Properties

Label 1840.3.k.e.321.5
Level $1840$
Weight $3$
Character 1840.321
Analytic conductor $50.136$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1840,3,Mod(321,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("1840.321");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1840.k (of order \(2\), degree \(1\), not 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: \(50.1363686423\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: no (minimal twist has level 920)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 321.5
Character \(\chi\) \(=\) 1840.321
Dual form 1840.3.k.e.321.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.84326 q^{3} +2.23607i q^{5} -4.25633i q^{7} +14.4572 q^{9} +O(q^{10})\) \(q-4.84326 q^{3} +2.23607i q^{5} -4.25633i q^{7} +14.4572 q^{9} +7.80230i q^{11} -9.89073 q^{13} -10.8299i q^{15} +23.5076i q^{17} -22.1948i q^{19} +20.6145i q^{21} +(-19.3379 - 12.4518i) q^{23} -5.00000 q^{25} -26.4307 q^{27} +57.8083 q^{29} -38.0375 q^{31} -37.7886i q^{33} +9.51744 q^{35} +14.1751i q^{37} +47.9034 q^{39} -18.7893 q^{41} +56.4998i q^{43} +32.3273i q^{45} +62.7323 q^{47} +30.8837 q^{49} -113.854i q^{51} +90.9388i q^{53} -17.4465 q^{55} +107.495i q^{57} -56.2233 q^{59} -4.12348i q^{61} -61.5346i q^{63} -22.1164i q^{65} +116.768i q^{67} +(93.6585 + 60.3072i) q^{69} +84.7000 q^{71} -105.079 q^{73} +24.2163 q^{75} +33.2092 q^{77} -72.7655i q^{79} -2.10402 q^{81} +29.1890i q^{83} -52.5647 q^{85} -279.981 q^{87} -106.120i q^{89} +42.0982i q^{91} +184.226 q^{93} +49.6292 q^{95} -94.8741i q^{97} +112.799i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 128 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 48 q + 128 q^{9} - 8 q^{23} - 240 q^{25} + 72 q^{29} - 32 q^{31} + 40 q^{35} + 96 q^{39} - 104 q^{41} - 128 q^{47} - 344 q^{49} - 80 q^{55} - 248 q^{59} + 292 q^{69} - 208 q^{71} + 224 q^{73} - 288 q^{77} + 184 q^{81} + 48 q^{87} - 672 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −4.84326 −1.61442 −0.807211 0.590263i \(-0.799024\pi\)
−0.807211 + 0.590263i \(0.799024\pi\)
\(4\) 0 0
\(5\) 2.23607i 0.447214i
\(6\) 0 0
\(7\) 4.25633i 0.608047i −0.952665 0.304024i \(-0.901670\pi\)
0.952665 0.304024i \(-0.0983302\pi\)
\(8\) 0 0
\(9\) 14.4572 1.60636
\(10\) 0 0
\(11\) 7.80230i 0.709300i 0.934999 + 0.354650i \(0.115400\pi\)
−0.934999 + 0.354650i \(0.884600\pi\)
\(12\) 0 0
\(13\) −9.89073 −0.760826 −0.380413 0.924817i \(-0.624218\pi\)
−0.380413 + 0.924817i \(0.624218\pi\)
\(14\) 0 0
\(15\) 10.8299i 0.721991i
\(16\) 0 0
\(17\) 23.5076i 1.38280i 0.722471 + 0.691401i \(0.243006\pi\)
−0.722471 + 0.691401i \(0.756994\pi\)
\(18\) 0 0
\(19\) 22.1948i 1.16815i −0.811700 0.584075i \(-0.801457\pi\)
0.811700 0.584075i \(-0.198543\pi\)
\(20\) 0 0
\(21\) 20.6145i 0.981644i
\(22\) 0 0
\(23\) −19.3379 12.4518i −0.840777 0.541381i
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) 0 0
\(27\) −26.4307 −0.978915
\(28\) 0 0
\(29\) 57.8083 1.99339 0.996694 0.0812432i \(-0.0258891\pi\)
0.996694 + 0.0812432i \(0.0258891\pi\)
\(30\) 0 0
\(31\) −38.0375 −1.22702 −0.613508 0.789688i \(-0.710242\pi\)
−0.613508 + 0.789688i \(0.710242\pi\)
\(32\) 0 0
\(33\) 37.7886i 1.14511i
\(34\) 0 0
\(35\) 9.51744 0.271927
\(36\) 0 0
\(37\) 14.1751i 0.383112i 0.981482 + 0.191556i \(0.0613533\pi\)
−0.981482 + 0.191556i \(0.938647\pi\)
\(38\) 0 0
\(39\) 47.9034 1.22829
\(40\) 0 0
\(41\) −18.7893 −0.458275 −0.229137 0.973394i \(-0.573590\pi\)
−0.229137 + 0.973394i \(0.573590\pi\)
\(42\) 0 0
\(43\) 56.4998i 1.31395i 0.753913 + 0.656974i \(0.228164\pi\)
−0.753913 + 0.656974i \(0.771836\pi\)
\(44\) 0 0
\(45\) 32.3273i 0.718384i
\(46\) 0 0
\(47\) 62.7323 1.33473 0.667365 0.744731i \(-0.267422\pi\)
0.667365 + 0.744731i \(0.267422\pi\)
\(48\) 0 0
\(49\) 30.8837 0.630279
\(50\) 0 0
\(51\) 113.854i 2.23242i
\(52\) 0 0
\(53\) 90.9388i 1.71583i 0.513794 + 0.857913i \(0.328239\pi\)
−0.513794 + 0.857913i \(0.671761\pi\)
\(54\) 0 0
\(55\) −17.4465 −0.317209
\(56\) 0 0
\(57\) 107.495i 1.88589i
\(58\) 0 0
\(59\) −56.2233 −0.952937 −0.476469 0.879191i \(-0.658083\pi\)
−0.476469 + 0.879191i \(0.658083\pi\)
\(60\) 0 0
\(61\) 4.12348i 0.0675980i −0.999429 0.0337990i \(-0.989239\pi\)
0.999429 0.0337990i \(-0.0107606\pi\)
\(62\) 0 0
\(63\) 61.5346i 0.976740i
\(64\) 0 0
\(65\) 22.1164i 0.340252i
