Properties

Label 2200.2.a.x.1.4
Level $2200$
Weight $2$
Character 2200.1
Self dual yes
Analytic conductor $17.567$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2200,2,Mod(1,2200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2200, 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("2200.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2200.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: \(17.5670884447\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.54764.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 9x^{2} + 3x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-2.67673\) of defining polynomial
Character \(\chi\) \(=\) 2200.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.67673 q^{3} -4.38559 q^{7} +4.16490 q^{9} +O(q^{10})\) \(q+2.67673 q^{3} -4.38559 q^{7} +4.16490 q^{9} +1.00000 q^{11} -4.00000 q^{13} -5.87995 q^{17} -0.526486 q^{19} -11.7391 q^{21} -6.15025 q^{23} +3.11812 q^{27} -0.967873 q^{29} -9.60629 q^{31} +2.67673 q^{33} -1.76466 q^{37} -10.7069 q^{39} +2.50564 q^{41} +3.85911 q^{43} +4.91208 q^{47} +12.2334 q^{49} -15.7391 q^{51} +5.29767 q^{53} -1.40926 q^{57} -2.63841 q^{59} +8.96787 q^{61} -18.2655 q^{63} +7.97440 q^{67} -16.4626 q^{69} +13.8718 q^{71} -12.7712 q^{73} -4.38559 q^{77} -13.2768 q^{79} -4.14832 q^{81} +7.61901 q^{83} -2.59074 q^{87} +2.83510 q^{89} +17.5424 q^{91} -25.7135 q^{93} +10.4800 q^{97} +4.16490 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} - q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{3} - q^{7} + 7 q^{9} + 4 q^{11} - 16 q^{13} - 7 q^{17} - 9 q^{19} - 7 q^{21} - 6 q^{23} - 13 q^{27} + 3 q^{29} - 15 q^{31} - q^{33} - 5 q^{37} + 4 q^{39} + 10 q^{41} - 8 q^{43} + 10 q^{47} + 9 q^{49} - 23 q^{51} - 5 q^{53} + 15 q^{57} + 6 q^{59} + 29 q^{61} - 40 q^{63} - 6 q^{67} - 30 q^{69} - q^{71} - 18 q^{73} - q^{77} - 20 q^{79} + 44 q^{81} - 26 q^{83} - 31 q^{87} + 21 q^{89} + 4 q^{91} - 25 q^{93} + 4 q^{97} + 7 q^{99}+O(q^{100}) \) 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
\(3\) 2.67673 1.54541 0.772706 0.634764i \(-0.218903\pi\)
0.772706 + 0.634764i \(0.218903\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −4.38559 −1.65760 −0.828799 0.559546i \(-0.810975\pi\)
−0.828799 + 0.559546i \(0.810975\pi\)
\(8\) 0 0
\(9\) 4.16490 1.38830
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.87995 −1.42610 −0.713049 0.701114i \(-0.752686\pi\)
−0.713049 + 0.701114i \(0.752686\pi\)
\(18\) 0 0
\(19\) −0.526486 −0.120784 −0.0603920 0.998175i \(-0.519235\pi\)
−0.0603920 + 0.998175i \(0.519235\pi\)
\(20\) 0 0
\(21\) −11.7391 −2.56167
\(22\) 0 0
\(23\) −6.15025 −1.28242 −0.641208 0.767367i \(-0.721566\pi\)
−0.641208 + 0.767367i \(0.721566\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3.11812 0.600083
\(28\) 0 0
\(29\) −0.967873 −0.179730 −0.0898648 0.995954i \(-0.528643\pi\)
−0.0898648 + 0.995954i \(0.528643\pi\)
\(30\) 0 0
\(31\) −9.60629 −1.72534 −0.862670 0.505767i \(-0.831209\pi\)
−0.862670 + 0.505767i \(0.831209\pi\)
\(32\) 0 0
\(33\) 2.67673 0.465959
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.76466 −0.290108 −0.145054 0.989424i \(-0.546336\pi\)
−0.145054 + 0.989424i \(0.546336\pi\)
\(38\) 0 0
\(39\) −10.7069 −1.71448
\(40\) 0 0
\(41\) 2.50564 0.391315 0.195658 0.980672i \(-0.437316\pi\)
0.195658 + 0.980672i \(0.437316\pi\)
\(42\) 0 0
\(43\) 3.85911 0.588508 0.294254 0.955727i \(-0.404929\pi\)
0.294254 + 0.955727i \(0.404929\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.91208 0.716500 0.358250 0.933626i \(-0.383374\pi\)
0.358250 + 0.933626i \(0.383374\pi\)
\(48\) 0 0
\(49\) 12.2334 1.74763
\(50\) 0 0
\(51\) −15.7391 −2.20391
\(52\) 0 0
\(53\) 5.29767 0.727691 0.363845 0.931459i \(-0.381464\pi\)
0.363845 + 0.931459i \(0.381464\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.40926 −0.186661
\(58\) 0 0
\(59\) −2.63841 −0.343492 −0.171746 0.985141i \(-0.554941\pi\)
−0.171746 + 0.985141i \(0.554941\pi\)
\(60\) 0 0
\(61\) 8.96787 1.14822 0.574109 0.818779i \(-0.305348\pi\)
0.574109 + 0.818779i \(0.305348\pi\)
\(62\) 0 0
\(63\) −18.2655 −2.30124
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 7.97440 0.974228 0.487114 0.873338i \(-0.338050\pi\)
