Properties

Label 1089.4.a.s.1.2
Level $1089$
Weight $4$
Character 1089.1
Self dual yes
Analytic conductor $64.253$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1089,4,Mod(1,1089)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1089, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1089.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1089.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: \(64.2530799963\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 363)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1089.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.19615 q^{2} +19.0000 q^{4} -9.00000 q^{5} -24.2487 q^{7} +57.1577 q^{8} +O(q^{10})\) \(q+5.19615 q^{2} +19.0000 q^{4} -9.00000 q^{5} -24.2487 q^{7} +57.1577 q^{8} -46.7654 q^{10} +71.0141 q^{13} -126.000 q^{14} +145.000 q^{16} -88.3346 q^{17} -145.492 q^{19} -171.000 q^{20} -90.0000 q^{23} -44.0000 q^{25} +369.000 q^{26} -460.726 q^{28} +88.3346 q^{29} -188.000 q^{31} +296.181 q^{32} -459.000 q^{34} +218.238 q^{35} +133.000 q^{37} -756.000 q^{38} -514.419 q^{40} -36.3731 q^{41} -72.7461 q^{43} -467.654 q^{46} -72.0000 q^{47} +245.000 q^{49} -228.631 q^{50} +1349.27 q^{52} +45.0000 q^{53} -1386.00 q^{56} +459.000 q^{58} -378.000 q^{59} +623.538 q^{61} -976.877 q^{62} +379.000 q^{64} -639.127 q^{65} -386.000 q^{67} -1678.36 q^{68} +1134.00 q^{70} +198.000 q^{71} +76.2102 q^{73} +691.088 q^{74} -2764.35 q^{76} -152.420 q^{79} -1305.00 q^{80} -189.000 q^{82} -1247.08 q^{83} +795.011 q^{85} -378.000 q^{86} -45.0000 q^{89} -1722.00 q^{91} -1710.00 q^{92} -374.123 q^{94} +1309.43 q^{95} +89.0000 q^{97} +1273.06 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 38 q^{4} - 18 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 38 q^{4} - 18 q^{5} - 252 q^{14} + 290 q^{16} - 342 q^{20} - 180 q^{23} - 88 q^{25} + 738 q^{26} - 376 q^{31} - 918 q^{34} + 266 q^{37} - 1512 q^{38} - 144 q^{47} + 490 q^{49} + 90 q^{53} - 2772 q^{56} + 918 q^{58} - 756 q^{59} + 758 q^{64} - 772 q^{67} + 2268 q^{70} + 396 q^{71} - 2610 q^{80} - 378 q^{82} - 756 q^{86} - 90 q^{89} - 3444 q^{91} - 3420 q^{92} + 178 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.19615 1.83712 0.918559 0.395285i \(-0.129354\pi\)
0.918559 + 0.395285i \(0.129354\pi\)
\(3\) 0 0
\(4\) 19.0000 2.37500
\(5\) −9.00000 −0.804984 −0.402492 0.915423i \(-0.631856\pi\)
−0.402492 + 0.915423i \(0.631856\pi\)
\(6\) 0 0
\(7\) −24.2487 −1.30931 −0.654654 0.755929i \(-0.727186\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 57.1577 2.52604
\(9\) 0 0
\(10\) −46.7654 −1.47885
\(11\) 0 0
\(12\) 0 0
\(13\) 71.0141 1.51506 0.757529 0.652801i \(-0.226406\pi\)
0.757529 + 0.652801i \(0.226406\pi\)
\(14\) −126.000 −2.40535
\(15\) 0 0
\(16\) 145.000 2.26562
\(17\) −88.3346 −1.26025 −0.630126 0.776493i \(-0.716997\pi\)
−0.630126 + 0.776493i \(0.716997\pi\)
\(18\) 0 0
\(19\) −145.492 −1.75675 −0.878374 0.477974i \(-0.841371\pi\)
−0.878374 + 0.477974i \(0.841371\pi\)
\(20\) −171.000 −1.91184
\(21\) 0 0
\(22\) 0 0
\(23\) −90.0000 −0.815926 −0.407963 0.912998i \(-0.633761\pi\)
−0.407963 + 0.912998i \(0.633761\pi\)
\(24\) 0 0
\(25\) −44.0000 −0.352000
\(26\) 369.000 2.78334
\(27\) 0 0
\(28\) −460.726 −3.10960
\(29\) 88.3346 0.565632 0.282816 0.959174i \(-0.408731\pi\)
0.282816 + 0.959174i \(0.408731\pi\)
\(30\) 0 0
\(31\) −188.000 −1.08922 −0.544610 0.838690i \(-0.683322\pi\)
−0.544610 + 0.838690i \(0.683322\pi\)
\(32\) 296.181 1.63618
\(33\) 0 0
\(34\) −459.000 −2.31523
\(35\) 218.238 1.05397
\(36\) 0 0
\(37\) 133.000 0.590948 0.295474 0.955351i \(-0.404522\pi\)
0.295474 + 0.955351i \(0.404522\pi\)
\(38\) −756.000 −3.22735
\(39\) 0 0
\(40\) −514.419 −2.03342
\(41\) −36.3731 −0.138549 −0.0692746 0.997598i \(-0.522068\pi\)
−0.0692746 + 0.997598i \(0.522068\pi\)
\(42\) 0 0
\(43\) −72.7461 −0.257993 −0.128996 0.991645i \(-0.541176\pi\)
−0.128996 + 0.991645i \(0.541176\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −467.654 −1.49895
\(47\) −72.0000 −0.223453 −0.111726 0.993739i \(-0.535638\pi\)
−0.111726 + 0.993739i \(0.535638\pi\)
\(48\) 0 0
\(49\) 245.000 0.714286
\(50\) −228.631 −0.646665
\(51\) 0 0
\(52\) 1349.27 3.59826
\(53\) 45.0000 0.116627 0.0583134 0.998298i \(-0.481428\pi\)
0.0583134 + 0.998298i \(0.481428\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1386.00 −3.30736
\(57\) 0 0
\(58\) 459.000 1.03913
\(59\) −378.000 −0.834092 −0.417046 0.908885i \(-0.636935\pi\)
