Properties

Label 1089.3.c.d.604.1
Level $1089$
Weight $3$
Character 1089.604
Analytic conductor $29.673$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1089,3,Mod(604,1089)]
 
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, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1089.604");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1089.c (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(29.6731007888\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) 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)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 604.1
Root \(1.93185i\) of defining polynomial
Character \(\chi\) \(=\) 1089.604
Dual form 1089.3.c.d.604.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.34607i q^{2} -7.19615 q^{4} +8.66025 q^{5} +8.10634i q^{7} +10.6945i q^{8} +O(q^{10})\) \(q-3.34607i q^{2} -7.19615 q^{4} +8.66025 q^{5} +8.10634i q^{7} +10.6945i q^{8} -28.9778i q^{10} +0.619174i q^{13} +27.1244 q^{14} +7.00000 q^{16} +25.2156i q^{17} -5.10205i q^{19} -62.3205 q^{20} +7.26795 q^{23} +50.0000 q^{25} +2.07180 q^{26} -58.3345i q^{28} +21.8052i q^{29} -15.4641 q^{31} +19.3557i q^{32} +84.3731 q^{34} +70.2030i q^{35} -24.8038 q^{37} -17.0718 q^{38} +92.6174i q^{40} -23.2466i q^{41} +36.4649i q^{43} -24.3190i q^{46} +86.1051 q^{47} -16.7128 q^{49} -167.303i q^{50} -4.45567i q^{52} +53.1051 q^{53} -86.6936 q^{56} +72.9615 q^{58} +66.4064 q^{59} +100.632i q^{61} +51.7439i q^{62} +92.7654 q^{64} +5.36220i q^{65} +74.0141 q^{67} -181.455i q^{68} +234.904 q^{70} +66.8372 q^{71} +43.1571i q^{73} +82.9953i q^{74} +36.7151i q^{76} -99.7528i q^{79} +60.6218 q^{80} -77.7846 q^{82} -20.5569i q^{83} +218.374i q^{85} +122.014 q^{86} -17.0141 q^{89} -5.01924 q^{91} -52.3013 q^{92} -288.113i q^{94} -44.1851i q^{95} -161.158 q^{97} +55.9222i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} + 60 q^{14} + 28 q^{16} - 180 q^{20} + 36 q^{23} + 200 q^{25} + 36 q^{26} - 48 q^{31} + 192 q^{34} - 120 q^{37} - 96 q^{38} + 192 q^{47} + 44 q^{49} + 60 q^{53} - 132 q^{56} + 84 q^{58} - 60 q^{59} + 184 q^{64} + 12 q^{67} + 420 q^{70} + 108 q^{71} - 228 q^{82} + 204 q^{86} + 216 q^{89} - 124 q^{91} - 36 q^{92} - 416 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.34607i − 1.67303i −0.547942 0.836516i \(-0.684589\pi\)
0.547942 0.836516i \(-0.315411\pi\)
\(3\) 0 0
\(4\) −7.19615 −1.79904
\(5\) 8.66025 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) 8.10634i 1.15805i 0.815310 + 0.579025i \(0.196566\pi\)
−0.815310 + 0.579025i \(0.803434\pi\)
\(8\) 10.6945i 1.33682i
\(9\) 0 0
\(10\) − 28.9778i − 2.89778i
\(11\) 0 0
\(12\) 0 0
\(13\) 0.619174i 0.0476288i 0.999716 + 0.0238144i \(0.00758107\pi\)
−0.999716 + 0.0238144i \(0.992419\pi\)
\(14\) 27.1244 1.93745
\(15\) 0 0
\(16\) 7.00000 0.437500
\(17\) 25.2156i 1.48327i 0.670803 + 0.741636i \(0.265950\pi\)
−0.670803 + 0.741636i \(0.734050\pi\)
\(18\) 0 0
\(19\) − 5.10205i − 0.268529i −0.990946 0.134265i \(-0.957133\pi\)
0.990946 0.134265i \(-0.0428672\pi\)
\(20\) −62.3205 −3.11603
\(21\) 0 0
\(22\) 0 0
\(23\) 7.26795 0.315998 0.157999 0.987439i \(-0.449496\pi\)
0.157999 + 0.987439i \(0.449496\pi\)
\(24\) 0 0
\(25\) 50.0000 2.00000
\(26\) 2.07180 0.0796845
\(27\) 0 0
\(28\) − 58.3345i − 2.08337i
\(29\) 21.8052i 0.751902i 0.926639 + 0.375951i \(0.122684\pi\)
−0.926639 + 0.375951i \(0.877316\pi\)
\(30\) 0 0
\(31\) −15.4641 −0.498842 −0.249421 0.968395i \(-0.580240\pi\)
−0.249421 + 0.968395i \(0.580240\pi\)
\(32\) 19.3557i 0.604865i
\(33\) 0 0
\(34\) 84.3731 2.48156
\(35\) 70.2030i 2.00580i
\(36\) 0 0
\(37\) −24.8038 −0.670374 −0.335187 0.942152i \(-0.608800\pi\)
−0.335187 + 0.942152i \(0.608800\pi\)
\(38\) −17.0718 −0.449258
\(39\) 0 0
\(40\) 92.6174i 2.31543i
\(41\) − 23.2466i − 0.566990i −0.958974 0.283495i \(-0.908506\pi\)
0.958974 0.283495i \(-0.0914939\pi\)
\(42\) 0 0
\(43\) 36.4649i 0.848022i 0.905657 + 0.424011i \(0.139378\pi\)
−0.905657 + 0.424011i \(0.860622\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) − 24.3190i − 0.528675i
\(47\) 86.1051 1.83202 0.916012 0.401151i \(-0.131390\pi\)
0.916012 + 0.401151i \(0.131390\pi\)
\(48\) 0 0
\(49\) −16.7128 −0.341078
\(50\) − 167.303i − 3.34607i
\(51\) 0 0
\(52\) − 4.45567i − 0.0856860i
\(53\) 53.1051 1.00198 0.500992 0.865452i \(-0.332969\pi\)
0.500992 + 0.865452i \(0.332969\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −86.6936 −1.54810
\(57\) 0 0
\(58\) 72.9615 1.25796
\(59\) 66.4064 1.12553 0.562766 0.826616i \(-0.309737\pi\)
0.562766 + 0.826616i \(0.309737\pi\)
\(60\) 0 0
\(61\) 100.632i 1.64971i 0.565346 + 0.824854i \(0.308743\pi\)
−0.565346 + 0.824854i \(0.691257\pi\)
