Properties

Label 3800.2.a.z.1.3
Level $3800$
Weight $2$
Character 3800.1
Self dual yes
Analytic conductor $30.343$
Analytic rank $1$
Dimension $6$
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] = [3800,2,Mod(1,3800)]
 
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, 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("3800.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.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: \(30.3431527681\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.253565184.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 11x^{4} + 20x^{3} + 22x^{2} - 32x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 760)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.664406\) of defining polynomial
Character \(\chi\) \(=\) 3800.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.664406 q^{3} +0.345799 q^{7} -2.55856 q^{9} +O(q^{10})\) \(q-0.664406 q^{3} +0.345799 q^{7} -2.55856 q^{9} +3.51609 q^{11} +3.22452 q^{13} -4.88550 q^{17} +1.00000 q^{19} -0.229751 q^{21} -7.62861 q^{23} +3.69315 q^{27} -1.73968 q^{29} -4.76164 q^{31} -2.33611 q^{33} +1.86749 q^{37} -2.14239 q^{39} +6.76164 q^{41} -0.606115 q^{43} -3.34735 q^{47} -6.88042 q^{49} +3.24596 q^{51} +0.107390 q^{53} -0.664406 q^{57} +12.3233 q^{59} -8.41231 q^{61} -0.884749 q^{63} -2.23888 q^{67} +5.06850 q^{69} -0.536605 q^{71} -5.57710 q^{73} +1.21586 q^{77} -2.67119 q^{79} +5.22194 q^{81} -1.79847 q^{83} +1.15586 q^{87} +8.94482 q^{89} +1.11504 q^{91} +3.16367 q^{93} -10.1560 q^{97} -8.99614 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} - 6 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{3} - 6 q^{7} + 8 q^{9} - 2 q^{11} - 14 q^{13} - 10 q^{17} + 6 q^{19} + 18 q^{21} - 2 q^{23} - 2 q^{27} - 2 q^{29} + 8 q^{31} - 8 q^{33} - 4 q^{37} - 18 q^{39} + 4 q^{41} - 4 q^{43} - 4 q^{47} + 2 q^{49} - 34 q^{51} + 14 q^{53} - 2 q^{57} - 2 q^{59} + 10 q^{61} - 44 q^{63} - 42 q^{67} + 18 q^{69} - 8 q^{71} + 2 q^{73} - 24 q^{77} - 20 q^{79} + 6 q^{81} - 16 q^{83} - 30 q^{87} + 32 q^{89} + 10 q^{91} - 12 q^{93} - 40 q^{97} - 22 q^{99}+O(q^{100}) \) 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
\(3\) −0.664406 −0.383595 −0.191798 0.981434i \(-0.561432\pi\)
−0.191798 + 0.981434i \(0.561432\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.345799 0.130700 0.0653499 0.997862i \(-0.479184\pi\)
0.0653499 + 0.997862i \(0.479184\pi\)
\(8\) 0 0
\(9\) −2.55856 −0.852855
\(10\) 0 0
\(11\) 3.51609 1.06014 0.530071 0.847954i \(-0.322165\pi\)
0.530071 + 0.847954i \(0.322165\pi\)
\(12\) 0 0
\(13\) 3.22452 0.894320 0.447160 0.894454i \(-0.352435\pi\)
0.447160 + 0.894454i \(0.352435\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.88550 −1.18491 −0.592454 0.805604i \(-0.701841\pi\)
−0.592454 + 0.805604i \(0.701841\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −0.229751 −0.0501358
\(22\) 0 0
\(23\) −7.62861 −1.59067 −0.795337 0.606167i \(-0.792706\pi\)
−0.795337 + 0.606167i \(0.792706\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3.69315 0.710746
\(28\) 0 0
\(29\) −1.73968 −0.323051 −0.161526 0.986869i \(-0.551641\pi\)
−0.161526 + 0.986869i \(0.551641\pi\)
\(30\) 0 0
\(31\) −4.76164 −0.855216 −0.427608 0.903964i \(-0.640644\pi\)
−0.427608 + 0.903964i \(0.640644\pi\)
\(32\) 0 0
\(33\) −2.33611 −0.406665
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.86749 0.307013 0.153506 0.988148i \(-0.450943\pi\)
0.153506 + 0.988148i \(0.450943\pi\)
\(38\) 0 0
\(39\) −2.14239 −0.343057
\(40\) 0 0
\(41\) 6.76164 1.05599 0.527996 0.849247i \(-0.322944\pi\)
0.527996 + 0.849247i \(0.322944\pi\)
\(42\) 0 0
\(43\) −0.606115 −0.0924317 −0.0462158 0.998931i \(-0.514716\pi\)
−0.0462158 + 0.998931i \(0.514716\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.34735 −0.488261 −0.244130 0.969742i \(-0.578502\pi\)
−0.244130 + 0.969742i \(0.578502\pi\)
\(48\) 0 0
\(49\) −6.88042 −0.982918
\(50\) 0 0
\(51\) 3.24596 0.454525
\(52\) 0 0
\(53\) 0.107390 0.0147512 0.00737559 0.999973i \(-0.497652\pi\)
0.00737559 + 0.999973i \(0.497652\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.664406 −0.0880028
\(58\) 0 0
\(59\) 12.3233 1.60436 0.802179 0.597084i \(-0.203674\pi\)