0.487114 + 0.873338i \(0.338050\pi\)
\(68\) 0 0
\(69\) −16.4626 −1.98186
\(70\) 0 0
\(71\) 13.8718 1.64628 0.823142 0.567836i \(-0.192219\pi\)
0.823142 + 0.567836i \(0.192219\pi\)
\(72\) 0 0
\(73\) −12.7712 −1.49475 −0.747377 0.664400i \(-0.768687\pi\)
−0.747377 + 0.664400i \(0.768687\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.38559 −0.499785
\(78\) 0 0
\(79\) −13.2768 −1.49376 −0.746880 0.664959i \(-0.768449\pi\)
−0.746880 + 0.664959i \(0.768449\pi\)
\(80\) 0 0
\(81\) −4.14832 −0.460924
\(82\) 0 0
\(83\) 7.61901 0.836295 0.418147 0.908379i \(-0.362680\pi\)
0.418147 + 0.908379i \(0.362680\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −2.59074 −0.277756
\(88\) 0 0
\(89\) 2.83510 0.300520 0.150260 0.988646i \(-0.451989\pi\)
0.150260 + 0.988646i \(0.451989\pi\)
\(90\) 0 0
\(91\) 17.5424 1.83894
\(92\) 0 0
\(93\) −25.7135 −2.66636
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.4800 1.06409 0.532044 0.846717i \(-0.321424\pi\)
0.532044 + 0.846717i \(0.321424\pi\)
\(98\) 0 0
\(99\) 4.16490 0.418588
\(100\) 0 0
\(101\) 8.70693 0.866372 0.433186 0.901305i \(-0.357389\pi\)
0.433186 + 0.901305i \(0.357389\pi\)
\(102\) 0 0
\(103\) −15.6190 −1.53899 −0.769493 0.638655i \(-0.779491\pi\)
−0.769493 + 0.638655i \(0.779491\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −7.08792 −0.685215 −0.342608 0.939479i \(-0.611310\pi\)
−0.342608 + 0.939479i \(0.611310\pi\)
\(108\) 0 0
\(109\) −15.5424 −1.48869 −0.744344 0.667796i \(-0.767238\pi\)
−0.744344 + 0.667796i \(0.767238\pi\)
\(110\) 0 0
\(111\) −4.72351 −0.448336
\(112\) 0 0
\(113\) −8.11530 −0.763423 −0.381711 0.924282i \(-0.624665\pi\)
−0.381711 + 0.924282i \(0.624665\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −16.6596 −1.54018
\(118\) 0 0
\(119\) 25.7871 2.36390
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 6.70693 0.604744
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −4.32980 −0.384207 −0.192104 0.981375i \(-0.561531\pi\)
−0.192104 + 0.981375i \(0.561531\pi\)
\(128\) 0 0
\(129\) 10.3298 0.909488
\(130\) 0 0
\(131\) −17.6748 −1.54425 −0.772127 0.635468i \(-0.780807\pi\)
−0.772127 + 0.635468i \(0.780807\pi\)
\(132\) 0 0
\(133\) 2.30895 0.200211
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −18.4800 −1.57886 −0.789428 0.613843i \(-0.789623\pi\)
−0.789428 + 0.613843i \(0.789623\pi\)
\(138\) 0 0
\(139\) −5.55861 −0.471475 −0.235738 0.971817i \(-0.575751\pi\)
−0.235738 + 0.971817i \(0.575751\pi\)
\(140\) 0 0
\(141\) 13.1483 1.10729
\(142\) 0 0
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 32.7456 2.70081
\(148\) 0 0
\(149\) 20.9516 1.71642 0.858212 0.513295i \(-0.171575\pi\)
0.858212 + 0.513295i \(0.171575\pi\)
\(150\) 0 0
\(151\) −1.67020 −0.135919 −0.0679596 0.997688i \(-0.521649\pi\)
−0.0679596 + 0.997688i \(0.521649\pi\)
\(152\) 0 0
\(153\) −24.4894 −1.97985
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 8.94228 0.713671 0.356836 0.934167i \(-0.383856\pi\)
0.356836 + 0.934167i \(0.383856\pi\)
\(158\) 0 0
\(159\) 14.1804 1.12458
\(160\) 0 0
\(161\) 26.9725 2.12573
\(162\) 0 0
\(163\) 9.43856 0.739285 0.369643 0.929174i \(-0.379480\pi\)
0.369643 + 0.929174i \(0.379480\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −8.96787 −0.693955 −0.346977 0.937873i \(-0.612792\pi\)
−0.346977 + 0.937873i \(0.612792\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −2.19276 −0.167684
\(172\) 0 0
\(173\) −2.64653 −0.201212 −0.100606 0.994926i \(-0.532078\pi\)
−0.100606 + 0.994926i \(0.532078\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −7.06232 −0.530837
\(178\) 0 0
\(179\) 20.2225 1.51150 0.755749 0.654861i \(-0.227273\pi\)
0.755749 + 0.654861i \(0.227273\pi\)
\(180\) 0 0
\(181\) 15.3453 1.14061 0.570305 0.821433i \(-0.306825\pi\)
0.570305 + 0.821433i \(0.306825\pi\)
\(182\) 0 0
\(183\) 24.0046 1.77447
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −5.87995 −0.429985
\(188\) 0 0
\(189\) −13.6748 −0.994696
\(190\) 0 0
\(191\) −5.62713 −0.407165 −0.203582 0.979058i \(-0.565258\pi\)
−0.203582 + 0.979058i \(0.565258\pi\)
\(192\) 0 0
\(193\) 7.23342 0.520673 0.260336 0.965518i \(-0.416167\pi\)