−0.417046 + 0.908885i \(0.636935\pi\)
\(60\) 0 0
\(61\) 623.538 1.30879 0.654393 0.756155i \(-0.272924\pi\)
0.654393 + 0.756155i \(0.272924\pi\)
\(62\) −976.877 −2.00102
\(63\) 0 0
\(64\) 379.000 0.740234
\(65\) −639.127 −1.21960
\(66\) 0 0
\(67\) −386.000 −0.703842 −0.351921 0.936030i \(-0.614471\pi\)
−0.351921 + 0.936030i \(0.614471\pi\)
\(68\) −1678.36 −2.99310
\(69\) 0 0
\(70\) 1134.00 1.93627
\(71\) 198.000 0.330962 0.165481 0.986213i \(-0.447082\pi\)
0.165481 + 0.986213i \(0.447082\pi\)
\(72\) 0 0
\(73\) 76.2102 0.122188 0.0610941 0.998132i \(-0.480541\pi\)
0.0610941 + 0.998132i \(0.480541\pi\)
\(74\) 691.088 1.08564
\(75\) 0 0
\(76\) −2764.35 −4.17228
\(77\) 0 0
\(78\) 0 0
\(79\) −152.420 −0.217071 −0.108536 0.994093i \(-0.534616\pi\)
−0.108536 + 0.994093i \(0.534616\pi\)
\(80\) −1305.00 −1.82379
\(81\) 0 0
\(82\) −189.000 −0.254531
\(83\) −1247.08 −1.64921 −0.824605 0.565709i \(-0.808603\pi\)
−0.824605 + 0.565709i \(0.808603\pi\)
\(84\) 0 0
\(85\) 795.011 1.01448
\(86\) −378.000 −0.473963
\(87\) 0 0
\(88\) 0 0
\(89\) −45.0000 −0.0535954 −0.0267977 0.999641i \(-0.508531\pi\)
−0.0267977 + 0.999641i \(0.508531\pi\)
\(90\) 0 0
\(91\) −1722.00 −1.98368
\(92\) −1710.00 −1.93782
\(93\) 0 0
\(94\) −374.123 −0.410509
\(95\) 1309.43 1.41416
\(96\) 0 0
\(97\) 89.0000 0.0931606 0.0465803 0.998915i \(-0.485168\pi\)
0.0465803 + 0.998915i \(0.485168\pi\)
\(98\) 1273.06 1.31223
\(99\) 0 0
\(100\) −836.000 −0.836000
\(101\) −1288.65 −1.26955 −0.634777 0.772695i \(-0.718908\pi\)
−0.634777 + 0.772695i \(0.718908\pi\)
\(102\) 0 0
\(103\) 1358.00 1.29910 0.649552 0.760317i \(-0.274956\pi\)
0.649552 + 0.760317i \(0.274956\pi\)
\(104\) 4059.00 3.82709
\(105\) 0 0
\(106\) 233.827 0.214257
\(107\) −1299.04 −1.17367 −0.586835 0.809706i \(-0.699626\pi\)
−0.586835 + 0.809706i \(0.699626\pi\)
\(108\) 0 0
\(109\) 1203.78 1.05781 0.528903 0.848683i \(-0.322604\pi\)
0.528903 + 0.848683i \(0.322604\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3516.06 −2.96640
\(113\) −711.000 −0.591905 −0.295952 0.955203i \(-0.595637\pi\)
−0.295952 + 0.955203i \(0.595637\pi\)
\(114\) 0 0
\(115\) 810.000 0.656808
\(116\) 1678.36 1.34338
\(117\) 0 0
\(118\) −1964.15 −1.53232
\(119\) 2142.00 1.65006
\(120\) 0 0
\(121\) 0 0
\(122\) 3240.00 2.40439
\(123\) 0 0
\(124\) −3572.00 −2.58690
\(125\) 1521.00 1.08834
\(126\) 0 0
\(127\) 651.251 0.455033 0.227516 0.973774i \(-0.426939\pi\)
0.227516 + 0.973774i \(0.426939\pi\)
\(128\) −400.104 −0.276285
\(129\) 0 0
\(130\) −3321.00 −2.24055
\(131\) −2608.47 −1.73972 −0.869859 0.493301i \(-0.835790\pi\)
−0.869859 + 0.493301i \(0.835790\pi\)
\(132\) 0 0
\(133\) 3528.00 2.30012
\(134\) −2005.71 −1.29304
\(135\) 0 0
\(136\) −5049.00 −3.18344
\(137\) −2394.00 −1.49294 −0.746472 0.665417i \(-0.768254\pi\)
−0.746472 + 0.665417i \(0.768254\pi\)
\(138\) 0 0
\(139\) 183.597 0.112033 0.0560163 0.998430i \(-0.482160\pi\)
0.0560163 + 0.998430i \(0.482160\pi\)
\(140\) 4146.53 2.50318
\(141\) 0 0
\(142\) 1028.84 0.608015
\(143\) 0 0
\(144\) 0 0
\(145\) −795.011 −0.455325
\(146\) 396.000 0.224474
\(147\) 0 0
\(148\) 2527.00 1.40350
\(149\) 3611.33 1.98558 0.992790 0.119869i \(-0.0382473\pi\)
0.992790 + 0.119869i \(0.0382473\pi\)
\(150\) 0 0
\(151\) 1357.93 0.731832 0.365916 0.930648i \(-0.380756\pi\)
0.365916 + 0.930648i \(0.380756\pi\)
\(152\) −8316.00 −4.43761
\(153\) 0 0
\(154\) 0 0
\(155\) 1692.00 0.876805
\(156\) 0 0
\(157\) 1306.00 0.663886 0.331943 0.943299i \(-0.392296\pi\)
0.331943 + 0.943299i \(0.392296\pi\)
\(158\) −792.000 −0.398786
\(159\) 0 0
\(160\) −2665.63 −1.31710
\(161\) 2182.38 1.06830
\(162\) 0 0
\(163\) 2546.00 1.22342 0.611712 0.791081i \(-0.290481\pi\)
0.611712 + 0.791081i \(0.290481\pi\)
\(164\) −691.088 −0.329054
\(165\) 0 0
\(166\) −6480.00 −3.02979
\(167\) 675.500 0.313004 0.156502 0.987678i \(-0.449978\pi\)
0.156502 + 0.987678i \(0.449978\pi\)
\(168\) 0 0
\(169\) 2846.00 1.29540
\(170\) 4131.00 1.86372
\(171\) 0 0
\(172\) −1382.18 −0.612732
\(173\) 3180.05 1.39754 0.698770 0.715347i \(-0.253731\pi\)
0.698770 + 0.715347i \(0.253731\pi\)
\(174\) 0 0
\(175\) 1066.94 0.460876
\(176\) 0 0
\(177\) 0 0
\(178\) −233.827 −0.0984610
\(179\) 2556.00 1.06729 0.533644 0.845709i \(-0.320822\pi\)
0.533644 + 0.845709i \(0.320822\pi\)
\(180\) 0 0
\(181\) −775.000 −0.318261 −0.159131 0.987258i \(-0.550869\pi\)
−0.159131 + 0.987258i \(0.550869\pi\)
\(182\) −8947.77 −3.64425
\(183\) 0 0
\(184\) −5144.19 −2.06106