0.802179 + 0.597084i \(0.203674\pi\)
\(60\) 0 0
\(61\) −8.41231 −1.07709 −0.538543 0.842598i \(-0.681025\pi\)
−0.538543 + 0.842598i \(0.681025\pi\)
\(62\) 0 0
\(63\) −0.884749 −0.111468
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −2.23888 −0.273522 −0.136761 0.990604i \(-0.543669\pi\)
−0.136761 + 0.990604i \(0.543669\pi\)
\(68\) 0 0
\(69\) 5.06850 0.610175
\(70\) 0 0
\(71\) −0.536605 −0.0636833 −0.0318417 0.999493i \(-0.510137\pi\)
−0.0318417 + 0.999493i \(0.510137\pi\)
\(72\) 0 0
\(73\) −5.57710 −0.652750 −0.326375 0.945240i \(-0.605827\pi\)
−0.326375 + 0.945240i \(0.605827\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.21586 0.138560
\(78\) 0 0
\(79\) −2.67119 −0.300532 −0.150266 0.988646i \(-0.548013\pi\)
−0.150266 + 0.988646i \(0.548013\pi\)
\(80\) 0 0
\(81\) 5.22194 0.580216
\(82\) 0 0
\(83\) −1.79847 −0.197408 −0.0987038 0.995117i \(-0.531470\pi\)
−0.0987038 + 0.995117i \(0.531470\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.15586 0.123921
\(88\) 0 0
\(89\) 8.94482 0.948149 0.474075 0.880485i \(-0.342783\pi\)
0.474075 + 0.880485i \(0.342783\pi\)
\(90\) 0 0
\(91\) 1.11504 0.116887
\(92\) 0 0
\(93\) 3.16367 0.328057
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −10.1560 −1.03119 −0.515594 0.856833i \(-0.672429\pi\)
−0.515594 + 0.856833i \(0.672429\pi\)
\(98\) 0 0
\(99\) −8.99614 −0.904146
\(100\) 0 0
\(101\) 16.5089 1.64270 0.821350 0.570425i \(-0.193221\pi\)
0.821350 + 0.570425i \(0.193221\pi\)
\(102\) 0 0
\(103\) −11.5212 −1.13522 −0.567610 0.823298i \(-0.692132\pi\)
−0.567610 + 0.823298i \(0.692132\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −10.0340 −0.970027 −0.485014 0.874507i \(-0.661185\pi\)
−0.485014 + 0.874507i \(0.661185\pi\)
\(108\) 0 0
\(109\) 12.2737 1.17560 0.587802 0.809005i \(-0.299993\pi\)
0.587802 + 0.809005i \(0.299993\pi\)
\(110\) 0 0
\(111\) −1.24077 −0.117769
\(112\) 0 0
\(113\) 0.700723 0.0659185 0.0329593 0.999457i \(-0.489507\pi\)
0.0329593 + 0.999457i \(0.489507\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −8.25014 −0.762725
\(118\) 0 0
\(119\) −1.68940 −0.154867
\(120\) 0 0
\(121\) 1.36289 0.123899
\(122\) 0 0
\(123\) −4.49248 −0.405073
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −11.8369 −1.05036 −0.525178 0.850992i \(-0.676001\pi\)
−0.525178 + 0.850992i \(0.676001\pi\)
\(128\) 0 0
\(129\) 0.402707 0.0354563
\(130\) 0 0
\(131\) −22.0556 −1.92701 −0.963503 0.267699i \(-0.913737\pi\)
−0.963503 + 0.267699i \(0.913737\pi\)
\(132\) 0 0
\(133\) 0.345799 0.0299846
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −19.4858 −1.66478 −0.832390 0.554190i \(-0.813028\pi\)
−0.832390 + 0.554190i \(0.813028\pi\)
\(138\) 0 0
\(139\) −14.5867 −1.23723 −0.618613 0.785696i \(-0.712305\pi\)
−0.618613 + 0.785696i \(0.712305\pi\)
\(140\) 0 0
\(141\) 2.22400 0.187294
\(142\) 0 0
\(143\) 11.3377 0.948106
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 4.57140 0.377043
\(148\) 0 0
\(149\) 15.3918 1.26094 0.630471 0.776213i \(-0.282862\pi\)
0.630471 + 0.776213i \(0.282862\pi\)
\(150\) 0 0
\(151\) −3.55281 −0.289124 −0.144562 0.989496i \(-0.546177\pi\)
−0.144562 + 0.989496i \(0.546177\pi\)
\(152\) 0 0
\(153\) 12.4999 1.01055
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −10.9812 −0.876393 −0.438196 0.898879i \(-0.644383\pi\)
−0.438196 + 0.898879i \(0.644383\pi\)
\(158\) 0 0
\(159\) −0.0713507 −0.00565848
\(160\) 0 0
\(161\) −2.63797 −0.207901
\(162\) 0 0
\(163\) −7.88687 −0.617747 −0.308874 0.951103i \(-0.599952\pi\)
−0.308874 + 0.951103i \(0.599952\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −14.1924 −1.09824 −0.549121 0.835743i \(-0.685037\pi\)
−0.549121 + 0.835743i \(0.685037\pi\)
\(168\) 0 0
\(169\) −2.60248 −0.200191
\(170\) 0 0
\(171\) −2.55856 −0.195658
\(172\) 0 0
\(173\) 0.654185 0.0497368 0.0248684 0.999691i \(-0.492083\pi\)
0.0248684 + 0.999691i \(0.492083\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −8.18768 −0.615424
\(178\) 0 0
\(179\) 0.893129 0.0667556 0.0333778 0.999443i \(-0.489374\pi\)
0.0333778 + 0.999443i \(0.489374\pi\)
\(180\) 0 0
\(181\) −20.8542 −1.55008 −0.775039 0.631914i \(-0.782270\pi\)
−0.775039 + 0.631914i \(0.782270\pi\)
\(182\) 0 0
\(183\) 5.58920 0.413165