\(66\) 0 0
\(67\) 116.768i 1.74280i 0.490572 + 0.871401i \(0.336788\pi\)
−0.490572 + 0.871401i \(0.663212\pi\)
\(68\) 0 0
\(69\) 93.6585 + 60.3072i 1.35737 + 0.874017i
\(70\) 0 0
\(71\) 84.7000 1.19296 0.596479 0.802629i \(-0.296566\pi\)
0.596479 + 0.802629i \(0.296566\pi\)
\(72\) 0 0
\(73\) −105.079 −1.43944 −0.719720 0.694265i \(-0.755730\pi\)
−0.719720 + 0.694265i \(0.755730\pi\)
\(74\) 0 0
\(75\) 24.2163 0.322884
\(76\) 0 0
\(77\) 33.2092 0.431288
\(78\) 0 0
\(79\) 72.7655i 0.921083i −0.887638 0.460541i \(-0.847655\pi\)
0.887638 0.460541i \(-0.152345\pi\)
\(80\) 0 0
\(81\) −2.10402 −0.0259756
\(82\) 0 0
\(83\) 29.1890i 0.351675i 0.984419 + 0.175838i \(0.0562633\pi\)
−0.984419 + 0.175838i \(0.943737\pi\)
\(84\) 0 0
\(85\) −52.5647 −0.618408
\(86\) 0 0
\(87\) −279.981 −3.21817
\(88\) 0 0
\(89\) 106.120i 1.19236i −0.802851 0.596180i \(-0.796685\pi\)
0.802851 0.596180i \(-0.203315\pi\)
\(90\) 0 0
\(91\) 42.0982i 0.462618i
\(92\) 0 0
\(93\) 184.226 1.98092
\(94\) 0 0
\(95\) 49.6292 0.522412
\(96\) 0 0
\(97\) 94.8741i 0.978084i −0.872260 0.489042i \(-0.837346\pi\)
0.872260 0.489042i \(-0.162654\pi\)
\(98\) 0 0
\(99\) 112.799i 1.13939i
\(100\) 0 0
\(101\) −28.3320 −0.280514 −0.140257 0.990115i \(-0.544793\pi\)
−0.140257 + 0.990115i \(0.544793\pi\)
\(102\) 0 0
\(103\) 72.5869i 0.704727i −0.935863 0.352363i \(-0.885378\pi\)
0.935863 0.352363i \(-0.114622\pi\)
\(104\) 0 0
\(105\) −46.0955 −0.439005
\(106\) 0 0
\(107\) 4.27988i 0.0399989i 0.999800 + 0.0199994i \(0.00636644\pi\)
−0.999800 + 0.0199994i \(0.993634\pi\)
\(108\) 0 0
\(109\) 114.253i 1.04819i 0.851659 + 0.524096i \(0.175597\pi\)
−0.851659 + 0.524096i \(0.824403\pi\)
\(110\) 0 0
\(111\) 68.6539i 0.618504i
\(112\) 0 0
\(113\) 116.414i 1.03021i −0.857126 0.515106i \(-0.827753\pi\)
0.857126 0.515106i \(-0.172247\pi\)
\(114\) 0 0
\(115\) 27.8430 43.2408i 0.242113 0.376007i
\(116\) 0 0
\(117\) −142.992 −1.22216
\(118\) 0 0
\(119\) 100.056 0.840809
\(120\) 0 0
\(121\) 60.1241 0.496894
\(122\) 0 0
\(123\) 91.0013 0.739848
\(124\) 0 0
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) −168.737 −1.32864 −0.664318 0.747450i \(-0.731278\pi\)
−0.664318 + 0.747450i \(0.731278\pi\)
\(128\) 0 0
\(129\) 273.643i 2.12127i
\(130\) 0 0
\(131\) 64.8148 0.494769 0.247385 0.968917i \(-0.420429\pi\)
0.247385 + 0.968917i \(0.420429\pi\)
\(132\) 0 0
\(133\) −94.4686 −0.710290
\(134\) 0 0
\(135\) 59.1008i 0.437784i
\(136\) 0 0
\(137\) 238.587i 1.74151i −0.491717 0.870755i \(-0.663631\pi\)
0.491717 0.870755i \(-0.336369\pi\)
\(138\) 0 0
\(139\) −124.256 −0.893929 −0.446965 0.894552i \(-0.647495\pi\)
−0.446965 + 0.894552i \(0.647495\pi\)
\(140\) 0 0
\(141\) −303.829 −2.15482
\(142\) 0 0
\(143\) 77.1705i 0.539654i
\(144\) 0 0
\(145\) 129.263i 0.891470i
\(146\) 0 0
\(147\) −149.578 −1.01754
\(148\) 0 0
\(149\) 206.750i 1.38759i −0.720174 0.693794i \(-0.755938\pi\)
0.720174 0.693794i \(-0.244062\pi\)
\(150\) 0 0
\(151\) −18.4800 −0.122384 −0.0611919 0.998126i \(-0.519490\pi\)
−0.0611919 + 0.998126i \(0.519490\pi\)
\(152\) 0 0
\(153\) 339.855i 2.22127i
\(154\) 0 0
\(155\) 85.0545i 0.548739i
\(156\) 0 0
\(157\) 239.367i 1.52463i −0.647205 0.762316i \(-0.724062\pi\)
0.647205 0.762316i \(-0.275938\pi\)
\(158\) 0 0
\(159\) 440.441i 2.77007i
\(160\) 0 0
\(161\) −52.9988 + 82.3084i −0.329185 + 0.511232i
\(162\) 0 0
\(163\) 17.6526 0.108298 0.0541489 0.998533i \(-0.482755\pi\)
0.0541489 + 0.998533i \(0.482755\pi\)
\(164\) 0 0
\(165\) 84.4979 0.512108
\(166\) 0 0
\(167\) −144.926 −0.867819 −0.433909 0.900956i \(-0.642866\pi\)
−0.433909 + 0.900956i \(0.642866\pi\)
\(168\) 0 0
\(169\) −71.1734 −0.421144
\(170\) 0 0
\(171\) 320.875i 1.87646i
\(172\) 0 0
\(173\) −7.90140 −0.0456729 −0.0228364 0.999739i \(-0.507270\pi\)
−0.0228364 + 0.999739i \(0.507270\pi\)
\(174\) 0 0
\(175\) 21.2816i 0.121609i
\(176\) 0 0
\(177\) 272.304 1.53844
\(178\) 0 0
\(179\) −4.93306 −0.0275590 −0.0137795 0.999905i \(-0.504386\pi\)
−0.0137795 + 0.999905i \(0.504386\pi\)
\(180\) 0 0
\(181\) 6.10508i 0.0337297i 0.999858 + 0.0168649i \(0.00536850\pi\)
−0.999858 + 0.0168649i \(0.994631\pi\)
\(182\) 0 0
\(183\) 19.9711i 0.109132i
\(184\) 0 0
\(185\) −31.6966 −0.171333
\(186\) 0 0
\(187\) −183.414 −0.980821
\(188\) 0 0
\(189\) 112.498i 0.595226i
\(190\) 0 0
\(191\) 136.948i 0.717004i −0.933529 0.358502i \(-0.883288\pi\)