\(62\) 51.7439i 0.834579i
\(63\) 0 0
\(64\) 92.7654 1.44946
\(65\) 5.36220i 0.0824955i
\(66\) 0 0
\(67\) 74.0141 1.10469 0.552344 0.833616i \(-0.313734\pi\)
0.552344 + 0.833616i \(0.313734\pi\)
\(68\) − 181.455i − 2.66846i
\(69\) 0 0
\(70\) 234.904 3.35577
\(71\) 66.8372 0.941369 0.470684 0.882302i \(-0.344007\pi\)
0.470684 + 0.882302i \(0.344007\pi\)
\(72\) 0 0
\(73\) 43.1571i 0.591193i 0.955313 + 0.295596i \(0.0955184\pi\)
−0.955313 + 0.295596i \(0.904482\pi\)
\(74\) 82.9953i 1.12156i
\(75\) 0 0
\(76\) 36.7151i 0.483094i
\(77\) 0 0
\(78\) 0 0
\(79\) − 99.7528i − 1.26269i −0.775500 0.631347i \(-0.782502\pi\)
0.775500 0.631347i \(-0.217498\pi\)
\(80\) 60.6218 0.757772
\(81\) 0 0
\(82\) −77.7846 −0.948593
\(83\) − 20.5569i − 0.247673i −0.992303 0.123837i \(-0.960480\pi\)
0.992303 0.123837i \(-0.0395198\pi\)
\(84\) 0 0
\(85\) 218.374i 2.56910i
\(86\) 122.014 1.41877
\(87\) 0 0
\(88\) 0 0
\(89\) −17.0141 −0.191169 −0.0955847 0.995421i \(-0.530472\pi\)
−0.0955847 + 0.995421i \(0.530472\pi\)
\(90\) 0 0
\(91\) −5.01924 −0.0551565
\(92\) −52.3013 −0.568492
\(93\) 0 0
\(94\) − 288.113i − 3.06504i
\(95\) − 44.1851i − 0.465106i
\(96\) 0 0
\(97\) −161.158 −1.66142 −0.830710 0.556706i \(-0.812065\pi\)
−0.830710 + 0.556706i \(0.812065\pi\)
\(98\) 55.9222i 0.570634i
\(99\) 0 0
\(100\) −359.808 −3.59808
\(101\) − 107.250i − 1.06188i −0.847409 0.530940i \(-0.821839\pi\)
0.847409 0.530940i \(-0.178161\pi\)
\(102\) 0 0
\(103\) −102.550 −0.995631 −0.497815 0.867283i \(-0.665864\pi\)
−0.497815 + 0.867283i \(0.665864\pi\)
\(104\) −6.62178 −0.0636709
\(105\) 0 0
\(106\) − 177.693i − 1.67635i
\(107\) − 3.89091i − 0.0363636i −0.999835 0.0181818i \(-0.994212\pi\)
0.999835 0.0181818i \(-0.00578777\pi\)
\(108\) 0 0
\(109\) 42.3348i 0.388393i 0.980963 + 0.194196i \(0.0622099\pi\)
−0.980963 + 0.194196i \(0.937790\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 56.7444i 0.506646i
\(113\) −129.813 −1.14879 −0.574393 0.818580i \(-0.694762\pi\)
−0.574393 + 0.818580i \(0.694762\pi\)
\(114\) 0 0
\(115\) 62.9423 0.547324
\(116\) − 156.913i − 1.35270i
\(117\) 0 0
\(118\) − 222.200i − 1.88305i
\(119\) −204.406 −1.71770
\(120\) 0 0
\(121\) 0 0
\(122\) 336.722 2.76001
\(123\) 0 0
\(124\) 111.282 0.897436
\(125\) 216.506 1.73205
\(126\) 0 0
\(127\) − 12.3490i − 0.0972361i −0.998817 0.0486180i \(-0.984518\pi\)
0.998817 0.0486180i \(-0.0154817\pi\)
\(128\) − 232.976i − 1.82013i
\(129\) 0 0
\(130\) 17.9423 0.138018
\(131\) 18.2361i 0.139207i 0.997575 + 0.0696035i \(0.0221734\pi\)
−0.997575 + 0.0696035i \(0.977827\pi\)
\(132\) 0 0
\(133\) 41.3590 0.310970
\(134\) − 247.656i − 1.84818i
\(135\) 0 0
\(136\) −269.669 −1.98286
\(137\) −51.6462 −0.376979 −0.188490 0.982075i \(-0.560359\pi\)
−0.188490 + 0.982075i \(0.560359\pi\)
\(138\) 0 0
\(139\) − 78.8714i − 0.567420i −0.958910 0.283710i \(-0.908435\pi\)
0.958910 0.283710i \(-0.0915654\pi\)
\(140\) − 505.191i − 3.60851i
\(141\) 0 0
\(142\) − 223.642i − 1.57494i
\(143\) 0 0
\(144\) 0 0
\(145\) 188.838i 1.30233i
\(146\) 144.406 0.989085
\(147\) 0 0
\(148\) 178.492 1.20603
\(149\) 156.656i 1.05138i 0.850676 + 0.525691i \(0.176193\pi\)
−0.850676 + 0.525691i \(0.823807\pi\)
\(150\) 0 0
\(151\) − 152.884i − 1.01247i −0.862394 0.506237i \(-0.831036\pi\)
0.862394 0.506237i \(-0.168964\pi\)
\(152\) 54.5641 0.358974
\(153\) 0 0
\(154\) 0 0
\(155\) −133.923 −0.864020
\(156\) 0 0
\(157\) −276.813 −1.76314 −0.881569 0.472055i \(-0.843513\pi\)
−0.881569 + 0.472055i \(0.843513\pi\)
\(158\) −333.779 −2.11253
\(159\) 0 0
\(160\) 167.625i 1.04766i
\(161\) 58.9165i 0.365941i
\(162\) 0 0
\(163\) −100.158 −0.614464 −0.307232 0.951635i \(-0.599403\pi\)
−0.307232 + 0.951635i \(0.599403\pi\)
\(164\) 167.286i 1.02004i
\(165\) 0 0
\(166\) −68.7846 −0.414365
\(167\) 68.4443i 0.409846i 0.978778 + 0.204923i \(0.0656945\pi\)
−0.978778 + 0.204923i \(0.934306\pi\)
\(168\) 0 0
\(169\) 168.617 0.997731
\(170\) 730.692 4.29819
\(171\) 0 0
\(172\) − 262.407i − 1.52562i
\(173\) 60.4050i 0.349162i 0.984643 + 0.174581i \(0.0558571\pi\)
−0.984643 + 0.174581i \(0.944143\pi\)
\(174\) 0 0
\(175\) 405.317i 2.31610i
\(176\) 0 0
\(177\) 0 0
\(178\) 56.9302i 0.319833i
\(179\) 207.962 1.16180 0.580898 0.813976i \(-0.302701\pi\)
0.580898 + 0.813976i \(0.302701\pi\)
\(180\) 0 0
\(181\) 314.650 1.73840 0.869199 0.494463i \(-0.164635\pi\)
0.869199 + 0.494463i \(0.164635\pi\)
\(182\) 16.7947i 0.0922786i
\(183\) 0 0
\(184\) 77.7273i 0.422431i
\(185\) −214.808 −1.16112
\(186\) 0 0
\(187\) 0 0
\(188\) −619.626 −3.29588
\(189\) 0 0
\(190\) −147.846 −0.778137
\(191\) −28.7077 −0.150302 −0.0751509 0.997172i \(-0.523944\pi\)
−0.0751509 + 0.997172i \(0.523944\pi\)
