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Properties

Label 1800.4.a.i.1.1
Level $1800$
Weight $4$
Character 1800.1
Self dual yes
Analytic conductor $106.203$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1800,4,Mod(1,1800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1800.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1800.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: \(106.203438010\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1800.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-10.0000 q^{7} +14.0000 q^{11} +82.0000 q^{13} +18.0000 q^{17} -136.000 q^{19} -140.000 q^{23} -112.000 q^{29} +72.0000 q^{31} -26.0000 q^{37} +446.000 q^{41} -396.000 q^{43} -144.000 q^{47} -243.000 q^{49} +158.000 q^{53} +342.000 q^{59} +314.000 q^{61} +152.000 q^{67} +932.000 q^{71} +548.000 q^{73} -140.000 q^{77} -512.000 q^{79} +284.000 q^{83} +810.000 q^{89} -820.000 q^{91} -1304.00 q^{97} +O(q^{100})\)

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 0
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −10.0000 −0.539949 −0.269975 − 0.962867i \(-0.587015\pi\)
−0.269975 + 0.962867i \(0.587015\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 14.0000 0.383742 0.191871 − 0.981420i \(-0.438545\pi\)
0.191871 + 0.981420i \(0.438545\pi\)
\(12\) 0 0
\(13\) 82.0000 1.74944 0.874720 − 0.484629i \(-0.161046\pi\)
0.874720 + 0.484629i \(0.161046\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 18.0000 0.256802 0.128401 − 0.991722i \(-0.459015\pi\)
0.128401 + 0.991722i \(0.459015\pi\)
\(18\) 0 0
\(19\) −136.000 −1.64213 −0.821067 − 0.570832i \(-0.806621\pi\)
−0.821067 + 0.570832i \(0.806621\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −140.000 −1.26922 −0.634609 − 0.772833i \(-0.718839\pi\)
−0.634609 + 0.772833i \(0.718839\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −112.000 −0.717168 −0.358584 − 0.933497i \(-0.616740\pi\)
−0.358584 + 0.933497i \(0.616740\pi\)
\(30\) 0 0
\(31\) 72.0000 0.417148 0.208574 − 0.978007i \(-0.433118\pi\)
0.208574 + 0.978007i \(0.433118\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −26.0000 −0.115524 −0.0577618 − 0.998330i \(-0.518396\pi\)
−0.0577618 + 0.998330i \(0.518396\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 446.000 1.69887 0.849433 − 0.527697i \(-0.176944\pi\)
0.849433 + 0.527697i \(0.176944\pi\)
\(42\) 0 0
\(43\) −396.000 −1.40441 −0.702203 − 0.711977i \(-0.747800\pi\)
−0.702203 + 0.711977i \(0.747800\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −144.000 −0.446906 −0.223453 − 0.974715i \(-0.571733\pi\)
−0.223453 + 0.974715i \(0.571733\pi\)
\(48\) 0 0
\(49\) −243.000 −0.708455
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 158.000 0.409490 0.204745 − 0.978815i \(-0.434363\pi\)
0.204745 + 0.978815i \(0.434363\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 342.000 0.754654 0.377327 − 0.926080i \(-0.376843\pi\)
0.377327 + 0.926080i \(0.376843\pi\)
\(60\) 0 0
\(61\) 314.000 0.659075 0.329538 − 0.944142i \(-0.393107\pi\)
0.329538 + 0.944142i \(0.393107\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 152.000 0.277161 0.138580 − 0.990351i \(-0.455746\pi\)
0.138580 + 0.990351i \(0.455746\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 932.000 1.55786 0.778930 − 0.627111i \(-0.215763\pi\)
0.778930 + 0.627111i \(0.215763\pi\)
\(72\) 0 0
\(73\) 548.000 0.878610 0.439305 − 0.898338i \(-0.355225\pi\)
0.439305 + 0.898338i \(0.355225\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −140.000 −0.207201
\(78\) 0 0
\(79\) −512.000 −0.729171 −0.364585 − 0.931170i \(-0.618789\pi\)
−0.364585 + 0.931170i \(0.618789\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 284.000 0.375579 0.187789 − 0.982209i \(-0.439868\pi\)
