Properties

Label 704.5.h.i.65.3
Level $704$
Weight $5$
Character 704.65
Analytic conductor $72.772$
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] = [704,5,Mod(65,704)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(704, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("704.65");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 704 = 2^{6} \cdot 11 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 704.h (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: \(72.7724540110\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{553})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 271x^{2} + 272x + 19602 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 22)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 65.3
Root \(-11.2580 + 1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 704.65
Dual form 704.5.h.i.65.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+11.2580 q^{3} +19.2580 q^{5} -33.9411i q^{7} +45.7420 q^{9} +O(q^{10})\) \(q+11.2580 q^{3} +19.2580 q^{5} -33.9411i q^{7} +45.7420 q^{9} +(-2.03190 + 120.983i) q^{11} +72.2602i q^{13} +216.806 q^{15} +517.873i q^{17} +411.672i q^{19} -382.108i q^{21} -967.641 q^{23} -254.130 q^{25} -396.933 q^{27} +640.503i q^{29} +716.487 q^{31} +(-22.8751 + 1362.02i) q^{33} -653.637i q^{35} -1209.64 q^{37} +813.504i q^{39} -966.781i q^{41} +2720.75i q^{43} +880.899 q^{45} -177.085 q^{47} +1249.00 q^{49} +5830.20i q^{51} +4023.28 q^{53} +(-39.1304 + 2329.89i) q^{55} +4634.59i q^{57} +1988.02 q^{59} -2396.69i q^{61} -1552.54i q^{63} +1391.59i q^{65} +2007.13 q^{67} -10893.7 q^{69} +4255.04 q^{71} -1288.68i q^{73} -2860.99 q^{75} +(4106.30 + 68.9651i) q^{77} -1480.23i q^{79} -8173.77 q^{81} -10124.2i q^{83} +9973.18i q^{85} +7210.77i q^{87} +5133.72 q^{89} +2452.59 q^{91} +8066.19 q^{93} +7927.96i q^{95} +2239.34 q^{97} +(-92.9434 + 5534.00i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 30 q^{5} + 230 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} + 30 q^{5} + 230 q^{9} + 180 q^{11} + 538 q^{15} - 1566 q^{23} - 1722 q^{25} - 506 q^{27} + 4418 q^{31} - 2302 q^{33} + 382 q^{37} + 1172 q^{45} + 5688 q^{47} + 4996 q^{49} + 8568 q^{53} - 862 q^{55} + 3390 q^{59} + 8734 q^{67} - 26314 q^{69} + 3522 q^{71} + 9156 q^{75} + 2880 q^{77} - 31096 q^{81} - 8766 q^{89} - 17280 q^{91} - 20458 q^{93} + 17282 q^{97} + 12562 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/704\mathbb{Z}\right)^\times\).

\(n\) \(133\) \(321\) \(639\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 11.2580 1.25089 0.625443 0.780270i \(-0.284918\pi\)