\(192\) 0 0
\(193\) 197.661i 1.02415i 0.858941 + 0.512074i \(0.171123\pi\)
−0.858941 + 0.512074i \(0.828877\pi\)
\(194\) 539.244i 2.77961i
\(195\) 0 0
\(196\) 120.268 0.613612
\(197\) − 184.514i − 0.936620i −0.883564 0.468310i \(-0.844863\pi\)
0.883564 0.468310i \(-0.155137\pi\)
\(198\) 0 0
\(199\) −41.8423 −0.210263 −0.105131 0.994458i \(-0.533526\pi\)
−0.105131 + 0.994458i \(0.533526\pi\)
\(200\) 534.727i 2.67363i
\(201\) 0 0
\(202\) −358.865 −1.77656
\(203\) −176.760 −0.870740
\(204\) 0 0
\(205\) − 201.321i − 0.982056i
\(206\) 343.139i 1.66572i
\(207\) 0 0
\(208\) 4.33422i 0.0208376i
\(209\) 0 0
\(210\) 0 0
\(211\) − 299.122i − 1.41764i −0.705388 0.708821i \(-0.749227\pi\)
0.705388 0.708821i \(-0.250773\pi\)
\(212\) −382.153 −1.80261
\(213\) 0 0
\(214\) −13.0192 −0.0608376
\(215\) 315.796i 1.46882i
\(216\) 0 0
\(217\) − 125.357i − 0.577684i
\(218\) 141.655 0.649794
\(219\) 0 0
\(220\) 0 0
\(221\) −15.6128 −0.0706464
\(222\) 0 0
\(223\) 195.023 0.874543 0.437271 0.899330i \(-0.355945\pi\)
0.437271 + 0.899330i \(0.355945\pi\)
\(224\) −156.904 −0.700463
\(225\) 0 0
\(226\) 434.362i 1.92196i
\(227\) − 418.580i − 1.84396i −0.387232 0.921982i \(-0.626569\pi\)
0.387232 0.921982i \(-0.373431\pi\)
\(228\) 0 0
\(229\) −185.942 −0.811975 −0.405988 0.913879i \(-0.633072\pi\)
−0.405988 + 0.913879i \(0.633072\pi\)
\(230\) − 210.609i − 0.915691i
\(231\) 0 0
\(232\) −233.196 −1.00516
\(233\) 54.2233i 0.232718i 0.993207 + 0.116359i \(0.0371223\pi\)
−0.993207 + 0.116359i \(0.962878\pi\)
\(234\) 0 0
\(235\) 745.692 3.17316
\(236\) −477.870 −2.02487
\(237\) 0 0
\(238\) 683.957i 2.87377i
\(239\) − 258.110i − 1.07996i −0.841678 0.539980i \(-0.818432\pi\)
0.841678 0.539980i \(-0.181568\pi\)
\(240\) 0 0
\(241\) − 244.178i − 1.01319i −0.862185 0.506594i \(-0.830904\pi\)
0.862185 0.506594i \(-0.169096\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) − 724.164i − 2.96789i
\(245\) −144.737 −0.590764
\(246\) 0 0
\(247\) 3.15906 0.0127897
\(248\) − 165.381i − 0.666860i
\(249\) 0 0
\(250\) − 724.444i − 2.89778i
\(251\) 151.019 0.601670 0.300835 0.953676i \(-0.402735\pi\)
0.300835 + 0.953676i \(0.402735\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −41.3205 −0.162679
\(255\) 0 0
\(256\) −408.492 −1.59567
\(257\) −55.2346 −0.214921 −0.107460 0.994209i \(-0.534272\pi\)
−0.107460 + 0.994209i \(0.534272\pi\)
\(258\) 0 0
\(259\) − 201.069i − 0.776326i
\(260\) − 38.5872i − 0.148412i
\(261\) 0 0
\(262\) 61.0192 0.232898
\(263\) 402.008i 1.52855i 0.644891 + 0.764274i \(0.276903\pi\)
−0.644891 + 0.764274i \(0.723097\pi\)
\(264\) 0 0
\(265\) 459.904 1.73549
\(266\) − 138.390i − 0.520263i
\(267\) 0 0
\(268\) −532.617 −1.98738
\(269\) −435.267 −1.61809 −0.809046 0.587746i \(-0.800016\pi\)
−0.809046 + 0.587746i \(0.800016\pi\)
\(270\) 0 0
\(271\) 112.461i 0.414985i 0.978237 + 0.207492i \(0.0665302\pi\)
−0.978237 + 0.207492i \(0.933470\pi\)
\(272\) 176.509i 0.648931i
\(273\) 0 0
\(274\) 172.811i 0.630699i
\(275\) 0 0
\(276\) 0 0
\(277\) 304.438i 1.09905i 0.835476 + 0.549526i \(0.185192\pi\)
−0.835476 + 0.549526i \(0.814808\pi\)
\(278\) −263.909 −0.949313
\(279\) 0 0
\(280\) −750.788 −2.68139
\(281\) − 307.053i − 1.09272i −0.837552 0.546358i \(-0.816014\pi\)
0.837552 0.546358i \(-0.183986\pi\)
\(282\) 0 0
\(283\) 505.726i 1.78702i 0.449044 + 0.893509i \(0.351765\pi\)
−0.449044 + 0.893509i \(0.648235\pi\)
\(284\) −480.970 −1.69356
\(285\) 0 0
\(286\) 0 0
\(287\) 188.445 0.656602
\(288\) 0 0
\(289\) −346.827 −1.20009
\(290\) 631.865 2.17885
\(291\) 0 0
\(292\) − 310.565i − 1.06358i
\(293\) − 484.102i − 1.65223i −0.563504 0.826113i \(-0.690547\pi\)
0.563504 0.826113i \(-0.309453\pi\)
\(294\) 0 0
\(295\) 575.096 1.94948
\(296\) − 265.266i − 0.896168i
\(297\) 0 0
\(298\) 524.181 1.75900
\(299\) 4.50013i 0.0150506i
\(300\) 0 0
\(301\) −295.597 −0.982051
\(302\) −511.559 −1.69390
\(303\) 0 0
\(304\) − 35.7144i − 0.117481i
\(305\) 871.500i 2.85738i
\(306\) 0 0
\(307\) 473.965i 1.54386i 0.635708 + 0.771929i \(0.280708\pi\)
−0.635708 + 0.771929i \(0.719292\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 448.115i 1.44553i
\(311\) 516.449 1.66061 0.830303 0.557312i \(-0.188167\pi\)
0.830303 + 0.557312i \(0.188167\pi\)
\(312\) 0 0
\(313\) −418.626 −1.33746 −0.668731 0.743505i \(-0.733162\pi\)
−0.668731 + 0.743505i \(0.733162\pi\)
\(314\) 926.234i 2.94979i
\(315\) 0 0
\(316\) 717.837i 2.27163i
\(317\) −70.5744 −0.222632 −0.111316 0.993785i \(-0.535507\pi\)
−0.111316 + 0.993785i \(0.535507\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 803.372 2.51054
\(321\) 0 0
\(322\) 197.138 0.612231
\(323\) 128.651 0.398301