0.187789 + 0.982209i \(0.439868\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 810.000 0.964717 0.482359 − 0.875974i \(-0.339780\pi\)
0.482359 + 0.875974i \(0.339780\pi\)
\(90\) 0 0
\(91\) −820.000 −0.944608
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1304.00 −1.36496 −0.682480 − 0.730904i \(-0.739099\pi\)
−0.682480 + 0.730904i \(0.739099\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −936.000 −0.922133 −0.461067 − 0.887365i \(-0.652533\pi\)
−0.461067 + 0.887365i \(0.652533\pi\)
\(102\) 0 0
\(103\) −1450.00 −1.38711 −0.693557 − 0.720402i \(-0.743957\pi\)
−0.693557 + 0.720402i \(0.743957\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1292.00 1.16731 0.583656 − 0.812001i \(-0.301622\pi\)
0.583656 + 0.812001i \(0.301622\pi\)
\(108\) 0 0
\(109\) −2142.00 −1.88226 −0.941130 − 0.338044i \(-0.890235\pi\)
−0.941130 + 0.338044i \(0.890235\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1418.00 −1.18048 −0.590240 − 0.807228i \(-0.700967\pi\)
−0.590240 + 0.807228i \(0.700967\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −180.000 −0.138660
\(120\) 0 0
\(121\) −1135.00 −0.852742
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −1674.00 −1.16963 −0.584817 − 0.811165i \(-0.698834\pi\)
−0.584817 + 0.811165i \(0.698834\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1134.00 −0.756321 −0.378160 − 0.925740i \(-0.623443\pi\)
−0.378160 + 0.925740i \(0.623443\pi\)
\(132\) 0 0
\(133\) 1360.00 0.886669
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2866.00 −1.78729 −0.893646 − 0.448773i \(-0.851861\pi\)
−0.893646 + 0.448773i \(0.851861\pi\)
\(138\) 0 0
\(139\) −764.000 −0.466199 −0.233099 − 0.972453i \(-0.574887\pi\)
−0.233099 + 0.972453i \(0.574887\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1148.00 0.671333
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3060.00 1.68245 0.841225 − 0.540686i \(-0.181835\pi\)
0.841225 + 0.540686i \(0.181835\pi\)
\(150\) 0 0
\(151\) −1304.00 −0.702768 −0.351384 − 0.936231i \(-0.614289\pi\)
−0.351384 + 0.936231i \(0.614289\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −1582.00 −0.804187 −0.402093 − 0.915599i \(-0.631717\pi\)
−0.402093 + 0.915599i \(0.631717\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1400.00 0.685313
\(162\) 0 0
\(163\) −3232.00 −1.55307 −0.776533 − 0.630077i \(-0.783024\pi\)
−0.776533 + 0.630077i \(0.783024\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2988.00 −1.38454 −0.692271 − 0.721638i \(-0.743390\pi\)
−0.692271 + 0.721638i \(0.743390\pi\)
\(168\) 0 0
\(169\) 4527.00 2.06054
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −918.000 −0.403435 −0.201717 − 0.979444i \(-0.564652\pi\)
−0.201717 + 0.979444i \(0.564652\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1238.00 0.516941 0.258471 − 0.966019i \(-0.416781\pi\)
0.258471 + 0.966019i \(0.416781\pi\)
\(180\) 0 0
\(181\) 1450.00 0.595457 0.297728 − 0.954651i \(-0.403771\pi\)
0.297728 + 0.954651i \(0.403771\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 252.000 0.0985458
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4860.00 −1.84114 −0.920569 − 0.390581i \(-0.872274\pi\)
−0.920569 + 0.390581i \(0.872274\pi\)
\(192\) 0 0
\(193\) 1412.00 0.526622 0.263311 − 0.964711i \(-0.415186\pi\)
0.263311 + 0.964711i \(0.415186\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1170.00 −0.423142 −0.211571 − 0.977363i \(-0.567858\pi\)
−0.211571 + 0.977363i \(0.567858\pi\)
\(198\) 0 0
\(199\) 2080.00 0.740941 0.370471 − 0.928844i \(-0.379196\pi\)
0.370471 + 0.928844i \(0.379196\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1120.00 0.387234
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1904.00 −0.630155
\(210\) 0 0
\(211\) 2692.00 0.878317 0.439159 − 0.898410i \(-0.355277\pi\)