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −17.1779 −1.25617
\(188\) 0 0
\(189\) 1.27709 0.0928944
\(190\) 0 0
\(191\) −13.8797 −1.00430 −0.502151 0.864780i \(-0.667458\pi\)
−0.502151 + 0.864780i \(0.667458\pi\)
\(192\) 0 0
\(193\) −9.98613 −0.718817 −0.359409 0.933180i \(-0.617022\pi\)
−0.359409 + 0.933180i \(0.617022\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.3247 0.878099 0.439050 0.898463i \(-0.355315\pi\)
0.439050 + 0.898463i \(0.355315\pi\)
\(198\) 0 0
\(199\) −16.0086 −1.13482 −0.567410 0.823435i \(-0.692055\pi\)
−0.567410 + 0.823435i \(0.692055\pi\)
\(200\) 0 0
\(201\) 1.48752 0.104922
\(202\) 0 0
\(203\) −0.601581 −0.0422227
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 19.5183 1.35661
\(208\) 0 0
\(209\) 3.51609 0.243213
\(210\) 0 0
\(211\) 19.0558 1.31185 0.655926 0.754825i \(-0.272278\pi\)
0.655926 + 0.754825i \(0.272278\pi\)
\(212\) 0 0
\(213\) 0.356524 0.0244286
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −1.64657 −0.111777
\(218\) 0 0
\(219\) 3.70546 0.250392
\(220\) 0 0
\(221\) −15.7534 −1.05969
\(222\) 0 0
\(223\) 8.68798 0.581790 0.290895 0.956755i \(-0.406047\pi\)
0.290895 + 0.956755i \(0.406047\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −28.8077 −1.91203 −0.956017 0.293311i \(-0.905243\pi\)
−0.956017 + 0.293311i \(0.905243\pi\)
\(228\) 0 0
\(229\) 8.27464 0.546803 0.273402 0.961900i \(-0.411851\pi\)
0.273402 + 0.961900i \(0.411851\pi\)
\(230\) 0 0
\(231\) −0.807826 −0.0531510
\(232\) 0 0
\(233\) 29.0199 1.90116 0.950580 0.310480i \(-0.100490\pi\)
0.950580 + 0.310480i \(0.100490\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.77475 0.115283
\(238\) 0 0
\(239\) −15.3744 −0.994486 −0.497243 0.867611i \(-0.665654\pi\)
−0.497243 + 0.867611i \(0.665654\pi\)
\(240\) 0 0
\(241\) 24.8583 1.60126 0.800632 0.599157i \(-0.204497\pi\)
0.800632 + 0.599157i \(0.204497\pi\)
\(242\) 0 0
\(243\) −14.5489 −0.933314
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.22452 0.205171
\(248\) 0 0
\(249\) 1.19491 0.0757246
\(250\) 0 0
\(251\) −14.3807 −0.907702 −0.453851 0.891077i \(-0.649950\pi\)
−0.453851 + 0.891077i \(0.649950\pi\)
\(252\) 0 0
\(253\) −26.8229 −1.68634
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −21.9784 −1.37097 −0.685486 0.728086i \(-0.740410\pi\)
−0.685486 + 0.728086i \(0.740410\pi\)
\(258\) 0 0
\(259\) 0.645775 0.0401265
\(260\) 0 0
\(261\) 4.45109 0.275516
\(262\) 0 0
\(263\) 2.24444 0.138398 0.0691990 0.997603i \(-0.477956\pi\)
0.0691990 + 0.997603i \(0.477956\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −5.94300 −0.363705
\(268\) 0 0
\(269\) 19.3526 1.17995 0.589975 0.807421i \(-0.299137\pi\)
0.589975 + 0.807421i \(0.299137\pi\)
\(270\) 0 0
\(271\) −4.18631 −0.254300 −0.127150 0.991883i \(-0.540583\pi\)
−0.127150 + 0.991883i \(0.540583\pi\)
\(272\) 0 0
\(273\) −0.740837 −0.0448375
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 15.2206 0.914518 0.457259 0.889334i \(-0.348831\pi\)
0.457259 + 0.889334i \(0.348831\pi\)
\(278\) 0 0
\(279\) 12.1830 0.729375
\(280\) 0 0
\(281\) 7.96393 0.475088 0.237544 0.971377i \(-0.423658\pi\)
0.237544 + 0.971377i \(0.423658\pi\)
\(282\) 0 0
\(283\) 31.6286 1.88013 0.940064 0.340998i \(-0.110765\pi\)
0.940064 + 0.340998i \(0.110765\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.33817 0.138018
\(288\) 0 0
\(289\) 6.86811 0.404006
\(290\) 0 0
\(291\) 6.74773 0.395559
\(292\) 0 0
\(293\) −32.6323 −1.90640 −0.953199 0.302342i \(-0.902231\pi\)
−0.953199 + 0.302342i \(0.902231\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 12.9854 0.753491
\(298\) 0 0
\(299\) −24.5986 −1.42257
\(300\) 0 0
\(301\) −0.209594 −0.0120808
\(302\) 0 0
\(303\) −10.9686 −0.630132
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −4.85468 −0.277071 −0.138536 0.990357i \(-0.544240\pi\)
−0.138536 + 0.990357i \(0.544240\pi\)
\(308\) 0 0
\(309\) 7.65477 0.435465
\(310\) 0 0
\(311\) −0.509761 −0.0289059 −0.0144530 0.999896i \(-0.504601\pi\)
−0.0144530 + 0.999896i \(0.504601\pi\)
\(312\) 0 0
\(313\) −5.33353 −0.301469 −0.150734 0.988574i \(-0.548164\pi\)
−0.150734 + 0.988574i \(0.548164\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 15.1325 0.849927 0.424963 0.905211i \(-0.360287\pi\)