0.933529 0.358502i \(-0.116712\pi\)
\(192\) 0 0
\(193\) 63.4162 0.328582 0.164291 0.986412i \(-0.447466\pi\)
0.164291 + 0.986412i \(0.447466\pi\)
\(194\) 0 0
\(195\) 107.115i 0.549309i
\(196\) 0 0
\(197\) 30.3151 0.153884 0.0769418 0.997036i \(-0.475484\pi\)
0.0769418 + 0.997036i \(0.475484\pi\)
\(198\) 0 0
\(199\) 170.671i 0.857646i −0.903389 0.428823i \(-0.858928\pi\)
0.903389 0.428823i \(-0.141072\pi\)
\(200\) 0 0
\(201\) 565.537i 2.81362i
\(202\) 0 0
\(203\) 246.051i 1.21207i
\(204\) 0 0
\(205\) 42.0141i 0.204947i
\(206\) 0 0
\(207\) −279.572 180.018i −1.35059 0.869651i
\(208\) 0 0
\(209\) 173.171 0.828568
\(210\) 0 0
\(211\) −70.1721 −0.332569 −0.166285 0.986078i \(-0.553177\pi\)
−0.166285 + 0.986078i \(0.553177\pi\)
\(212\) 0 0
\(213\) −410.224 −1.92594
\(214\) 0 0
\(215\) −126.337 −0.587616
\(216\) 0 0
\(217\) 161.900i 0.746084i
\(218\) 0 0
\(219\) 508.926 2.32386
\(220\) 0 0
\(221\) 232.508i 1.05207i
\(222\) 0 0
\(223\) 42.2122 0.189292 0.0946461 0.995511i \(-0.469828\pi\)
0.0946461 + 0.995511i \(0.469828\pi\)
\(224\) 0 0
\(225\) −72.2860 −0.321271
\(226\) 0 0
\(227\) 134.443i 0.592258i 0.955148 + 0.296129i \(0.0956958\pi\)
−0.955148 + 0.296129i \(0.904304\pi\)
\(228\) 0 0
\(229\) 329.622i 1.43940i −0.694288 0.719698i \(-0.744280\pi\)
0.694288 0.719698i \(-0.255720\pi\)
\(230\) 0 0
\(231\) −160.841 −0.696280
\(232\) 0 0
\(233\) −205.310 −0.881157 −0.440579 0.897714i \(-0.645227\pi\)
−0.440579 + 0.897714i \(0.645227\pi\)
\(234\) 0 0
\(235\) 140.274i 0.596910i
\(236\) 0 0
\(237\) 352.423i 1.48702i
\(238\) 0 0
\(239\) −345.287 −1.44471 −0.722357 0.691520i \(-0.756941\pi\)
−0.722357 + 0.691520i \(0.756941\pi\)
\(240\) 0 0
\(241\) 478.152i 1.98403i 0.126109 + 0.992016i \(0.459751\pi\)
−0.126109 + 0.992016i \(0.540249\pi\)
\(242\) 0 0
\(243\) 248.067 1.02085
\(244\) 0 0
\(245\) 69.0580i 0.281869i
\(246\) 0 0
\(247\) 219.523i 0.888758i
\(248\) 0 0
\(249\) 141.370i 0.567752i
\(250\) 0 0
\(251\) 39.5773i 0.157679i 0.996887 + 0.0788393i \(0.0251214\pi\)
−0.996887 + 0.0788393i \(0.974879\pi\)
\(252\) 0 0
\(253\) 97.1524 150.880i 0.384001 0.596363i
\(254\) 0 0
\(255\) 254.585 0.998371
\(256\) 0 0
\(257\) −327.429 −1.27404 −0.637022 0.770846i \(-0.719834\pi\)
−0.637022 + 0.770846i \(0.719834\pi\)
\(258\) 0 0
\(259\) 60.3341 0.232950
\(260\) 0 0
\(261\) 835.746 3.20209
\(262\) 0 0
\(263\) 262.464i 0.997961i 0.866613 + 0.498980i \(0.166292\pi\)
−0.866613 + 0.498980i \(0.833708\pi\)
\(264\) 0 0
\(265\) −203.345 −0.767341
\(266\) 0 0
\(267\) 513.967i 1.92497i
\(268\) 0 0
\(269\) −78.9203 −0.293384 −0.146692 0.989182i \(-0.546863\pi\)
−0.146692 + 0.989182i \(0.546863\pi\)
\(270\) 0 0
\(271\) 297.037 1.09608 0.548039 0.836453i \(-0.315375\pi\)
0.548039 + 0.836453i \(0.315375\pi\)
\(272\) 0 0
\(273\) 203.893i 0.746860i
\(274\) 0 0
\(275\) 39.0115i 0.141860i
\(276\) 0 0
\(277\) −522.010 −1.88451 −0.942256 0.334893i \(-0.891300\pi\)
−0.942256 + 0.334893i \(0.891300\pi\)
\(278\) 0 0
\(279\) −549.916 −1.97103
\(280\) 0 0
\(281\) 374.374i 1.33229i −0.745821 0.666146i \(-0.767943\pi\)
0.745821 0.666146i \(-0.232057\pi\)
\(282\) 0 0
\(283\) 561.471i 1.98400i −0.126255 0.991998i \(-0.540296\pi\)
0.126255 0.991998i \(-0.459704\pi\)
\(284\) 0 0
\(285\) −240.367 −0.843394
\(286\) 0 0
\(287\) 79.9733i 0.278653i
\(288\) 0 0
\(289\) −263.609 −0.912141
\(290\) 0 0
\(291\) 459.501i 1.57904i
\(292\) 0 0
\(293\) 114.848i 0.391972i −0.980607 0.195986i \(-0.937209\pi\)
0.980607 0.195986i \(-0.0627907\pi\)
\(294\) 0 0
\(295\) 125.719i 0.426166i
\(296\) 0 0
\(297\) 206.220i 0.694344i
\(298\) 0 0
\(299\) 191.266 + 123.157i 0.639685 + 0.411896i
\(300\) 0 0
\(301\) 240.482 0.798943
\(302\) 0 0
\(303\) 137.219 0.452868
\(304\) 0 0
\(305\) 9.22038 0.0302307
\(306\) 0 0
\(307\) 204.415 0.665848 0.332924 0.942954i \(-0.391965\pi\)
0.332924 + 0.942954i \(0.391965\pi\)
\(308\) 0 0
\(309\) 351.557i 1.13773i
\(310\) 0 0
\(311\) −57.5890 −0.185173 −0.0925867 0.995705i \(-0.529514\pi\)
−0.0925867 + 0.995705i \(0.529514\pi\)
\(312\) 0 0
\(313\) 174.612i 0.557865i −0.960311 0.278933i \(-0.910019\pi\)
0.960311 0.278933i \(-0.0899806\pi\)
\(314\) 0 0
\(315\) 137.596 0.436812
\(316\) 0 0
\(317\) −302.355 −0.953802 −0.476901 0.878957i \(-0.658240\pi\)
−0.476901 + 0.878957i \(0.658240\pi\)
\(318\) 0 0