0.439159 + 0.898410i \(0.355277\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −720.000 −0.225239
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1476.00 0.449260
\(222\) 0 0
\(223\) 846.000 0.254046 0.127023 − 0.991900i \(-0.459458\pi\)
0.127023 + 0.991900i \(0.459458\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2884.00 0.843250 0.421625 − 0.906770i \(-0.361460\pi\)
0.421625 + 0.906770i \(0.361460\pi\)
\(228\) 0 0
\(229\) 3150.00 0.908986 0.454493 − 0.890750i \(-0.349820\pi\)
0.454493 + 0.890750i \(0.349820\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −4014.00 −1.12861 −0.564304 − 0.825567i \(-0.690856\pi\)
−0.564304 + 0.825567i \(0.690856\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −4900.00 −1.32617 −0.663085 − 0.748544i \(-0.730753\pi\)
−0.663085 + 0.748544i \(0.730753\pi\)
\(240\) 0 0
\(241\) −2314.00 −0.618497 −0.309249 − 0.950981i \(-0.600078\pi\)
−0.309249 + 0.950981i \(0.600078\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −11152.0 −2.87281
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 2002.00 0.503447 0.251723 − 0.967799i \(-0.419003\pi\)
0.251723 + 0.967799i \(0.419003\pi\)
\(252\) 0 0
\(253\) −1960.00 −0.487052
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −450.000 −0.109223 −0.0546113 − 0.998508i \(-0.517392\pi\)
−0.0546113 + 0.998508i \(0.517392\pi\)
\(258\) 0 0
\(259\) 260.000 0.0623769
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 180.000 0.0422026 0.0211013 − 0.999777i \(-0.493283\pi\)
0.0211013 + 0.999777i \(0.493283\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −2448.00 −0.554859 −0.277430 − 0.960746i \(-0.589483\pi\)
−0.277430 + 0.960746i \(0.589483\pi\)
\(270\) 0 0
\(271\) 6776.00 1.51887 0.759433 − 0.650586i \(-0.225476\pi\)
0.759433 + 0.650586i \(0.225476\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −6426.00 −1.39387 −0.696933 − 0.717136i \(-0.745453\pi\)
−0.696933 + 0.717136i \(0.745453\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2718.00 −0.577019 −0.288509 − 0.957477i \(-0.593160\pi\)
−0.288509 + 0.957477i \(0.593160\pi\)
\(282\) 0 0
\(283\) −6048.00 −1.27038 −0.635188 − 0.772358i \(-0.719077\pi\)
−0.635188 + 0.772358i \(0.719077\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −4460.00 −0.917301
\(288\) 0 0
\(289\) −4589.00 −0.934053
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1246.00 0.248437 0.124219 − 0.992255i \(-0.460358\pi\)
0.124219 + 0.992255i \(0.460358\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −11480.0 −2.22042
\(300\) 0 0
\(301\) 3960.00 0.758308
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −1244.00 −0.231267 −0.115633 − 0.993292i \(-0.536890\pi\)
−0.115633 + 0.993292i \(0.536890\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 6372.00 1.16181 0.580905 − 0.813971i \(-0.302699\pi\)
0.580905 + 0.813971i \(0.302699\pi\)
\(312\) 0 0
\(313\) −5500.00 −0.993222 −0.496611 − 0.867973i \(-0.665422\pi\)
−0.496611 + 0.867973i \(0.665422\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 378.000 0.0669735 0.0334867 − 0.999439i \(-0.489339\pi\)
0.0334867 + 0.999439i \(0.489339\pi\)
\(318\) 0 0
\(319\) −1568.00 −0.275207
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −2448.00 −0.421704
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 1440.00 0.241306
\(330\) 0 0
\(331\) −11888.0 −1.97409 −0.987045 − 0.160446i \(-0.948707\pi\)
−0.987045 + 0.160446i \(0.948707\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 9116.00 1.47353 0.736766 − 0.676148i \(-0.236352\pi\)
0.736766 + 0.676148i \(0.236352\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1008.00 0.160077
\(342\) 0 0
\(343\) 5860.00 0.922479
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −4676.00 −0.723403 −0.361701 − 0.932294i \(-0.617804\pi\)