0.260336 + 0.965518i \(0.416167\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −5.35347 −0.381419 −0.190709 0.981647i \(-0.561079\pi\)
−0.190709 + 0.981647i \(0.561079\pi\)
\(198\) 0 0
\(199\) 7.23342 0.512763 0.256382 0.966576i \(-0.417470\pi\)
0.256382 + 0.966576i \(0.417470\pi\)
\(200\) 0 0
\(201\) 21.3453 1.50558
\(202\) 0 0
\(203\) 4.24470 0.297919
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −25.6152 −1.78038
\(208\) 0 0
\(209\) −0.526486 −0.0364178
\(210\) 0 0
\(211\) −12.2447 −0.842960 −0.421480 0.906838i \(-0.638489\pi\)
−0.421480 + 0.906838i \(0.638489\pi\)
\(212\) 0 0
\(213\) 37.1312 2.54419
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 42.1292 2.85992
\(218\) 0 0
\(219\) −34.1850 −2.31001
\(220\) 0 0
\(221\) 23.5198 1.58211
\(222\) 0 0
\(223\) −19.4525 −1.30264 −0.651318 0.758805i \(-0.725784\pi\)
−0.651318 + 0.758805i \(0.725784\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −3.90080 −0.258905 −0.129452 0.991586i \(-0.541322\pi\)
−0.129452 + 0.991586i \(0.541322\pi\)
\(228\) 0 0
\(229\) 13.4096 0.886131 0.443066 0.896489i \(-0.353891\pi\)
0.443066 + 0.896489i \(0.353891\pi\)
\(230\) 0 0
\(231\) −11.7391 −0.772373
\(232\) 0 0
\(233\) 3.36192 0.220247 0.110123 0.993918i \(-0.464875\pi\)
0.110123 + 0.993918i \(0.464875\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −35.5385 −2.30847
\(238\) 0 0
\(239\) −13.2768 −0.858806 −0.429403 0.903113i \(-0.641276\pi\)
−0.429403 + 0.903113i \(0.641276\pi\)
\(240\) 0 0
\(241\) −17.7437 −1.14297 −0.571485 0.820613i \(-0.693632\pi\)
−0.571485 + 0.820613i \(0.693632\pi\)
\(242\) 0 0
\(243\) −20.4583 −1.31240
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2.10594 0.133998
\(248\) 0 0
\(249\) 20.3940 1.29242
\(250\) 0 0
\(251\) 8.57416 0.541196 0.270598 0.962692i \(-0.412779\pi\)
0.270598 + 0.962692i \(0.412779\pi\)
\(252\) 0 0
\(253\) −6.15025 −0.386663
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.70693 0.168854 0.0844269 0.996430i \(-0.473094\pi\)
0.0844269 + 0.996430i \(0.473094\pi\)
\(258\) 0 0
\(259\) 7.73906 0.480882
\(260\) 0 0
\(261\) −4.03109 −0.249518
\(262\) 0 0
\(263\) −14.3213 −0.883092 −0.441546 0.897239i \(-0.645570\pi\)
−0.441546 + 0.897239i \(0.645570\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 7.58881 0.464428
\(268\) 0 0
\(269\) −11.9357 −0.727735 −0.363868 0.931451i \(-0.618544\pi\)
−0.363868 + 0.931451i \(0.618544\pi\)
\(270\) 0 0
\(271\) −24.4195 −1.48338 −0.741689 0.670744i \(-0.765975\pi\)
−0.741689 + 0.670744i \(0.765975\pi\)
\(272\) 0 0
\(273\) 46.9562 2.84192
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 23.0074 1.38238 0.691191 0.722672i \(-0.257086\pi\)
0.691191 + 0.722672i \(0.257086\pi\)
\(278\) 0 0
\(279\) −40.0092 −2.39529
\(280\) 0 0
\(281\) −10.6596 −0.635898 −0.317949 0.948108i \(-0.602994\pi\)
−0.317949 + 0.948108i \(0.602994\pi\)
\(282\) 0 0
\(283\) −10.3771 −0.616857 −0.308428 0.951248i \(-0.599803\pi\)
−0.308428 + 0.951248i \(0.599803\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −10.9887 −0.648644
\(288\) 0 0
\(289\) 17.5738 1.03375
\(290\) 0 0
\(291\) 28.0523 1.64445
\(292\) 0 0
\(293\) 22.7069 1.32655 0.663277 0.748374i \(-0.269165\pi\)
0.663277 + 0.748374i \(0.269165\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 3.11812 0.180932
\(298\) 0 0
\(299\) 24.6010 1.42271
\(300\) 0 0
\(301\) −16.9245 −0.975510
\(302\) 0 0
\(303\) 23.3061 1.33890
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 5.49436 0.313580 0.156790 0.987632i \(-0.449885\pi\)
0.156790 + 0.987632i \(0.449885\pi\)
\(308\) 0 0
\(309\) −41.8079 −2.37837
\(310\) 0 0
\(311\) −24.2447 −1.37479 −0.687395 0.726283i \(-0.741246\pi\)
−0.687395 + 0.726283i \(0.741246\pi\)
\(312\) 0 0
\(313\) 11.7562 0.664500 0.332250 0.943191i \(-0.392192\pi\)
0.332250 + 0.943191i \(0.392192\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.05387 0.508516 0.254258 0.967136i \(-0.418169\pi\)
0.254258 + 0.967136i \(0.418169\pi\)
\(318\) 0 0
\(319\) −0.967873 −0.0541905
\(320\) 0 0
\(321\) −18.9725 −1.05894
\(322\) 0 0
\(323\) 3.09571 0.172250
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −41.6028 −2.30064