\(319\) 451.037i 1.41391i
\(320\) 0 0
\(321\) 20.7286i 0.0645750i
\(322\) 0 0
\(323\) 521.748 1.61532
\(324\) 0 0
\(325\) 49.4537 0.152165
\(326\) 0 0
\(327\) 553.357i 1.69222i
\(328\) 0 0
\(329\) 267.009i 0.811579i
\(330\) 0 0
\(331\) −534.227 −1.61398 −0.806989 0.590566i \(-0.798905\pi\)
−0.806989 + 0.590566i \(0.798905\pi\)
\(332\) 0 0
\(333\) 204.933i 0.615414i
\(334\) 0 0
\(335\) −261.100 −0.779404
\(336\) 0 0
\(337\) 72.7809i 0.215967i 0.994153 + 0.107984i \(0.0344394\pi\)
−0.994153 + 0.107984i \(0.965561\pi\)
\(338\) 0 0
\(339\) 563.824i 1.66320i
\(340\) 0 0
\(341\) 296.780i 0.870323i
\(342\) 0 0
\(343\) 340.011i 0.991286i
\(344\) 0 0
\(345\) −134.851 + 209.427i −0.390872 + 0.607034i
\(346\) 0 0
\(347\) −511.199 −1.47319 −0.736597 0.676332i \(-0.763569\pi\)
−0.736597 + 0.676332i \(0.763569\pi\)
\(348\) 0 0
\(349\) 303.181 0.868715 0.434357 0.900741i \(-0.356975\pi\)
0.434357 + 0.900741i \(0.356975\pi\)
\(350\) 0 0
\(351\) 261.419 0.744783
\(352\) 0 0
\(353\) 134.926 0.382226 0.191113 0.981568i \(-0.438790\pi\)
0.191113 + 0.981568i \(0.438790\pi\)
\(354\) 0 0
\(355\) 189.395i 0.533507i
\(356\) 0 0
\(357\) −484.599 −1.35742
\(358\) 0 0
\(359\) 697.585i 1.94313i 0.236765 + 0.971567i \(0.423913\pi\)
−0.236765 + 0.971567i \(0.576087\pi\)
\(360\) 0 0
\(361\) −131.611 −0.364573
\(362\) 0 0
\(363\) −291.197 −0.802196
\(364\) 0 0
\(365\) 234.964i 0.643737i
\(366\) 0 0
\(367\) 0.324838i 0.000885116i −1.00000 0.000442558i \(-0.999859\pi\)
1.00000 0.000442558i \(-0.000140871\pi\)
\(368\) 0 0
\(369\) −271.640 −0.736152
\(370\) 0 0
\(371\) 387.066 1.04330
\(372\) 0 0
\(373\) 458.513i 1.22926i −0.788817 0.614629i \(-0.789306\pi\)
0.788817 0.614629i \(-0.210694\pi\)
\(374\) 0 0
\(375\) 54.1493i 0.144398i
\(376\) 0 0
\(377\) −571.766 −1.51662
\(378\) 0 0
\(379\) 336.529i 0.887940i −0.896042 0.443970i \(-0.853570\pi\)
0.896042 0.443970i \(-0.146430\pi\)
\(380\) 0 0
\(381\) 817.237 2.14498
\(382\) 0 0
\(383\) 25.7590i 0.0672558i 0.999434 + 0.0336279i \(0.0107061\pi\)
−0.999434 + 0.0336279i \(0.989294\pi\)
\(384\) 0 0
\(385\) 74.2579i 0.192878i
\(386\) 0 0
\(387\) 816.829i 2.11067i
\(388\) 0 0
\(389\) 236.383i 0.607669i 0.952725 + 0.303834i \(0.0982669\pi\)
−0.952725 + 0.303834i \(0.901733\pi\)
\(390\) 0 0
\(391\) 292.711 454.588i 0.748622 1.16263i
\(392\) 0 0
\(393\) −313.915 −0.798766
\(394\) 0 0
\(395\) 162.709 0.411921
\(396\) 0 0
\(397\) 231.062 0.582021 0.291010 0.956720i \(-0.406009\pi\)
0.291010 + 0.956720i \(0.406009\pi\)
\(398\) 0 0
\(399\) 457.536 1.14671
\(400\) 0 0
\(401\) 391.881i 0.977260i −0.872491 0.488630i \(-0.837497\pi\)
0.872491 0.488630i \(-0.162503\pi\)
\(402\) 0 0
\(403\) 376.219 0.933546
\(404\) 0 0
\(405\) 4.70474i 0.0116166i
\(406\) 0 0
\(407\) −110.599 −0.271741
\(408\) 0 0
\(409\) 20.5445 0.0502309 0.0251155 0.999685i \(-0.492005\pi\)
0.0251155 + 0.999685i \(0.492005\pi\)
\(410\) 0 0
\(411\) 1155.54i 2.81153i
\(412\) 0 0
\(413\) 239.305i 0.579431i
\(414\) 0 0
\(415\) −65.2687 −0.157274
\(416\) 0 0
\(417\) 601.805 1.44318
\(418\) 0 0
\(419\) 70.9780i 0.169399i −0.996407 0.0846993i \(-0.973007\pi\)
0.996407 0.0846993i \(-0.0269930\pi\)
\(420\) 0 0
\(421\) 218.160i 0.518195i −0.965851 0.259098i \(-0.916575\pi\)
0.965851 0.259098i \(-0.0834252\pi\)
\(422\) 0 0
\(423\) 906.934 2.14405
\(424\) 0 0
\(425\) 117.538i 0.276560i
\(426\) 0 0
\(427\) −17.5509 −0.0411028
\(428\) 0 0
\(429\) 373.757i 0.871228i
\(430\) 0 0
\(431\) 13.6479i 0.0316658i 0.999875 + 0.0158329i \(0.00503997\pi\)
−0.999875 + 0.0158329i \(0.994960\pi\)
\(432\) 0 0
\(433\) 596.772i 1.37823i −0.724654 0.689113i \(-0.758000\pi\)
0.724654 0.689113i \(-0.242000\pi\)
\(434\) 0 0
\(435\) 626.056i 1.43921i
\(436\) 0 0
\(437\) −276.365 + 429.201i −0.632414 + 0.982154i
\(438\) 0 0
\(439\) 395.203 0.900234 0.450117 0.892970i \(-0.351382\pi\)
0.450117 + 0.892970i \(0.351382\pi\)
\(440\) 0 0
\(441\) 446.491 1.01245
\(442\) 0 0
\(443\) 641.179 1.44736 0.723678 0.690137i \(-0.242450\pi\)
0.723678 + 0.690137i \(0.242450\pi\)
\(444\) 0 0
\(445\) 237.292 0.533240
\(446\) 0 0
\(447\) 1001.35i 2.24015i
\(448\) 0 0
\(449\) 457.160 1.01817 0.509087 0.860715i \(-0.329983\pi\)
0.509087 + 0.860715i \(0.329983\pi\)
\(450\) 0 0
\(451\) 146.599i 0.325054i
\(452\) 0 0
\(453\) 89.5033 0.197579
\(454\) 0 0
\(455\) −94.1345 −0.206889
\(456\) 0 0