−0.361701 + 0.932294i \(0.617804\pi\)
\(348\) 0 0
\(349\) 11906.0 1.82611 0.913057 − 0.407833i \(-0.133715\pi\)
0.913057 + 0.407833i \(0.133715\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −2142.00 −0.322966 −0.161483 − 0.986875i \(-0.551628\pi\)
−0.161483 + 0.986875i \(0.551628\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −9824.00 −1.44426 −0.722132 − 0.691755i \(-0.756838\pi\)
−0.722132 + 0.691755i \(0.756838\pi\)
\(360\) 0 0
\(361\) 11637.0 1.69660
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −5354.00 −0.761516 −0.380758 − 0.924675i \(-0.624337\pi\)
−0.380758 + 0.924675i \(0.624337\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1580.00 −0.221104
\(372\) 0 0
\(373\) −7694.00 −1.06804 −0.534022 − 0.845471i \(-0.679320\pi\)
−0.534022 + 0.845471i \(0.679320\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −9184.00 −1.25464
\(378\) 0 0
\(379\) −6004.00 −0.813733 −0.406866 − 0.913488i \(-0.633379\pi\)
−0.406866 + 0.913488i \(0.633379\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 9432.00 1.25836 0.629181 − 0.777259i \(-0.283390\pi\)
0.629181 + 0.777259i \(0.283390\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 6156.00 0.802369 0.401185 − 0.915997i \(-0.368599\pi\)
0.401185 + 0.915997i \(0.368599\pi\)
\(390\) 0 0
\(391\) −2520.00 −0.325938
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −7866.00 −0.994416 −0.497208 − 0.867631i \(-0.665641\pi\)
−0.497208 + 0.867631i \(0.665641\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2074.00 0.258281 0.129140 − 0.991626i \(-0.458778\pi\)
0.129140 + 0.991626i \(0.458778\pi\)
\(402\) 0 0
\(403\) 5904.00 0.729775
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −364.000 −0.0443312
\(408\) 0 0
\(409\) −2746.00 −0.331983 −0.165991 − 0.986127i \(-0.553082\pi\)
−0.165991 + 0.986127i \(0.553082\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −3420.00 −0.407475
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 6426.00 0.749238 0.374619 − 0.927179i \(-0.377774\pi\)
0.374619 + 0.927179i \(0.377774\pi\)
\(420\) 0 0
\(421\) −10610.0 −1.22827 −0.614133 − 0.789203i \(-0.710494\pi\)
−0.614133 + 0.789203i \(0.710494\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −3140.00 −0.355867
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −8384.00 −0.936991 −0.468495 − 0.883466i \(-0.655204\pi\)
−0.468495 + 0.883466i \(0.655204\pi\)
\(432\) 0 0
\(433\) −1980.00 −0.219752 −0.109876 − 0.993945i \(-0.535045\pi\)
−0.109876 + 0.993945i \(0.535045\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 19040.0 2.08423
\(438\) 0 0
\(439\) 864.000 0.0939327 0.0469664 − 0.998896i \(-0.485045\pi\)
0.0469664 + 0.998896i \(0.485045\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −13212.0 −1.41698 −0.708489 − 0.705722i \(-0.750623\pi\)
−0.708489 + 0.705722i \(0.750623\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 16290.0 1.71219 0.856094 − 0.516820i \(-0.172884\pi\)
0.856094 + 0.516820i \(0.172884\pi\)
\(450\) 0 0
\(451\) 6244.00 0.651926
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −6336.00 −0.648546 −0.324273 − 0.945964i \(-0.605120\pi\)
−0.324273 + 0.945964i \(0.605120\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 13716.0 1.38572 0.692861 − 0.721071i \(-0.256350\pi\)
0.692861 + 0.721071i \(0.256350\pi\)
\(462\) 0 0
\(463\) 14626.0 1.46809 0.734047 − 0.679098i \(-0.237629\pi\)
0.734047 + 0.679098i \(0.237629\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −5796.00 −0.574319 −0.287159 − 0.957883i \(-0.592711\pi\)
−0.287159 + 0.957883i \(0.592711\pi\)
\(468\) 0 0
\(469\) −1520.00 −0.149653
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −5544.00 −0.538929
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 8348.00 0.796305 0.398152 − 0.917319i \(-0.369652\pi\)
