Properties

Label 4056.2.a.z.1.3
Level $4056$
Weight $2$
Character 4056.1
Self dual yes
Analytic conductor $32.387$
Analytic rank $0$
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] = [4056,2,Mod(1,4056)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4056, 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("4056.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4056.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: \(32.3873230598\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.837.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 312)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.167449\) of defining polynomial
Character \(\chi\) \(=\) 4056.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.80451 q^{5} +5.13941 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +3.80451 q^{5} +5.13941 q^{7} +1.00000 q^{9} +0.334898 q^{11} -3.80451 q^{15} +4.13941 q^{17} -5.94392 q^{19} -5.13941 q^{21} +0.334898 q^{23} +9.47431 q^{25} -1.00000 q^{27} -0.195488 q^{29} +4.80451 q^{31} -0.334898 q^{33} +19.5529 q^{35} -2.13941 q^{37} +3.46961 q^{41} -2.86059 q^{43} +3.80451 q^{45} -3.66510 q^{47} +19.4135 q^{49} -4.13941 q^{51} +9.41353 q^{53} +1.27412 q^{55} +5.94392 q^{57} -6.27882 q^{59} -6.94392 q^{61} +5.13941 q^{63} +13.1394 q^{67} -0.334898 q^{69} +4.33490 q^{71} -10.6090 q^{73} -9.47431 q^{75} +1.72118 q^{77} -0.134715 q^{79} +1.00000 q^{81} -13.2741 q^{83} +15.7484 q^{85} +0.195488 q^{87} +0.390977 q^{89} -4.80451 q^{93} -22.6137 q^{95} +10.8045 q^{97} +0.334898 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9} + 6 q^{19} - 3 q^{21} + 15 q^{25} - 3 q^{27} - 12 q^{29} + 3 q^{31} + 12 q^{35} + 6 q^{37} - 21 q^{43} - 12 q^{47} + 24 q^{49} - 6 q^{53} - 18 q^{55} - 6 q^{57} + 6 q^{59} + 3 q^{61} + 3 q^{63} + 27 q^{67} + 12 q^{71} - 9 q^{73} - 15 q^{75} + 30 q^{77} + 9 q^{79} + 3 q^{81} - 18 q^{83} + 12 q^{85} + 12 q^{87} + 24 q^{89} - 3 q^{93} - 42 q^{95} + 21 q^{97}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 3.80451 1.70143 0.850715 0.525628i \(-0.176170\pi\)
0.850715 + 0.525628i \(0.176170\pi\)
\(6\) 0 0
\(7\) 5.13941 1.94251 0.971257 0.238033i \(-0.0765025\pi\)
0.971257 + 0.238033i \(0.0765025\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.334898 0.100976 0.0504878 0.998725i \(-0.483922\pi\)
0.0504878 + 0.998725i \(0.483922\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) −3.80451 −0.982321
\(16\) 0 0
\(17\) 4.13941 1.00395 0.501977 0.864881i \(-0.332606\pi\)
0.501977 + 0.864881i \(0.332606\pi\)
\(18\) 0 0
\(19\) −5.94392 −1.36363 −0.681815 0.731525i \(-0.738809\pi\)
−0.681815 + 0.731525i \(0.738809\pi\)
\(20\) 0 0
\(21\) −5.13941 −1.12151
\(22\) 0 0
\(23\) 0.334898 0.0698311 0.0349156 0.999390i \(-0.488884\pi\)
0.0349156 + 0.999390i \(0.488884\pi\)
\(24\) 0 0
\(25\) 9.47431 1.89486
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −0.195488 −0.0363013 −0.0181506 0.999835i \(-0.505778\pi\)
−0.0181506 + 0.999835i \(0.505778\pi\)
\(30\) 0 0
\(31\) 4.80451 0.862916 0.431458 0.902133i \(-0.357999\pi\)
0.431458 + 0.902133i \(0.357999\pi\)
\(32\) 0 0
\(33\) −0.334898 −0.0582983
\(34\) 0 0
\(35\) 19.5529 3.30505
\(36\) 0 0
\(37\) −2.13941 −0.351717 −0.175858 0.984415i \(-0.556270\pi\)
−0.175858 + 0.984415i \(0.556270\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.46961 0.541863 0.270931 0.962599i \(-0.412668\pi\)
0.270931 + 0.962599i \(0.412668\pi\)
\(42\) 0 0
\(43\) −2.86059 −0.436236 −0.218118 0.975922i \(-0.569992\pi\)
−0.218118 + 0.975922i \(0.569992\pi\)
\(44\) 0 0
\(45\) 3.80451 0.567143
\(46\) 0 0
\(47\) −3.66510 −0.534610 −0.267305 0.963612i \(-0.586133\pi\)
−0.267305 + 0.963612i \(0.586133\pi\)
\(48\) 0 0
\(49\) 19.4135 2.77336
\(50\) 0 0
\(51\) −4.13941 −0.579633
\(52\) 0 0
\(53\) 9.41353 1.29305 0.646524 0.762893i \(-0.276222\pi\)
0.646524 + 0.762893i \(0.276222\pi\)
\(54\) 0 0
\(55\) 1.27412 0.171803
\(56\) 0 0
\(57\) 5.94392 0.787292
\(58\) 0 0
\(59\) −6.27882 −0.817433 −0.408716 0.912661i \(-0.634023\pi\)
−0.408716 + 0.912661i \(0.634023\pi\)
\(60\) 0 0
\(61\) −6.94392 −0.889078 −0.444539 0.895759i \(-0.646632\pi\)
−0.444539 + 0.895759i \(0.646632\pi\)
\(62\) 0 0
\(63\) 5.13941 0.647505
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 13.1394 1.60523 0.802617 0.596494i \(-0.203440\pi\)