\(185\) −1197.00 −0.475704
\(186\) 0 0
\(187\) 0 0
\(188\) −1368.00 −0.530700
\(189\) 0 0
\(190\) 6804.00 2.59797
\(191\) −2628.00 −0.995578 −0.497789 0.867298i \(-0.665855\pi\)
−0.497789 + 0.867298i \(0.665855\pi\)
\(192\) 0 0
\(193\) 2790.33 1.04069 0.520344 0.853957i \(-0.325804\pi\)
0.520344 + 0.853957i \(0.325804\pi\)
\(194\) 462.458 0.171147
\(195\) 0 0
\(196\) 4655.00 1.69643
\(197\) 3590.54 1.29856 0.649278 0.760551i \(-0.275071\pi\)
0.649278 + 0.760551i \(0.275071\pi\)
\(198\) 0 0
\(199\) 1618.00 0.576367 0.288183 0.957575i \(-0.406949\pi\)
0.288183 + 0.957575i \(0.406949\pi\)
\(200\) −2514.94 −0.889165
\(201\) 0 0
\(202\) −6696.00 −2.33232
\(203\) −2142.00 −0.740586
\(204\) 0 0
\(205\) 327.358 0.111530
\(206\) 7056.37 2.38661
\(207\) 0 0
\(208\) 10297.0 3.43255
\(209\) 0 0
\(210\) 0 0
\(211\) 1860.22 0.606934 0.303467 0.952842i \(-0.401856\pi\)
0.303467 + 0.952842i \(0.401856\pi\)
\(212\) 855.000 0.276989
\(213\) 0 0
\(214\) −6750.00 −2.15617
\(215\) 654.715 0.207680
\(216\) 0 0
\(217\) 4558.76 1.42612
\(218\) 6255.00 1.94331
\(219\) 0 0
\(220\) 0 0
\(221\) −6273.00 −1.90936
\(222\) 0 0
\(223\) −964.000 −0.289481 −0.144740 0.989470i \(-0.546235\pi\)
−0.144740 + 0.989470i \(0.546235\pi\)
\(224\) −7182.00 −2.14227
\(225\) 0 0
\(226\) −3694.46 −1.08740
\(227\) 2857.88 0.835614 0.417807 0.908536i \(-0.362799\pi\)
0.417807 + 0.908536i \(0.362799\pi\)
\(228\) 0 0
\(229\) −5519.00 −1.59260 −0.796301 0.604901i \(-0.793213\pi\)
−0.796301 + 0.604901i \(0.793213\pi\)
\(230\) 4208.88 1.20663
\(231\) 0 0
\(232\) 5049.00 1.42881
\(233\) 3611.33 1.01539 0.507695 0.861537i \(-0.330498\pi\)
0.507695 + 0.861537i \(0.330498\pi\)
\(234\) 0 0
\(235\) 648.000 0.179876
\(236\) −7182.00 −1.98097
\(237\) 0 0
\(238\) 11130.2 3.03135
\(239\) −3263.18 −0.883171 −0.441585 0.897219i \(-0.645584\pi\)
−0.441585 + 0.897219i \(0.645584\pi\)
\(240\) 0 0
\(241\) 3457.17 0.924050 0.462025 0.886867i \(-0.347123\pi\)
0.462025 + 0.886867i \(0.347123\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 11847.2 3.10836
\(245\) −2205.00 −0.574989
\(246\) 0 0
\(247\) −10332.0 −2.66158
\(248\) −10745.6 −2.75141
\(249\) 0 0
\(250\) 7903.35 1.99941
\(251\) −6714.00 −1.68838 −0.844191 0.536042i \(-0.819919\pi\)
−0.844191 + 0.536042i \(0.819919\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 3384.00 0.835949
\(255\) 0 0
\(256\) −5111.00 −1.24780
\(257\) −1341.00 −0.325484 −0.162742 0.986669i \(-0.552034\pi\)
−0.162742 + 0.986669i \(0.552034\pi\)
\(258\) 0 0
\(259\) −3225.08 −0.773732
\(260\) −12143.4 −2.89655
\(261\) 0 0
\(262\) −13554.0 −3.19606
\(263\) −956.092 −0.224164 −0.112082 0.993699i \(-0.535752\pi\)
−0.112082 + 0.993699i \(0.535752\pi\)
\(264\) 0 0
\(265\) −405.000 −0.0938828
\(266\) 18332.0 4.22560
\(267\) 0 0
\(268\) −7334.00 −1.67162
\(269\) −3087.00 −0.699694 −0.349847 0.936807i \(-0.613766\pi\)
−0.349847 + 0.936807i \(0.613766\pi\)
\(270\) 0 0
\(271\) 4035.68 0.904613 0.452306 0.891863i \(-0.350601\pi\)
0.452306 + 0.891863i \(0.350601\pi\)
\(272\) −12808.5 −2.85526
\(273\) 0 0
\(274\) −12439.6 −2.74271
\(275\) 0 0
\(276\) 0 0
\(277\) −3791.46 −0.822407 −0.411203 0.911544i \(-0.634891\pi\)
−0.411203 + 0.911544i \(0.634891\pi\)
\(278\) 954.000 0.205817
\(279\) 0 0
\(280\) 12474.0 2.66237
\(281\) 2722.78 0.578034 0.289017 0.957324i \(-0.406672\pi\)
0.289017 + 0.957324i \(0.406672\pi\)
\(282\) 0 0
\(283\) −3238.94 −0.680335 −0.340167 0.940365i \(-0.610484\pi\)
−0.340167 + 0.940365i \(0.610484\pi\)
\(284\) 3762.00 0.786034
\(285\) 0 0
\(286\) 0 0
\(287\) 882.000 0.181404
\(288\) 0 0
\(289\) 2890.00 0.588235
\(290\) −4131.00 −0.836485
\(291\) 0 0
\(292\) 1447.99 0.290197
\(293\) −4338.79 −0.865101 −0.432551 0.901610i \(-0.642386\pi\)
−0.432551 + 0.901610i \(0.642386\pi\)
\(294\) 0 0
\(295\) 3402.00 0.671431
\(296\) 7601.97 1.49276
\(297\) 0 0
\(298\) 18765.0 3.64774
\(299\) −6391.27 −1.23618
\(300\) 0 0
\(301\) 1764.00 0.337792
\(302\) 7056.00 1.34446
\(303\) 0 0
\(304\) −21096.4 −3.98013
\(305\) −5611.84 −1.05355
\(306\) 0 0
\(307\) −5781.59 −1.07483 −0.537415 0.843318i \(-0.680599\pi\)
−0.537415 + 0.843318i \(0.680599\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 8791.89 1.61079
\(311\) −5220.00 −0.951765 −0.475883 0.879509i \(-0.657871\pi\)
−0.475883 + 0.879509i \(0.657871\pi\)
\(312\) 0 0
\(313\) −3977.00 −0.718190 −0.359095 0.933301i \(-0.616915\pi\)
−0.359095 + 0.933301i \(0.616915\pi\)
\(314\) 6786.18 1.21964
\(315\) 0 0