\(324\) 0 0
\(325\) 30.9587i 0.0952575i
\(326\) 335.134i 1.02802i
\(327\) 0 0
\(328\) 248.611 0.757962
\(329\) 697.998i 2.12157i
\(330\) 0 0
\(331\) −442.315 −1.33630 −0.668150 0.744027i \(-0.732914\pi\)
−0.668150 + 0.744027i \(0.732914\pi\)
\(332\) 147.930i 0.445573i
\(333\) 0 0
\(334\) 229.019 0.685686
\(335\) 640.981 1.91338
\(336\) 0 0
\(337\) − 442.956i − 1.31441i −0.753712 0.657205i \(-0.771739\pi\)
0.753712 0.657205i \(-0.228261\pi\)
\(338\) − 564.202i − 1.66924i
\(339\) 0 0
\(340\) − 1571.45i − 4.62191i
\(341\) 0 0
\(342\) 0 0
\(343\) 261.731i 0.763064i
\(344\) −389.976 −1.13365
\(345\) 0 0
\(346\) 202.119 0.584160
\(347\) − 469.153i − 1.35202i −0.736890 0.676012i \(-0.763707\pi\)
0.736890 0.676012i \(-0.236293\pi\)
\(348\) 0 0
\(349\) 251.591i 0.720892i 0.932780 + 0.360446i \(0.117375\pi\)
−0.932780 + 0.360446i \(0.882625\pi\)
\(350\) 1356.22 3.87491
\(351\) 0 0
\(352\) 0 0
\(353\) −29.7424 −0.0842560 −0.0421280 0.999112i \(-0.513414\pi\)
−0.0421280 + 0.999112i \(0.513414\pi\)
\(354\) 0 0
\(355\) 578.827 1.63050
\(356\) 122.436 0.343921
\(357\) 0 0
\(358\) − 695.853i − 1.94372i
\(359\) − 196.041i − 0.546075i −0.962003 0.273037i \(-0.911972\pi\)
0.962003 0.273037i \(-0.0880283\pi\)
\(360\) 0 0
\(361\) 334.969 0.927892
\(362\) − 1052.84i − 2.90840i
\(363\) 0 0
\(364\) 36.1192 0.0992286
\(365\) 373.751i 1.02398i
\(366\) 0 0
\(367\) 123.881 0.337550 0.168775 0.985655i \(-0.446019\pi\)
0.168775 + 0.985655i \(0.446019\pi\)
\(368\) 50.8756 0.138249
\(369\) 0 0
\(370\) 718.760i 1.94260i
\(371\) 430.488i 1.16035i
\(372\) 0 0
\(373\) − 101.316i − 0.271624i −0.990735 0.135812i \(-0.956636\pi\)
0.990735 0.135812i \(-0.0433643\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 920.854i 2.44908i
\(377\) −13.5012 −0.0358122
\(378\) 0 0
\(379\) −44.8653 −0.118378 −0.0591891 0.998247i \(-0.518851\pi\)
−0.0591891 + 0.998247i \(0.518851\pi\)
\(380\) 317.962i 0.836743i
\(381\) 0 0
\(382\) 96.0577i 0.251460i
\(383\) 102.340 0.267206 0.133603 0.991035i \(-0.457345\pi\)
0.133603 + 0.991035i \(0.457345\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 661.386 1.71343
\(387\) 0 0
\(388\) 1159.72 2.98896
\(389\) 141.109 0.362748 0.181374 0.983414i \(-0.441946\pi\)
0.181374 + 0.983414i \(0.441946\pi\)
\(390\) 0 0
\(391\) 183.266i 0.468710i
\(392\) − 178.736i − 0.455959i
\(393\) 0 0
\(394\) −617.396 −1.56700
\(395\) − 863.885i − 2.18705i
\(396\) 0 0
\(397\) 485.431 1.22275 0.611374 0.791342i \(-0.290617\pi\)
0.611374 + 0.791342i \(0.290617\pi\)
\(398\) 140.007i 0.351777i
\(399\) 0 0
\(400\) 350.000 0.875000
\(401\) −230.023 −0.573623 −0.286812 0.957987i \(-0.592595\pi\)
−0.286812 + 0.957987i \(0.592595\pi\)
\(402\) 0 0
\(403\) − 9.57497i − 0.0237592i
\(404\) 771.787i 1.91036i
\(405\) 0 0
\(406\) 591.451i 1.45678i
\(407\) 0 0
\(408\) 0 0
\(409\) − 204.705i − 0.500500i −0.968181 0.250250i \(-0.919487\pi\)
0.968181 0.250250i \(-0.0805129\pi\)
\(410\) −673.634 −1.64301
\(411\) 0 0
\(412\) 737.965 1.79118
\(413\) 538.313i 1.30342i
\(414\) 0 0
\(415\) − 178.028i − 0.428982i
\(416\) −11.9845 −0.0288090
\(417\) 0 0
\(418\) 0 0
\(419\) 68.6448 0.163830 0.0819150 0.996639i \(-0.473896\pi\)
0.0819150 + 0.996639i \(0.473896\pi\)
\(420\) 0 0
\(421\) −40.2436 −0.0955904 −0.0477952 0.998857i \(-0.515219\pi\)
−0.0477952 + 0.998857i \(0.515219\pi\)
\(422\) −1000.88 −2.37176
\(423\) 0 0
\(424\) 567.935i 1.33947i
\(425\) 1260.78i 2.96654i
\(426\) 0 0
\(427\) −815.759 −1.91044
\(428\) 27.9996i 0.0654196i
\(429\) 0 0
\(430\) 1056.67 2.45738
\(431\) − 6.13004i − 0.0142228i −0.999975 0.00711141i \(-0.997736\pi\)
0.999975 0.00711141i \(-0.00226365\pi\)
\(432\) 0 0
\(433\) 266.173 0.614719 0.307359 0.951594i \(-0.400555\pi\)
0.307359 + 0.951594i \(0.400555\pi\)
\(434\) −419.454 −0.966483
\(435\) 0 0
\(436\) − 304.648i − 0.698734i
\(437\) − 37.0815i − 0.0848546i
\(438\) 0 0
\(439\) − 382.432i − 0.871144i −0.900154 0.435572i \(-0.856546\pi\)
0.900154 0.435572i \(-0.143454\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 52.2416i 0.118194i
\(443\) −469.965 −1.06087 −0.530435 0.847726i \(-0.677971\pi\)
−0.530435 + 0.847726i \(0.677971\pi\)
\(444\) 0 0
\(445\) −147.346 −0.331115
\(446\) − 652.560i − 1.46314i
\(447\) 0 0
\(448\) 751.988i 1.67854i
\(449\) −95.4627 −0.212612 −0.106306 0.994333i \(-0.533902\pi\)
−0.106306 + 0.994333i \(0.533902\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 934.153 2.06671
\(453\) 0 0
\(454\) −1400.60 −3.08501
\(455\) −43.4679 −0.0955338
\(456\) 0 0
\(457\) − 32.4154i − 0.0709309i −0.999371 0.0354654i \(-0.988709\pi\)
0.999371 0.0354654i \(-0.0112914\pi\)
\(458\) 622.175i 1.35846i
\(459\) 0 0
\(460\) −452.942 −0.984657