0.625443 + 0.780270i \(0.284918\pi\)
\(4\) 0 0
\(5\) 19.2580 0.770319 0.385160 0.922850i \(-0.374146\pi\)
0.385160 + 0.922850i \(0.374146\pi\)
\(6\) 0 0
\(7\) 33.9411i 0.692676i −0.938110 0.346338i \(-0.887425\pi\)
0.938110 0.346338i \(-0.112575\pi\)
\(8\) 0 0
\(9\) 45.7420 0.564716
\(10\) 0 0
\(11\) −2.03190 + 120.983i −0.0167926 + 0.999859i
\(12\) 0 0
\(13\) 72.2602i 0.427575i 0.976880 + 0.213788i \(0.0685801\pi\)
−0.976880 + 0.213788i \(0.931420\pi\)
\(14\) 0 0
\(15\) 216.806 0.963581
\(16\) 0 0
\(17\) 517.873i 1.79195i 0.444107 + 0.895974i \(0.353521\pi\)
−0.444107 + 0.895974i \(0.646479\pi\)
\(18\) 0 0
\(19\) 411.672i 1.14036i 0.821518 + 0.570182i \(0.193127\pi\)
−0.821518 + 0.570182i \(0.806873\pi\)
\(20\) 0 0
\(21\) 382.108i 0.866459i
\(22\) 0 0
\(23\) −967.641 −1.82919 −0.914594 0.404373i \(-0.867490\pi\)
−0.914594 + 0.404373i \(0.867490\pi\)
\(24\) 0 0
\(25\) −254.130 −0.406609
\(26\) 0 0
\(27\) −396.933 −0.544490
\(28\) 0 0
\(29\) 640.503i 0.761597i 0.924658 + 0.380799i \(0.124351\pi\)
−0.924658 + 0.380799i \(0.875649\pi\)
\(30\) 0 0
\(31\) 716.487 0.745564 0.372782 0.927919i \(-0.378404\pi\)
0.372782 + 0.927919i \(0.378404\pi\)
\(32\) 0 0
\(33\) −22.8751 + 1362.02i −0.0210056 + 1.25071i
\(34\) 0 0
\(35\) 653.637i 0.533582i
\(36\) 0 0
\(37\) −1209.64 −0.883590 −0.441795 0.897116i \(-0.645658\pi\)
−0.441795 + 0.897116i \(0.645658\pi\)
\(38\) 0 0
\(39\) 813.504i 0.534848i
\(40\) 0 0
\(41\) 966.781i 0.575122i −0.957762 0.287561i \(-0.907156\pi\)
0.957762 0.287561i \(-0.0928445\pi\)
\(42\) 0 0
\(43\) 2720.75i 1.47147i 0.677269 + 0.735736i \(0.263163\pi\)
−0.677269 + 0.735736i \(0.736837\pi\)
\(44\) 0 0
\(45\) 880.899 0.435012
\(46\) 0 0
\(47\) −177.085 −0.0801651 −0.0400826 0.999196i \(-0.512762\pi\)
−0.0400826 + 0.999196i \(0.512762\pi\)
\(48\) 0 0
\(49\) 1249.00 0.520200
\(50\) 0 0
\(51\) 5830.20i 2.24152i
\(52\) 0 0
\(53\) 4023.28 1.43228 0.716140 0.697956i \(-0.245907\pi\)
0.716140 + 0.697956i \(0.245907\pi\)
\(54\) 0 0
\(55\) −39.1304 + 2329.89i −0.0129357 + 0.770210i
\(56\) 0 0
\(57\) 4634.59i 1.42647i
\(58\) 0 0
\(59\) 1988.02 0.571107 0.285554 0.958363i \(-0.407823\pi\)
0.285554 + 0.958363i \(0.407823\pi\)
\(60\) 0 0
\(61\) 2396.69i 0.644097i −0.946723 0.322049i \(-0.895629\pi\)
0.946723 0.322049i \(-0.104371\pi\)
\(62\) 0 0
\(63\) 1552.54i 0.391165i
\(64\) 0 0
\(65\) 1391.59i 0.329370i
\(66\) 0 0
\(67\) 2007.13 0.447122 0.223561 0.974690i \(-0.428232\pi\)
0.223561 + 0.974690i \(0.428232\pi\)
\(68\) 0 0
\(69\) −10893.7 −2.28811