\(316\) −2895.99 −0.515545
\(317\) −9918.00 −1.75726 −0.878628 0.477506i \(-0.841541\pi\)
−0.878628 + 0.477506i \(0.841541\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −3411.00 −0.595877
\(321\) 0 0
\(322\) 11340.0 1.96259
\(323\) 12852.0 2.21395
\(324\) 0 0
\(325\) −3124.62 −0.533301
\(326\) 13229.4 2.24757
\(327\) 0 0
\(328\) −2079.00 −0.349980
\(329\) 1745.91 0.292568
\(330\) 0 0
\(331\) 8756.00 1.45400 0.726999 0.686639i \(-0.240915\pi\)
0.726999 + 0.686639i \(0.240915\pi\)
\(332\) −23694.5 −3.91687
\(333\) 0 0
\(334\) 3510.00 0.575026
\(335\) 3474.00 0.566582
\(336\) 0 0
\(337\) 9919.45 1.60340 0.801702 0.597724i \(-0.203928\pi\)
0.801702 + 0.597724i \(0.203928\pi\)
\(338\) 14788.2 2.37981
\(339\) 0 0
\(340\) 15105.2 2.40940
\(341\) 0 0
\(342\) 0 0
\(343\) 2376.37 0.374088
\(344\) −4158.00 −0.651699
\(345\) 0 0
\(346\) 16524.0 2.56744
\(347\) 1330.22 0.205792 0.102896 0.994692i \(-0.467189\pi\)
0.102896 + 0.994692i \(0.467189\pi\)
\(348\) 0 0
\(349\) 2842.30 0.435944 0.217972 0.975955i \(-0.430056\pi\)
0.217972 + 0.975955i \(0.430056\pi\)
\(350\) 5544.00 0.846684
\(351\) 0 0
\(352\) 0 0
\(353\) 1431.00 0.215763 0.107882 0.994164i \(-0.465593\pi\)
0.107882 + 0.994164i \(0.465593\pi\)
\(354\) 0 0
\(355\) −1782.00 −0.266419
\(356\) −855.000 −0.127289
\(357\) 0 0
\(358\) 13281.4 1.96073
\(359\) 5393.61 0.792935 0.396467 0.918049i \(-0.370236\pi\)
0.396467 + 0.918049i \(0.370236\pi\)
\(360\) 0 0
\(361\) 14309.0 2.08616
\(362\) −4027.02 −0.584683
\(363\) 0 0
\(364\) −32718.0 −4.71123
\(365\) −685.892 −0.0983595
\(366\) 0 0
\(367\) 3554.00 0.505497 0.252748 0.967532i \(-0.418666\pi\)
0.252748 + 0.967532i \(0.418666\pi\)
\(368\) −13050.0 −1.84858
\(369\) 0 0
\(370\) −6219.79 −0.873924
\(371\) −1091.19 −0.152700
\(372\) 0 0
\(373\) −5577.20 −0.774200 −0.387100 0.922038i \(-0.626523\pi\)
−0.387100 + 0.922038i \(0.626523\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −4115.35 −0.564450
\(377\) 6273.00 0.856965
\(378\) 0 0
\(379\) −13186.0 −1.78712 −0.893561 0.448942i \(-0.851801\pi\)
−0.893561 + 0.448942i \(0.851801\pi\)
\(380\) 24879.2 3.35862
\(381\) 0 0
\(382\) −13655.5 −1.82899
\(383\) 3330.00 0.444269 0.222135 0.975016i \(-0.428698\pi\)
0.222135 + 0.975016i \(0.428698\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 14499.0 1.91186
\(387\) 0 0
\(388\) 1691.00 0.221256
\(389\) −3537.00 −0.461010 −0.230505 0.973071i \(-0.574038\pi\)
−0.230505 + 0.973071i \(0.574038\pi\)
\(390\) 0 0
\(391\) 7950.11 1.02827
\(392\) 14003.6 1.80431
\(393\) 0 0
\(394\) 18657.0 2.38560
\(395\) 1371.78 0.174739
\(396\) 0 0
\(397\) −4501.00 −0.569014 −0.284507 0.958674i \(-0.591830\pi\)
−0.284507 + 0.958674i \(0.591830\pi\)
\(398\) 8407.37 1.05885
\(399\) 0 0
\(400\) −6380.00 −0.797500
\(401\) −3303.00 −0.411332 −0.205666 0.978622i \(-0.565936\pi\)
−0.205666 + 0.978622i \(0.565936\pi\)
\(402\) 0 0
\(403\) −13350.6 −1.65023
\(404\) −24484.3 −3.01519
\(405\) 0 0
\(406\) −11130.2 −1.36054
\(407\) 0 0
\(408\) 0 0
\(409\) −9576.51 −1.15777 −0.578885 0.815409i \(-0.696512\pi\)
−0.578885 + 0.815409i \(0.696512\pi\)
\(410\) 1701.00 0.204894
\(411\) 0 0
\(412\) 25802.0 3.08537
\(413\) 9166.01 1.09208
\(414\) 0 0
\(415\) 11223.7 1.32759
\(416\) 21033.0 2.47891
\(417\) 0 0
\(418\) 0 0
\(419\) 6750.00 0.787015 0.393507 0.919322i \(-0.371262\pi\)
0.393507 + 0.919322i \(0.371262\pi\)
\(420\) 0 0
\(421\) −13481.0 −1.56063 −0.780313 0.625389i \(-0.784940\pi\)
−0.780313 + 0.625389i \(0.784940\pi\)
\(422\) 9666.00 1.11501
\(423\) 0 0
\(424\) 2572.10 0.294604
\(425\) 3886.72 0.443609
\(426\) 0 0
\(427\) −15120.0 −1.71360
\(428\) −24681.7 −2.78747
\(429\) 0 0
\(430\) 3402.00 0.381533
\(431\) 1382.18 0.154471 0.0772356 0.997013i \(-0.475391\pi\)
0.0772356 + 0.997013i \(0.475391\pi\)
\(432\) 0 0
\(433\) −1531.00 −0.169920 −0.0849598 0.996384i \(-0.527076\pi\)
−0.0849598 + 0.996384i \(0.527076\pi\)
\(434\) 23688.0 2.61995
\(435\) 0 0
\(436\) 22871.7 2.51229
\(437\) 13094.3 1.43338
\(438\) 0 0
\(439\) 1919.11 0.208643 0.104321 0.994544i \(-0.466733\pi\)
0.104321 + 0.994544i \(0.466733\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −32595.5 −3.50771
\(443\) −14598.0 −1.56563 −0.782813 0.622258i \(-0.786216\pi\)
−0.782813 + 0.622258i \(0.786216\pi\)
\(444\) 0 0
\(445\) 405.000 0.0431435
\(446\) −5009.09 −0.531810
\(447\) 0 0
\(448\) −9190.26 −0.969194
\(449\) 12591.0 1.32340 0.661699 0.749769i \(-0.269836\pi\)