0.424963 + 0.905211i \(0.360287\pi\)
\(318\) 0 0
\(319\) −6.11689 −0.342480
\(320\) 0 0
\(321\) 6.66668 0.372098
\(322\) 0 0
\(323\) −4.88550 −0.271836
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −8.15470 −0.450956
\(328\) 0 0
\(329\) −1.15751 −0.0638156
\(330\) 0 0
\(331\) 4.21378 0.231610 0.115805 0.993272i \(-0.463055\pi\)
0.115805 + 0.993272i \(0.463055\pi\)
\(332\) 0 0
\(333\) −4.77808 −0.261837
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 16.5707 0.902664 0.451332 0.892356i \(-0.350949\pi\)
0.451332 + 0.892356i \(0.350949\pi\)
\(338\) 0 0
\(339\) −0.465565 −0.0252860
\(340\) 0 0
\(341\) −16.7424 −0.906650
\(342\) 0 0
\(343\) −4.79984 −0.259167
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −5.17123 −0.277606 −0.138803 0.990320i \(-0.544325\pi\)
−0.138803 + 0.990320i \(0.544325\pi\)
\(348\) 0 0
\(349\) −12.5843 −0.673624 −0.336812 0.941572i \(-0.609349\pi\)
−0.336812 + 0.941572i \(0.609349\pi\)
\(350\) 0 0
\(351\) 11.9086 0.635635
\(352\) 0 0
\(353\) 13.2242 0.703853 0.351927 0.936028i \(-0.385527\pi\)
0.351927 + 0.936028i \(0.385527\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 1.12245 0.0594063
\(358\) 0 0
\(359\) −9.77950 −0.516142 −0.258071 0.966126i \(-0.583087\pi\)
−0.258071 + 0.966126i \(0.583087\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −0.905512 −0.0475271
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 19.7113 1.02892 0.514460 0.857515i \(-0.327993\pi\)
0.514460 + 0.857515i \(0.327993\pi\)
\(368\) 0 0
\(369\) −17.3001 −0.900607
\(370\) 0 0
\(371\) 0.0371354 0.00192798
\(372\) 0 0
\(373\) −33.0128 −1.70934 −0.854670 0.519172i \(-0.826240\pi\)
−0.854670 + 0.519172i \(0.826240\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −5.60964 −0.288911
\(378\) 0 0
\(379\) −29.3422 −1.50721 −0.753604 0.657329i \(-0.771686\pi\)
−0.753604 + 0.657329i \(0.771686\pi\)
\(380\) 0 0
\(381\) 7.86453 0.402912
\(382\) 0 0
\(383\) 29.9938 1.53261 0.766305 0.642477i \(-0.222093\pi\)
0.766305 + 0.642477i \(0.222093\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.55078 0.0788308
\(388\) 0 0
\(389\) 7.64392 0.387562 0.193781 0.981045i \(-0.437925\pi\)
0.193781 + 0.981045i \(0.437925\pi\)
\(390\) 0 0
\(391\) 37.2696 1.88480
\(392\) 0 0
\(393\) 14.6539 0.739190
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 4.48509 0.225100 0.112550 0.993646i \(-0.464098\pi\)
0.112550 + 0.993646i \(0.464098\pi\)
\(398\) 0 0
\(399\) −0.229751 −0.0115019
\(400\) 0 0
\(401\) 3.11631 0.155621 0.0778105 0.996968i \(-0.475207\pi\)
0.0778105 + 0.996968i \(0.475207\pi\)
\(402\) 0 0
\(403\) −15.3540 −0.764837
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6.56625 0.325477
\(408\) 0 0
\(409\) 9.88207 0.488637 0.244319 0.969695i \(-0.421436\pi\)
0.244319 + 0.969695i \(0.421436\pi\)
\(410\) 0 0
\(411\) 12.9465 0.638602
\(412\) 0 0
\(413\) 4.26139 0.209689
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 9.69148 0.474594
\(418\) 0 0
\(419\) −4.05273 −0.197989 −0.0989944 0.995088i \(-0.531563\pi\)
−0.0989944 + 0.995088i \(0.531563\pi\)
\(420\) 0 0
\(421\) −34.5916 −1.68589 −0.842945 0.538000i \(-0.819180\pi\)
−0.842945 + 0.538000i \(0.819180\pi\)
\(422\) 0 0
\(423\) 8.56440 0.416415
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −2.90897 −0.140775
\(428\) 0 0
\(429\) −7.53284 −0.363689
\(430\) 0 0
\(431\) −22.8014 −1.09831 −0.549153 0.835722i \(-0.685050\pi\)
−0.549153 + 0.835722i \(0.685050\pi\)
\(432\) 0 0
\(433\) −19.8043 −0.951733 −0.475867 0.879517i \(-0.657866\pi\)
−0.475867 + 0.879517i \(0.657866\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −7.62861 −0.364926
\(438\) 0 0
\(439\) −24.5069 −1.16965 −0.584825 0.811160i \(-0.698837\pi\)
−0.584825 + 0.811160i \(0.698837\pi\)
\(440\) 0 0
\(441\) 17.6040 0.838286
\(442\) 0 0
\(443\) −29.8226 −1.41692 −0.708458 0.705753i \(-0.750609\pi\)
−0.708458 + 0.705753i \(0.750609\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −10.2264 −0.483692
\(448\) 0 0
\(449\) −10.8675 −0.512869 −0.256435 0.966562i \(-0.582548\pi\)
−0.256435 + 0.966562i \(0.582548\pi\)
\(450\) 0 0
\(451\) 23.7745 1.11950
\(452\) 0 0
\(453\) 2.36051 0.110906