\(70\) 0 0
\(71\) 4255.04 0.844086 0.422043 0.906576i \(-0.361313\pi\)
0.422043 + 0.906576i \(0.361313\pi\)
\(72\) 0 0
\(73\) 1288.68i 0.241824i −0.992663 0.120912i \(-0.961418\pi\)
0.992663 0.120912i \(-0.0385819\pi\)
\(74\) 0 0
\(75\) −2860.99 −0.508621
\(76\) 0 0
\(77\) 4106.30 + 68.9651i 0.692578 + 0.0116318i
\(78\) 0 0
\(79\) 1480.23i 0.237178i −0.992943 0.118589i \(-0.962163\pi\)
0.992943 0.118589i \(-0.0378371\pi\)
\(80\) 0 0
\(81\) −8173.77 −1.24581
\(82\) 0 0
\(83\) 10124.2i 1.46963i −0.678270 0.734813i \(-0.737270\pi\)
0.678270 0.734813i \(-0.262730\pi\)
\(84\) 0 0
\(85\) 9973.18i 1.38037i
\(86\) 0 0
\(87\) 7210.77i 0.952672i
\(88\) 0 0
\(89\) 5133.72 0.648115 0.324058 0.946037i \(-0.394953\pi\)
0.324058 + 0.946037i \(0.394953\pi\)
\(90\) 0 0
\(91\) 2452.59 0.296171
\(92\) 0 0
\(93\) 8066.19 0.932615
\(94\) 0 0
\(95\) 7927.96i 0.878444i
\(96\) 0 0
\(97\) 2239.34 0.238000 0.119000 0.992894i \(-0.462031\pi\)
0.119000 + 0.992894i \(0.462031\pi\)
\(98\) 0 0
\(99\) −92.9434 + 5534.00i −0.00948305 + 0.564637i
\(100\) 0 0
\(101\) 15580.0i 1.52730i 0.645629 + 0.763651i \(0.276595\pi\)
−0.645629 + 0.763651i \(0.723405\pi\)
\(102\) 0 0
\(103\) −9119.47 −0.859597 −0.429799 0.902925i \(-0.641415\pi\)
−0.429799 + 0.902925i \(0.641415\pi\)
\(104\) 0 0
\(105\) 7358.63i 0.667450i
\(106\) 0 0
\(107\) 11204.9i 0.978684i −0.872092 0.489342i \(-0.837237\pi\)
0.872092 0.489342i \(-0.162763\pi\)
\(108\) 0 0
\(109\) 10286.3i 0.865776i 0.901448 + 0.432888i \(0.142505\pi\)
−0.901448 + 0.432888i \(0.857495\pi\)
\(110\) 0 0
\(111\) −13618.0 −1.10527
\(112\) 0 0
\(113\) 17026.9 1.33345 0.666727 0.745302i \(-0.267695\pi\)
0.666727 + 0.745302i \(0.267695\pi\)
\(114\) 0 0
\(115\) −18634.8 −1.40906
\(116\) 0 0
\(117\) 3305.33i 0.241459i
\(118\) 0 0
\(119\) 17577.2 1.24124
\(120\) 0 0
\(121\) −14632.7 491.651i −0.999436 0.0335805i
\(122\) 0 0
\(123\) 10884.0i 0.719413i
\(124\) 0 0
\(125\) −16930.3 −1.08354
\(126\) 0 0
\(127\) 27813.0i 1.72441i −0.506558 0.862206i \(-0.669082\pi\)
0.506558 0.862206i \(-0.330918\pi\)
\(128\) 0 0
\(129\) 30630.1i 1.84064i
\(130\) 0 0
\(131\) 6310.74i 0.367738i 0.982951 + 0.183869i \(0.0588621\pi\)
−0.982951 + 0.183869i \(0.941138\pi\)
\(132\) 0 0
\(133\) 13972.6 0.789903
\(134\) 0 0
\(135\) −7644.13 −0.419431
\(136\) 0 0
\(137\) −14207.1 −0.756944 −0.378472 0.925613i \(-0.623550\pi\)
−0.378472 + 0.925613i \(0.623550\pi\)
\(138\) 0 0
\(139\) 27772.6i 1.43743i −0.695306 0.718714i \(-0.744731\pi\)
0.695306 0.718714i \(-0.255269\pi\)
\(140\) 0 0
\(141\) −1993.62 −0.100277