\(461\) 383.378i 0.831622i 0.909451 + 0.415811i \(0.136502\pi\)
−0.909451 + 0.415811i \(0.863498\pi\)
\(462\) 0 0
\(463\) −491.804 −1.06221 −0.531106 0.847306i \(-0.678223\pi\)
−0.531106 + 0.847306i \(0.678223\pi\)
\(464\) 152.636i 0.328957i
\(465\) 0 0
\(466\) 181.435 0.389345
\(467\) 788.396 1.68821 0.844107 0.536175i \(-0.180131\pi\)
0.844107 + 0.536175i \(0.180131\pi\)
\(468\) 0 0
\(469\) 599.984i 1.27928i
\(470\) − 2495.13i − 5.30880i
\(471\) 0 0
\(472\) 710.185i 1.50463i
\(473\) 0 0
\(474\) 0 0
\(475\) − 255.103i − 0.537058i
\(476\) 1470.94 3.09021
\(477\) 0 0
\(478\) −863.654 −1.80681
\(479\) − 370.105i − 0.772662i −0.922360 0.386331i \(-0.873742\pi\)
0.922360 0.386331i \(-0.126258\pi\)
\(480\) 0 0
\(481\) − 15.3579i − 0.0319291i
\(482\) −817.037 −1.69510
\(483\) 0 0
\(484\) 0 0
\(485\) −1395.67 −2.87766
\(486\) 0 0
\(487\) 388.732 0.798218 0.399109 0.916904i \(-0.369320\pi\)
0.399109 + 0.916904i \(0.369320\pi\)
\(488\) −1076.21 −2.20536
\(489\) 0 0
\(490\) 484.300i 0.988368i
\(491\) 40.5643i 0.0826156i 0.999146 + 0.0413078i \(0.0131524\pi\)
−0.999146 + 0.0413078i \(0.986848\pi\)
\(492\) 0 0
\(493\) −549.831 −1.11528
\(494\) − 10.5704i − 0.0213976i
\(495\) 0 0
\(496\) −108.249 −0.218243
\(497\) 541.805i 1.09015i
\(498\) 0 0
\(499\) 203.678 0.408173 0.204086 0.978953i \(-0.434578\pi\)
0.204086 + 0.978953i \(0.434578\pi\)
\(500\) −1558.01 −3.11603
\(501\) 0 0
\(502\) − 505.320i − 1.00661i
\(503\) − 303.501i − 0.603382i −0.953406 0.301691i \(-0.902449\pi\)
0.953406 0.301691i \(-0.0975511\pi\)
\(504\) 0 0
\(505\) − 928.812i − 1.83923i
\(506\) 0 0
\(507\) 0 0
\(508\) 88.8652i 0.174931i
\(509\) −639.895 −1.25716 −0.628580 0.777745i \(-0.716364\pi\)
−0.628580 + 0.777745i \(0.716364\pi\)
\(510\) 0 0
\(511\) −349.846 −0.684630
\(512\) 434.937i 0.849486i
\(513\) 0 0
\(514\) 184.819i 0.359569i
\(515\) −888.109 −1.72448
\(516\) 0 0
\(517\) 0 0
\(518\) −672.788 −1.29882
\(519\) 0 0
\(520\) −57.3463 −0.110281
\(521\) −70.3744 −0.135076 −0.0675379 0.997717i \(-0.521514\pi\)
−0.0675379 + 0.997717i \(0.521514\pi\)
\(522\) 0 0
\(523\) 135.046i 0.258215i 0.991631 + 0.129108i \(0.0412112\pi\)
−0.991631 + 0.129108i \(0.958789\pi\)
\(524\) − 131.230i − 0.250439i
\(525\) 0 0
\(526\) 1345.15 2.55731
\(527\) − 389.937i − 0.739918i
\(528\) 0 0
\(529\) −476.177 −0.900145
\(530\) − 1538.87i − 2.90352i
\(531\) 0 0
\(532\) −297.626 −0.559447
\(533\) 14.3937 0.0270050
\(534\) 0 0
\(535\) − 33.6963i − 0.0629837i
\(536\) 791.546i 1.47677i
\(537\) 0 0
\(538\) 1456.43i 2.70712i
\(539\) 0 0
\(540\) 0 0
\(541\) 463.429i 0.856615i 0.903633 + 0.428307i \(0.140890\pi\)
−0.903633 + 0.428307i \(0.859110\pi\)
\(542\) 376.301 0.694283
\(543\) 0 0
\(544\) −488.065 −0.897179
\(545\) 366.630i 0.672716i
\(546\) 0 0
\(547\) − 424.973i − 0.776916i −0.921467 0.388458i \(-0.873008\pi\)
0.921467 0.388458i \(-0.126992\pi\)
\(548\) 371.654 0.678200
\(549\) 0 0
\(550\) 0 0
\(551\) 111.251 0.201908
\(552\) 0 0
\(553\) 808.631 1.46226
\(554\) 1018.67 1.83875
\(555\) 0 0
\(556\) 567.571i 1.02081i
\(557\) 157.458i 0.282690i 0.989960 + 0.141345i \(0.0451426\pi\)
−0.989960 + 0.141345i \(0.954857\pi\)
\(558\) 0 0
\(559\) −22.5781 −0.0403902
\(560\) 491.421i 0.877537i
\(561\) 0 0
\(562\) −1027.42 −1.82815
\(563\) 1026.05i 1.82248i 0.411881 + 0.911238i \(0.364872\pi\)
−0.411881 + 0.911238i \(0.635128\pi\)
\(564\) 0 0
\(565\) −1124.21 −1.98976
\(566\) 1692.19 2.98974
\(567\) 0 0
\(568\) 714.792i 1.25844i
\(569\) 853.945i 1.50078i 0.660994 + 0.750391i \(0.270135\pi\)
−0.660994 + 0.750391i \(0.729865\pi\)
\(570\) 0 0
\(571\) − 155.457i − 0.272253i −0.990691 0.136127i \(-0.956535\pi\)
0.990691 0.136127i \(-0.0434654\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) − 630.549i − 1.09852i
\(575\) 363.397 0.631996
\(576\) 0 0
\(577\) 466.219 0.808005 0.404003 0.914758i \(-0.367619\pi\)
0.404003 + 0.914758i \(0.367619\pi\)
\(578\) 1160.51i 2.00779i
\(579\) 0 0
\(580\) − 1358.91i − 2.34295i
\(581\) 166.641 0.286818
\(582\) 0 0
\(583\) 0 0
\(584\) −461.545 −0.790316
\(585\) 0 0
\(586\) −1619.84 −2.76423
\(587\) 62.6025 0.106648 0.0533241 0.998577i \(-0.483018\pi\)
0.0533241 + 0.998577i \(0.483018\pi\)
\(588\) 0 0
\(589\) 78.8986i 0.133954i
\(590\) − 1924.31i − 3.26154i
\(591\) 0 0
\(592\) −173.627 −0.293289
\(593\) 685.312i 1.15567i 0.816154 + 0.577835i \(0.196102\pi\)
−0.816154 + 0.577835i \(0.803898\pi\)
\(594\) 0 0
\(595\) −1770.21 −2.97514
\(596\) − 1127.32i − 1.89148i
\(597\) 0 0
\(598\) 15.0577 0.0251801
\(599\) −736.939 −1.23028 −0.615141 0.788417i \(-0.710901\pi\)
−0.615141 + 0.788417i \(0.710901\pi\)
\(600\) 0 0