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −0.604334 −0.0282696 −0.0141348 0.999900i \(-0.504499\pi\)
−0.0141348 + 0.999900i \(0.504499\pi\)
\(458\) 0 0
\(459\) −18.0429 −0.842169
\(460\) 0 0
\(461\) −31.2349 −1.45475 −0.727376 0.686239i \(-0.759261\pi\)
−0.727376 + 0.686239i \(0.759261\pi\)
\(462\) 0 0
\(463\) −33.5246 −1.55802 −0.779010 0.627011i \(-0.784278\pi\)
−0.779010 + 0.627011i \(0.784278\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 37.0124 1.71273 0.856364 0.516372i \(-0.172718\pi\)
0.856364 + 0.516372i \(0.172718\pi\)
\(468\) 0 0
\(469\) −0.774201 −0.0357493
\(470\) 0 0
\(471\) 7.29596 0.336180
\(472\) 0 0
\(473\) −2.13115 −0.0979906
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −0.274765 −0.0125806
\(478\) 0 0
\(479\) 14.1388 0.646020 0.323010 0.946396i \(-0.395305\pi\)
0.323010 + 0.946396i \(0.395305\pi\)
\(480\) 0 0
\(481\) 6.02174 0.274568
\(482\) 0 0
\(483\) 1.75268 0.0797498
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −36.2708 −1.64359 −0.821793 0.569785i \(-0.807026\pi\)
−0.821793 + 0.569785i \(0.807026\pi\)
\(488\) 0 0
\(489\) 5.24009 0.236965
\(490\) 0 0
\(491\) −12.8630 −0.580498 −0.290249 0.956951i \(-0.593738\pi\)
−0.290249 + 0.956951i \(0.593738\pi\)
\(492\) 0 0
\(493\) 8.49923 0.382786
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −0.185558 −0.00832339
\(498\) 0 0
\(499\) 30.9889 1.38725 0.693626 0.720336i \(-0.256012\pi\)
0.693626 + 0.720336i \(0.256012\pi\)
\(500\) 0 0
\(501\) 9.42953 0.421280
\(502\) 0 0
\(503\) 3.17853 0.141724 0.0708619 0.997486i \(-0.477425\pi\)
0.0708619 + 0.997486i \(0.477425\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.72911 0.0767923
\(508\) 0 0
\(509\) −10.6322 −0.471265 −0.235632 0.971842i \(-0.575716\pi\)
−0.235632 + 0.971842i \(0.575716\pi\)
\(510\) 0 0
\(511\) −1.92856 −0.0853143
\(512\) 0 0
\(513\) 3.69315 0.163056
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −11.7696 −0.517625
\(518\) 0 0
\(519\) −0.434645 −0.0190788
\(520\) 0 0
\(521\) 31.4792 1.37913 0.689564 0.724225i \(-0.257802\pi\)
0.689564 + 0.724225i \(0.257802\pi\)
\(522\) 0 0
\(523\) 2.60906 0.114086 0.0570430 0.998372i \(-0.481833\pi\)
0.0570430 + 0.998372i \(0.481833\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 23.2630 1.01335
\(528\) 0 0
\(529\) 35.1957 1.53025
\(530\) 0 0
\(531\) −31.5300 −1.36828
\(532\) 0 0
\(533\) 21.8030 0.944395
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −0.593401 −0.0256071
\(538\) 0 0
\(539\) −24.1922 −1.04203
\(540\) 0 0
\(541\) 8.22279 0.353525 0.176763 0.984254i \(-0.443438\pi\)
0.176763 + 0.984254i \(0.443438\pi\)
\(542\) 0 0
\(543\) 13.8556 0.594602
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 14.9776 0.640394 0.320197 0.947351i \(-0.396251\pi\)
0.320197 + 0.947351i \(0.396251\pi\)
\(548\) 0 0
\(549\) 21.5234 0.918598
\(550\) 0 0
\(551\) −1.73968 −0.0741130
\(552\) 0 0
\(553\) −0.923694 −0.0392795
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −22.5965 −0.957443 −0.478722 0.877967i \(-0.658900\pi\)
−0.478722 + 0.877967i \(0.658900\pi\)
\(558\) 0 0
\(559\) −1.95443 −0.0826635
\(560\) 0 0
\(561\) 11.4131 0.481861
\(562\) 0 0
\(563\) 21.7518 0.916728 0.458364 0.888765i \(-0.348436\pi\)
0.458364 + 0.888765i \(0.348436\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.80574 0.0758341
\(568\) 0 0
\(569\) 32.5606 1.36501 0.682506 0.730880i \(-0.260890\pi\)
0.682506 + 0.730880i \(0.260890\pi\)
\(570\) 0 0
\(571\) −11.4285 −0.478267 −0.239134 0.970987i \(-0.576863\pi\)
−0.239134 + 0.970987i \(0.576863\pi\)
\(572\) 0 0
\(573\) 9.22179 0.385246
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 23.4277 0.975307 0.487654 0.873037i \(-0.337853\pi\)
0.487654 + 0.873037i \(0.337853\pi\)
\(578\) 0 0
\(579\) 6.63485 0.275735
\(580\) 0 0
\(581\) −0.621909 −0.0258011
\(582\) 0 0
\(583\) 0.377594 0.0156383
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.85818 −0.0766953 −0.0383477 0.999264i \(-0.512209\pi\)
−0.0383477 + 0.999264i \(0.512209\pi\)
\(588\) 0 0
\(589\) −4.76164 −0.196200
\(590\) 0 0
\(591\) −8.18862 −0.336835
\(592\) 0 0
\(593\) −1.37788 −0.0565829 −0.0282914 0.999600i \(-0.509007\pi\)
−0.0282914 + 0.999600i \(0.509007\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 10.6362 0.435312