\(328\) 0 0
\(329\) −21.5424 −1.18767
\(330\) 0 0
\(331\) −3.29733 −0.181238 −0.0906190 0.995886i \(-0.528885\pi\)
−0.0906190 + 0.995886i \(0.528885\pi\)
\(332\) 0 0
\(333\) −7.34961 −0.402756
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 3.17302 0.172845 0.0864227 0.996259i \(-0.472456\pi\)
0.0864227 + 0.996259i \(0.472456\pi\)
\(338\) 0 0
\(339\) −21.7225 −1.17980
\(340\) 0 0
\(341\) −9.60629 −0.520210
\(342\) 0 0
\(343\) −22.9516 −1.23927
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4.44139 0.238426 0.119213 0.992869i \(-0.461963\pi\)
0.119213 + 0.992869i \(0.461963\pi\)
\(348\) 0 0
\(349\) 29.2380 1.56508 0.782538 0.622603i \(-0.213925\pi\)
0.782538 + 0.622603i \(0.213925\pi\)
\(350\) 0 0
\(351\) −12.4725 −0.665732
\(352\) 0 0
\(353\) −1.04927 −0.0558469 −0.0279234 0.999610i \(-0.508889\pi\)
−0.0279234 + 0.999610i \(0.508889\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 69.0251 3.65319
\(358\) 0 0
\(359\) 9.69565 0.511717 0.255858 0.966714i \(-0.417642\pi\)
0.255858 + 0.966714i \(0.417642\pi\)
\(360\) 0 0
\(361\) −18.7228 −0.985411
\(362\) 0 0
\(363\) 2.67673 0.140492
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −11.3921 −0.594664 −0.297332 0.954774i \(-0.596097\pi\)
−0.297332 + 0.954774i \(0.596097\pi\)
\(368\) 0 0
\(369\) 10.4357 0.543263
\(370\) 0 0
\(371\) −23.2334 −1.20622
\(372\) 0 0
\(373\) 6.47634 0.335332 0.167666 0.985844i \(-0.446377\pi\)
0.167666 + 0.985844i \(0.446377\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.87149 0.199392
\(378\) 0 0
\(379\) −33.4993 −1.72074 −0.860372 0.509667i \(-0.829768\pi\)
−0.860372 + 0.509667i \(0.829768\pi\)
\(380\) 0 0
\(381\) −11.5897 −0.593759
\(382\) 0 0
\(383\) −30.8702 −1.57740 −0.788698 0.614781i \(-0.789244\pi\)
−0.788698 + 0.614781i \(0.789244\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 16.0728 0.817026
\(388\) 0 0
\(389\) −23.3453 −1.18366 −0.591828 0.806064i \(-0.701594\pi\)
−0.591828 + 0.806064i \(0.701594\pi\)
\(390\) 0 0
\(391\) 36.1632 1.82885
\(392\) 0 0
\(393\) −47.3107 −2.38651
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −19.4781 −0.977579 −0.488789 0.872402i \(-0.662561\pi\)
−0.488789 + 0.872402i \(0.662561\pi\)
\(398\) 0 0
\(399\) 6.18045 0.309409
\(400\) 0 0
\(401\) 2.24470 0.112095 0.0560474 0.998428i \(-0.482150\pi\)
0.0560474 + 0.998428i \(0.482150\pi\)
\(402\) 0 0
\(403\) 38.4251 1.91409
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1.76466 −0.0874707
\(408\) 0 0
\(409\) 29.4781 1.45760 0.728799 0.684727i \(-0.240079\pi\)
0.728799 + 0.684727i \(0.240079\pi\)
\(410\) 0 0
\(411\) −49.4661 −2.43998
\(412\) 0 0
\(413\) 11.5710 0.569372
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −14.8789 −0.728624
\(418\) 0 0
\(419\) 7.64831 0.373644 0.186822 0.982394i \(-0.440181\pi\)
0.186822 + 0.982394i \(0.440181\pi\)
\(420\) 0 0
\(421\) 29.4781 1.43668 0.718338 0.695695i \(-0.244903\pi\)
0.718338 + 0.695695i \(0.244903\pi\)
\(422\) 0 0
\(423\) 20.4583 0.994717
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −39.3294 −1.90328
\(428\) 0 0
\(429\) −10.7069 −0.516935
\(430\) 0 0
\(431\) −2.08970 −0.100657 −0.0503286 0.998733i \(-0.516027\pi\)
−0.0503286 + 0.998733i \(0.516027\pi\)
\(432\) 0 0
\(433\) 5.17777 0.248828 0.124414 0.992230i \(-0.460295\pi\)
0.124414 + 0.992230i \(0.460295\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.23802 0.154895
\(438\) 0 0
\(439\) 27.6066 1.31759 0.658796 0.752322i \(-0.271066\pi\)
0.658796 + 0.752322i \(0.271066\pi\)
\(440\) 0 0
\(441\) 50.9509 2.42623
\(442\) 0 0
\(443\) 14.1502 0.672299 0.336149 0.941809i \(-0.390875\pi\)
0.336149 + 0.941809i \(0.390875\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 56.0819 2.65258
\(448\) 0 0
\(449\) −6.46257 −0.304987 −0.152494 0.988304i \(-0.548730\pi\)
−0.152494 + 0.988304i \(0.548730\pi\)
\(450\) 0 0
\(451\) 2.50564 0.117986
\(452\) 0 0
\(453\) −4.47069 −0.210051
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 38.5582 1.80368 0.901839 0.432071i \(-0.142217\pi\)
0.901839 + 0.432071i \(0.142217\pi\)
\(458\) 0 0
\(459\) −18.3344 −0.855776
\(460\) 0 0
\(461\) −18.4234 −0.858064 −0.429032 0.903289i \(-0.641145\pi\)