\(601\) − 725.519i − 1.20719i −0.797292 0.603594i \(-0.793735\pi\)
0.797292 0.603594i \(-0.206265\pi\)
\(602\) 989.088i 1.64300i
\(603\) 0 0
\(604\) 1100.17i 1.82148i
\(605\) 0 0
\(606\) 0 0
\(607\) − 733.163i − 1.20785i −0.797043 0.603923i \(-0.793603\pi\)
0.797043 0.603923i \(-0.206397\pi\)
\(608\) 98.7537 0.162424
\(609\) 0 0
\(610\) 2916.10 4.78048
\(611\) 53.3141i 0.0872570i
\(612\) 0 0
\(613\) − 262.717i − 0.428575i −0.976771 0.214288i \(-0.931257\pi\)
0.976771 0.214288i \(-0.0687430\pi\)
\(614\) 1585.92 2.58293
\(615\) 0 0
\(616\) 0 0
\(617\) 451.077 0.731081 0.365540 0.930795i \(-0.380884\pi\)
0.365540 + 0.930795i \(0.380884\pi\)
\(618\) 0 0
\(619\) −363.212 −0.586772 −0.293386 0.955994i \(-0.594782\pi\)
−0.293386 + 0.955994i \(0.594782\pi\)
\(620\) 963.731 1.55440
\(621\) 0 0
\(622\) − 1728.07i − 2.77825i
\(623\) − 137.922i − 0.221384i
\(624\) 0 0
\(625\) 625.000 1.00000
\(626\) 1400.75i 2.23762i
\(627\) 0 0
\(628\) 1991.99 3.17195
\(629\) − 625.444i − 0.994347i
\(630\) 0 0
\(631\) −1151.68 −1.82516 −0.912582 0.408894i \(-0.865914\pi\)
−0.912582 + 0.408894i \(0.865914\pi\)
\(632\) 1066.81 1.68799
\(633\) 0 0
\(634\) 236.146i 0.372471i
\(635\) − 106.945i − 0.168418i
\(636\) 0 0
\(637\) − 10.3481i − 0.0162451i
\(638\) 0 0
\(639\) 0 0
\(640\) − 2017.63i − 3.15255i
\(641\) −582.324 −0.908462 −0.454231 0.890884i \(-0.650086\pi\)
−0.454231 + 0.890884i \(0.650086\pi\)
\(642\) 0 0
\(643\) −735.061 −1.14317 −0.571587 0.820541i \(-0.693672\pi\)
−0.571587 + 0.820541i \(0.693672\pi\)
\(644\) − 423.972i − 0.658342i
\(645\) 0 0
\(646\) − 430.476i − 0.666371i
\(647\) 164.074 0.253592 0.126796 0.991929i \(-0.459531\pi\)
0.126796 + 0.991929i \(0.459531\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 103.590 0.159369
\(651\) 0 0
\(652\) 720.750 1.10544
\(653\) −415.328 −0.636031 −0.318015 0.948086i \(-0.603016\pi\)
−0.318015 + 0.948086i \(0.603016\pi\)
\(654\) 0 0
\(655\) 157.929i 0.241114i
\(656\) − 162.726i − 0.248058i
\(657\) 0 0
\(658\) 2335.55 3.54946
\(659\) − 267.591i − 0.406056i −0.979173 0.203028i \(-0.934922\pi\)
0.979173 0.203028i \(-0.0650783\pi\)
\(660\) 0 0
\(661\) −312.979 −0.473494 −0.236747 0.971571i \(-0.576081\pi\)
−0.236747 + 0.971571i \(0.576081\pi\)
\(662\) 1480.02i 2.23567i
\(663\) 0 0
\(664\) 219.846 0.331094
\(665\) 358.179 0.538615
\(666\) 0 0
\(667\) 158.479i 0.237600i
\(668\) − 492.536i − 0.737329i
\(669\) 0 0
\(670\) − 2144.76i − 3.20114i
\(671\) 0 0
\(672\) 0 0
\(673\) 1010.71i 1.50179i 0.660419 + 0.750897i \(0.270379\pi\)
−0.660419 + 0.750897i \(0.729621\pi\)
\(674\) −1482.16 −2.19905
\(675\) 0 0
\(676\) −1213.39 −1.79496
\(677\) − 1104.82i − 1.63193i −0.578103 0.815964i \(-0.696207\pi\)
0.578103 0.815964i \(-0.303793\pi\)
\(678\) 0 0
\(679\) − 1306.40i − 1.92401i
\(680\) −2335.40 −3.43442
\(681\) 0 0
\(682\) 0 0
\(683\) −697.322 −1.02097 −0.510484 0.859887i \(-0.670534\pi\)
−0.510484 + 0.859887i \(0.670534\pi\)
\(684\) 0 0
\(685\) −447.269 −0.652947
\(686\) 875.769 1.27663
\(687\) 0 0
\(688\) 255.255i 0.371010i
\(689\) 32.8813i 0.0477232i
\(690\) 0 0
\(691\) 11.8883 0.0172046 0.00860228 0.999963i \(-0.497262\pi\)
0.00860228 + 0.999963i \(0.497262\pi\)
\(692\) − 434.684i − 0.628156i
\(693\) 0 0
\(694\) −1569.82 −2.26198
\(695\) − 683.047i − 0.982801i
\(696\) 0 0
\(697\) 586.177 0.841000
\(698\) 841.841 1.20608
\(699\) 0 0
\(700\) − 2916.72i − 4.16675i
\(701\) − 1075.30i − 1.53395i −0.641679 0.766973i \(-0.721762\pi\)
0.641679 0.766973i \(-0.278238\pi\)
\(702\) 0 0
\(703\) 126.551i 0.180015i
\(704\) 0 0
\(705\) 0 0
\(706\) 99.5199i 0.140963i
\(707\) 869.405 1.22971
\(708\) 0 0
\(709\) 742.841 1.04773 0.523865 0.851801i \(-0.324490\pi\)
0.523865 + 0.851801i \(0.324490\pi\)
\(710\) − 1936.79i − 2.72788i
\(711\) 0 0
\(712\) − 181.958i − 0.255559i
\(713\) −112.392 −0.157633
\(714\) 0 0
\(715\) 0 0
\(716\) −1496.52 −2.09012
\(717\) 0 0
\(718\) −655.965 −0.913601
\(719\) 934.852 1.30021 0.650106 0.759844i \(-0.274725\pi\)
0.650106 + 0.759844i \(0.274725\pi\)
\(720\) 0 0
\(721\) − 831.305i − 1.15299i
\(722\) − 1120.83i − 1.55239i
\(723\) 0 0
\(724\) −2264.27 −3.12744
\(725\) 1090.26i 1.50380i
\(726\) 0 0
\(727\) 1351.59 1.85913 0.929565 0.368657i \(-0.120182\pi\)
0.929565 + 0.368657i \(0.120182\pi\)
\(728\) − 53.6784i − 0.0737341i
\(729\) 0 0
\(730\) 1250.60 1.71315
\(731\) −919.486 −1.25785
\(732\) 0 0
\(733\) − 696.688i − 0.950461i −0.879861 0.475230i \(-0.842365\pi\)
0.879861 0.475230i \(-0.157635\pi\)
\(734\) − 414.513i − 0.564732i
\(735\) 0 0
\(736\) 140.676i 0.191136i
\(737\) 0 0
\(738\) 0 0
\(739\) 950.312i 1.28594i 0.765890 + 0.642971i \(0.222299\pi\)
−0.765890 + 0.642971i \(0.777701\pi\)
\(740\) 1545.79 2.08890