\(598\) 0 0
\(599\) 33.9275 1.38624 0.693120 0.720822i \(-0.256235\pi\)
0.693120 + 0.720822i \(0.256235\pi\)
\(600\) 0 0
\(601\) −15.9368 −0.650076 −0.325038 0.945701i \(-0.605377\pi\)
−0.325038 + 0.945701i \(0.605377\pi\)
\(602\) 0 0
\(603\) 5.72831 0.233275
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 32.3921 1.31475 0.657377 0.753562i \(-0.271666\pi\)
0.657377 + 0.753562i \(0.271666\pi\)
\(608\) 0 0
\(609\) 0.399694 0.0161964
\(610\) 0 0
\(611\) −10.7936 −0.436662
\(612\) 0 0
\(613\) −8.44582 −0.341123 −0.170562 0.985347i \(-0.554558\pi\)
−0.170562 + 0.985347i \(0.554558\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 37.4621 1.50817 0.754084 0.656778i \(-0.228081\pi\)
0.754084 + 0.656778i \(0.228081\pi\)
\(618\) 0 0
\(619\) −7.35835 −0.295757 −0.147878 0.989006i \(-0.547244\pi\)
−0.147878 + 0.989006i \(0.547244\pi\)
\(620\) 0 0
\(621\) −28.1736 −1.13057
\(622\) 0 0
\(623\) 3.09311 0.123923
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −2.33611 −0.0932954
\(628\) 0 0
\(629\) −9.12360 −0.363782
\(630\) 0 0
\(631\) 35.8314 1.42642 0.713212 0.700948i \(-0.247240\pi\)
0.713212 + 0.700948i \(0.247240\pi\)
\(632\) 0 0
\(633\) −12.6608 −0.503220
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −22.1860 −0.879043
\(638\) 0 0
\(639\) 1.37294 0.0543126
\(640\) 0 0
\(641\) 15.5827 0.615479 0.307740 0.951471i \(-0.400427\pi\)
0.307740 + 0.951471i \(0.400427\pi\)
\(642\) 0 0
\(643\) 44.5775 1.75797 0.878983 0.476852i \(-0.158222\pi\)
0.878983 + 0.476852i \(0.158222\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 13.9836 0.549752 0.274876 0.961480i \(-0.411363\pi\)
0.274876 + 0.961480i \(0.411363\pi\)
\(648\) 0 0
\(649\) 43.3298 1.70085
\(650\) 0 0
\(651\) 1.09399 0.0428770
\(652\) 0 0
\(653\) 18.9179 0.740314 0.370157 0.928969i \(-0.379304\pi\)
0.370157 + 0.928969i \(0.379304\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 14.2694 0.556701
\(658\) 0 0
\(659\) 35.9816 1.40164 0.700822 0.713337i \(-0.252817\pi\)
0.700822 + 0.713337i \(0.252817\pi\)
\(660\) 0 0
\(661\) 7.69177 0.299175 0.149588 0.988748i \(-0.452205\pi\)
0.149588 + 0.988748i \(0.452205\pi\)
\(662\) 0 0
\(663\) 10.4667 0.406491
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 13.2714 0.513870
\(668\) 0 0
\(669\) −5.77235 −0.223172
\(670\) 0 0
\(671\) −29.5785 −1.14186
\(672\) 0 0
\(673\) −19.0426 −0.734040 −0.367020 0.930213i \(-0.619622\pi\)
−0.367020 + 0.930213i \(0.619622\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4.61140 0.177230 0.0886152 0.996066i \(-0.471756\pi\)
0.0886152 + 0.996066i \(0.471756\pi\)
\(678\) 0 0
\(679\) −3.51194 −0.134776
\(680\) 0 0
\(681\) 19.1400 0.733447
\(682\) 0 0
\(683\) 37.5087 1.43523 0.717614 0.696441i \(-0.245234\pi\)
0.717614 + 0.696441i \(0.245234\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −5.49772 −0.209751
\(688\) 0 0
\(689\) 0.346282 0.0131923
\(690\) 0 0
\(691\) −18.3081 −0.696473 −0.348236 0.937407i \(-0.613219\pi\)
−0.348236 + 0.937407i \(0.613219\pi\)
\(692\) 0 0
\(693\) −3.11086 −0.118172
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −33.0340 −1.25125
\(698\) 0 0
\(699\) −19.2810 −0.729276
\(700\) 0 0
\(701\) 6.22677 0.235182 0.117591 0.993062i \(-0.462483\pi\)
0.117591 + 0.993062i \(0.462483\pi\)
\(702\) 0 0
\(703\) 1.86749 0.0704335
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 5.70877 0.214700
\(708\) 0 0
\(709\) −30.0878 −1.12997 −0.564986 0.825100i \(-0.691119\pi\)
−0.564986 + 0.825100i \(0.691119\pi\)
\(710\) 0 0
\(711\) 6.83440 0.256310
\(712\) 0 0
\(713\) 36.3247 1.36037
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 10.2148 0.381480
\(718\) 0 0
\(719\) −32.7649 −1.22193 −0.610963 0.791659i \(-0.709218\pi\)
−0.610963 + 0.791659i \(0.709218\pi\)
\(720\) 0 0
\(721\) −3.98403 −0.148373
\(722\) 0 0
\(723\) −16.5160 −0.614237
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −24.9335 −0.924733 −0.462367 0.886689i \(-0.653000\pi\)
−0.462367 + 0.886689i \(0.653000\pi\)
\(728\) 0 0
\(729\) −5.99942 −0.222201
\(730\) 0 0
\(731\) 2.96117 0.109523
\(732\) 0 0
\(733\) −16.3618 −0.604338 −0.302169 0.953254i \(-0.597711\pi\)