0.802617 + 0.596494i \(0.203440\pi\)
\(68\) 0 0
\(69\) −0.334898 −0.0403170
\(70\) 0 0
\(71\) 4.33490 0.514458 0.257229 0.966351i \(-0.417191\pi\)
0.257229 + 0.966351i \(0.417191\pi\)
\(72\) 0 0
\(73\) −10.6090 −1.24169 −0.620846 0.783932i \(-0.713211\pi\)
−0.620846 + 0.783932i \(0.713211\pi\)
\(74\) 0 0
\(75\) −9.47431 −1.09400
\(76\) 0 0
\(77\) 1.72118 0.196147
\(78\) 0 0
\(79\) −0.134715 −0.0151566 −0.00757830 0.999971i \(-0.502412\pi\)
−0.00757830 + 0.999971i \(0.502412\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −13.2741 −1.45702 −0.728512 0.685033i \(-0.759788\pi\)
−0.728512 + 0.685033i \(0.759788\pi\)
\(84\) 0 0
\(85\) 15.7484 1.70816
\(86\) 0 0
\(87\) 0.195488 0.0209586
\(88\) 0 0
\(89\) 0.390977 0.0414435 0.0207217 0.999785i \(-0.493404\pi\)
0.0207217 + 0.999785i \(0.493404\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −4.80451 −0.498205
\(94\) 0 0
\(95\) −22.6137 −2.32012
\(96\) 0 0
\(97\) 10.8045 1.09703 0.548516 0.836140i \(-0.315193\pi\)
0.548516 + 0.836140i \(0.315193\pi\)
\(98\) 0 0
\(99\) 0.334898 0.0336586
\(100\) 0 0
\(101\) −16.0833 −1.60035 −0.800176 0.599766i \(-0.795260\pi\)
−0.800176 + 0.599766i \(0.795260\pi\)
\(102\) 0 0
\(103\) 4.46961 0.440404 0.220202 0.975454i \(-0.429328\pi\)
0.220202 + 0.975454i \(0.429328\pi\)
\(104\) 0 0
\(105\) −19.5529 −1.90817
\(106\) 0 0
\(107\) −3.66510 −0.354319 −0.177159 0.984182i \(-0.556691\pi\)
−0.177159 + 0.984182i \(0.556691\pi\)
\(108\) 0 0
\(109\) −9.08333 −0.870025 −0.435013 0.900424i \(-0.643256\pi\)
−0.435013 + 0.900424i \(0.643256\pi\)
\(110\) 0 0
\(111\) 2.13941 0.203064
\(112\) 0 0
\(113\) 9.86059 0.927606 0.463803 0.885938i \(-0.346484\pi\)
0.463803 + 0.885938i \(0.346484\pi\)
\(114\) 0 0
\(115\) 1.27412 0.118813
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 21.2741 1.95020
\(120\) 0 0
\(121\) −10.8878 −0.989804
\(122\) 0 0
\(123\) −3.46961 −0.312844
\(124\) 0 0
\(125\) 17.0226 1.52254
\(126\) 0 0
\(127\) −0.134715 −0.0119540 −0.00597700 0.999982i \(-0.501903\pi\)
−0.00597700 + 0.999982i \(0.501903\pi\)
\(128\) 0 0
\(129\) 2.86059 0.251861
\(130\) 0 0
\(131\) −18.9486 −1.65555 −0.827774 0.561061i \(-0.810393\pi\)
−0.827774 + 0.561061i \(0.810393\pi\)
\(132\) 0 0
\(133\) −30.5482 −2.64887
\(134\) 0 0
\(135\) −3.80451 −0.327440
\(136\) 0 0
\(137\) −15.0786 −1.28825 −0.644127 0.764918i \(-0.722779\pi\)
−0.644127 + 0.764918i \(0.722779\pi\)
\(138\) 0 0
\(139\) −14.4135 −1.22254 −0.611270 0.791422i \(-0.709341\pi\)
−0.611270 + 0.791422i \(0.709341\pi\)
\(140\) 0 0
\(141\) 3.66510 0.308657
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −0.743738 −0.0617641
\(146\) 0 0
\(147\) −19.4135 −1.60120
\(148\) 0 0
\(149\) 18.0833 1.48144 0.740722 0.671812i \(-0.234484\pi\)
0.740722 + 0.671812i \(0.234484\pi\)
\(150\) 0 0
\(151\) 5.27412 0.429202 0.214601 0.976702i \(-0.431155\pi\)
0.214601 + 0.976702i \(0.431155\pi\)
\(152\) 0 0
\(153\) 4.13941 0.334651
\(154\) 0 0
\(155\) 18.2788 1.46819
\(156\) 0 0
\(157\) 4.27412 0.341112 0.170556 0.985348i \(-0.445444\pi\)
0.170556 + 0.985348i \(0.445444\pi\)
\(158\) 0 0
\(159\) −9.41353 −0.746542
\(160\) 0 0
\(161\) 1.72118 0.135648
\(162\) 0 0
\(163\) 3.19549 0.250290 0.125145 0.992138i \(-0.460060\pi\)
0.125145 + 0.992138i \(0.460060\pi\)
\(164\) 0 0
\(165\) −1.27412 −0.0991905
\(166\) 0 0
\(167\) −25.2274 −1.95216 −0.976079 0.217417i \(-0.930237\pi\)
−0.976079 + 0.217417i \(0.930237\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) −5.94392 −0.454543
\(172\) 0 0
\(173\) 21.8878 1.66410 0.832051 0.554699i \(-0.187167\pi\)
0.832051 + 0.554699i \(0.187167\pi\)
\(174\) 0 0
\(175\) 48.6924 3.68080
\(176\) 0 0
\(177\) 6.27882 0.471945
\(178\) 0 0
\(179\) −13.9439 −1.04222 −0.521109 0.853490i \(-0.674481\pi\)
−0.521109 + 0.853490i \(0.674481\pi\)
\(180\) 0 0
\(181\) 9.86059 0.732932 0.366466 0.930431i \(-0.380568\pi\)
0.366466 + 0.930431i \(0.380568\pi\)
\(182\) 0 0
\(183\) 6.94392 0.513309
\(184\) 0 0
\(185\) −8.13941 −0.598421
\(186\) 0 0
\(187\) 1.38628 0.101375
\(188\) 0 0
\(189\) −5.13941 −0.373837
\(190\) 0 0
\(191\) −10.2788 −0.743749 −0.371875 0.928283i \(-0.621285\pi\)
−0.371875 + 0.928283i \(0.621285\pi\)
\(192\) 0 0
\(193\) 3.66980 0.264158 0.132079 0.991239i \(-0.457835\pi\)