\(741\) 0 0
\(742\) 1440.44 1.94130
\(743\) − 785.944i − 1.05780i −0.848685 0.528899i \(-0.822605\pi\)
0.848685 0.528899i \(-0.177395\pi\)
\(744\) 0 0
\(745\) 1356.68i 1.82105i
\(746\) −339.009 −0.454436
\(747\) 0 0
\(748\) 0 0
\(749\) 31.5411 0.0421109
\(750\) 0 0
\(751\) −1115.51 −1.48537 −0.742686 0.669640i \(-0.766448\pi\)
−0.742686 + 0.669640i \(0.766448\pi\)
\(752\) 602.736 0.801510
\(753\) 0 0
\(754\) 45.1759i 0.0599150i
\(755\) − 1324.01i − 1.75366i
\(756\) 0 0
\(757\) −336.149 −0.444054 −0.222027 0.975041i \(-0.571267\pi\)
−0.222027 + 0.975041i \(0.571267\pi\)
\(758\) 150.122i 0.198051i
\(759\) 0 0
\(760\) 472.539 0.621761
\(761\) 107.035i 0.140650i 0.997524 + 0.0703252i \(0.0224037\pi\)
−0.997524 + 0.0703252i \(0.977596\pi\)
\(762\) 0 0
\(763\) −343.181 −0.449778
\(764\) 206.585 0.270399
\(765\) 0 0
\(766\) − 342.435i − 0.447044i
\(767\) 41.1171i 0.0536077i
\(768\) 0 0
\(769\) − 39.4845i − 0.0513453i −0.999670 0.0256726i \(-0.991827\pi\)
0.999670 0.0256726i \(-0.00817276\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 1422.40i − 1.84248i
\(773\) 670.228 0.867048 0.433524 0.901142i \(-0.357270\pi\)
0.433524 + 0.901142i \(0.357270\pi\)
\(774\) 0 0
\(775\) −773.205 −0.997684
\(776\) − 1723.51i − 2.22101i
\(777\) 0 0
\(778\) − 472.160i − 0.606889i
\(779\) −118.605 −0.152253
\(780\) 0 0
\(781\) 0 0
\(782\) 613.219 0.784168
\(783\) 0 0
\(784\) −116.990 −0.149222
\(785\) −2397.27 −3.05385
\(786\) 0 0
\(787\) − 149.600i − 0.190089i −0.995473 0.0950445i \(-0.969701\pi\)
0.995473 0.0950445i \(-0.0302993\pi\)
\(788\) 1327.79i 1.68501i
\(789\) 0 0
\(790\) −2890.61 −3.65901
\(791\) − 1052.31i − 1.33035i
\(792\) 0 0
\(793\) −62.3088 −0.0785735
\(794\) − 1624.28i − 2.04570i
\(795\) 0 0
\(796\) 301.104 0.378271
\(797\) 379.528 0.476196 0.238098 0.971241i \(-0.423476\pi\)
0.238098 + 0.971241i \(0.423476\pi\)
\(798\) 0 0
\(799\) 2171.19i 2.71739i
\(800\) 967.784i 1.20973i
\(801\) 0 0
\(802\) 769.672i 0.959691i
\(803\) 0 0
\(804\) 0 0
\(805\) 510.232i 0.633828i
\(806\) −32.0385 −0.0397500
\(807\) 0 0
\(808\) 1146.99 1.41954
\(809\) 70.1432i 0.0867036i 0.999060 + 0.0433518i \(0.0138036\pi\)
−0.999060 + 0.0433518i \(0.986196\pi\)
\(810\) 0 0
\(811\) 13.2954i 0.0163938i 0.999966 + 0.00819689i \(0.00260918\pi\)
−0.999966 + 0.00819689i \(0.997391\pi\)
\(812\) 1271.99 1.56649
\(813\) 0 0
\(814\) 0 0
\(815\) −867.391 −1.06428
\(816\) 0 0
\(817\) 186.046 0.227719
\(818\) −684.955 −0.837353
\(819\) 0 0
\(820\) 1448.74i 1.76676i
\(821\) − 614.891i − 0.748954i −0.927236 0.374477i \(-0.877822\pi\)
0.927236 0.374477i \(-0.122178\pi\)
\(822\) 0 0
\(823\) 1537.20 1.86780 0.933898 0.357540i \(-0.116384\pi\)
0.933898 + 0.357540i \(0.116384\pi\)
\(824\) − 1096.72i − 1.33098i
\(825\) 0 0
\(826\) 1801.23 2.18067
\(827\) 1143.12i 1.38225i 0.722734 + 0.691127i \(0.242885\pi\)
−0.722734 + 0.691127i \(0.757115\pi\)
\(828\) 0 0
\(829\) −940.623 −1.13465 −0.567324 0.823495i \(-0.692021\pi\)
−0.567324 + 0.823495i \(0.692021\pi\)
\(830\) −595.692 −0.717701
\(831\) 0 0
\(832\) 57.4379i 0.0690360i
\(833\) − 421.424i − 0.505911i
\(834\) 0 0
\(835\) 592.745i 0.709875i
\(836\) 0 0
\(837\) 0 0
\(838\) − 229.690i − 0.274093i
\(839\) −795.384 −0.948015 −0.474007 0.880521i \(-0.657193\pi\)
−0.474007 + 0.880521i \(0.657193\pi\)
\(840\) 0 0
\(841\) 365.535 0.434643
\(842\) 134.658i 0.159926i
\(843\) 0 0
\(844\) 2152.53i 2.55039i
\(845\) 1460.26 1.72812
\(846\) 0 0
\(847\) 0 0
\(848\) 371.736 0.438368
\(849\) 0 0
\(850\) 4218.65 4.96312
\(851\) −180.273 −0.211837
\(852\) 0 0
\(853\) − 1410.47i − 1.65354i −0.562543 0.826768i \(-0.690177\pi\)
0.562543 0.826768i \(-0.309823\pi\)
\(854\) 2729.58i 3.19623i
\(855\) 0 0
\(856\) 41.6115 0.0486115
\(857\) − 628.919i − 0.733861i −0.930248 0.366931i \(-0.880409\pi\)
0.930248 0.366931i \(-0.119591\pi\)
\(858\) 0 0
\(859\) −255.965 −0.297981 −0.148990 0.988839i \(-0.547602\pi\)
−0.148990 + 0.988839i \(0.547602\pi\)
\(860\) − 2272.51i − 2.64246i
\(861\) 0 0
\(862\) −20.5115 −0.0237953
\(863\) −295.104 −0.341951 −0.170976 0.985275i \(-0.554692\pi\)
−0.170976 + 0.985275i \(0.554692\pi\)
\(864\) 0 0
\(865\) 523.123i 0.604766i
\(866\) − 890.633i − 1.02844i
\(867\) 0 0
\(868\) 902.090i 1.03927i
\(869\) 0 0
\(870\) 0 0
\(871\) 45.8276i 0.0526149i
\(872\) −452.751 −0.519210
\(873\) 0 0
\(874\) −124.077 −0.141964
\(875\) 1755.07i 2.00580i
\(876\) 0 0
\(877\) − 241.360i − 0.275211i −0.990487 0.137605i \(-0.956059\pi\)
0.990487 0.137605i \(-0.0439406\pi\)
\(878\) −1279.64 −1.45745
\(879\) 0 0
\(880\) 0 0
\(881\) 708.997 0.804764 0.402382 0.915472i \(-0.368182\pi\)
0.402382 + 0.915472i \(0.368182\pi\)
\(882\) 0 0