0.398152 + 0.917319i \(0.369652\pi\)
\(480\) 0 0
\(481\) −2132.00 −0.202102
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 1898.00 0.176605 0.0883025 − 0.996094i \(-0.471856\pi\)
0.0883025 + 0.996094i \(0.471856\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −16654.0 −1.53072 −0.765361 − 0.643601i \(-0.777440\pi\)
−0.765361 + 0.643601i \(0.777440\pi\)
\(492\) 0 0
\(493\) −2016.00 −0.184171
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −9320.00 −0.841165
\(498\) 0 0
\(499\) −10600.0 −0.950944 −0.475472 − 0.879731i \(-0.657723\pi\)
−0.475472 + 0.879731i \(0.657723\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −10048.0 −0.890692 −0.445346 − 0.895359i \(-0.646919\pi\)
−0.445346 + 0.895359i \(0.646919\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 15088.0 1.31388 0.656939 − 0.753944i \(-0.271851\pi\)
0.656939 + 0.753944i \(0.271851\pi\)
\(510\) 0 0
\(511\) −5480.00 −0.474405
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −2016.00 −0.171496
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −8582.00 −0.721659 −0.360829 − 0.932632i \(-0.617506\pi\)
−0.360829 + 0.932632i \(0.617506\pi\)
\(522\) 0 0
\(523\) −16928.0 −1.41532 −0.707658 − 0.706556i \(-0.750248\pi\)
−0.707658 + 0.706556i \(0.750248\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1296.00 0.107125
\(528\) 0 0
\(529\) 7433.00 0.610915
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 36572.0 2.97206
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −3402.00 −0.271864
\(540\) 0 0
\(541\) −11150.0 −0.886092 −0.443046 − 0.896499i \(-0.646102\pi\)
−0.443046 + 0.896499i \(0.646102\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 19628.0 1.53425 0.767123 − 0.641500i \(-0.221688\pi\)
0.767123 + 0.641500i \(0.221688\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 15232.0 1.17769
\(552\) 0 0
\(553\) 5120.00 0.393715
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 8694.00 0.661358 0.330679 − 0.943743i \(-0.392722\pi\)
0.330679 + 0.943743i \(0.392722\pi\)
\(558\) 0 0
\(559\) −32472.0 −2.45692
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −828.000 −0.0619823 −0.0309912 − 0.999520i \(-0.509866\pi\)
−0.0309912 + 0.999520i \(0.509866\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 17514.0 1.29038 0.645189 − 0.764023i \(-0.276779\pi\)
0.645189 + 0.764023i \(0.276779\pi\)
\(570\) 0 0
\(571\) −5552.00 −0.406907 −0.203454 − 0.979085i \(-0.565217\pi\)
−0.203454 + 0.979085i \(0.565217\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 12896.0 0.930446 0.465223 − 0.885193i \(-0.345974\pi\)
0.465223 + 0.885193i \(0.345974\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −2840.00 −0.202794
\(582\) 0 0
\(583\) 2212.00 0.157138
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1588.00 −0.111659 −0.0558295 − 0.998440i \(-0.517780\pi\)
−0.0558295 + 0.998440i \(0.517780\pi\)
\(588\) 0 0
\(589\) −9792.00 −0.685012
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 22262.0 1.54164 0.770819 − 0.637055i \(-0.219848\pi\)
0.770819 + 0.637055i \(0.219848\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −4640.00 −0.316503 −0.158251 − 0.987399i \(-0.550586\pi\)
−0.158251 + 0.987399i \(0.550586\pi\)
\(600\) 0 0
\(601\) −2574.00 −0.174702 −0.0873508 − 0.996178i \(-0.527840\pi\)
−0.0873508 + 0.996178i \(0.527840\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 11170.0 0.746913 0.373457 − 0.927648i \(-0.378172\pi\)
0.373457 + 0.927648i \(0.378172\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −11808.0 −0.781834
\(612\) 0 0
\(613\) 3646.00 0.240229 0.120115 − 0.992760i \(-0.461674\pi\)
0.120115 + 0.992760i \(0.461674\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −7646.00 −0.498892 −0.249446 − 0.968389i \(-0.580249\pi\)