0.661699 + 0.749769i \(0.269836\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −13509.0 −1.40577
\(453\) 0 0
\(454\) 14850.0 1.53512
\(455\) 15498.0 1.59683
\(456\) 0 0
\(457\) −16778.4 −1.71742 −0.858708 0.512465i \(-0.828733\pi\)
−0.858708 + 0.512465i \(0.828733\pi\)
\(458\) −28677.6 −2.92580
\(459\) 0 0
\(460\) 15390.0 1.55992
\(461\) 14138.7 1.42843 0.714215 0.699926i \(-0.246784\pi\)
0.714215 + 0.699926i \(0.246784\pi\)
\(462\) 0 0
\(463\) 6902.00 0.692793 0.346396 0.938088i \(-0.387405\pi\)
0.346396 + 0.938088i \(0.387405\pi\)
\(464\) 12808.5 1.28151
\(465\) 0 0
\(466\) 18765.0 1.86539
\(467\) −15894.0 −1.57492 −0.787459 0.616367i \(-0.788604\pi\)
−0.787459 + 0.616367i \(0.788604\pi\)
\(468\) 0 0
\(469\) 9360.00 0.921545
\(470\) 3367.11 0.330453
\(471\) 0 0
\(472\) −21605.6 −2.10695
\(473\) 0 0
\(474\) 0 0
\(475\) 6401.66 0.618375
\(476\) 40698.0 3.91889
\(477\) 0 0
\(478\) −16956.0 −1.62249
\(479\) −9716.81 −0.926873 −0.463436 0.886130i \(-0.653384\pi\)
−0.463436 + 0.886130i \(0.653384\pi\)
\(480\) 0 0
\(481\) 9444.87 0.895320
\(482\) 17964.0 1.69759
\(483\) 0 0
\(484\) 0 0
\(485\) −801.000 −0.0749929
\(486\) 0 0
\(487\) −7622.00 −0.709211 −0.354606 0.935016i \(-0.615385\pi\)
−0.354606 + 0.935016i \(0.615385\pi\)
\(488\) 35640.0 3.30604
\(489\) 0 0
\(490\) −11457.5 −1.05632
\(491\) 14902.6 1.36974 0.684871 0.728664i \(-0.259859\pi\)
0.684871 + 0.728664i \(0.259859\pi\)
\(492\) 0 0
\(493\) −7803.00 −0.712839
\(494\) −53686.6 −4.88963
\(495\) 0 0
\(496\) −27260.0 −2.46776
\(497\) −4801.24 −0.433331
\(498\) 0 0
\(499\) 13006.0 1.16679 0.583395 0.812188i \(-0.301724\pi\)
0.583395 + 0.812188i \(0.301724\pi\)
\(500\) 28899.0 2.58481
\(501\) 0 0
\(502\) −34887.0 −3.10176
\(503\) −14185.5 −1.25746 −0.628728 0.777626i \(-0.716424\pi\)
−0.628728 + 0.777626i \(0.716424\pi\)
\(504\) 0 0
\(505\) 11597.8 1.02197
\(506\) 0 0
\(507\) 0 0
\(508\) 12373.8 1.08070
\(509\) −18234.0 −1.58783 −0.793917 0.608026i \(-0.791962\pi\)
−0.793917 + 0.608026i \(0.791962\pi\)
\(510\) 0 0
\(511\) −1848.00 −0.159982
\(512\) −23356.7 −2.01607
\(513\) 0 0
\(514\) −6968.04 −0.597952
\(515\) −12222.0 −1.04576
\(516\) 0 0
\(517\) 0 0
\(518\) −16758.0 −1.42144
\(519\) 0 0
\(520\) −36531.0 −3.08075
\(521\) −20502.0 −1.72401 −0.862005 0.506900i \(-0.830791\pi\)
−0.862005 + 0.506900i \(0.830791\pi\)
\(522\) 0 0
\(523\) 10600.2 0.886257 0.443128 0.896458i \(-0.353869\pi\)
0.443128 + 0.896458i \(0.353869\pi\)
\(524\) −49560.9 −4.13183
\(525\) 0 0
\(526\) −4968.00 −0.411816
\(527\) 16606.9 1.37269
\(528\) 0 0
\(529\) −4067.00 −0.334265
\(530\) −2104.44 −0.172474
\(531\) 0 0
\(532\) 67032.0 5.46279
\(533\) −2583.00 −0.209910
\(534\) 0 0
\(535\) 11691.3 0.944787
\(536\) −22062.9 −1.77793
\(537\) 0 0
\(538\) −16040.5 −1.28542
\(539\) 0 0
\(540\) 0 0
\(541\) −6082.96 −0.483414 −0.241707 0.970349i \(-0.577707\pi\)
−0.241707 + 0.970349i \(0.577707\pi\)
\(542\) 20970.0 1.66188
\(543\) 0 0
\(544\) −26163.0 −2.06200
\(545\) −10834.0 −0.851517
\(546\) 0 0
\(547\) −19378.2 −1.51472 −0.757360 0.652998i \(-0.773511\pi\)
−0.757360 + 0.652998i \(0.773511\pi\)
\(548\) −45486.0 −3.54574
\(549\) 0 0
\(550\) 0 0
\(551\) −12852.0 −0.993673
\(552\) 0 0
\(553\) 3696.00 0.284213
\(554\) −19701.0 −1.51086
\(555\) 0 0
\(556\) 3488.35 0.266077
\(557\) −14216.7 −1.08147 −0.540736 0.841192i \(-0.681854\pi\)
−0.540736 + 0.841192i \(0.681854\pi\)
\(558\) 0 0
\(559\) −5166.00 −0.390874
\(560\) 31644.6 2.38791
\(561\) 0 0
\(562\) 14148.0 1.06192
\(563\) −2130.42 −0.159479 −0.0797394 0.996816i \(-0.525409\pi\)
−0.0797394 + 0.996816i \(0.525409\pi\)
\(564\) 0 0
\(565\) 6399.00 0.476474
\(566\) −16830.0 −1.24985
\(567\) 0 0
\(568\) 11317.2 0.836021
\(569\) 17251.2 1.27102 0.635509 0.772094i \(-0.280790\pi\)
0.635509 + 0.772094i \(0.280790\pi\)
\(570\) 0 0
\(571\) −9079.41 −0.665432 −0.332716 0.943027i \(-0.607965\pi\)
−0.332716 + 0.943027i \(0.607965\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 4583.01 0.333260
\(575\) 3960.00 0.287206
\(576\) 0 0
\(577\) −7427.00 −0.535858 −0.267929 0.963439i \(-0.586339\pi\)
−0.267929 + 0.963439i \(0.586339\pi\)
\(578\) 15016.9 1.08066
\(579\) 0 0
\(580\) −15105.2 −1.08140
\(581\) 30240.0 2.15932
\(582\) 0 0
\(583\) 0 0
\(584\) 4356.00 0.308652
\(585\) 0 0
\(586\) −22545.0 −1.58929
\(587\) 9972.00 0.701173 0.350586 0.936530i \(-0.385982\pi\)
0.350586 + 0.936530i \(0.385982\pi\)
\(588\) 0 0
\(589\) 27352.5 1.91348
\(590\) 17677.3 1.23350
\(591\) 0 0