0.132079 + 0.991239i \(0.457835\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 20.1667 1.43682 0.718408 0.695622i \(-0.244871\pi\)
0.718408 + 0.695622i \(0.244871\pi\)
\(198\) 0 0
\(199\) 2.74843 0.194831 0.0974156 0.995244i \(-0.468942\pi\)
0.0974156 + 0.995244i \(0.468942\pi\)
\(200\) 0 0
\(201\) −13.1394 −0.926783
\(202\) 0 0
\(203\) −1.00470 −0.0705158
\(204\) 0 0
\(205\) 13.2002 0.921941
\(206\) 0 0
\(207\) 0.334898 0.0232770
\(208\) 0 0
\(209\) −1.99061 −0.137693
\(210\) 0 0
\(211\) −12.1347 −0.835388 −0.417694 0.908588i \(-0.637162\pi\)
−0.417694 + 0.908588i \(0.637162\pi\)
\(212\) 0 0
\(213\) −4.33490 −0.297022
\(214\) 0 0
\(215\) −10.8831 −0.742225
\(216\) 0 0
\(217\) 24.6924 1.67623
\(218\) 0 0
\(219\) 10.6090 0.716891
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) 9.47431 0.631621
\(226\) 0 0
\(227\) −11.2835 −0.748913 −0.374457 0.927244i \(-0.622171\pi\)
−0.374457 + 0.927244i \(0.622171\pi\)
\(228\) 0 0
\(229\) 2.66980 0.176425 0.0882126 0.996102i \(-0.471885\pi\)
0.0882126 + 0.996102i \(0.471885\pi\)
\(230\) 0 0
\(231\) −1.72118 −0.113245
\(232\) 0 0
\(233\) 21.2180 1.39004 0.695020 0.718990i \(-0.255395\pi\)
0.695020 + 0.718990i \(0.255395\pi\)
\(234\) 0 0
\(235\) −13.9439 −0.909601
\(236\) 0 0
\(237\) 0.134715 0.00875067
\(238\) 0 0
\(239\) −13.2741 −0.858632 −0.429316 0.903154i \(-0.641245\pi\)
−0.429316 + 0.903154i \(0.641245\pi\)
\(240\) 0 0
\(241\) 0.586465 0.0377775 0.0188888 0.999822i \(-0.493987\pi\)
0.0188888 + 0.999822i \(0.493987\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 73.8590 4.71868
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 13.2741 0.841213
\(250\) 0 0
\(251\) 0.669797 0.0422772 0.0211386 0.999777i \(-0.493271\pi\)
0.0211386 + 0.999777i \(0.493271\pi\)
\(252\) 0 0
\(253\) 0.112157 0.00705125
\(254\) 0 0
\(255\) −15.7484 −0.986205
\(256\) 0 0
\(257\) −0.530387 −0.0330846 −0.0165423 0.999863i \(-0.505266\pi\)
−0.0165423 + 0.999863i \(0.505266\pi\)
\(258\) 0 0
\(259\) −10.9953 −0.683215
\(260\) 0 0
\(261\) −0.195488 −0.0121004
\(262\) 0 0
\(263\) −2.61372 −0.161169 −0.0805844 0.996748i \(-0.525679\pi\)
−0.0805844 + 0.996748i \(0.525679\pi\)
\(264\) 0 0
\(265\) 35.8139 2.20003
\(266\) 0 0
\(267\) −0.390977 −0.0239274
\(268\) 0 0
\(269\) 12.2788 0.748653 0.374326 0.927297i \(-0.377874\pi\)
0.374326 + 0.927297i \(0.377874\pi\)
\(270\) 0 0
\(271\) 2.14411 0.130245 0.0651226 0.997877i \(-0.479256\pi\)
0.0651226 + 0.997877i \(0.479256\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.17293 0.191335
\(276\) 0 0
\(277\) 6.53039 0.392373 0.196186 0.980567i \(-0.437144\pi\)
0.196186 + 0.980567i \(0.437144\pi\)
\(278\) 0 0
\(279\) 4.80451 0.287639
\(280\) 0 0
\(281\) −29.3575 −1.75132 −0.875660 0.482929i \(-0.839573\pi\)
−0.875660 + 0.482929i \(0.839573\pi\)
\(282\) 0 0
\(283\) −12.0880 −0.718559 −0.359279 0.933230i \(-0.616977\pi\)
−0.359279 + 0.933230i \(0.616977\pi\)
\(284\) 0 0
\(285\) 22.6137 1.33952
\(286\) 0 0
\(287\) 17.8318 1.05258
\(288\) 0 0
\(289\) 0.134715 0.00792440
\(290\) 0 0
\(291\) −10.8045 −0.633372
\(292\) 0 0
\(293\) 3.69235 0.215710 0.107855 0.994167i \(-0.465602\pi\)
0.107855 + 0.994167i \(0.465602\pi\)
\(294\) 0 0
\(295\) −23.8878 −1.39080
\(296\) 0 0
\(297\) −0.334898 −0.0194328
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −14.7017 −0.847394
\(302\) 0 0
\(303\) 16.0833 0.923963
\(304\) 0 0
\(305\) −26.4182 −1.51270
\(306\) 0 0
\(307\) −6.07864 −0.346926 −0.173463 0.984840i \(-0.555496\pi\)
−0.173463 + 0.984840i \(0.555496\pi\)
\(308\) 0 0
\(309\) −4.46961 −0.254267
\(310\) 0 0
\(311\) 10.0561 0.570228 0.285114 0.958494i \(-0.407969\pi\)
0.285114 + 0.958494i \(0.407969\pi\)
\(312\) 0 0
\(313\) −11.3622 −0.642227 −0.321113 0.947041i \(-0.604057\pi\)
−0.321113 + 0.947041i \(0.604057\pi\)
\(314\) 0 0
\(315\) 19.5529 1.10168
\(316\) 0 0
\(317\) 19.4229 1.09090 0.545450 0.838143i \(-0.316359\pi\)
0.545450 + 0.838143i \(0.316359\pi\)
\(318\) 0 0
\(319\) −0.0654688 −0.00366555
\(320\) 0 0
\(321\) 3.66510 0.204566
\(322\) 0 0
\(323\) −24.6043 −1.36902
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 9.08333 0.502309
\(328\) 0 0
\(329\) −18.8365 −1.03849
\(330\) 0 0
\(331\) 28.8045 1.58324 0.791620 0.611014i \(-0.209238\pi\)