−0.249446 + 0.968389i \(0.580249\pi\)
\(618\) 0 0
\(619\) 26668.0 1.73163 0.865814 − 0.500366i \(-0.166801\pi\)
0.865814 + 0.500366i \(0.166801\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −8100.00 −0.520898
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −468.000 −0.0296667
\(630\) 0 0
\(631\) 7712.00 0.486545 0.243272 − 0.969958i \(-0.421779\pi\)
0.243272 + 0.969958i \(0.421779\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −19926.0 −1.23940
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 22302.0 1.37422 0.687111 − 0.726553i \(-0.258879\pi\)
0.687111 + 0.726553i \(0.258879\pi\)
\(642\) 0 0
\(643\) −2232.00 −0.136892 −0.0684459 − 0.997655i \(-0.521804\pi\)
−0.0684459 + 0.997655i \(0.521804\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 9464.00 0.575067 0.287533 − 0.957771i \(-0.407165\pi\)
0.287533 + 0.957771i \(0.407165\pi\)
\(648\) 0 0
\(649\) 4788.00 0.289592
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −4878.00 −0.292329 −0.146165 − 0.989260i \(-0.546693\pi\)
−0.146165 + 0.989260i \(0.546693\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −10206.0 −0.603292 −0.301646 − 0.953420i \(-0.597536\pi\)
−0.301646 + 0.953420i \(0.597536\pi\)
\(660\) 0 0
\(661\) 20906.0 1.23018 0.615090 − 0.788457i \(-0.289120\pi\)
0.615090 + 0.788457i \(0.289120\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 15680.0 0.910243
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4396.00 0.252915
\(672\) 0 0
\(673\) 6812.00 0.390168 0.195084 − 0.980787i \(-0.437502\pi\)
0.195084 + 0.980787i \(0.437502\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 10026.0 0.569174 0.284587 − 0.958650i \(-0.408144\pi\)
0.284587 + 0.958650i \(0.408144\pi\)
\(678\) 0 0
\(679\) 13040.0 0.737009
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −21236.0 −1.18971 −0.594856 − 0.803832i \(-0.702791\pi\)
−0.594856 + 0.803832i \(0.702791\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 12956.0 0.716378
\(690\) 0 0
\(691\) −11520.0 −0.634213 −0.317107 − 0.948390i \(-0.602711\pi\)
−0.317107 + 0.948390i \(0.602711\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 8028.00 0.436273
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 400.000 0.0215518 0.0107759 − 0.999942i \(-0.496570\pi\)
0.0107759 + 0.999942i \(0.496570\pi\)
\(702\) 0 0
\(703\) 3536.00 0.189705
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9360.00 0.497905
\(708\) 0 0
\(709\) −5930.00 −0.314113 −0.157056 − 0.987590i \(-0.550200\pi\)
−0.157056 + 0.987590i \(0.550200\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −10080.0 −0.529452
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −7160.00 −0.371381 −0.185691 − 0.982608i \(-0.559452\pi\)
−0.185691 + 0.982608i \(0.559452\pi\)
\(720\) 0 0
\(721\) 14500.0 0.748971
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 8874.00 0.452708 0.226354 − 0.974045i \(-0.427319\pi\)
0.226354 + 0.974045i \(0.427319\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −7128.00 −0.360655
\(732\) 0 0
\(733\) −5562.00 −0.280269 −0.140134 − 0.990132i \(-0.544753\pi\)
−0.140134 + 0.990132i \(0.544753\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2128.00 0.106358
\(738\) 0 0
\(739\) −12096.0 −0.602109 −0.301055 − 0.953607i \(-0.597339\pi\)
−0.301055 + 0.953607i \(0.597339\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 21312.0 1.05230 0.526152 − 0.850391i \(-0.323634\pi\)
0.526152 + 0.850391i \(0.323634\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −12920.0 −0.630289
\(750\) 0 0
\(751\) −6832.00 −0.331962 −0.165981 − 0.986129i \(-0.553079\pi\)
−0.165981 + 0.986129i \(0.553079\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 7174.00 0.344443 0.172222 − 0.985058i \(-0.444905\pi\)