\(457\) 478.890i 1.04790i 0.851749 + 0.523949i \(0.175542\pi\)
−0.851749 + 0.523949i \(0.824458\pi\)
\(458\) 0 0
\(459\) 621.323i 1.35364i
\(460\) 0 0
\(461\) 734.375 1.59300 0.796502 0.604636i \(-0.206681\pi\)
0.796502 + 0.604636i \(0.206681\pi\)
\(462\) 0 0
\(463\) −500.834 −1.08171 −0.540857 0.841115i \(-0.681900\pi\)
−0.540857 + 0.841115i \(0.681900\pi\)
\(464\) 0 0
\(465\) 411.941i 0.885895i
\(466\) 0 0
\(467\) 21.4457i 0.0459223i 0.999736 + 0.0229612i \(0.00730941\pi\)
−0.999736 + 0.0229612i \(0.992691\pi\)
\(468\) 0 0
\(469\) 497.002 1.05971
\(470\) 0 0
\(471\) 1159.32i 2.46140i
\(472\) 0 0
\(473\) −440.828 −0.931984
\(474\) 0 0
\(475\) 110.974i 0.233630i
\(476\) 0 0
\(477\) 1314.72i 2.75623i
\(478\) 0 0
\(479\) 333.257i 0.695735i −0.937544 0.347867i \(-0.886906\pi\)
0.937544 0.347867i \(-0.113094\pi\)
\(480\) 0 0
\(481\) 140.202i 0.291481i
\(482\) 0 0
\(483\) 256.687 398.641i 0.531443 0.825344i
\(484\) 0 0
\(485\) 212.145 0.437412
\(486\) 0 0
\(487\) 522.920 1.07376 0.536879 0.843659i \(-0.319603\pi\)
0.536879 + 0.843659i \(0.319603\pi\)
\(488\) 0 0
\(489\) −85.4960 −0.174838
\(490\) 0 0
\(491\) −107.040 −0.218005 −0.109002 0.994041i \(-0.534766\pi\)
−0.109002 + 0.994041i \(0.534766\pi\)
\(492\) 0 0
\(493\) 1358.94i 2.75646i
\(494\) 0 0
\(495\) −252.227 −0.509550
\(496\) 0 0
\(497\) 360.511i 0.725374i
\(498\) 0 0
\(499\) −655.185 −1.31300 −0.656498 0.754328i \(-0.727963\pi\)
−0.656498 + 0.754328i \(0.727963\pi\)
\(500\) 0 0
\(501\) 701.914 1.40103
\(502\) 0 0
\(503\) 241.825i 0.480766i 0.970678 + 0.240383i \(0.0772731\pi\)
−0.970678 + 0.240383i \(0.922727\pi\)
\(504\) 0 0
\(505\) 63.3522i 0.125450i
\(506\) 0 0
\(507\) 344.712 0.679905
\(508\) 0 0
\(509\) −432.806 −0.850307 −0.425153 0.905121i \(-0.639780\pi\)
−0.425153 + 0.905121i \(0.639780\pi\)
\(510\) 0 0
\(511\) 447.251i 0.875247i
\(512\) 0 0
\(513\) 586.625i 1.14352i
\(514\) 0 0
\(515\) 162.309 0.315163
\(516\) 0 0
\(517\) 489.456i 0.946724i
\(518\) 0 0
\(519\) 38.2686 0.0737352
\(520\) 0 0
\(521\) 49.2636i 0.0945558i 0.998882 + 0.0472779i \(0.0150546\pi\)
−0.998882 + 0.0472779i \(0.984945\pi\)
\(522\) 0 0
\(523\) 184.254i 0.352303i 0.984363 + 0.176151i \(0.0563649\pi\)
−0.984363 + 0.176151i \(0.943635\pi\)
\(524\) 0 0
\(525\) 103.073i 0.196329i
\(526\) 0 0
\(527\) 894.172i 1.69672i
\(528\) 0 0
\(529\) 218.907 + 481.581i 0.413813 + 0.910362i
\(530\) 0 0
\(531\) −812.832 −1.53076
\(532\) 0 0
\(533\) 185.840 0.348667
\(534\) 0 0
\(535\) −9.57010 −0.0178880
\(536\) 0 0
\(537\) 23.8921 0.0444918
\(538\) 0 0
\(539\) 240.963i 0.447057i
\(540\) 0 0
\(541\) 411.725 0.761044 0.380522 0.924772i \(-0.375744\pi\)
0.380522 + 0.924772i \(0.375744\pi\)
\(542\) 0 0
\(543\) 29.5685i 0.0544540i
\(544\) 0 0
\(545\) −255.477 −0.468765
\(546\) 0 0
\(547\) 388.425 0.710101 0.355050 0.934847i \(-0.384464\pi\)
0.355050 + 0.934847i \(0.384464\pi\)
\(548\) 0 0
\(549\) 59.6140i 0.108586i
\(550\) 0 0
\(551\) 1283.05i 2.32858i
\(552\) 0 0
\(553\) −309.714 −0.560062
\(554\) 0 0
\(555\) 153.515 0.276603
\(556\) 0 0
\(557\) 87.2673i 0.156674i −0.996927 0.0783369i \(-0.975039\pi\)
0.996927 0.0783369i \(-0.0249610\pi\)
\(558\) 0 0
\(559\) 558.824i 0.999686i
\(560\) 0 0
\(561\) 888.320 1.58346
\(562\) 0 0
\(563\) 1061.94i 1.88621i −0.332495 0.943105i \(-0.607891\pi\)
0.332495 0.943105i \(-0.392109\pi\)
\(564\) 0 0
\(565\) 260.310 0.460725
\(566\) 0 0
\(567\) 8.95542i 0.0157944i
\(568\) 0 0
\(569\) 89.2879i 0.156921i −0.996917 0.0784604i \(-0.975000\pi\)
0.996917 0.0784604i \(-0.0250004\pi\)
\(570\) 0 0
\(571\) 378.800i 0.663398i 0.943385 + 0.331699i \(0.107622\pi\)
−0.943385 + 0.331699i \(0.892378\pi\)
\(572\) 0 0
\(573\) 663.274i 1.15755i
\(574\) 0 0
\(575\) 96.6894 + 62.2588i 0.168155 + 0.108276i
\(576\) 0 0
\(577\) 512.166 0.887635 0.443818 0.896117i \(-0.353624\pi\)
0.443818 + 0.896117i \(0.353624\pi\)
\(578\) 0 0
\(579\) −307.142 −0.530469
\(580\) 0 0
\(581\) 124.238 0.213835
\(582\) 0 0
\(583\) −709.532 −1.21704
\(584\) 0 0
\(585\) 319.741i 0.546565i
\(586\) 0 0
\(587\) −915.337 −1.55935 −0.779674 0.626186i \(-0.784615\pi\)
−0.779674 + 0.626186i \(0.784615\pi\)
\(588\) 0 0
\(589\) 844.237i 1.43334i
\(590\) 0 0
\(591\) −146.824 −0.248433
\(592\) 0 0
\(593\) −522.001 −0.880272 −0.440136 0.897931i \(-0.645070\pi\)
−0.440136 + 0.897931i \(0.645070\pi\)