0.791620 + 0.611014i \(0.209238\pi\)
\(332\) 0 0
\(333\) −2.13941 −0.117239
\(334\) 0 0
\(335\) 49.9890 2.73119
\(336\) 0 0
\(337\) 0.0513833 0.00279903 0.00139951 0.999999i \(-0.499555\pi\)
0.00139951 + 0.999999i \(0.499555\pi\)
\(338\) 0 0
\(339\) −9.86059 −0.535554
\(340\) 0 0
\(341\) 1.60902 0.0871335
\(342\) 0 0
\(343\) 63.7982 3.44478
\(344\) 0 0
\(345\) −1.27412 −0.0685966
\(346\) 0 0
\(347\) 2.72588 0.146333 0.0731663 0.997320i \(-0.476690\pi\)
0.0731663 + 0.997320i \(0.476690\pi\)
\(348\) 0 0
\(349\) 27.7437 1.48509 0.742544 0.669797i \(-0.233619\pi\)
0.742544 + 0.669797i \(0.233619\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −22.0273 −1.17239 −0.586196 0.810169i \(-0.699375\pi\)
−0.586196 + 0.810169i \(0.699375\pi\)
\(354\) 0 0
\(355\) 16.4922 0.875314
\(356\) 0 0
\(357\) −21.2741 −1.12595
\(358\) 0 0
\(359\) −1.94392 −0.102596 −0.0512981 0.998683i \(-0.516336\pi\)
−0.0512981 + 0.998683i \(0.516336\pi\)
\(360\) 0 0
\(361\) 16.3302 0.859484
\(362\) 0 0
\(363\) 10.8878 0.571464
\(364\) 0 0
\(365\) −40.3622 −2.11265
\(366\) 0 0
\(367\) −1.13941 −0.0594767 −0.0297384 0.999558i \(-0.509467\pi\)
−0.0297384 + 0.999558i \(0.509467\pi\)
\(368\) 0 0
\(369\) 3.46961 0.180621
\(370\) 0 0
\(371\) 48.3800 2.51177
\(372\) 0 0
\(373\) 4.83176 0.250179 0.125090 0.992145i \(-0.460078\pi\)
0.125090 + 0.992145i \(0.460078\pi\)
\(374\) 0 0
\(375\) −17.0226 −0.879041
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 28.9618 1.48767 0.743833 0.668365i \(-0.233006\pi\)
0.743833 + 0.668365i \(0.233006\pi\)
\(380\) 0 0
\(381\) 0.134715 0.00690165
\(382\) 0 0
\(383\) 9.33959 0.477231 0.238615 0.971114i \(-0.423306\pi\)
0.238615 + 0.971114i \(0.423306\pi\)
\(384\) 0 0
\(385\) 6.54825 0.333730
\(386\) 0 0
\(387\) −2.86059 −0.145412
\(388\) 0 0
\(389\) 0.204879 0.0103878 0.00519388 0.999987i \(-0.498347\pi\)
0.00519388 + 0.999987i \(0.498347\pi\)
\(390\) 0 0
\(391\) 1.38628 0.0701073
\(392\) 0 0
\(393\) 18.9486 0.955831
\(394\) 0 0
\(395\) −0.512524 −0.0257879
\(396\) 0 0
\(397\) −8.81390 −0.442357 −0.221179 0.975233i \(-0.570990\pi\)
−0.221179 + 0.975233i \(0.570990\pi\)
\(398\) 0 0
\(399\) 30.5482 1.52933
\(400\) 0 0
\(401\) −23.6363 −1.18034 −0.590170 0.807279i \(-0.700939\pi\)
−0.590170 + 0.807279i \(0.700939\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 3.80451 0.189048
\(406\) 0 0
\(407\) −0.716485 −0.0355148
\(408\) 0 0
\(409\) −28.4875 −1.40862 −0.704308 0.709895i \(-0.748742\pi\)
−0.704308 + 0.709895i \(0.748742\pi\)
\(410\) 0 0
\(411\) 15.0786 0.743774
\(412\) 0 0
\(413\) −32.2694 −1.58787
\(414\) 0 0
\(415\) −50.5016 −2.47902
\(416\) 0 0
\(417\) 14.4135 0.705834
\(418\) 0 0
\(419\) 20.5576 1.00431 0.502153 0.864779i \(-0.332541\pi\)
0.502153 + 0.864779i \(0.332541\pi\)
\(420\) 0 0
\(421\) −8.55294 −0.416845 −0.208423 0.978039i \(-0.566833\pi\)
−0.208423 + 0.978039i \(0.566833\pi\)
\(422\) 0 0
\(423\) −3.66510 −0.178203
\(424\) 0 0
\(425\) 39.2180 1.90235
\(426\) 0 0
\(427\) −35.6877 −1.72705
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −25.2741 −1.21741 −0.608706 0.793396i \(-0.708311\pi\)
−0.608706 + 0.793396i \(0.708311\pi\)
\(432\) 0 0
\(433\) 16.3396 0.785231 0.392615 0.919703i \(-0.371570\pi\)
0.392615 + 0.919703i \(0.371570\pi\)
\(434\) 0 0
\(435\) 0.743738 0.0356595
\(436\) 0 0
\(437\) −1.99061 −0.0952238
\(438\) 0 0
\(439\) −3.41823 −0.163143 −0.0815716 0.996667i \(-0.525994\pi\)
−0.0815716 + 0.996667i \(0.525994\pi\)
\(440\) 0 0
\(441\) 19.4135 0.924454
\(442\) 0 0
\(443\) 0.669797 0.0318230 0.0159115 0.999873i \(-0.494935\pi\)
0.0159115 + 0.999873i \(0.494935\pi\)
\(444\) 0 0
\(445\) 1.48748 0.0705131
\(446\) 0 0
\(447\) −18.0833 −0.855312
\(448\) 0 0
\(449\) 21.2180 1.00134 0.500671 0.865638i \(-0.333087\pi\)
0.500671 + 0.865638i \(0.333087\pi\)
\(450\) 0 0
\(451\) 1.16197 0.0547149
\(452\) 0 0
\(453\) −5.27412 −0.247800
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 6.61841 0.309596 0.154798 0.987946i \(-0.450527\pi\)
0.154798 + 0.987946i \(0.450527\pi\)
\(458\) 0 0
\(459\) −4.13941 −0.193211
\(460\) 0 0
\(461\) −29.6924 −1.38291 −0.691455 0.722419i \(-0.743030\pi\)
−0.691455 + 0.722419i \(0.743030\pi\)
\(462\) 0 0
\(463\) 18.7484 0.871314 0.435657 0.900113i \(-0.356516\pi\)
0.435657 + 0.900113i \(0.356516\pi\)