\(592\) 19285.0 1.33887
\(593\) −21922.6 −1.51813 −0.759066 0.651014i \(-0.774344\pi\)
−0.759066 + 0.651014i \(0.774344\pi\)
\(594\) 0 0
\(595\) −19278.0 −1.32827
\(596\) 68615.2 4.71575
\(597\) 0 0
\(598\) −33210.0 −2.27100
\(599\) −576.000 −0.0392900 −0.0196450 0.999807i \(-0.506254\pi\)
−0.0196450 + 0.999807i \(0.506254\pi\)
\(600\) 0 0
\(601\) 7740.54 0.525363 0.262681 0.964883i \(-0.415393\pi\)
0.262681 + 0.964883i \(0.415393\pi\)
\(602\) 9166.01 0.620563
\(603\) 0 0
\(604\) 25800.6 1.73810
\(605\) 0 0
\(606\) 0 0
\(607\) 14500.7 0.969632 0.484816 0.874616i \(-0.338887\pi\)
0.484816 + 0.874616i \(0.338887\pi\)
\(608\) −43092.0 −2.87436
\(609\) 0 0
\(610\) −29160.0 −1.93550
\(611\) −5113.01 −0.338544
\(612\) 0 0
\(613\) −5298.34 −0.349100 −0.174550 0.984648i \(-0.555847\pi\)
−0.174550 + 0.984648i \(0.555847\pi\)
\(614\) −30042.0 −1.97459
\(615\) 0 0
\(616\) 0 0
\(617\) −6939.00 −0.452761 −0.226381 0.974039i \(-0.572689\pi\)
−0.226381 + 0.974039i \(0.572689\pi\)
\(618\) 0 0
\(619\) 1286.00 0.0835036 0.0417518 0.999128i \(-0.486706\pi\)
0.0417518 + 0.999128i \(0.486706\pi\)
\(620\) 32148.0 2.08241
\(621\) 0 0
\(622\) −27123.9 −1.74850
\(623\) 1091.19 0.0701728
\(624\) 0 0
\(625\) −8189.00 −0.524096
\(626\) −20665.1 −1.31940
\(627\) 0 0
\(628\) 24814.0 1.57673
\(629\) −11748.5 −0.744743
\(630\) 0 0
\(631\) −15110.0 −0.953280 −0.476640 0.879099i \(-0.658145\pi\)
−0.476640 + 0.879099i \(0.658145\pi\)
\(632\) −8712.00 −0.548330
\(633\) 0 0
\(634\) −51535.4 −3.22829
\(635\) −5861.26 −0.366294
\(636\) 0 0
\(637\) 17398.5 1.08218
\(638\) 0 0
\(639\) 0 0
\(640\) 3600.93 0.222405
\(641\) −13293.0 −0.819098 −0.409549 0.912288i \(-0.634314\pi\)
−0.409549 + 0.912288i \(0.634314\pi\)
\(642\) 0 0
\(643\) 20528.0 1.25901 0.629506 0.776995i \(-0.283257\pi\)
0.629506 + 0.776995i \(0.283257\pi\)
\(644\) 41465.3 2.53721
\(645\) 0 0
\(646\) 66781.0 4.06728
\(647\) 11124.0 0.675934 0.337967 0.941158i \(-0.390261\pi\)
0.337967 + 0.941158i \(0.390261\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −16236.0 −0.979736
\(651\) 0 0
\(652\) 48374.0 2.90563
\(653\) −5562.00 −0.333320 −0.166660 0.986014i \(-0.553298\pi\)
−0.166660 + 0.986014i \(0.553298\pi\)
\(654\) 0 0
\(655\) 23476.2 1.40045
\(656\) −5274.09 −0.313901
\(657\) 0 0
\(658\) 9072.00 0.537482
\(659\) 16378.3 0.968144 0.484072 0.875028i \(-0.339157\pi\)
0.484072 + 0.875028i \(0.339157\pi\)
\(660\) 0 0
\(661\) −28385.0 −1.67027 −0.835135 0.550045i \(-0.814611\pi\)
−0.835135 + 0.550045i \(0.814611\pi\)
\(662\) 45497.5 2.67116
\(663\) 0 0
\(664\) −71280.0 −4.16596
\(665\) −31752.0 −1.85156
\(666\) 0 0
\(667\) −7950.11 −0.461514
\(668\) 12834.5 0.743386
\(669\) 0 0
\(670\) 18051.4 1.04088
\(671\) 0 0
\(672\) 0 0
\(673\) 6948.99 0.398015 0.199007 0.979998i \(-0.436228\pi\)
0.199007 + 0.979998i \(0.436228\pi\)
\(674\) 51543.0 2.94564
\(675\) 0 0
\(676\) 54074.0 3.07658
\(677\) −3923.10 −0.222713 −0.111357 0.993781i \(-0.535520\pi\)
−0.111357 + 0.993781i \(0.535520\pi\)
\(678\) 0 0
\(679\) −2158.14 −0.121976
\(680\) 45441.0 2.56262
\(681\) 0 0
\(682\) 0 0
\(683\) −2034.00 −0.113951 −0.0569757 0.998376i \(-0.518146\pi\)
−0.0569757 + 0.998376i \(0.518146\pi\)
\(684\) 0 0
\(685\) 21546.0 1.20180
\(686\) 12348.0 0.687243
\(687\) 0 0
\(688\) −10548.2 −0.584514
\(689\) 3195.63 0.176697
\(690\) 0 0
\(691\) 4598.00 0.253135 0.126567 0.991958i \(-0.459604\pi\)
0.126567 + 0.991958i \(0.459604\pi\)
\(692\) 60420.9 3.31916
\(693\) 0 0
\(694\) 6912.00 0.378063
\(695\) −1652.38 −0.0901845
\(696\) 0 0
\(697\) 3213.00 0.174607
\(698\) 14769.0 0.800881
\(699\) 0 0
\(700\) 20271.9 1.09458
\(701\) −5949.59 −0.320561 −0.160280 0.987072i \(-0.551240\pi\)
−0.160280 + 0.987072i \(0.551240\pi\)
\(702\) 0 0
\(703\) −19350.5 −1.03815
\(704\) 0 0
\(705\) 0 0
\(706\) 7435.69 0.396382
\(707\) 31248.0 1.66224
\(708\) 0 0
\(709\) 24814.0 1.31440 0.657200 0.753716i \(-0.271741\pi\)
0.657200 + 0.753716i \(0.271741\pi\)
\(710\) −9259.54 −0.489443
\(711\) 0 0
\(712\) −2572.10 −0.135384
\(713\) 16920.0 0.888722
\(714\) 0 0
\(715\) 0 0
\(716\) 48564.0 2.53481
\(717\) 0 0
\(718\) 28026.0 1.45671
\(719\) −26442.0 −1.37152 −0.685758 0.727829i \(-0.740529\pi\)
−0.685758 + 0.727829i \(0.740529\pi\)
\(720\) 0 0
\(721\) −32929.7 −1.70093
\(722\) 74351.7 3.83253
\(723\) 0 0
\(724\) −14725.0 −0.755871
\(725\) −3886.72 −0.199102
\(726\) 0 0
\(727\) −28370.0 −1.44730 −0.723649 0.690169i \(-0.757536\pi\)