\(594\) 0 0
\(595\) 223.733i 0.376021i
\(596\) 0 0
\(597\) 826.607i 1.38460i
\(598\) 0 0
\(599\) 431.271 0.719985 0.359993 0.932955i \(-0.382779\pi\)
0.359993 + 0.932955i \(0.382779\pi\)
\(600\) 0 0
\(601\) −24.3935 −0.0405882 −0.0202941 0.999794i \(-0.506460\pi\)
−0.0202941 + 0.999794i \(0.506460\pi\)
\(602\) 0 0
\(603\) 1688.13i 2.79956i
\(604\) 0 0
\(605\) 134.442i 0.222218i
\(606\) 0 0
\(607\) −618.034 −1.01818 −0.509089 0.860714i \(-0.670018\pi\)
−0.509089 + 0.860714i \(0.670018\pi\)
\(608\) 0 0
\(609\) 1191.69i 1.95680i
\(610\) 0 0
\(611\) −620.469 −1.01550
\(612\) 0 0
\(613\) 997.907i 1.62791i 0.580930 + 0.813954i \(0.302689\pi\)
−0.580930 + 0.813954i \(0.697311\pi\)
\(614\) 0 0
\(615\) 203.485i 0.330870i
\(616\) 0 0
\(617\) 797.346i 1.29230i −0.763212 0.646148i \(-0.776379\pi\)
0.763212 0.646148i \(-0.223621\pi\)
\(618\) 0 0
\(619\) 667.584i 1.07849i −0.842150 0.539244i \(-0.818710\pi\)
0.842150 0.539244i \(-0.181290\pi\)
\(620\) 0 0
\(621\) 511.114 + 329.109i 0.823049 + 0.529966i
\(622\) 0 0
\(623\) −451.682 −0.725011
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) −838.712 −1.33766
\(628\) 0 0
\(629\) −333.224 −0.529768
\(630\) 0 0
\(631\) 2.47756i 0.00392641i −0.999998 0.00196320i \(-0.999375\pi\)
0.999998 0.00196320i \(-0.000624908\pi\)
\(632\) 0 0
\(633\) 339.862 0.536907
\(634\) 0 0
\(635\) 377.307i 0.594184i
\(636\) 0 0
\(637\) −305.462 −0.479532
\(638\) 0 0
\(639\) 1224.53 1.91631
\(640\) 0 0
\(641\) 171.378i 0.267361i 0.991024 + 0.133680i \(0.0426796\pi\)
−0.991024 + 0.133680i \(0.957320\pi\)
\(642\) 0 0
\(643\) 879.530i 1.36785i −0.729551 0.683927i \(-0.760271\pi\)
0.729551 0.683927i \(-0.239729\pi\)
\(644\) 0 0
\(645\) 611.885 0.948660
\(646\) 0 0
\(647\) 465.470 0.719428 0.359714 0.933063i \(-0.382874\pi\)
0.359714 + 0.933063i \(0.382874\pi\)
\(648\) 0 0
\(649\) 438.671i 0.675918i
\(650\) 0 0
\(651\) 784.126i 1.20449i
\(652\) 0 0
\(653\) −868.579 −1.33014 −0.665068 0.746782i \(-0.731598\pi\)
−0.665068 + 0.746782i \(0.731598\pi\)
\(654\) 0 0
\(655\) 144.930i 0.221268i
\(656\) 0 0
\(657\) −1519.15 −2.31225
\(658\) 0 0
\(659\) 216.285i 0.328202i −0.986444 0.164101i \(-0.947528\pi\)
0.986444 0.164101i \(-0.0524723\pi\)
\(660\) 0 0
\(661\) 650.905i 0.984727i −0.870390 0.492363i \(-0.836133\pi\)
0.870390 0.492363i \(-0.163867\pi\)
\(662\) 0 0
\(663\) 1126.10i 1.69849i
\(664\) 0 0
\(665\) 211.238i 0.317651i
\(666\) 0 0
\(667\) −1117.89 719.815i −1.67600 1.07918i
\(668\) 0 0
\(669\) −204.445 −0.305597
\(670\) 0 0
\(671\) 32.1726 0.0479473
\(672\) 0 0
\(673\) −431.470 −0.641114 −0.320557 0.947229i \(-0.603870\pi\)
−0.320557 + 0.947229i \(0.603870\pi\)
\(674\) 0 0
\(675\) 132.153 0.195783
\(676\) 0 0
\(677\) 912.137i 1.34732i −0.739041 0.673661i \(-0.764721\pi\)
0.739041 0.673661i \(-0.235279\pi\)
\(678\) 0 0
\(679\) −403.816 −0.594721
\(680\) 0 0
\(681\) 651.141i 0.956154i
\(682\) 0 0
\(683\) 635.519 0.930481 0.465241 0.885184i \(-0.345968\pi\)
0.465241 + 0.885184i \(0.345968\pi\)
\(684\) 0 0
\(685\) 533.496 0.778827
\(686\) 0 0
\(687\) 1596.44i 2.32379i
\(688\) 0 0
\(689\) 899.452i 1.30544i
\(690\) 0 0
\(691\) 157.061 0.227295 0.113648 0.993521i \(-0.463747\pi\)
0.113648 + 0.993521i \(0.463747\pi\)
\(692\) 0 0
\(693\) 480.112 0.692802
\(694\) 0 0
\(695\) 277.845i 0.399777i
\(696\) 0 0
\(697\) 441.691i 0.633703i
\(698\) 0 0
\(699\) 994.369 1.42256
\(700\) 0 0
\(701\) 458.204i 0.653643i 0.945086 + 0.326821i \(0.105978\pi\)
−0.945086 + 0.326821i \(0.894022\pi\)
\(702\) 0 0
\(703\) 314.615 0.447532
\(704\) 0 0
\(705\) 679.383i 0.963664i
\(706\) 0 0
\(707\) 120.590i 0.170566i
\(708\) 0 0
\(709\) 745.366i 1.05129i −0.850703 0.525646i \(-0.823824\pi\)
0.850703 0.525646i \(-0.176176\pi\)
\(710\) 0 0
\(711\) 1051.99i 1.47959i
\(712\) 0 0
\(713\) 735.565 + 473.634i 1.03165 + 0.664283i
\(714\) 0 0
\(715\) 172.558 0.241340
\(716\) 0 0
\(717\) 1672.31 2.33238
\(718\) 0 0
\(719\) −308.243 −0.428711 −0.214355 0.976756i \(-0.568765\pi\)
−0.214355 + 0.976756i \(0.568765\pi\)
\(720\) 0 0
\(721\) −308.954 −0.428507
\(722\) 0 0
\(723\) 2315.82i 3.20307i
\(724\) 0 0
\(725\) −289.041 −0.398678
\(726\) 0 0
\(727\) 221.410i 0.304553i −0.988338 0.152276i \(-0.951340\pi\)
0.988338 0.152276i \(-0.0486604\pi\)
\(728\) 0 0
\(729\) −1182.52 −1.62211
\(730\) 0 0
\(731\) −1328.18 −1.81693