\(464\) 0 0
\(465\) −18.2788 −0.847660
\(466\) 0 0
\(467\) −1.00470 −0.0464917 −0.0232459 0.999730i \(-0.507400\pi\)
−0.0232459 + 0.999730i \(0.507400\pi\)
\(468\) 0 0
\(469\) 67.5288 3.11819
\(470\) 0 0
\(471\) −4.27412 −0.196941
\(472\) 0 0
\(473\) −0.958007 −0.0440492
\(474\) 0 0
\(475\) −56.3145 −2.58389
\(476\) 0 0
\(477\) 9.41353 0.431016
\(478\) 0 0
\(479\) 23.8878 1.09146 0.545732 0.837960i \(-0.316252\pi\)
0.545732 + 0.837960i \(0.316252\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −1.72118 −0.0783164
\(484\) 0 0
\(485\) 41.1059 1.86652
\(486\) 0 0
\(487\) 29.9439 1.35689 0.678444 0.734652i \(-0.262654\pi\)
0.678444 + 0.734652i \(0.262654\pi\)
\(488\) 0 0
\(489\) −3.19549 −0.144505
\(490\) 0 0
\(491\) −22.2134 −1.00247 −0.501237 0.865310i \(-0.667122\pi\)
−0.501237 + 0.865310i \(0.667122\pi\)
\(492\) 0 0
\(493\) −0.809207 −0.0364448
\(494\) 0 0
\(495\) 1.27412 0.0572676
\(496\) 0 0
\(497\) 22.2788 0.999342
\(498\) 0 0
\(499\) 4.66980 0.209049 0.104524 0.994522i \(-0.466668\pi\)
0.104524 + 0.994522i \(0.466668\pi\)
\(500\) 0 0
\(501\) 25.2274 1.12708
\(502\) 0 0
\(503\) −34.2134 −1.52550 −0.762749 0.646695i \(-0.776151\pi\)
−0.762749 + 0.646695i \(0.776151\pi\)
\(504\) 0 0
\(505\) −61.1892 −2.72288
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 1.81390 0.0803998 0.0401999 0.999192i \(-0.487201\pi\)
0.0401999 + 0.999192i \(0.487201\pi\)
\(510\) 0 0
\(511\) −54.5241 −2.41201
\(512\) 0 0
\(513\) 5.94392 0.262431
\(514\) 0 0
\(515\) 17.0047 0.749316
\(516\) 0 0
\(517\) −1.22744 −0.0539826
\(518\) 0 0
\(519\) −21.8878 −0.960770
\(520\) 0 0
\(521\) −24.9759 −1.09421 −0.547106 0.837063i \(-0.684271\pi\)
−0.547106 + 0.837063i \(0.684271\pi\)
\(522\) 0 0
\(523\) −3.39567 −0.148482 −0.0742412 0.997240i \(-0.523653\pi\)
−0.0742412 + 0.997240i \(0.523653\pi\)
\(524\) 0 0
\(525\) −48.6924 −2.12511
\(526\) 0 0
\(527\) 19.8878 0.866328
\(528\) 0 0
\(529\) −22.8878 −0.995124
\(530\) 0 0
\(531\) −6.27882 −0.272478
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −13.9439 −0.602848
\(536\) 0 0
\(537\) 13.9439 0.601725
\(538\) 0 0
\(539\) 6.50156 0.280042
\(540\) 0 0
\(541\) −29.7804 −1.28036 −0.640179 0.768226i \(-0.721140\pi\)
−0.640179 + 0.768226i \(0.721140\pi\)
\(542\) 0 0
\(543\) −9.86059 −0.423158
\(544\) 0 0
\(545\) −34.5576 −1.48029
\(546\) 0 0
\(547\) 22.6363 0.967857 0.483929 0.875107i \(-0.339209\pi\)
0.483929 + 0.875107i \(0.339209\pi\)
\(548\) 0 0
\(549\) −6.94392 −0.296359
\(550\) 0 0
\(551\) 1.16197 0.0495015
\(552\) 0 0
\(553\) −0.692355 −0.0294419
\(554\) 0 0
\(555\) 8.13941 0.345499
\(556\) 0 0
\(557\) −19.7017 −0.834790 −0.417395 0.908725i \(-0.637057\pi\)
−0.417395 + 0.908725i \(0.637057\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −1.38628 −0.0585289
\(562\) 0 0
\(563\) −38.0545 −1.60381 −0.801903 0.597454i \(-0.796179\pi\)
−0.801903 + 0.597454i \(0.796179\pi\)
\(564\) 0 0
\(565\) 37.5147 1.57826
\(566\) 0 0
\(567\) 5.13941 0.215835
\(568\) 0 0
\(569\) 11.2274 0.470679 0.235339 0.971913i \(-0.424380\pi\)
0.235339 + 0.971913i \(0.424380\pi\)
\(570\) 0 0
\(571\) −37.9439 −1.58790 −0.793952 0.607981i \(-0.791980\pi\)
−0.793952 + 0.607981i \(0.791980\pi\)
\(572\) 0 0
\(573\) 10.2788 0.429404
\(574\) 0 0
\(575\) 3.17293 0.132320
\(576\) 0 0
\(577\) 33.6830 1.40224 0.701120 0.713043i \(-0.252684\pi\)
0.701120 + 0.713043i \(0.252684\pi\)
\(578\) 0 0
\(579\) −3.66980 −0.152512
\(580\) 0 0
\(581\) −68.2212 −2.83029
\(582\) 0 0
\(583\) 3.15258 0.130566
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 9.60902 0.396607 0.198303 0.980141i \(-0.436457\pi\)
0.198303 + 0.980141i \(0.436457\pi\)
\(588\) 0 0
\(589\) −28.5576 −1.17670
\(590\) 0 0
\(591\) −20.1667 −0.829546
\(592\) 0 0
\(593\) −38.0273 −1.56159 −0.780796 0.624786i \(-0.785186\pi\)
−0.780796 + 0.624786i \(0.785186\pi\)
\(594\) 0 0
\(595\) 80.9377 3.31812
\(596\) 0 0
\(597\) −2.74843 −0.112486
\(598\) 0 0
\(599\) −4.82707 −0.197229 −0.0986144 0.995126i \(-0.531441\pi\)
−0.0986144 + 0.995126i \(0.531441\pi\)
\(600\) 0 0
\(601\) −2.74374 −0.111919 −0.0559597 0.998433i \(-0.517822\pi\)
−0.0559597 + 0.998433i \(0.517822\pi\)
\(602\) 0 0
\(603\) 13.1394 0.535078
\(604\) 0 0
\(605\) −41.4229 −1.68408
\(606\) 0 0
\(607\) 35.1059 1.42490 0.712452 0.701721i \(-0.247585\pi\)