0.172222 + 0.985058i \(0.444905\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 15394.0 0.733288 0.366644 − 0.930361i \(-0.380507\pi\)
0.366644 + 0.930361i \(0.380507\pi\)
\(762\) 0 0
\(763\) 21420.0 1.01633
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 28044.0 1.32022
\(768\) 0 0
\(769\) 9730.00 0.456271 0.228136 − 0.973629i \(-0.426737\pi\)
0.228136 + 0.973629i \(0.426737\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −24206.0 −1.12630 −0.563150 − 0.826355i \(-0.690411\pi\)
−0.563150 + 0.826355i \(0.690411\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −60656.0 −2.78976
\(780\) 0 0
\(781\) 13048.0 0.597816
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 31312.0 1.41824 0.709118 − 0.705089i \(-0.249093\pi\)
0.709118 + 0.705089i \(0.249093\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 14180.0 0.637399
\(792\) 0 0
\(793\) 25748.0 1.15301
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 10746.0 0.477595 0.238797 − 0.971069i \(-0.423247\pi\)
0.238797 + 0.971069i \(0.423247\pi\)
\(798\) 0 0
\(799\) −2592.00 −0.114766
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 7672.00 0.337159
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −30546.0 −1.32749 −0.663745 − 0.747959i \(-0.731034\pi\)
−0.663745 + 0.747959i \(0.731034\pi\)
\(810\) 0 0
\(811\) −2628.00 −0.113787 −0.0568937 − 0.998380i \(-0.518120\pi\)
−0.0568937 + 0.998380i \(0.518120\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 53856.0 2.30622
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −26280.0 −1.11715 −0.558574 − 0.829455i \(-0.688651\pi\)
−0.558574 + 0.829455i \(0.688651\pi\)
\(822\) 0 0
\(823\) 26146.0 1.10740 0.553701 − 0.832715i \(-0.313215\pi\)
0.553701 + 0.832715i \(0.313215\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 29268.0 1.23065 0.615325 − 0.788273i \(-0.289025\pi\)
0.615325 + 0.788273i \(0.289025\pi\)
\(828\) 0 0
\(829\) 6202.00 0.259836 0.129918 − 0.991525i \(-0.458529\pi\)
0.129918 + 0.991525i \(0.458529\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −4374.00 −0.181933
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −4680.00 −0.192576 −0.0962882 − 0.995353i \(-0.530697\pi\)
−0.0962882 + 0.995353i \(0.530697\pi\)
\(840\) 0 0
\(841\) −11845.0 −0.485670
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 11350.0 0.460438
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 3640.00 0.146625
\(852\) 0 0
\(853\) −12122.0 −0.486576 −0.243288 − 0.969954i \(-0.578226\pi\)
−0.243288 + 0.969954i \(0.578226\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −26006.0 −1.03658 −0.518289 − 0.855205i \(-0.673431\pi\)
−0.518289 + 0.855205i \(0.673431\pi\)
\(858\) 0 0
\(859\) −17684.0 −0.702410 −0.351205 − 0.936299i \(-0.614228\pi\)
−0.351205 + 0.936299i \(0.614228\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 15084.0 0.594977 0.297489 − 0.954725i \(-0.403851\pi\)
0.297489 + 0.954725i \(0.403851\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −7168.00 −0.279813
\(870\) 0 0
\(871\) 12464.0 0.484875
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −17614.0 −0.678201 −0.339101 − 0.940750i \(-0.610123\pi\)
−0.339101 + 0.940750i \(0.610123\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 3298.00 0.126121 0.0630604 − 0.998010i \(-0.479914\pi\)
0.0630604 + 0.998010i \(0.479914\pi\)
\(882\) 0 0
\(883\) −9496.00 −0.361909 −0.180955 − 0.983491i \(-0.557919\pi\)
−0.180955 + 0.983491i \(0.557919\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −50220.0 −1.90104 −0.950520 − 0.310663i \(-0.899449\pi\)
−0.950520 + 0.310663i \(0.899449\pi\)
\(888\) 0 0
\(889\) 16740.0 0.631543
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 19584.0 0.733879
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −8064.00 −0.299165