\(883\) 682.264 0.772666 0.386333 0.922359i \(-0.373742\pi\)
0.386333 + 0.922359i \(0.373742\pi\)
\(884\) 112.352 0.127096
\(885\) 0 0
\(886\) 1572.53i 1.77487i
\(887\) 87.3115i 0.0984347i 0.998788 + 0.0492173i \(0.0156727\pi\)
−0.998788 + 0.0492173i \(0.984327\pi\)
\(888\) 0 0
\(889\) 100.105 0.112604
\(890\) 493.030i 0.553967i
\(891\) 0 0
\(892\) −1403.42 −1.57334
\(893\) − 439.313i − 0.491952i
\(894\) 0 0
\(895\) 1801.00 2.01229
\(896\) 1888.59 2.10780
\(897\) 0 0
\(898\) 319.424i 0.355707i
\(899\) − 337.197i − 0.375081i
\(900\) 0 0
\(901\) 1339.08i 1.48621i
\(902\) 0 0
\(903\) 0 0
\(904\) − 1388.29i − 1.53572i
\(905\) 2724.95 3.01099
\(906\) 0 0
\(907\) 590.750 0.651323 0.325661 0.945486i \(-0.394413\pi\)
0.325661 + 0.945486i \(0.394413\pi\)
\(908\) 3012.17i 3.31736i
\(909\) 0 0
\(910\) 145.446i 0.159831i
\(911\) −240.655 −0.264166 −0.132083 0.991239i \(-0.542167\pi\)
−0.132083 + 0.991239i \(0.542167\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −108.464 −0.118670
\(915\) 0 0
\(916\) 1338.07 1.46077
\(917\) −147.828 −0.161209
\(918\) 0 0
\(919\) 15.4874i 0.0168524i 0.999964 + 0.00842620i \(0.00268217\pi\)
−0.999964 + 0.00842620i \(0.997318\pi\)
\(920\) 673.138i 0.731672i
\(921\) 0 0
\(922\) 1282.81 1.39133
\(923\) 41.3838i 0.0448362i
\(924\) 0 0
\(925\) −1240.19 −1.34075
\(926\) 1645.61i 1.77711i
\(927\) 0 0
\(928\) −422.054 −0.454800
\(929\) −382.741 −0.411992 −0.205996 0.978553i \(-0.566043\pi\)
−0.205996 + 0.978553i \(0.566043\pi\)
\(930\) 0 0
\(931\) 85.2696i 0.0915893i
\(932\) − 390.199i − 0.418668i
\(933\) 0 0
\(934\) − 2638.02i − 2.82444i
\(935\) 0 0
\(936\) 0 0
\(937\) − 720.608i − 0.769059i −0.923113 0.384529i \(-0.874364\pi\)
0.923113 0.384529i \(-0.125636\pi\)
\(938\) 2007.58 2.14028
\(939\) 0 0
\(940\) −5366.11 −5.70863
\(941\) − 93.3461i − 0.0991988i −0.998769 0.0495994i \(-0.984206\pi\)
0.998769 0.0495994i \(-0.0157945\pi\)
\(942\) 0 0
\(943\) − 168.955i − 0.179168i
\(944\) 464.845 0.492420
\(945\) 0 0
\(946\) 0 0
\(947\) 306.176 0.323311 0.161656 0.986847i \(-0.448317\pi\)
0.161656 + 0.986847i \(0.448317\pi\)
\(948\) 0 0
\(949\) −26.7217 −0.0281578
\(950\) −853.590 −0.898516
\(951\) 0 0
\(952\) − 2186.03i − 2.29625i
\(953\) 1825.42i 1.91545i 0.287691 + 0.957723i \(0.407112\pi\)
−0.287691 + 0.957723i \(0.592888\pi\)
\(954\) 0 0
\(955\) −248.616 −0.260330
\(956\) 1857.40i 1.94289i
\(957\) 0 0
\(958\) −1238.40 −1.29269
\(959\) − 418.662i − 0.436561i
\(960\) 0 0
\(961\) −721.862 −0.751157
\(962\) −51.3885 −0.0534184
\(963\) 0 0
\(964\) 1757.15i 1.82277i
\(965\) 1711.79i 1.77388i
\(966\) 0 0
\(967\) − 1563.66i − 1.61702i −0.588483 0.808510i \(-0.700275\pi\)
0.588483 0.808510i \(-0.299725\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 4669.99i 4.81442i
\(971\) 107.545 0.110757 0.0553784 0.998465i \(-0.482363\pi\)
0.0553784 + 0.998465i \(0.482363\pi\)
\(972\) 0 0
\(973\) 639.359 0.657101
\(974\) − 1300.72i − 1.33544i
\(975\) 0 0
\(976\) 704.425i 0.721747i
\(977\) 265.623 0.271876 0.135938 0.990717i \(-0.456595\pi\)
0.135938 + 0.990717i \(0.456595\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 1041.55 1.06281
\(981\) 0 0
\(982\) 135.731 0.138219
\(983\) 767.587 0.780862 0.390431 0.920632i \(-0.372326\pi\)
0.390431 + 0.920632i \(0.372326\pi\)
\(984\) 0 0
\(985\) − 1597.94i − 1.62227i
\(986\) 1839.77i 1.86589i
\(987\) 0 0
\(988\) −22.7331 −0.0230092
\(989\) 265.025i 0.267973i
\(990\) 0 0
\(991\) 1137.26 1.14759 0.573796 0.818998i \(-0.305470\pi\)
0.573796 + 0.818998i \(0.305470\pi\)
\(992\) − 299.318i − 0.301732i
\(993\) 0 0
\(994\) 1812.92 1.82386
\(995\) −362.365 −0.364186
\(996\) 0 0
\(997\) 280.401i 0.281245i 0.990063 + 0.140622i \(0.0449104\pi\)
−0.990063 + 0.140622i \(0.955090\pi\)
\(998\) − 681.520i − 0.682886i
\(999\) 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 1089.3.c.d.604.1 4
3.2 odd 2 363.3.c.b.241.4 yes 4
11.10 odd 2 inner 1089.3.c.d.604.4 4
33.2 even 10 363.3.g.c.40.1 16
33.5 odd 10 363.3.g.c.118.1 16
33.8 even 10 363.3.g.c.112.1 16
33.14 odd 10 363.3.g.c.112.4 16
33.17 even 10 363.3.g.c.118.4 16
33.20 odd 10 363.3.g.c.40.4 16
33.26 odd 10 363.3.g.c.94.1 16
33.29 even 10 363.3.g.c.94.4 16
33.32 even 2 363.3.c.b.241.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
363.3.c.b.241.1 4 33.32 even 2
363.3.c.b.241.4 yes 4 3.2 odd 2
363.3.g.c.40.1 16 33.2 even 10
363.3.g.c.40.4 16 33.20 odd 10
363.3.g.c.94.1 16 33.26 odd 10
363.3.g.c.94.4 16 33.29 even 10
363.3.g.c.112.1 16 33.8 even 10
363.3.g.c.112.4 16 33.14 odd 10
363.3.g.c.118.1 16 33.5 odd 10
363.3.g.c.118.4 16 33.17 even 10
1089.3.c.d.604.1 4 1.1 even 1 trivial
1089.3.c.d.604.4 4 11.10 odd 2 inner