−0.429032 + 0.903289i \(0.641145\pi\)
\(462\) 0 0
\(463\) −7.09163 −0.329576 −0.164788 0.986329i \(-0.552694\pi\)
−0.164788 + 0.986329i \(0.552694\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −40.4328 −1.87101 −0.935503 0.353319i \(-0.885053\pi\)
−0.935503 + 0.353319i \(0.885053\pi\)
\(468\) 0 0
\(469\) −34.9725 −1.61488
\(470\) 0 0
\(471\) 23.9361 1.10292
\(472\) 0 0
\(473\) 3.85911 0.177442
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 22.0643 1.01025
\(478\) 0 0
\(479\) −10.7486 −0.491117 −0.245559 0.969382i \(-0.578971\pi\)
−0.245559 + 0.969382i \(0.578971\pi\)
\(480\) 0 0
\(481\) 7.05862 0.321845
\(482\) 0 0
\(483\) 72.1981 3.28513
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 38.6396 1.75093 0.875465 0.483282i \(-0.160555\pi\)
0.875465 + 0.483282i \(0.160555\pi\)
\(488\) 0 0
\(489\) 25.2645 1.14250
\(490\) 0 0
\(491\) 32.9460 1.48683 0.743416 0.668830i \(-0.233204\pi\)
0.743416 + 0.668830i \(0.233204\pi\)
\(492\) 0 0
\(493\) 5.69105 0.256312
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −60.8362 −2.72888
\(498\) 0 0
\(499\) 26.9562 1.20673 0.603363 0.797466i \(-0.293827\pi\)
0.603363 + 0.797466i \(0.293827\pi\)
\(500\) 0 0
\(501\) −24.0046 −1.07245
\(502\) 0 0
\(503\) −14.0766 −0.627646 −0.313823 0.949481i \(-0.601610\pi\)
−0.313823 + 0.949481i \(0.601610\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 8.03020 0.356634
\(508\) 0 0
\(509\) 1.28109 0.0567833 0.0283917 0.999597i \(-0.490961\pi\)
0.0283917 + 0.999597i \(0.490961\pi\)
\(510\) 0 0
\(511\) 56.0092 2.47770
\(512\) 0 0
\(513\) −1.64165 −0.0724804
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 4.91208 0.216033
\(518\) 0 0
\(519\) −7.08407 −0.310956
\(520\) 0 0
\(521\) −29.2394 −1.28100 −0.640501 0.767958i \(-0.721273\pi\)
−0.640501 + 0.767958i \(0.721273\pi\)
\(522\) 0 0
\(523\) 2.50564 0.109564 0.0547820 0.998498i \(-0.482554\pi\)
0.0547820 + 0.998498i \(0.482554\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 56.4845 2.46050
\(528\) 0 0
\(529\) 14.8255 0.644589
\(530\) 0 0
\(531\) −10.9887 −0.476870
\(532\) 0 0
\(533\) −10.0226 −0.434125
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 54.1301 2.33589
\(538\) 0 0
\(539\) 12.2334 0.526931
\(540\) 0 0
\(541\) −3.27222 −0.140684 −0.0703420 0.997523i \(-0.522409\pi\)
−0.0703420 + 0.997523i \(0.522409\pi\)
\(542\) 0 0
\(543\) 41.0754 1.76271
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −3.03673 −0.129841 −0.0649205 0.997890i \(-0.520679\pi\)
−0.0649205 + 0.997890i \(0.520679\pi\)
\(548\) 0 0
\(549\) 37.3503 1.59407
\(550\) 0 0
\(551\) 0.509572 0.0217085
\(552\) 0 0
\(553\) 58.2267 2.47605
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.52366 0.0645596 0.0322798 0.999479i \(-0.489723\pi\)
0.0322798 + 0.999479i \(0.489723\pi\)
\(558\) 0 0
\(559\) −15.4364 −0.652891
\(560\) 0 0
\(561\) −15.7391 −0.664504
\(562\) 0 0
\(563\) −27.9195 −1.17667 −0.588333 0.808618i \(-0.700216\pi\)
−0.588333 + 0.808618i \(0.700216\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 18.1928 0.764027
\(568\) 0 0
\(569\) −13.7437 −0.576164 −0.288082 0.957606i \(-0.593018\pi\)
−0.288082 + 0.957606i \(0.593018\pi\)
\(570\) 0 0
\(571\) −1.42618 −0.0596836 −0.0298418 0.999555i \(-0.509500\pi\)
−0.0298418 + 0.999555i \(0.509500\pi\)
\(572\) 0 0
\(573\) −15.0623 −0.629238
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −6.35154 −0.264418 −0.132209 0.991222i \(-0.542207\pi\)
−0.132209 + 0.991222i \(0.542207\pi\)
\(578\) 0 0
\(579\) 19.3619 0.804654
\(580\) 0 0
\(581\) −33.4139 −1.38624
\(582\) 0 0
\(583\) 5.29767 0.219407
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −22.1681 −0.914974 −0.457487 0.889216i \(-0.651250\pi\)
−0.457487 + 0.889216i \(0.651250\pi\)
\(588\) 0 0
\(589\) 5.05757 0.208394
\(590\) 0 0
\(591\) −14.3298 −0.589449
\(592\) 0 0
\(593\) −1.05297 −0.0432403 −0.0216202 0.999766i \(-0.506882\pi\)
−0.0216202 + 0.999766i \(0.506882\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 19.3619 0.792431
\(598\) 0 0
\(599\) −5.12747 −0.209503 −0.104751 0.994498i \(-0.533405\pi\)
−0.104751 + 0.994498i \(0.533405\pi\)
\(600\) 0 0