−0.302169 + 0.953254i \(0.597711\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −7.87209 −0.289972
\(738\) 0 0
\(739\) 14.0239 0.515876 0.257938 0.966162i \(-0.416957\pi\)
0.257938 + 0.966162i \(0.416957\pi\)
\(740\) 0 0
\(741\) −2.14239 −0.0787027
\(742\) 0 0
\(743\) 48.2145 1.76882 0.884408 0.466714i \(-0.154562\pi\)
0.884408 + 0.466714i \(0.154562\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 4.60150 0.168360
\(748\) 0 0
\(749\) −3.46976 −0.126782
\(750\) 0 0
\(751\) −29.4639 −1.07515 −0.537577 0.843214i \(-0.680660\pi\)
−0.537577 + 0.843214i \(0.680660\pi\)
\(752\) 0 0
\(753\) 9.55464 0.348190
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 17.8966 0.650464 0.325232 0.945634i \(-0.394558\pi\)
0.325232 + 0.945634i \(0.394558\pi\)
\(758\) 0 0
\(759\) 17.8213 0.646872
\(760\) 0 0
\(761\) 38.1996 1.38474 0.692368 0.721545i \(-0.256568\pi\)
0.692368 + 0.721545i \(0.256568\pi\)
\(762\) 0 0
\(763\) 4.24422 0.153651
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 39.7367 1.43481
\(768\) 0 0
\(769\) 39.5374 1.42576 0.712878 0.701289i \(-0.247392\pi\)
0.712878 + 0.701289i \(0.247392\pi\)
\(770\) 0 0
\(771\) 14.6026 0.525898
\(772\) 0 0
\(773\) 51.3427 1.84667 0.923335 0.383995i \(-0.125452\pi\)
0.923335 + 0.383995i \(0.125452\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −0.429057 −0.0153923
\(778\) 0 0
\(779\) 6.76164 0.242261
\(780\) 0 0
\(781\) −1.88675 −0.0675133
\(782\) 0 0
\(783\) −6.42491 −0.229607
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 41.6629 1.48512 0.742560 0.669779i \(-0.233611\pi\)
0.742560 + 0.669779i \(0.233611\pi\)
\(788\) 0 0
\(789\) −1.49122 −0.0530888
\(790\) 0 0
\(791\) 0.242309 0.00861553
\(792\) 0 0
\(793\) −27.1257 −0.963261
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −12.4446 −0.440812 −0.220406 0.975408i \(-0.570738\pi\)
−0.220406 + 0.975408i \(0.570738\pi\)
\(798\) 0 0
\(799\) 16.3535 0.578544
\(800\) 0 0
\(801\) −22.8859 −0.808633
\(802\) 0 0
\(803\) −19.6096 −0.692007
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −12.8580 −0.452624
\(808\) 0 0
\(809\) 0.283694 0.00997414 0.00498707 0.999988i \(-0.498413\pi\)
0.00498707 + 0.999988i \(0.498413\pi\)
\(810\) 0 0
\(811\) −46.2273 −1.62326 −0.811631 0.584171i \(-0.801420\pi\)
−0.811631 + 0.584171i \(0.801420\pi\)
\(812\) 0 0
\(813\) 2.78141 0.0975483
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −0.606115 −0.0212053
\(818\) 0 0
\(819\) −2.85289 −0.0996880
\(820\) 0 0
\(821\) 24.7448 0.863600 0.431800 0.901969i \(-0.357879\pi\)
0.431800 + 0.901969i \(0.357879\pi\)
\(822\) 0 0
\(823\) −29.5937 −1.03157 −0.515786 0.856717i \(-0.672500\pi\)
−0.515786 + 0.856717i \(0.672500\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 52.8976 1.83943 0.919715 0.392587i \(-0.128420\pi\)
0.919715 + 0.392587i \(0.128420\pi\)
\(828\) 0 0
\(829\) 7.44924 0.258723 0.129361 0.991598i \(-0.458707\pi\)
0.129361 + 0.991598i \(0.458707\pi\)
\(830\) 0 0
\(831\) −10.1127 −0.350805
\(832\) 0 0
\(833\) 33.6143 1.16467
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −17.5854 −0.607842
\(838\) 0 0
\(839\) 7.89398 0.272531 0.136265 0.990672i \(-0.456490\pi\)
0.136265 + 0.990672i \(0.456490\pi\)
\(840\) 0 0
\(841\) −25.9735 −0.895638
\(842\) 0 0
\(843\) −5.29128 −0.182242
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0.471286 0.0161936
\(848\) 0 0
\(849\) −21.0143 −0.721208
\(850\) 0 0
\(851\) −14.2463 −0.488357
\(852\) 0 0
\(853\) 46.6687 1.59791 0.798953 0.601393i \(-0.205387\pi\)
0.798953 + 0.601393i \(0.205387\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 7.92083 0.270570 0.135285 0.990807i \(-0.456805\pi\)
0.135285 + 0.990807i \(0.456805\pi\)
\(858\) 0 0
\(859\) 8.42793 0.287557 0.143779 0.989610i \(-0.454075\pi\)
0.143779 + 0.989610i \(0.454075\pi\)
\(860\) 0 0
\(861\) −1.55350 −0.0529430
\(862\) 0 0
\(863\) −28.2751 −0.962495 −0.481248 0.876585i \(-0.659816\pi\)
−0.481248 + 0.876585i \(0.659816\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −4.56322 −0.154975
\(868\) 0 0
\(869\) −9.39213 −0.318606
\(870\) 0 0
\(871\) −7.21930 −0.244617
\(872\) 0 0
\(873\) 25.9848 0.879453
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 32.0169 1.08113 0.540567 0.841301i \(-0.318210\pi\)