\(732\) 0 0
\(733\) 1103.22i 1.50508i 0.658548 + 0.752539i \(0.271171\pi\)
−0.658548 + 0.752539i \(0.728829\pi\)
\(734\) 0 0
\(735\) 334.466i 0.455056i
\(736\) 0 0
\(737\) −911.056 −1.23617
\(738\) 0 0
\(739\) −1170.21 −1.58351 −0.791754 0.610841i \(-0.790832\pi\)
−0.791754 + 0.610841i \(0.790832\pi\)
\(740\) 0 0
\(741\) 1063.21i 1.43483i
\(742\) 0 0
\(743\) 1463.99i 1.97038i 0.171460 + 0.985191i \(0.445151\pi\)
−0.171460 + 0.985191i \(0.554849\pi\)
\(744\) 0 0
\(745\) 462.308 0.620548
\(746\) 0 0
\(747\) 421.992i 0.564916i
\(748\) 0 0
\(749\) 18.2166 0.0243212
\(750\) 0 0
\(751\) 127.395i 0.169633i −0.996397 0.0848166i \(-0.972970\pi\)
0.996397 0.0848166i \(-0.0270304\pi\)
\(752\) 0 0
\(753\) 191.683i 0.254560i
\(754\) 0 0
\(755\) 41.3224i 0.0547317i
\(756\) 0 0
\(757\) 896.048i 1.18368i 0.806055 + 0.591841i \(0.201599\pi\)
−0.806055 + 0.591841i \(0.798401\pi\)
\(758\) 0 0
\(759\) −470.535 + 730.751i −0.619940 + 0.962782i
\(760\) 0 0
\(761\) −1141.19 −1.49959 −0.749797 0.661667i \(-0.769849\pi\)
−0.749797 + 0.661667i \(0.769849\pi\)
\(762\) 0 0
\(763\) 486.298 0.637350
\(764\) 0 0
\(765\) −759.938 −0.993383
\(766\) 0 0
\(767\) 556.090 0.725019
\(768\) 0 0
\(769\) 15.1581i 0.0197115i −0.999951 0.00985575i \(-0.996863\pi\)
0.999951 0.00985575i \(-0.00313723\pi\)
\(770\) 0 0
\(771\) 1585.83 2.05684
\(772\) 0 0
\(773\) 923.266i 1.19439i −0.802095 0.597197i \(-0.796281\pi\)
0.802095 0.597197i \(-0.203719\pi\)
\(774\) 0 0
\(775\) 190.188 0.245403
\(776\) 0 0
\(777\) −292.214 −0.376079
\(778\) 0 0
\(779\) 417.025i 0.535333i
\(780\) 0 0
\(781\) 660.855i 0.846165i
\(782\) 0 0
\(783\) −1527.91 −1.95136
\(784\) 0 0
\(785\) 535.241 0.681836
\(786\) 0 0
\(787\) 679.117i 0.862919i −0.902132 0.431459i \(-0.857999\pi\)
0.902132 0.431459i \(-0.142001\pi\)
\(788\) 0 0
\(789\) 1271.18i 1.61113i
\(790\) 0 0
\(791\) −495.496 −0.626417
\(792\) 0 0
\(793\) 40.7842i 0.0514303i
\(794\) 0 0
\(795\) 984.855 1.23881
\(796\) 0 0
\(797\) 962.620i 1.20780i 0.797059 + 0.603902i \(0.206388\pi\)
−0.797059 + 0.603902i \(0.793612\pi\)
\(798\) 0 0
\(799\) 1474.69i 1.84567i
\(800\) 0 0
\(801\) 1534.20i 1.91535i
\(802\) 0 0
\(803\) 819.858i 1.02099i
\(804\) 0 0
\(805\) −184.047 118.509i −0.228630 0.147216i
\(806\) 0 0
\(807\) 382.232 0.473645
\(808\) 0 0
\(809\) 263.301 0.325464 0.162732 0.986670i \(-0.447969\pi\)
0.162732 + 0.986670i \(0.447969\pi\)
\(810\) 0 0
\(811\) −352.086 −0.434139 −0.217069 0.976156i \(-0.569650\pi\)
−0.217069 + 0.976156i \(0.569650\pi\)
\(812\) 0 0
\(813\) −1438.63 −1.76953
\(814\) 0 0
\(815\) 39.4723i 0.0484323i
\(816\) 0 0
\(817\) 1254.00 1.53489
\(818\) 0 0
\(819\) 608.623i 0.743129i
\(820\) 0 0
\(821\) 134.116 0.163357 0.0816786 0.996659i \(-0.473972\pi\)
0.0816786 + 0.996659i \(0.473972\pi\)
\(822\) 0 0
\(823\) −526.603 −0.639858 −0.319929 0.947441i \(-0.603659\pi\)
−0.319929 + 0.947441i \(0.603659\pi\)
\(824\) 0 0
\(825\) 188.943i 0.229022i
\(826\) 0 0
\(827\) 533.202i 0.644743i −0.946613 0.322371i \(-0.895520\pi\)
0.946613 0.322371i \(-0.104480\pi\)
\(828\) 0 0
\(829\) 208.483 0.251487 0.125744 0.992063i \(-0.459868\pi\)
0.125744 + 0.992063i \(0.459868\pi\)
\(830\) 0 0
\(831\) 2528.23 3.04240
\(832\) 0 0
\(833\) 726.002i 0.871550i
\(834\) 0 0
\(835\) 324.064i 0.388100i
\(836\) 0 0
\(837\) 1005.36 1.20114
\(838\) 0 0
\(839\) 470.155i 0.560376i 0.959945 + 0.280188i \(0.0903968\pi\)
−0.959945 + 0.280188i \(0.909603\pi\)
\(840\) 0 0
\(841\) 2500.80 2.97360
\(842\) 0 0
\(843\) 1813.19i 2.15088i
\(844\) 0 0
\(845\) 159.149i 0.188341i
\(846\) 0 0
\(847\) 255.908i 0.302135i
\(848\) 0 0
\(849\) 2719.35i 3.20301i
\(850\) 0 0
\(851\) 176.505 274.117i 0.207409 0.322112i
\(852\) 0 0
\(853\) −808.878 −0.948275 −0.474137 0.880451i \(-0.657240\pi\)
−0.474137 + 0.880451i \(0.657240\pi\)
\(854\) 0 0
\(855\) 717.499 0.839180
\(856\) 0 0
\(857\) −416.626 −0.486145 −0.243072 0.970008i \(-0.578155\pi\)
−0.243072 + 0.970008i \(0.578155\pi\)
\(858\) 0 0
\(859\) −710.886 −0.827574 −0.413787 0.910374i \(-0.635794\pi\)
−0.413787 + 0.910374i \(0.635794\pi\)
\(860\) 0 0
\(861\) 387.332i 0.449863i
\(862\) 0 0
\(863\) 903.225 1.04661 0.523306 0.852145i \(-0.324699\pi\)
0.523306 + 0.852145i \(0.324699\pi\)
\(864\) 0 0
\(865\) 17.6681i 0.0204255i
\(866\) 0 0
\(867\) 1276.73 1.47258
\(868\) 0 0
\(869\) 567.739 0.653324