\(601\) −1.71821 −0.0700874 −0.0350437 0.999386i \(-0.511157\pi\)
−0.0350437 + 0.999386i \(0.511157\pi\)
\(602\) 0 0
\(603\) 33.2126 1.35252
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −4.42728 −0.179698 −0.0898489 0.995955i \(-0.528638\pi\)
−0.0898489 + 0.995955i \(0.528638\pi\)
\(608\) 0 0
\(609\) 11.3619 0.460408
\(610\) 0 0
\(611\) −19.6483 −0.794886
\(612\) 0 0
\(613\) 33.8334 1.36652 0.683258 0.730177i \(-0.260562\pi\)
0.683258 + 0.730177i \(0.260562\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −40.3778 −1.62555 −0.812775 0.582578i \(-0.802044\pi\)
−0.812775 + 0.582578i \(0.802044\pi\)
\(618\) 0 0
\(619\) −1.62713 −0.0653999 −0.0327000 0.999465i \(-0.510411\pi\)
−0.0327000 + 0.999465i \(0.510411\pi\)
\(620\) 0 0
\(621\) −19.1772 −0.769555
\(622\) 0 0
\(623\) −12.4336 −0.498142
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −1.40926 −0.0562805
\(628\) 0 0
\(629\) 10.3761 0.413722
\(630\) 0 0
\(631\) 3.71223 0.147781 0.0738907 0.997266i \(-0.476458\pi\)
0.0738907 + 0.997266i \(0.476458\pi\)
\(632\) 0 0
\(633\) −32.7758 −1.30272
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −48.9337 −1.93882
\(638\) 0 0
\(639\) 57.7748 2.28553
\(640\) 0 0
\(641\) 32.5562 1.28589 0.642946 0.765911i \(-0.277712\pi\)
0.642946 + 0.765911i \(0.277712\pi\)
\(642\) 0 0
\(643\) −27.9017 −1.10034 −0.550168 0.835054i \(-0.685436\pi\)
−0.550168 + 0.835054i \(0.685436\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −28.7873 −1.13174 −0.565872 0.824493i \(-0.691460\pi\)
−0.565872 + 0.824493i \(0.691460\pi\)
\(648\) 0 0
\(649\) −2.63841 −0.103567
\(650\) 0 0
\(651\) 112.769 4.41976
\(652\) 0 0
\(653\) −34.7152 −1.35851 −0.679256 0.733901i \(-0.737697\pi\)
−0.679256 + 0.733901i \(0.737697\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −53.1907 −2.07517
\(658\) 0 0
\(659\) −25.5632 −0.995801 −0.497901 0.867234i \(-0.665896\pi\)
−0.497901 + 0.867234i \(0.665896\pi\)
\(660\) 0 0
\(661\) 1.40960 0.0548269 0.0274135 0.999624i \(-0.491273\pi\)
0.0274135 + 0.999624i \(0.491273\pi\)
\(662\) 0 0
\(663\) 62.9562 2.44502
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 5.95266 0.230488
\(668\) 0 0
\(669\) −52.0692 −2.01311
\(670\) 0 0
\(671\) 8.96787 0.346201
\(672\) 0 0
\(673\) −0.0557959 −0.00215078 −0.00107539 0.999999i \(-0.500342\pi\)
−0.00107539 + 0.999999i \(0.500342\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −20.8316 −0.800623 −0.400311 0.916379i \(-0.631098\pi\)
−0.400311 + 0.916379i \(0.631098\pi\)
\(678\) 0 0
\(679\) −45.9612 −1.76383
\(680\) 0 0
\(681\) −10.4414 −0.400115
\(682\) 0 0
\(683\) −21.5085 −0.822999 −0.411499 0.911410i \(-0.634995\pi\)
−0.411499 + 0.911410i \(0.634995\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 35.8939 1.36944
\(688\) 0 0
\(689\) −21.1907 −0.807301
\(690\) 0 0
\(691\) 5.49862 0.209178 0.104589 0.994516i \(-0.466647\pi\)
0.104589 + 0.994516i \(0.466647\pi\)
\(692\) 0 0
\(693\) −18.2655 −0.693851
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −14.7330 −0.558054
\(698\) 0 0
\(699\) 8.99897 0.340372
\(700\) 0 0
\(701\) −6.00460 −0.226791 −0.113395 0.993550i \(-0.536173\pi\)
−0.113395 + 0.993550i \(0.536173\pi\)
\(702\) 0 0
\(703\) 0.929066 0.0350404
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −38.1850 −1.43610
\(708\) 0 0
\(709\) 17.7168 0.665369 0.332685 0.943038i \(-0.392046\pi\)
0.332685 + 0.943038i \(0.392046\pi\)
\(710\) 0 0
\(711\) −55.2966 −2.07379
\(712\) 0 0
\(713\) 59.0810 2.21260
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −35.5385 −1.32721
\(718\) 0 0
\(719\) 30.5308 1.13860 0.569302 0.822128i \(-0.307213\pi\)
0.569302 + 0.822128i \(0.307213\pi\)
\(720\) 0 0
\(721\) 68.4986 2.55102
\(722\) 0 0
\(723\) −47.4950 −1.76636
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 43.9063 1.62839 0.814197 0.580589i \(-0.197177\pi\)
0.814197 + 0.580589i \(0.197177\pi\)
\(728\) 0 0
\(729\) −42.3165 −1.56728
\(730\) 0 0
\(731\) −22.6914 −0.839270
\(732\) 0 0
\(733\) 20.6427 0.762455 0.381227 0.924481i \(-0.375502\pi\)
0.381227 + 0.924481i \(0.375502\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 7.97440 0.293741
\(738\) 0 0
\(739\) −7.73446 −0.284517 −0.142258 0.989830i \(-0.545436\pi\)