0.712452 + 0.701721i \(0.247585\pi\)
\(608\) 0 0
\(609\) 1.00470 0.0407123
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 24.1620 0.975893 0.487946 0.872874i \(-0.337746\pi\)
0.487946 + 0.872874i \(0.337746\pi\)
\(614\) 0 0
\(615\) −13.2002 −0.532283
\(616\) 0 0
\(617\) 17.6363 0.710010 0.355005 0.934864i \(-0.384479\pi\)
0.355005 + 0.934864i \(0.384479\pi\)
\(618\) 0 0
\(619\) 34.1441 1.37237 0.686184 0.727428i \(-0.259285\pi\)
0.686184 + 0.727428i \(0.259285\pi\)
\(620\) 0 0
\(621\) −0.334898 −0.0134390
\(622\) 0 0
\(623\) 2.00939 0.0805045
\(624\) 0 0
\(625\) 17.3910 0.695639
\(626\) 0 0
\(627\) 1.99061 0.0794973
\(628\) 0 0
\(629\) −8.85589 −0.353108
\(630\) 0 0
\(631\) 26.9712 1.07371 0.536853 0.843676i \(-0.319613\pi\)
0.536853 + 0.843676i \(0.319613\pi\)
\(632\) 0 0
\(633\) 12.1347 0.482312
\(634\) 0 0
\(635\) −0.512524 −0.0203389
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 4.33490 0.171486
\(640\) 0 0
\(641\) −28.6877 −1.13309 −0.566547 0.824029i \(-0.691721\pi\)
−0.566547 + 0.824029i \(0.691721\pi\)
\(642\) 0 0
\(643\) −20.0226 −0.789613 −0.394806 0.918764i \(-0.629188\pi\)
−0.394806 + 0.918764i \(0.629188\pi\)
\(644\) 0 0
\(645\) 10.8831 0.428524
\(646\) 0 0
\(647\) −6.16666 −0.242437 −0.121218 0.992626i \(-0.538680\pi\)
−0.121218 + 0.992626i \(0.538680\pi\)
\(648\) 0 0
\(649\) −2.10277 −0.0825408
\(650\) 0 0
\(651\) −24.6924 −0.967770
\(652\) 0 0
\(653\) −21.5996 −0.845259 −0.422629 0.906303i \(-0.638893\pi\)
−0.422629 + 0.906303i \(0.638893\pi\)
\(654\) 0 0
\(655\) −72.0902 −2.81680
\(656\) 0 0
\(657\) −10.6090 −0.413897
\(658\) 0 0
\(659\) −11.2180 −0.436993 −0.218497 0.975838i \(-0.570115\pi\)
−0.218497 + 0.975838i \(0.570115\pi\)
\(660\) 0 0
\(661\) 44.7196 1.73939 0.869696 0.493589i \(-0.164315\pi\)
0.869696 + 0.493589i \(0.164315\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −116.221 −4.50686
\(666\) 0 0
\(667\) −0.0654688 −0.00253496
\(668\) 0 0
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) −2.32551 −0.0897752
\(672\) 0 0
\(673\) −15.4361 −0.595018 −0.297509 0.954719i \(-0.596156\pi\)
−0.297509 + 0.954719i \(0.596156\pi\)
\(674\) 0 0
\(675\) −9.47431 −0.364666
\(676\) 0 0
\(677\) −35.4969 −1.36426 −0.682128 0.731233i \(-0.738945\pi\)
−0.682128 + 0.731233i \(0.738945\pi\)
\(678\) 0 0
\(679\) 55.5288 2.13100
\(680\) 0 0
\(681\) 11.2835 0.432385
\(682\) 0 0
\(683\) 9.60902 0.367679 0.183840 0.982956i \(-0.441147\pi\)
0.183840 + 0.982956i \(0.441147\pi\)
\(684\) 0 0
\(685\) −57.3668 −2.19187
\(686\) 0 0
\(687\) −2.66980 −0.101859
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −37.0273 −1.40858 −0.704292 0.709911i \(-0.748735\pi\)
−0.704292 + 0.709911i \(0.748735\pi\)
\(692\) 0 0
\(693\) 1.72118 0.0653822
\(694\) 0 0
\(695\) −54.8365 −2.08007
\(696\) 0 0
\(697\) 14.3622 0.544005
\(698\) 0 0
\(699\) −21.2180 −0.802540
\(700\) 0 0
\(701\) −19.7212 −0.744859 −0.372429 0.928061i \(-0.621475\pi\)
−0.372429 + 0.928061i \(0.621475\pi\)
\(702\) 0 0
\(703\) 12.7165 0.479611
\(704\) 0 0
\(705\) 13.9439 0.525158
\(706\) 0 0
\(707\) −82.6588 −3.10871
\(708\) 0 0
\(709\) 21.3443 0.801602 0.400801 0.916165i \(-0.368732\pi\)
0.400801 + 0.916165i \(0.368732\pi\)
\(710\) 0 0
\(711\) −0.134715 −0.00505220
\(712\) 0 0
\(713\) 1.60902 0.0602584
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 13.2741 0.495731
\(718\) 0 0
\(719\) 23.6184 0.880818 0.440409 0.897797i \(-0.354833\pi\)
0.440409 + 0.897797i \(0.354833\pi\)
\(720\) 0 0
\(721\) 22.9712 0.855491
\(722\) 0 0
\(723\) −0.586465 −0.0218109
\(724\) 0 0
\(725\) −1.85212 −0.0687859
\(726\) 0 0
\(727\) 40.8029 1.51330 0.756649 0.653822i \(-0.226835\pi\)
0.756649 + 0.653822i \(0.226835\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −11.8412 −0.437961
\(732\) 0 0
\(733\) −33.3349 −1.23125 −0.615626 0.788038i \(-0.711097\pi\)
−0.615626 + 0.788038i \(0.711097\pi\)
\(734\) 0 0
\(735\) −73.8590 −2.72433
\(736\) 0 0
\(737\) 4.40037 0.162090
\(738\) 0 0
\(739\) −3.21805 −0.118378 −0.0591889 0.998247i \(-0.518851\pi\)
−0.0591889 + 0.998247i \(0.518851\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 13.4969 0.495152 0.247576 0.968868i \(-0.420366\pi\)
0.247576 + 0.968868i \(0.420366\pi\)
\(744\) 0 0
\(745\) 68.7982 2.52057