\(900\) 0 0
\(901\) 2844.00 0.105158
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −20636.0 −0.755465 −0.377733 − 0.925915i \(-0.623296\pi\)
−0.377733 + 0.925915i \(0.623296\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −13680.0 −0.497518 −0.248759 − 0.968565i \(-0.580023\pi\)
−0.248759 + 0.968565i \(0.580023\pi\)
\(912\) 0 0
\(913\) 3976.00 0.144125
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 11340.0 0.408375
\(918\) 0 0
\(919\) 21456.0 0.770150 0.385075 − 0.922885i \(-0.374176\pi\)
0.385075 + 0.922885i \(0.374176\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 76424.0 2.72538
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −16510.0 −0.583074 −0.291537 − 0.956560i \(-0.594167\pi\)
−0.291537 + 0.956560i \(0.594167\pi\)
\(930\) 0 0
\(931\) 33048.0 1.16338
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 36296.0 1.26546 0.632731 − 0.774371i \(-0.281934\pi\)
0.632731 + 0.774371i \(0.281934\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −13540.0 −0.469066 −0.234533 − 0.972108i \(-0.575356\pi\)
−0.234533 + 0.972108i \(0.575356\pi\)
\(942\) 0 0
\(943\) −62440.0 −2.15623
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 32940.0 1.13031 0.565156 − 0.824984i \(-0.308816\pi\)
0.565156 + 0.824984i \(0.308816\pi\)
\(948\) 0 0
\(949\) 44936.0 1.53708
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −22482.0 −0.764180 −0.382090 − 0.924125i \(-0.624796\pi\)
−0.382090 + 0.924125i \(0.624796\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 28660.0 0.965047
\(960\) 0 0
\(961\) −24607.0 −0.825988
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 9566.00 0.318120 0.159060 − 0.987269i \(-0.449154\pi\)
0.159060 + 0.987269i \(0.449154\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −10062.0 −0.332549 −0.166274 − 0.986080i \(-0.553174\pi\)
−0.166274 + 0.986080i \(0.553174\pi\)
\(972\) 0 0
\(973\) 7640.00 0.251724
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 48506.0 1.58838 0.794189 − 0.607671i \(-0.207896\pi\)
0.794189 + 0.607671i \(0.207896\pi\)
\(978\) 0 0
\(979\) 11340.0 0.370202
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −41144.0 −1.33498 −0.667492 − 0.744617i \(-0.732632\pi\)
−0.667492 + 0.744617i \(0.732632\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 55440.0 1.78250
\(990\) 0 0
\(991\) 16120.0 0.516719 0.258360 − 0.966049i \(-0.416818\pi\)
0.258360 + 0.966049i \(0.416818\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 36666.0 1.16472 0.582359 − 0.812932i \(-0.302130\pi\)
0.582359 + 0.812932i \(0.302130\pi\)
\(998\) 0 0
\(999\) 0 0
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 1800.4.a.i.1.1 1
3.2 odd 2 600.4.a.c.1.1 1
5.2 odd 4 360.4.f.b.289.2 2
5.3 odd 4 360.4.f.b.289.1 2
5.4 even 2 1800.4.a.x.1.1 1
12.11 even 2 1200.4.a.bg.1.1 1
15.2 even 4 120.4.f.b.49.2 yes 2
15.8 even 4 120.4.f.b.49.1 ✓ 2
15.14 odd 2 600.4.a.p.1.1 1
20.3 even 4 720.4.f.e.289.1 2
20.7 even 4 720.4.f.e.289.2 2
60.23 odd 4 240.4.f.c.49.2 2
60.47 odd 4 240.4.f.c.49.1 2
60.59 even 2 1200.4.a.e.1.1 1
120.53 even 4 960.4.f.f.769.2 2
120.77 even 4 960.4.f.f.769.1 2
120.83 odd 4 960.4.f.e.769.1 2
120.107 odd 4 960.4.f.e.769.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.f.b.49.1 ✓ 2 15.8 even 4
120.4.f.b.49.2 yes 2 15.2 even 4
240.4.f.c.49.1 2 60.47 odd 4
240.4.f.c.49.2 2 60.23 odd 4
360.4.f.b.289.1 2 5.3 odd 4
360.4.f.b.289.2 2 5.2 odd 4
600.4.a.c.1.1 1 3.2 odd 2
600.4.a.p.1.1 1 15.14 odd 2
720.4.f.e.289.1 2 20.3 even 4
720.4.f.e.289.2 2 20.7 even 4
960.4.f.e.769.1 2 120.83 odd 4
960.4.f.e.769.2 2 120.107 odd 4
960.4.f.f.769.1 2 120.77 even 4
960.4.f.f.769.2 2 120.53 even 4
1200.4.a.e.1.1 1 60.59 even 2
1200.4.a.bg.1.1 1 12.11 even 2
1800.4.a.i.1.1 1 1.1 even 1 trivial
1800.4.a.x.1.1 1 5.4 even 2