−0.142258 + 0.989830i \(0.545436\pi\)
\(740\) 0 0
\(741\) 5.63705 0.207082
\(742\) 0 0
\(743\) 29.1755 1.07034 0.535172 0.844743i \(-0.320247\pi\)
0.535172 + 0.844743i \(0.320247\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 31.7324 1.16103
\(748\) 0 0
\(749\) 31.0847 1.13581
\(750\) 0 0
\(751\) −9.38310 −0.342394 −0.171197 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(752\) 0 0
\(753\) 22.9507 0.836371
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −12.8997 −0.468847 −0.234424 0.972135i \(-0.575320\pi\)
−0.234424 + 0.972135i \(0.575320\pi\)
\(758\) 0 0
\(759\) −16.4626 −0.597553
\(760\) 0 0
\(761\) 15.6963 0.568991 0.284496 0.958677i \(-0.408174\pi\)
0.284496 + 0.958677i \(0.408174\pi\)
\(762\) 0 0
\(763\) 68.1625 2.46765
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 10.5536 0.381070
\(768\) 0 0
\(769\) 34.1123 1.23012 0.615060 0.788480i \(-0.289132\pi\)
0.615060 + 0.788480i \(0.289132\pi\)
\(770\) 0 0
\(771\) 7.24573 0.260949
\(772\) 0 0
\(773\) 52.2539 1.87944 0.939721 0.341942i \(-0.111085\pi\)
0.939721 + 0.341942i \(0.111085\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 20.7154 0.743160
\(778\) 0 0
\(779\) −1.31918 −0.0472647
\(780\) 0 0
\(781\) 13.8718 0.496373
\(782\) 0 0
\(783\) −3.01795 −0.107853
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −6.89902 −0.245924 −0.122962 0.992411i \(-0.539239\pi\)
−0.122962 + 0.992411i \(0.539239\pi\)
\(788\) 0 0
\(789\) −38.3344 −1.36474
\(790\) 0 0
\(791\) 35.5904 1.26545
\(792\) 0 0
\(793\) −35.8715 −1.27383
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0.544296 0.0192800 0.00963998 0.999954i \(-0.496931\pi\)
0.00963998 + 0.999954i \(0.496931\pi\)
\(798\) 0 0
\(799\) −28.8828 −1.02180
\(800\) 0 0
\(801\) 11.8079 0.417212
\(802\) 0 0
\(803\) −12.7712 −0.450685
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −31.9488 −1.12465
\(808\) 0 0
\(809\) 26.3354 0.925905 0.462952 0.886383i \(-0.346790\pi\)
0.462952 + 0.886383i \(0.346790\pi\)
\(810\) 0 0
\(811\) −24.4686 −0.859207 −0.429604 0.903018i \(-0.641347\pi\)
−0.429604 + 0.903018i \(0.641347\pi\)
\(812\) 0 0
\(813\) −65.3645 −2.29243
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −2.03176 −0.0710824
\(818\) 0 0
\(819\) 73.0622 2.55300
\(820\) 0 0
\(821\) 42.8503 1.49549 0.747743 0.663989i \(-0.231138\pi\)
0.747743 + 0.663989i \(0.231138\pi\)
\(822\) 0 0
\(823\) −18.7417 −0.653296 −0.326648 0.945146i \(-0.605919\pi\)
−0.326648 + 0.945146i \(0.605919\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 30.6681 1.06644 0.533218 0.845978i \(-0.320983\pi\)
0.533218 + 0.845978i \(0.320983\pi\)
\(828\) 0 0
\(829\) −19.8758 −0.690314 −0.345157 0.938545i \(-0.612174\pi\)
−0.345157 + 0.938545i \(0.612174\pi\)
\(830\) 0 0
\(831\) 61.5847 2.13635
\(832\) 0 0
\(833\) −71.9319 −2.49229
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −29.9536 −1.03535
\(838\) 0 0
\(839\) −48.3178 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(840\) 0 0
\(841\) −28.0632 −0.967697
\(842\) 0 0
\(843\) −28.5329 −0.982725
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −4.38559 −0.150691
\(848\) 0 0
\(849\) −27.7768 −0.953298
\(850\) 0 0
\(851\) 10.8531 0.372038
\(852\) 0 0
\(853\) −39.4686 −1.35138 −0.675690 0.737186i \(-0.736154\pi\)
−0.675690 + 0.737186i \(0.736154\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −16.7666 −0.572736 −0.286368 0.958120i \(-0.592448\pi\)
−0.286368 + 0.958120i \(0.592448\pi\)
\(858\) 0 0
\(859\) 0.350976 0.0119751 0.00598757 0.999982i \(-0.498094\pi\)
0.00598757 + 0.999982i \(0.498094\pi\)
\(860\) 0 0
\(861\) −29.4139 −1.00242
\(862\) 0 0
\(863\) 35.2087 1.19852 0.599259 0.800555i \(-0.295462\pi\)
0.599259 + 0.800555i \(0.295462\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 47.0404 1.59758
\(868\) 0 0
\(869\) −13.2768 −0.450385
\(870\) 0 0
\(871\) −31.8976 −1.08081
\(872\) 0 0
\(873\) 43.6483 1.47727
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 8.61790 0.291006 0.145503 0.989358i \(-0.453520\pi\)
0.145503 + 0.989358i \(0.453520\pi\)
\(878\) 0 0
\(879\) 60.7804 2.05007
\(880\) 0 0
\(881\) −42.5162 −1.43241 −0.716204 0.697891i \(-0.754122\pi\)