\(746\) 0 0
\(747\) −13.2741 −0.485675
\(748\) 0 0
\(749\) −18.8365 −0.688269
\(750\) 0 0
\(751\) −15.8224 −0.577367 −0.288683 0.957425i \(-0.593217\pi\)
−0.288683 + 0.957425i \(0.593217\pi\)
\(752\) 0 0
\(753\) −0.669797 −0.0244088
\(754\) 0 0
\(755\) 20.0655 0.730257
\(756\) 0 0
\(757\) −0.660406 −0.0240029 −0.0120014 0.999928i \(-0.503820\pi\)
−0.0120014 + 0.999928i \(0.503820\pi\)
\(758\) 0 0
\(759\) −0.112157 −0.00407104
\(760\) 0 0
\(761\) −15.6090 −0.565827 −0.282913 0.959145i \(-0.591301\pi\)
−0.282913 + 0.959145i \(0.591301\pi\)
\(762\) 0 0
\(763\) −46.6830 −1.69004
\(764\) 0 0
\(765\) 15.7484 0.569386
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 40.3239 1.45412 0.727059 0.686575i \(-0.240887\pi\)
0.727059 + 0.686575i \(0.240887\pi\)
\(770\) 0 0
\(771\) 0.530387 0.0191014
\(772\) 0 0
\(773\) −13.3302 −0.479454 −0.239727 0.970840i \(-0.577058\pi\)
−0.239727 + 0.970840i \(0.577058\pi\)
\(774\) 0 0
\(775\) 45.5194 1.63511
\(776\) 0 0
\(777\) 10.9953 0.394454
\(778\) 0 0
\(779\) −20.6231 −0.738900
\(780\) 0 0
\(781\) 1.45175 0.0519477
\(782\) 0 0
\(783\) 0.195488 0.00698619
\(784\) 0 0
\(785\) 16.2610 0.580378
\(786\) 0 0
\(787\) 9.85589 0.351325 0.175662 0.984450i \(-0.443793\pi\)
0.175662 + 0.984450i \(0.443793\pi\)
\(788\) 0 0
\(789\) 2.61372 0.0930508
\(790\) 0 0
\(791\) 50.6776 1.80189
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −35.8139 −1.27019
\(796\) 0 0
\(797\) −45.4875 −1.61125 −0.805625 0.592426i \(-0.798170\pi\)
−0.805625 + 0.592426i \(0.798170\pi\)
\(798\) 0 0
\(799\) −15.1714 −0.536724
\(800\) 0 0
\(801\) 0.390977 0.0138145
\(802\) 0 0
\(803\) −3.55294 −0.125381
\(804\) 0 0
\(805\) 6.54825 0.230795
\(806\) 0 0
\(807\) −12.2788 −0.432235
\(808\) 0 0
\(809\) −19.5241 −0.686431 −0.343216 0.939257i \(-0.611516\pi\)
−0.343216 + 0.939257i \(0.611516\pi\)
\(810\) 0 0
\(811\) 33.3622 1.17150 0.585752 0.810490i \(-0.300799\pi\)
0.585752 + 0.810490i \(0.300799\pi\)
\(812\) 0 0
\(813\) −2.14411 −0.0751970
\(814\) 0 0
\(815\) 12.1573 0.425851
\(816\) 0 0
\(817\) 17.0031 0.594864
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −43.2274 −1.50865 −0.754324 0.656502i \(-0.772035\pi\)
−0.754324 + 0.656502i \(0.772035\pi\)
\(822\) 0 0
\(823\) 22.0094 0.767199 0.383600 0.923500i \(-0.374684\pi\)
0.383600 + 0.923500i \(0.374684\pi\)
\(824\) 0 0
\(825\) −3.17293 −0.110467
\(826\) 0 0
\(827\) 38.5482 1.34045 0.670227 0.742156i \(-0.266197\pi\)
0.670227 + 0.742156i \(0.266197\pi\)
\(828\) 0 0
\(829\) 30.4408 1.05725 0.528626 0.848855i \(-0.322707\pi\)
0.528626 + 0.848855i \(0.322707\pi\)
\(830\) 0 0
\(831\) −6.53039 −0.226537
\(832\) 0 0
\(833\) 80.3606 2.78433
\(834\) 0 0
\(835\) −95.9781 −3.32146
\(836\) 0 0
\(837\) −4.80451 −0.166068
\(838\) 0 0
\(839\) −13.3847 −0.462091 −0.231046 0.972943i \(-0.574215\pi\)
−0.231046 + 0.972943i \(0.574215\pi\)
\(840\) 0 0
\(841\) −28.9618 −0.998682
\(842\) 0 0
\(843\) 29.3575 1.01112
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −55.9571 −1.92271
\(848\) 0 0
\(849\) 12.0880 0.414860
\(850\) 0 0
\(851\) −0.716485 −0.0245608
\(852\) 0 0
\(853\) 44.7196 1.53117 0.765585 0.643335i \(-0.222450\pi\)
0.765585 + 0.643335i \(0.222450\pi\)
\(854\) 0 0
\(855\) −22.6137 −0.773373
\(856\) 0 0
\(857\) 17.6363 0.602444 0.301222 0.953554i \(-0.402606\pi\)
0.301222 + 0.953554i \(0.402606\pi\)
\(858\) 0 0
\(859\) 12.4696 0.425458 0.212729 0.977111i \(-0.431765\pi\)
0.212729 + 0.977111i \(0.431765\pi\)
\(860\) 0 0
\(861\) −17.8318 −0.607705
\(862\) 0 0
\(863\) −30.2134 −1.02847 −0.514237 0.857648i \(-0.671925\pi\)
−0.514237 + 0.857648i \(0.671925\pi\)
\(864\) 0 0
\(865\) 83.2726 2.83135
\(866\) 0 0
\(867\) −0.134715 −0.00457515
\(868\) 0 0
\(869\) −0.0451158 −0.00153045
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 10.8045 0.365677
\(874\) 0 0
\(875\) 87.4859 2.95756
\(876\) 0 0
\(877\) −15.3668 −0.518902 −0.259451 0.965756i \(-0.583542\pi\)
−0.259451 + 0.965756i \(0.583542\pi\)
\(878\) 0 0
\(879\) −3.69235 −0.124540
\(880\) 0 0
\(881\) 3.35746 0.113116 0.0565578 0.998399i \(-0.481987\pi\)
0.0565578 + 0.998399i \(0.481987\pi\)
\(882\) 0 0
\(883\) −29.6316 −0.997182 −0.498591 0.866837i \(-0.666149\pi\)
−0.498591 + 0.866837i \(0.666149\pi\)
\(884\) 0 0
\(885\) 23.8878 0.802981