0.540567 + 0.841301i \(0.318210\pi\)
\(878\) 0 0
\(879\) 21.6811 0.731286
\(880\) 0 0
\(881\) 55.6183 1.87383 0.936914 0.349560i \(-0.113669\pi\)
0.936914 + 0.349560i \(0.113669\pi\)
\(882\) 0 0
\(883\) −16.1272 −0.542723 −0.271362 0.962477i \(-0.587474\pi\)
−0.271362 + 0.962477i \(0.587474\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 14.9850 0.503146 0.251573 0.967838i \(-0.419052\pi\)
0.251573 + 0.967838i \(0.419052\pi\)
\(888\) 0 0
\(889\) −4.09320 −0.137281
\(890\) 0 0
\(891\) 18.3608 0.615111
\(892\) 0 0
\(893\) −3.34735 −0.112015
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 16.3435 0.545692
\(898\) 0 0
\(899\) 8.28375 0.276279
\(900\) 0 0
\(901\) −0.524655 −0.0174788
\(902\) 0 0
\(903\) 0.139256 0.00463414
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −45.0654 −1.49637 −0.748185 0.663490i \(-0.769075\pi\)
−0.748185 + 0.663490i \(0.769075\pi\)
\(908\) 0 0
\(909\) −42.2391 −1.40098
\(910\) 0 0
\(911\) −3.86130 −0.127930 −0.0639652 0.997952i \(-0.520375\pi\)
−0.0639652 + 0.997952i \(0.520375\pi\)
\(912\) 0 0
\(913\) −6.32358 −0.209280
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −7.62680 −0.251859
\(918\) 0 0
\(919\) −4.62296 −0.152497 −0.0762487 0.997089i \(-0.524294\pi\)
−0.0762487 + 0.997089i \(0.524294\pi\)
\(920\) 0 0
\(921\) 3.22548 0.106283
\(922\) 0 0
\(923\) −1.73029 −0.0569533
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 29.4778 0.968177
\(928\) 0 0
\(929\) −31.0613 −1.01909 −0.509543 0.860445i \(-0.670186\pi\)
−0.509543 + 0.860445i \(0.670186\pi\)
\(930\) 0 0
\(931\) −6.88042 −0.225497
\(932\) 0 0
\(933\) 0.338689 0.0110882
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 27.5290 0.899334 0.449667 0.893196i \(-0.351543\pi\)
0.449667 + 0.893196i \(0.351543\pi\)
\(938\) 0 0
\(939\) 3.54363 0.115642
\(940\) 0 0
\(941\) −40.2018 −1.31054 −0.655271 0.755394i \(-0.727445\pi\)
−0.655271 + 0.755394i \(0.727445\pi\)
\(942\) 0 0
\(943\) −51.5819 −1.67974
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −18.8753 −0.613363 −0.306682 0.951812i \(-0.599219\pi\)
−0.306682 + 0.951812i \(0.599219\pi\)
\(948\) 0 0
\(949\) −17.9835 −0.583768
\(950\) 0 0
\(951\) −10.0541 −0.326028
\(952\) 0 0
\(953\) −46.2885 −1.49943 −0.749715 0.661760i \(-0.769810\pi\)
−0.749715 + 0.661760i \(0.769810\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 4.06410 0.131374
\(958\) 0 0
\(959\) −6.73815 −0.217586
\(960\) 0 0
\(961\) −8.32676 −0.268605
\(962\) 0 0
\(963\) 25.6727 0.827292
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 6.98996 0.224782 0.112391 0.993664i \(-0.464149\pi\)
0.112391 + 0.993664i \(0.464149\pi\)
\(968\) 0 0
\(969\) 3.24596 0.104275
\(970\) 0 0
\(971\) 2.02077 0.0648496 0.0324248 0.999474i \(-0.489677\pi\)
0.0324248 + 0.999474i \(0.489677\pi\)
\(972\) 0 0
\(973\) −5.04406 −0.161705
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 6.12836 0.196064 0.0980318 0.995183i \(-0.468745\pi\)
0.0980318 + 0.995183i \(0.468745\pi\)
\(978\) 0 0
\(979\) 31.4508 1.00517
\(980\) 0 0
\(981\) −31.4030 −1.00262
\(982\) 0 0
\(983\) −30.8941 −0.985370 −0.492685 0.870208i \(-0.663984\pi\)
−0.492685 + 0.870208i \(0.663984\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0.769057 0.0244793
\(988\) 0 0
\(989\) 4.62381 0.147029
\(990\) 0 0
\(991\) −4.01803 −0.127637 −0.0638185 0.997962i \(-0.520328\pi\)
−0.0638185 + 0.997962i \(0.520328\pi\)
\(992\) 0 0
\(993\) −2.79966 −0.0888446
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −5.96552 −0.188930 −0.0944650 0.995528i \(-0.530114\pi\)
−0.0944650 + 0.995528i \(0.530114\pi\)
\(998\) 0 0
\(999\) 6.89690 0.218208
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.a.z.1.3 6
4.3 odd 2 7600.2.a.cn.1.4 6
5.2 odd 4 760.2.d.e.609.8 yes 12
5.3 odd 4 760.2.d.e.609.5 12
5.4 even 2 3800.2.a.be.1.4 6
20.3 even 4 1520.2.d.k.609.8 12
20.7 even 4 1520.2.d.k.609.5 12
20.19 odd 2 7600.2.a.cg.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.d.e.609.5 12 5.3 odd 4
760.2.d.e.609.8 yes 12 5.2 odd 4
1520.2.d.k.609.5 12 20.7 even 4
1520.2.d.k.609.8 12 20.3 even 4
3800.2.a.z.1.3 6 1.1 even 1 trivial
3800.2.a.be.1.4 6 5.4 even 2
7600.2.a.cg.1.3 6 20.19 odd 2
7600.2.a.cn.1.4 6 4.3 odd 2