−0.716204 + 0.697891i \(0.754122\pi\)
\(882\) 0 0
\(883\) −25.1568 −0.846593 −0.423296 0.905991i \(-0.639127\pi\)
−0.423296 + 0.905991i \(0.639127\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −6.27505 −0.210696 −0.105348 0.994435i \(-0.533596\pi\)
−0.105348 + 0.994435i \(0.533596\pi\)
\(888\) 0 0
\(889\) 18.9887 0.636861
\(890\) 0 0
\(891\) −4.14832 −0.138974
\(892\) 0 0
\(893\) −2.58614 −0.0865418
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 65.8503 2.19868
\(898\) 0 0
\(899\) 9.29767 0.310095
\(900\) 0 0
\(901\) −31.1500 −1.03776
\(902\) 0 0
\(903\) −45.3023 −1.50757
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −11.0925 −0.368321 −0.184161 0.982896i \(-0.558957\pi\)
−0.184161 + 0.982896i \(0.558957\pi\)
\(908\) 0 0
\(909\) 36.2635 1.20278
\(910\) 0 0
\(911\) −32.8175 −1.08729 −0.543646 0.839315i \(-0.682956\pi\)
−0.543646 + 0.839315i \(0.682956\pi\)
\(912\) 0 0
\(913\) 7.61901 0.252152
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 77.5145 2.55975
\(918\) 0 0
\(919\) 0.658922 0.0217358 0.0108679 0.999941i \(-0.496541\pi\)
0.0108679 + 0.999941i \(0.496541\pi\)
\(920\) 0 0
\(921\) 14.7069 0.484610
\(922\) 0 0
\(923\) −55.4873 −1.82639
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −65.0516 −2.13657
\(928\) 0 0
\(929\) −28.0519 −0.920354 −0.460177 0.887827i \(-0.652214\pi\)
−0.460177 + 0.887827i \(0.652214\pi\)
\(930\) 0 0
\(931\) −6.44072 −0.211086
\(932\) 0 0
\(933\) −64.8966 −2.12462
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −26.6652 −0.871115 −0.435558 0.900161i \(-0.643449\pi\)
−0.435558 + 0.900161i \(0.643449\pi\)
\(938\) 0 0
\(939\) 31.4682 1.02693
\(940\) 0 0
\(941\) −0.222135 −0.00724139 −0.00362069 0.999993i \(-0.501153\pi\)
−0.00362069 + 0.999993i \(0.501153\pi\)
\(942\) 0 0
\(943\) −15.4103 −0.501829
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 4.99028 0.162162 0.0810812 0.996707i \(-0.474163\pi\)
0.0810812 + 0.996707i \(0.474163\pi\)
\(948\) 0 0
\(949\) 51.0847 1.65828
\(950\) 0 0
\(951\) 24.2348 0.785867
\(952\) 0 0
\(953\) 9.66807 0.313179 0.156590 0.987664i \(-0.449950\pi\)
0.156590 + 0.987664i \(0.449950\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −2.59074 −0.0837467
\(958\) 0 0
\(959\) 81.0459 2.61711
\(960\) 0 0
\(961\) 61.2807 1.97680
\(962\) 0 0
\(963\) −29.5205 −0.951284
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 19.8033 0.636832 0.318416 0.947951i \(-0.396849\pi\)
0.318416 + 0.947951i \(0.396849\pi\)
\(968\) 0 0
\(969\) 8.28639 0.266197
\(970\) 0 0
\(971\) 39.0410 1.25289 0.626443 0.779468i \(-0.284510\pi\)
0.626443 + 0.779468i \(0.284510\pi\)
\(972\) 0 0
\(973\) 24.3778 0.781517
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 50.8314 1.62624 0.813121 0.582095i \(-0.197767\pi\)
0.813121 + 0.582095i \(0.197767\pi\)
\(978\) 0 0
\(979\) 2.83510 0.0906103
\(980\) 0 0
\(981\) −64.7324 −2.06675
\(982\) 0 0
\(983\) 21.7569 0.693936 0.346968 0.937877i \(-0.387211\pi\)
0.346968 + 0.937877i \(0.387211\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −57.6632 −1.83544
\(988\) 0 0
\(989\) −23.7345 −0.754712
\(990\) 0 0
\(991\) −33.0622 −1.05025 −0.525127 0.851024i \(-0.675982\pi\)
−0.525127 + 0.851024i \(0.675982\pi\)
\(992\) 0 0
\(993\) −8.82608 −0.280087
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 19.4008 0.614430 0.307215 0.951640i \(-0.400603\pi\)
0.307215 + 0.951640i \(0.400603\pi\)
\(998\) 0 0
\(999\) −5.50241 −0.174088
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 2200.2.a.x.1.4 4
4.3 odd 2 4400.2.a.ce.1.1 4
5.2 odd 4 440.2.b.d.89.2 8
5.3 odd 4 440.2.b.d.89.7 yes 8
5.4 even 2 2200.2.a.y.1.1 4
15.2 even 4 3960.2.d.f.3169.1 8
15.8 even 4 3960.2.d.f.3169.2 8
20.3 even 4 880.2.b.j.529.2 8
20.7 even 4 880.2.b.j.529.7 8
20.19 odd 2 4400.2.a.cb.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.b.d.89.2 8 5.2 odd 4
440.2.b.d.89.7 yes 8 5.3 odd 4
880.2.b.j.529.2 8 20.3 even 4
880.2.b.j.529.7 8 20.7 even 4
2200.2.a.x.1.4 4 1.1 even 1 trivial
2200.2.a.y.1.1 4 5.4 even 2
3960.2.d.f.3169.1 8 15.2 even 4
3960.2.d.f.3169.2 8 15.8 even 4
4400.2.a.cb.1.4 4 20.19 odd 2
4400.2.a.ce.1.1 4 4.3 odd 2