\(886\) 0 0
\(887\) 17.4969 0.587487 0.293744 0.955884i \(-0.405099\pi\)
0.293744 + 0.955884i \(0.405099\pi\)
\(888\) 0 0
\(889\) −0.692355 −0.0232208
\(890\) 0 0
\(891\) 0.334898 0.0112195
\(892\) 0 0
\(893\) 21.7851 0.729010
\(894\) 0 0
\(895\) −53.0498 −1.77326
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −0.939226 −0.0313250
\(900\) 0 0
\(901\) 38.9665 1.29816
\(902\) 0 0
\(903\) 14.7017 0.489243
\(904\) 0 0
\(905\) 37.5147 1.24703
\(906\) 0 0
\(907\) 3.34898 0.111201 0.0556006 0.998453i \(-0.482293\pi\)
0.0556006 + 0.998453i \(0.482293\pi\)
\(908\) 0 0
\(909\) −16.0833 −0.533450
\(910\) 0 0
\(911\) −22.0094 −0.729204 −0.364602 0.931164i \(-0.618795\pi\)
−0.364602 + 0.931164i \(0.618795\pi\)
\(912\) 0 0
\(913\) −4.44548 −0.147124
\(914\) 0 0
\(915\) 26.4182 0.873360
\(916\) 0 0
\(917\) −97.3847 −3.21593
\(918\) 0 0
\(919\) 43.1059 1.42193 0.710966 0.703226i \(-0.248258\pi\)
0.710966 + 0.703226i \(0.248258\pi\)
\(920\) 0 0
\(921\) 6.07864 0.200298
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −20.2694 −0.666455
\(926\) 0 0
\(927\) 4.46961 0.146801
\(928\) 0 0
\(929\) −44.3061 −1.45364 −0.726818 0.686831i \(-0.759001\pi\)
−0.726818 + 0.686831i \(0.759001\pi\)
\(930\) 0 0
\(931\) −115.393 −3.78184
\(932\) 0 0
\(933\) −10.0561 −0.329221
\(934\) 0 0
\(935\) 5.27412 0.172482
\(936\) 0 0
\(937\) 2.46492 0.0805254 0.0402627 0.999189i \(-0.487181\pi\)
0.0402627 + 0.999189i \(0.487181\pi\)
\(938\) 0 0
\(939\) 11.3622 0.370790
\(940\) 0 0
\(941\) 44.3239 1.44492 0.722460 0.691413i \(-0.243012\pi\)
0.722460 + 0.691413i \(0.243012\pi\)
\(942\) 0 0
\(943\) 1.16197 0.0378389
\(944\) 0 0
\(945\) −19.5529 −0.636057
\(946\) 0 0
\(947\) −8.11216 −0.263610 −0.131805 0.991276i \(-0.542077\pi\)
−0.131805 + 0.991276i \(0.542077\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −19.4229 −0.629831
\(952\) 0 0
\(953\) −39.0031 −1.26344 −0.631718 0.775199i \(-0.717650\pi\)
−0.631718 + 0.775199i \(0.717650\pi\)
\(954\) 0 0
\(955\) −39.1059 −1.26544
\(956\) 0 0
\(957\) 0.0654688 0.00211630
\(958\) 0 0
\(959\) −77.4953 −2.50245
\(960\) 0 0
\(961\) −7.91667 −0.255376
\(962\) 0 0
\(963\) −3.66510 −0.118106
\(964\) 0 0
\(965\) 13.9618 0.449446
\(966\) 0 0
\(967\) −33.1807 −1.06702 −0.533510 0.845793i \(-0.679128\pi\)
−0.533510 + 0.845793i \(0.679128\pi\)
\(968\) 0 0
\(969\) 24.6043 0.790405
\(970\) 0 0
\(971\) −27.2368 −0.874071 −0.437036 0.899444i \(-0.643972\pi\)
−0.437036 + 0.899444i \(0.643972\pi\)
\(972\) 0 0
\(973\) −74.0771 −2.37480
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 51.1331 1.63589 0.817947 0.575293i \(-0.195112\pi\)
0.817947 + 0.575293i \(0.195112\pi\)
\(978\) 0 0
\(979\) 0.130938 0.00418478
\(980\) 0 0
\(981\) −9.08333 −0.290008
\(982\) 0 0
\(983\) −54.8365 −1.74901 −0.874506 0.485015i \(-0.838814\pi\)
−0.874506 + 0.485015i \(0.838814\pi\)
\(984\) 0 0
\(985\) 76.7243 2.44464
\(986\) 0 0
\(987\) 18.8365 0.599571
\(988\) 0 0
\(989\) −0.958007 −0.0304628
\(990\) 0 0
\(991\) 40.3800 1.28271 0.641357 0.767243i \(-0.278372\pi\)
0.641357 + 0.767243i \(0.278372\pi\)
\(992\) 0 0
\(993\) −28.8045 −0.914084
\(994\) 0 0
\(995\) 10.4564 0.331492
\(996\) 0 0
\(997\) −14.2741 −0.452066 −0.226033 0.974120i \(-0.572576\pi\)
−0.226033 + 0.974120i \(0.572576\pi\)
\(998\) 0 0
\(999\) 2.13941 0.0676879
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 4056.2.a.z.1.3 3
4.3 odd 2 8112.2.a.ck.1.3 3
13.3 even 3 312.2.q.e.217.3 6
13.5 odd 4 4056.2.c.m.337.2 6
13.8 odd 4 4056.2.c.m.337.5 6
13.9 even 3 312.2.q.e.289.3 yes 6
13.12 even 2 4056.2.a.y.1.1 3
39.29 odd 6 936.2.t.h.217.1 6
39.35 odd 6 936.2.t.h.289.1 6
52.3 odd 6 624.2.q.j.529.3 6
52.35 odd 6 624.2.q.j.289.3 6
52.51 odd 2 8112.2.a.cl.1.1 3
156.35 even 6 1872.2.t.u.289.1 6
156.107 even 6 1872.2.t.u.1153.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
312.2.q.e.217.3 6 13.3 even 3
312.2.q.e.289.3 yes 6 13.9 even 3
624.2.q.j.289.3 6 52.35 odd 6
624.2.q.j.529.3 6 52.3 odd 6
936.2.t.h.217.1 6 39.29 odd 6
936.2.t.h.289.1 6 39.35 odd 6
1872.2.t.u.289.1 6 156.35 even 6
1872.2.t.u.1153.1 6 156.107 even 6
4056.2.a.y.1.1 3 13.12 even 2
4056.2.a.z.1.3 3 1.1 even 1 trivial
4056.2.c.m.337.2 6 13.5 odd 4
4056.2.c.m.337.5 6 13.8 odd 4
8112.2.a.ck.1.3 3 4.3 odd 2
8112.2.a.cl.1.1 3 52.51 odd 2