\) −9.19554 −0.323298 −0.161649 0.986848i \(-0.551681\pi\)
−0.161649 + 0.986848i \(0.551681\pi\)
\(810\) 0 0
\(811\) −18.9527 −0.665520 −0.332760 0.943011i \(-0.607980\pi\)
−0.332760 + 0.943011i \(0.607980\pi\)
\(812\) 0 0
\(813\) 14.6049i 0.512218i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 10.9817i − 0.384200i
\(818\) 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
\(823\) − 11.3738i − 0.396466i −0.980155 0.198233i \(-0.936480\pi\)
0.980155 0.198233i \(-0.0635202\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 11.8386i 0.411669i 0.978587 + 0.205835i \(0.0659909\pi\)
−0.978587 + 0.205835i \(0.934009\pi\)
\(828\) 0 0
\(829\) 26.9894 0.937380 0.468690 0.883363i \(-0.344726\pi\)
0.468690 + 0.883363i \(0.344726\pi\)
\(830\) 0 0
\(831\) −50.3265 −1.74581
\(832\) 0 0
\(833\) − 95.4286i − 3.30640i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 6.39108i 0.220908i
\(838\) 0 0
\(839\) −2.81215 −0.0970864 −0.0485432 0.998821i \(-0.515458\pi\)
−0.0485432 + 0.998821i \(0.515458\pi\)
\(840\) 0 0
\(841\) 33.0722 1.14042
\(842\) 0 0
\(843\) − 34.0279i − 1.17198i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 90.2330i 3.10044i
\(848\) 0 0
\(849\) −28.7110 −0.985358
\(850\) 0 0
\(851\) −1.60892 −0.0551530
\(852\) 0 0
\(853\) − 39.6068i − 1.35611i −0.735010 0.678056i \(-0.762823\pi\)
0.735010 0.678056i \(-0.237177\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 17.8742i − 0.610572i −0.952261 0.305286i \(-0.901248\pi\)
0.952261 0.305286i \(-0.0987520\pi\)
\(858\) 0 0
\(859\) −4.12910 −0.140883 −0.0704416 0.997516i \(-0.522441\pi\)
−0.0704416 + 0.997516i \(0.522441\pi\)
\(860\) 0 0
\(861\) −55.2158 −1.88175
\(862\) 0 0
\(863\) 14.7509i 0.502128i 0.967971 + 0.251064i \(0.0807804\pi\)
−0.967971 + 0.251064i \(0.919220\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 47.3973i − 1.60970i
\(868\) 0 0
\(869\) −16.4681 −0.558644
\(870\) 0 0
\(871\) 28.6628 0.971203
\(872\) 0 0
\(873\) − 1.41273i − 0.0478137i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 16.8357i − 0.568501i −0.958750 0.284250i \(-0.908255\pi\)
0.958750 0.284250i \(-0.0917447\pi\)
\(878\) 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
\(883\) 38.0837i 1.28162i 0.767700 + 0.640810i \(0.221401\pi\)
−0.767700 + 0.640810i \(0.778599\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 21.3049i − 0.715348i −0.933847 0.357674i \(-0.883570\pi\)
0.933847 0.357674i \(-0.116430\pi\)
\(888\) 0 0
\(889\) 61.8496 2.07437
\(890\) 0 0
\(891\) 51.3574 1.72054
\(892\) 0 0
\(893\) − 1.72928i − 0.0578682i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 40.1339i − 1.34003i
\(898\) 0 0
\(899\) 9.86712 0.329087
\(900\) 0 0
\(901\) −9.87859 −0.329104
\(902\) 0 0
\(903\) 89.4931i 2.97814i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 12.4600i − 0.413728i −0.978370 0.206864i \(-0.933674\pi\)
0.978370 0.206864i \(-0.0663257\pi\)
\(908\) 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
\(913\) 15.1753i 0.502230i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 18.5048i − 0.611082i
\(918\) 0 0
\(919\) 0.392124 0.0129350 0.00646749 0.999979i \(-0.497941\pi\)
0.00646749 + 0.999979i \(0.497941\pi\)
\(920\) 0 0
\(921\) −53.2158 −1.75352
\(922\) 0 0
\(923\) − 60.6531i − 1.99642i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 0.0678779i − 0.00222940i
\(928\) 0 0
\(929\) 13.4942 0.442729 0.221365 0.975191i \(-0.428949\pi\)
0.221365 + 0.975191i \(0.428949\pi\)
\(930\) 0 0
\(931\) −14.4017 −0.471997
\(932\) 0 0
\(933\) − 7.26009i − 0.237685i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 53.4296i 1.74547i 0.488195 + 0.872735i \(0.337656\pi\)
−0.488195 + 0.872735i \(0.662344\pi\)
\(938\) 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
\(943\) 28.1137i 0.915508i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 2.03664i − 0.0661820i −0.999452 0.0330910i \(-0.989465\pi\)
0.999452 0.0330910i \(-0.0105351\pi\)
\(948\) 0 0
\(949\) 26.5337 0.861322
\(950\) 0 0
\(951\) −23.4509 −0.760446
\(952\) 0 0
\(953\) 17.5352i 0.568020i 0.958821 + 0.284010i \(0.0916649\pi\)
−0.958821 + 0.284010i \(0.908335\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 76.6531i − 2.47784i
\(958\) 0 0
\(959\) −49.1589 −1.58742
\(960\) 0 0
\(961\) −29.4315 −0.949403
\(962\) 0 0
\(963\) − 0.538109i − 0.0173403i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 14.9904i 0.482059i 0.970518 + 0.241030i \(0.0774851\pi\)
−0.970518 + 0.241030i \(0.922515\pi\)
\(968\) 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
\(973\) 58.1483i 1.86415i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 24.0246i 0.768614i 0.923205 + 0.384307i \(0.125560\pi\)
−0.923205 + 0.384307i \(0.874440\pi\)
\(978\) 0 0
\(979\) 23.5877 0.753865
\(980\) 0 0
\(981\) −1.21784 −0.0388825
\(982\) 0 0
\(983\) − 30.9205i − 0.986209i −0.869970 0.493105i \(-0.835862\pi\)
0.869970 0.493105i \(-0.164138\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 14.0925i 0.448568i
\(988\) 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
\(993\) 29.8140i 0.946120i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 35.1878i − 1.11441i −0.830375 0.557205i \(-0.811874\pi\)
0.830375 0.557205i \(-0.188126\pi\)
\(998\) 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 3800.2.d.k.3649.5 6
5.2 odd 4 760.2.a.h.1.3 3
5.3 odd 4 3800.2.a.y.1.1 3
5.4 even 2 inner 3800.2.d.k.3649.2 6
15.2 even 4 6840.2.a.bj.1.1 3
20.3 even 4 7600.2.a.bo.1.3 3
20.7 even 4 1520.2.a.r.1.1 3
40.27 even 4 6080.2.a.bs.1.3 3
40.37 odd 4 6080.2.a.bw.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.h.1.3 3 5.2 odd 4
1520.2.a.r.1.1 3 20.7 even 4
3800.2.a.y.1.1 3 5.3 odd 4
3800.2.d.k.3649.2 6 5.4 even 2 inner
3800.2.d.k.3649.5 6 1.1 even 1 trivial
6080.2.a.bs.1.3 3 40.27 even 4
6080.2.a.bw.1.1 3 40.37 odd 4
6840.2.a.bj.1.1 3 15.2 even 4
7600.2.a.bo.1.3 3 20.3 even 4