−0.723649 + 0.690169i \(0.757536\pi\)
\(728\) −98425.5 −5.01084
\(729\) 0 0
\(730\) −3564.00 −0.180698
\(731\) 6426.00 0.325136
\(732\) 0 0
\(733\) 20332.5 1.02456 0.512278 0.858820i \(-0.328802\pi\)
0.512278 + 0.858820i \(0.328802\pi\)
\(734\) 18467.1 0.928657
\(735\) 0 0
\(736\) −26656.3 −1.33500
\(737\) 0 0
\(738\) 0 0
\(739\) −9048.23 −0.450399 −0.225199 0.974313i \(-0.572303\pi\)
−0.225199 + 0.974313i \(0.572303\pi\)
\(740\) −22743.0 −1.12980
\(741\) 0 0
\(742\) −5670.00 −0.280529
\(743\) 9758.37 0.481830 0.240915 0.970546i \(-0.422552\pi\)
0.240915 + 0.970546i \(0.422552\pi\)
\(744\) 0 0
\(745\) −32501.9 −1.59836
\(746\) −28980.0 −1.42230
\(747\) 0 0
\(748\) 0 0
\(749\) 31500.0 1.53670
\(750\) 0 0
\(751\) −34010.0 −1.65252 −0.826260 0.563289i \(-0.809536\pi\)
−0.826260 + 0.563289i \(0.809536\pi\)
\(752\) −10440.0 −0.506260
\(753\) 0 0
\(754\) 32595.5 1.57435
\(755\) −12221.4 −0.589113
\(756\) 0 0
\(757\) −5345.00 −0.256628 −0.128314 0.991734i \(-0.540957\pi\)
−0.128314 + 0.991734i \(0.540957\pi\)
\(758\) −68516.5 −3.28315
\(759\) 0 0
\(760\) 74844.0 3.57221
\(761\) −22629.2 −1.07794 −0.538968 0.842326i \(-0.681186\pi\)
−0.538968 + 0.842326i \(0.681186\pi\)
\(762\) 0 0
\(763\) −29190.0 −1.38499
\(764\) −49932.0 −2.36450
\(765\) 0 0
\(766\) 17303.2 0.816174
\(767\) −26843.3 −1.26370
\(768\) 0 0
\(769\) 34961.4 1.63946 0.819728 0.572753i \(-0.194124\pi\)
0.819728 + 0.572753i \(0.194124\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 53016.3 2.47163
\(773\) −29610.0 −1.37775 −0.688873 0.724882i \(-0.741894\pi\)
−0.688873 + 0.724882i \(0.741894\pi\)
\(774\) 0 0
\(775\) 8272.00 0.383405
\(776\) 5087.03 0.235327
\(777\) 0 0
\(778\) −18378.8 −0.846930
\(779\) 5292.00 0.243396
\(780\) 0 0
\(781\) 0 0
\(782\) 41310.0 1.88906
\(783\) 0 0
\(784\) 35525.0 1.61830
\(785\) −11754.0 −0.534418
\(786\) 0 0
\(787\) 5785.05 0.262026 0.131013 0.991381i \(-0.458177\pi\)
0.131013 + 0.991381i \(0.458177\pi\)
\(788\) 68220.3 3.08407
\(789\) 0 0
\(790\) 7128.00 0.321016
\(791\) 17240.8 0.774985
\(792\) 0 0
\(793\) 44280.0 1.98289
\(794\) −23387.9 −1.04535
\(795\) 0 0
\(796\) 30742.0 1.36887
\(797\) 4626.00 0.205598 0.102799 0.994702i \(-0.467220\pi\)
0.102799 + 0.994702i \(0.467220\pi\)
\(798\) 0 0
\(799\) 6360.09 0.281607
\(800\) −13032.0 −0.575936
\(801\) 0 0
\(802\) −17162.9 −0.755664
\(803\) 0 0
\(804\) 0 0
\(805\) −19641.5 −0.859963
\(806\) −69372.0 −3.03167
\(807\) 0 0
\(808\) −73656.0 −3.20694
\(809\) 31239.3 1.35762 0.678810 0.734314i \(-0.262496\pi\)
0.678810 + 0.734314i \(0.262496\pi\)
\(810\) 0 0
\(811\) −44340.5 −1.91986 −0.959929 0.280242i \(-0.909585\pi\)
−0.959929 + 0.280242i \(0.909585\pi\)
\(812\) −40698.0 −1.75889
\(813\) 0 0
\(814\) 0 0
\(815\) −22914.0 −0.984837
\(816\) 0 0
\(817\) 10584.0 0.453228
\(818\) −49761.0 −2.12696
\(819\) 0 0
\(820\) 6219.79 0.264884
\(821\) −38493.1 −1.63632 −0.818160 0.574991i \(-0.805006\pi\)
−0.818160 + 0.574991i \(0.805006\pi\)
\(822\) 0 0
\(823\) 8678.00 0.367553 0.183776 0.982968i \(-0.441168\pi\)
0.183776 + 0.982968i \(0.441168\pi\)
\(824\) 77620.1 3.28158
\(825\) 0 0
\(826\) 47628.0 2.00628
\(827\) −27238.2 −1.14530 −0.572652 0.819799i \(-0.694085\pi\)
−0.572652 + 0.819799i \(0.694085\pi\)
\(828\) 0 0
\(829\) −19789.0 −0.829072 −0.414536 0.910033i \(-0.636056\pi\)
−0.414536 + 0.910033i \(0.636056\pi\)
\(830\) 58320.0 2.43894
\(831\) 0 0
\(832\) 26914.3 1.12150
\(833\) −21642.0 −0.900180
\(834\) 0 0
\(835\) −6079.50 −0.251964
\(836\) 0 0
\(837\) 0 0
\(838\) 35074.0 1.44584
\(839\) 1800.00 0.0740678 0.0370339 0.999314i \(-0.488209\pi\)
0.0370339 + 0.999314i \(0.488209\pi\)
\(840\) 0 0
\(841\) −16586.0 −0.680061
\(842\) −70049.3 −2.86705
\(843\) 0 0
\(844\) 35344.2 1.44147
\(845\) −25614.0 −1.04278
\(846\) 0 0
\(847\) 0 0
\(848\) 6525.00 0.264233
\(849\) 0 0
\(850\) 20196.0 0.814961
\(851\) −11970.0 −0.482170
\(852\) 0 0
\(853\) −33513.5 −1.34523 −0.672614 0.739994i \(-0.734828\pi\)
−0.672614 + 0.739994i \(0.734828\pi\)
\(854\) −78565.8 −3.14809
\(855\) 0 0
\(856\) −74250.0 −2.96473
\(857\) 26687.4 1.06374 0.531870 0.846826i \(-0.321489\pi\)
0.531870 + 0.846826i \(0.321489\pi\)
\(858\) 0 0
\(859\) 46694.0 1.85469 0.927345 0.374207i \(-0.122085\pi\)
0.927345 + 0.374207i \(0.122085\pi\)
\(860\) 12439.6 0.493240
\(861\) 0 0
\(862\) 7182.00 0.283782
\(863\) 36018.0 1.42070 0.710352 0.703847i \(-0.248536\pi\)
0.710352 + 0.703847i \(0.248536\pi\)
\(864\) 0 0
\(865\) −28620.4 −1.12500