\(870\) 0 0
\(871\) 1154.92i 1.32597i
\(872\) 0 0
\(873\) 1371.62i 1.57115i
\(874\) 0 0
\(875\) −47.5872 −0.0543854
\(876\) 0 0
\(877\) 812.051 0.925941 0.462971 0.886374i \(-0.346784\pi\)
0.462971 + 0.886374i \(0.346784\pi\)
\(878\) 0 0
\(879\) 556.238i 0.632808i
\(880\) 0 0
\(881\) 530.228i 0.601848i 0.953648 + 0.300924i \(0.0972951\pi\)
−0.953648 + 0.300924i \(0.902705\pi\)
\(882\) 0 0
\(883\) −92.2733 −0.104500 −0.0522499 0.998634i \(-0.516639\pi\)
−0.0522499 + 0.998634i \(0.516639\pi\)
\(884\) 0 0
\(885\) 608.891i 0.688012i
\(886\) 0 0
\(887\) 1234.24 1.39147 0.695737 0.718296i \(-0.255078\pi\)
0.695737 + 0.718296i \(0.255078\pi\)
\(888\) 0 0
\(889\) 718.199i 0.807873i
\(890\) 0 0
\(891\) 16.4162i 0.0184245i
\(892\) 0 0
\(893\) 1392.33i 1.55916i
\(894\) 0 0
\(895\) 11.0307i 0.0123248i
\(896\) 0 0
\(897\) −926.351 596.482i −1.03272 0.664974i
\(898\) 0 0
\(899\) −2198.88 −2.44592
\(900\) 0 0
\(901\) −2137.76 −2.37265
\(902\) 0 0
\(903\) −1164.72 −1.28983
\(904\) 0 0
\(905\) −13.6514 −0.0150844
\(906\) 0 0
\(907\) 138.686i 0.152906i 0.997073 + 0.0764530i \(0.0243595\pi\)
−0.997073 + 0.0764530i \(0.975640\pi\)
\(908\) 0 0
\(909\) −409.601 −0.450606
\(910\) 0 0
\(911\) 1075.06i 1.18009i 0.807371 + 0.590044i \(0.200890\pi\)
−0.807371 + 0.590044i \(0.799110\pi\)
\(912\) 0 0
\(913\) −227.742 −0.249443
\(914\) 0 0
\(915\) −44.6567 −0.0488052
\(916\) 0 0
\(917\) 275.873i 0.300843i
\(918\) 0 0
\(919\) 924.661i 1.00616i −0.864240 0.503080i \(-0.832200\pi\)
0.864240 0.503080i \(-0.167800\pi\)
\(920\) 0 0
\(921\) −990.037 −1.07496
\(922\) 0 0
\(923\) −837.745 −0.907633
\(924\) 0 0
\(925\) 70.8757i 0.0766224i
\(926\) 0 0
\(927\) 1049.40i 1.13204i
\(928\) 0 0
\(929\) −804.685 −0.866184 −0.433092 0.901350i \(-0.642577\pi\)
−0.433092 + 0.901350i \(0.642577\pi\)
\(930\) 0 0
\(931\) 685.458i 0.736260i
\(932\) 0 0
\(933\) 278.919 0.298948
\(934\) 0 0
\(935\) 410.125i 0.438637i
\(936\) 0 0
\(937\) 1026.35i 1.09536i 0.836689 + 0.547678i \(0.184488\pi\)
−0.836689 + 0.547678i \(0.815512\pi\)
\(938\) 0 0
\(939\) 845.691i 0.900629i
\(940\) 0 0
\(941\) 1639.31i 1.74210i 0.491196 + 0.871049i \(0.336560\pi\)
−0.491196 + 0.871049i \(0.663440\pi\)
\(942\) 0 0
\(943\) 363.344 + 233.959i 0.385307 + 0.248101i
\(944\) 0 0
\(945\) −251.553 −0.266193
\(946\) 0 0
\(947\) −1301.61 −1.37446 −0.687228 0.726442i \(-0.741173\pi\)
−0.687228 + 0.726442i \(0.741173\pi\)
\(948\) 0 0
\(949\) 1039.31 1.09516
\(950\) 0 0
\(951\) 1464.39 1.53984
\(952\) 0 0
\(953\) 1341.73i 1.40790i −0.710248 0.703951i \(-0.751417\pi\)
0.710248 0.703951i \(-0.248583\pi\)
\(954\) 0 0
\(955\) 306.224 0.320654
\(956\) 0 0
\(957\) 2184.49i 2.28265i
\(958\) 0 0
\(959\) −1015.50 −1.05892
\(960\) 0 0
\(961\) 485.853 0.505570
\(962\) 0 0
\(963\) 61.8751i 0.0642524i
\(964\) 0 0
\(965\) 141.803i 0.146946i
\(966\) 0 0
\(967\) −1308.76 −1.35342 −0.676709 0.736250i \(-0.736595\pi\)
−0.676709 + 0.736250i \(0.736595\pi\)
\(968\) 0 0
\(969\) −2526.96 −2.60781
\(970\) 0 0
\(971\) 1483.52i 1.52783i 0.645316 + 0.763916i \(0.276726\pi\)
−0.645316 + 0.763916i \(0.723274\pi\)
\(972\) 0 0
\(973\) 528.875i 0.543551i
\(974\) 0 0
\(975\) −239.517 −0.245659
\(976\) 0 0
\(977\) 518.336i 0.530538i 0.964174 + 0.265269i \(0.0854608\pi\)
−0.964174 + 0.265269i \(0.914539\pi\)
\(978\) 0 0
\(979\) 827.980 0.845741
\(980\) 0 0
\(981\) 1651.78i 1.68377i
\(982\) 0 0
\(983\) 1079.22i 1.09789i 0.835859 + 0.548944i \(0.184970\pi\)
−0.835859 + 0.548944i \(0.815030\pi\)
\(984\) 0 0
\(985\) 67.7866i 0.0688189i
\(986\) 0 0
\(987\) 1293.20i 1.31023i
\(988\) 0 0
\(989\) 703.522 1092.59i 0.711347 1.10474i
\(990\) 0 0
\(991\) 1431.19 1.44419 0.722094 0.691795i \(-0.243180\pi\)
0.722094 + 0.691795i \(0.243180\pi\)
\(992\) 0 0
\(993\) 2587.40 2.60564
\(994\) 0 0
\(995\) 381.633 0.383551
\(996\) 0 0
\(997\) −582.437 −0.584189 −0.292095 0.956389i \(-0.594352\pi\)
−0.292095 + 0.956389i \(0.594352\pi\)
\(998\) 0 0
\(999\) 374.659i 0.375034i
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 1840.3.k.e.321.5 48
4.3 odd 2 920.3.k.a.321.43 48
23.22 odd 2 inner 1840.3.k.e.321.6 48
92.91 even 2 920.3.k.a.321.44 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.3.k.a.321.43 48 4.3 odd 2
920.3.k.a.321.44 yes 48 92.91 even 2
1840.3.k.e.321.5 48 1.1 even 1 trivial
1840.3.k.e.321.6 48 23.22 odd 2 inner