\(866\) −7955.31 −0.312162
\(867\) 0 0
\(868\) 86616.4 3.38704
\(869\) 0 0
\(870\) 0 0
\(871\) −27411.4 −1.06636
\(872\) 68805.0 2.67205
\(873\) 0 0
\(874\) 68040.0 2.63328
\(875\) −36882.3 −1.42497
\(876\) 0 0
\(877\) 9191.99 0.353924 0.176962 0.984218i \(-0.443373\pi\)
0.176962 + 0.984218i \(0.443373\pi\)
\(878\) 9972.00 0.383301
\(879\) 0 0
\(880\) 0 0
\(881\) 40005.0 1.52986 0.764928 0.644116i \(-0.222775\pi\)
0.764928 + 0.644116i \(0.222775\pi\)
\(882\) 0 0
\(883\) 4492.00 0.171198 0.0855990 0.996330i \(-0.472720\pi\)
0.0855990 + 0.996330i \(0.472720\pi\)
\(884\) −119187. −4.53472
\(885\) 0 0
\(886\) −75853.4 −2.87624
\(887\) 43554.1 1.64871 0.824355 0.566074i \(-0.191538\pi\)
0.824355 + 0.566074i \(0.191538\pi\)
\(888\) 0 0
\(889\) −15792.0 −0.595778
\(890\) 2104.44 0.0792596
\(891\) 0 0
\(892\) −18316.0 −0.687517
\(893\) 10475.4 0.392550
\(894\) 0 0
\(895\) −23004.0 −0.859150
\(896\) 9702.00 0.361742
\(897\) 0 0
\(898\) 65424.8 2.43124
\(899\) −16606.9 −0.616097
\(900\) 0 0
\(901\) −3975.06 −0.146979
\(902\) 0 0
\(903\) 0 0
\(904\) −40639.1 −1.49517
\(905\) 6975.00 0.256195
\(906\) 0 0
\(907\) 7634.00 0.279474 0.139737 0.990189i \(-0.455374\pi\)
0.139737 + 0.990189i \(0.455374\pi\)
\(908\) 54299.8 1.98458
\(909\) 0 0
\(910\) 80530.0 2.93356
\(911\) 43830.0 1.59402 0.797010 0.603966i \(-0.206414\pi\)
0.797010 + 0.603966i \(0.206414\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −87183.0 −3.15510
\(915\) 0 0
\(916\) −104861. −3.78243
\(917\) 63252.0 2.27782
\(918\) 0 0
\(919\) 32531.4 1.16769 0.583847 0.811864i \(-0.301547\pi\)
0.583847 + 0.811864i \(0.301547\pi\)
\(920\) 46297.7 1.65912
\(921\) 0 0
\(922\) 73467.0 2.62419
\(923\) 14060.8 0.501426
\(924\) 0 0
\(925\) −5852.00 −0.208014
\(926\) 35863.8 1.27274
\(927\) 0 0
\(928\) 26163.0 0.925477
\(929\) −15651.0 −0.552737 −0.276368 0.961052i \(-0.589131\pi\)
−0.276368 + 0.961052i \(0.589131\pi\)
\(930\) 0 0
\(931\) −35645.6 −1.25482
\(932\) 68615.2 2.41155
\(933\) 0 0
\(934\) −82587.6 −2.89331
\(935\) 0 0
\(936\) 0 0
\(937\) 6337.57 0.220960 0.110480 0.993878i \(-0.464761\pi\)
0.110480 + 0.993878i \(0.464761\pi\)
\(938\) 48636.0 1.69299
\(939\) 0 0
\(940\) 12312.0 0.427205
\(941\) 9846.71 0.341120 0.170560 0.985347i \(-0.445442\pi\)
0.170560 + 0.985347i \(0.445442\pi\)
\(942\) 0 0
\(943\) 3273.58 0.113046
\(944\) −54810.0 −1.88974
\(945\) 0 0
\(946\) 0 0
\(947\) 738.000 0.0253239 0.0126620 0.999920i \(-0.495969\pi\)
0.0126620 + 0.999920i \(0.495969\pi\)
\(948\) 0 0
\(949\) 5412.00 0.185122
\(950\) 33264.0 1.13603
\(951\) 0 0
\(952\) 122432. 4.16810
\(953\) −15541.7 −0.528274 −0.264137 0.964485i \(-0.585087\pi\)
−0.264137 + 0.964485i \(0.585087\pi\)
\(954\) 0 0
\(955\) 23652.0 0.801425
\(956\) −62000.5 −2.09753
\(957\) 0 0
\(958\) −50490.0 −1.70277
\(959\) 58051.4 1.95472
\(960\) 0 0
\(961\) 5553.00 0.186399
\(962\) 49077.0 1.64481
\(963\) 0 0
\(964\) 65686.3 2.19462
\(965\) −25113.0 −0.837737
\(966\) 0 0
\(967\) 37692.9 1.25349 0.626743 0.779226i \(-0.284387\pi\)
0.626743 + 0.779226i \(0.284387\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −4162.12 −0.137771
\(971\) 12402.0 0.409886 0.204943 0.978774i \(-0.434299\pi\)
0.204943 + 0.978774i \(0.434299\pi\)
\(972\) 0 0
\(973\) −4452.00 −0.146685
\(974\) −39605.1 −1.30290
\(975\) 0 0
\(976\) 90413.1 2.96522
\(977\) −31203.0 −1.02177 −0.510887 0.859648i \(-0.670683\pi\)
−0.510887 + 0.859648i \(0.670683\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −41895.0 −1.36560
\(981\) 0 0
\(982\) 77436.0 2.51638
\(983\) 36540.0 1.18560 0.592800 0.805350i \(-0.298022\pi\)
0.592800 + 0.805350i \(0.298022\pi\)
\(984\) 0 0
\(985\) −32314.9 −1.04532
\(986\) −40545.6 −1.30957
\(987\) 0 0
\(988\) −196308. −6.32124
\(989\) 6547.15 0.210503
\(990\) 0 0
\(991\) −56888.0 −1.82352 −0.911759 0.410725i \(-0.865276\pi\)
−0.911759 + 0.410725i \(0.865276\pi\)
\(992\) −55682.0 −1.78216
\(993\) 0 0
\(994\) −24948.0 −0.796079
\(995\) −14562.0 −0.463966
\(996\) 0 0
\(997\) −15711.4 −0.499083 −0.249542 0.968364i \(-0.580280\pi\)
−0.249542 + 0.968364i \(0.580280\pi\)
\(998\) 67581.2 2.14353
\(999\) 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 1089.4.a.s.1.2 2
3.2 odd 2 363.4.a.l.1.1 2
11.10 odd 2 inner 1089.4.a.s.1.1 2
33.32 even 2 363.4.a.l.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
363.4.a.l.1.1 2 3.2 odd 2
363.4.a.l.1.2 yes 2 33.32 even 2
1089.4.a.s.1.1 2 11.10 odd 2 inner
1089.4.a.s.1.2 2 1.1 even 1 trivial