\(142\) 0 0
\(143\) −8742.26 146.826i −0.427515 0.00718010i
\(144\) 0 0
\(145\) 12334.8i 0.586673i
\(146\) 0 0
\(147\) 14061.2 0.650711
\(148\) 0 0
\(149\) 23436.8i 1.05567i 0.849348 + 0.527833i \(0.176995\pi\)
−0.849348 + 0.527833i \(0.823005\pi\)
\(150\) 0 0
\(151\) 6556.14i 0.287538i −0.989611 0.143769i \(-0.954078\pi\)
0.989611 0.143769i \(-0.0459222\pi\)
\(152\) 0 0
\(153\) 23688.6i 1.01194i
\(154\) 0 0
\(155\) 13798.1 0.574322
\(156\) 0 0
\(157\) −2315.17 −0.0939254 −0.0469627 0.998897i \(-0.514954\pi\)
−0.0469627 + 0.998897i \(0.514954\pi\)
\(158\) 0 0
\(159\) 45293.9 1.79162
\(160\) 0 0
\(161\) 32842.8i 1.26704i
\(162\) 0 0
\(163\) 21728.0 0.817796 0.408898 0.912580i \(-0.365913\pi\)
0.408898 + 0.912580i \(0.365913\pi\)
\(164\) 0 0
\(165\) −440.529 + 26229.8i −0.0161810 + 0.963446i
\(166\) 0 0
\(167\) 26470.6i 0.949143i −0.880217 0.474571i \(-0.842603\pi\)
0.880217 0.474571i \(-0.157397\pi\)
\(168\) 0 0
\(169\) 23339.5 0.817179
\(170\) 0 0
\(171\) 18830.7i 0.643982i
\(172\) 0 0
\(173\) 6136.89i 0.205048i −0.994731 0.102524i \(-0.967308\pi\)
0.994731 0.102524i \(-0.0326919\pi\)
\(174\) 0 0
\(175\) 8625.47i 0.281648i
\(176\) 0 0
\(177\) 22381.1 0.714390
\(178\) 0 0
\(179\) 8164.31 0.254808 0.127404 0.991851i \(-0.459335\pi\)
0.127404 + 0.991851i \(0.459335\pi\)
\(180\) 0 0
\(181\) 10130.6 0.309228 0.154614 0.987975i \(-0.450587\pi\)
0.154614 + 0.987975i \(0.450587\pi\)
\(182\) 0 0
\(183\) 26981.8i 0.805692i
\(184\) 0 0
\(185\) −23295.1 −0.680647
\(186\) 0 0
\(187\) −62653.8 1052.27i −1.79170 0.0300915i
\(188\) 0 0
\(189\) 13472.4i 0.377155i
\(190\) 0 0
\(191\) 13853.6 0.379749 0.189875 0.981808i \(-0.439192\pi\)
0.189875 + 0.981808i \(0.439192\pi\)
\(192\) 0 0
\(193\) 55752.8i 1.49676i 0.663270 + 0.748380i \(0.269168\pi\)
−0.663270 + 0.748380i \(0.730832\pi\)
\(194\) 0 0
\(195\) 15666.4i 0.412004i
\(196\) 0 0
\(197\) 38577.9i 0.994046i 0.867737 + 0.497023i \(0.165574\pi\)
−0.867737 + 0.497023i \(0.834426\pi\)
\(198\) 0 0
\(199\) −43741.2 −1.10455 −0.552275 0.833662i \(-0.686240\pi\)
−0.552275 + 0.833662i \(0.686240\pi\)
\(200\) 0 0
\(201\) 22596.2 0.559299
\(202\) 0 0
\(203\) 21739.4 0.527540
\(204\) 0 0
\(205\) 18618.2i 0.443028i
\(206\) 0 0
\(207\) −44261.8 −1.03297
\(208\) 0 0
\(209\) −49805.2 836.477i −1.14020 0.0191497i
\(210\) 0 0
\(211\) 40460.8i 0.908803i 0.890797 + 0.454402i \(0.150147\pi\)
−0.890797 + 0.454402i \(0.849853\pi\)
\(212\) 0 0
\(213\) 47903.1 1.05586
\(214\) 0 0
\(215\) 52396.2i 1.13350i
\(216\) 0 0
\(217\) 24318.4i 0.516434i
\(218\) 0 0
\(219\) 14507.9i 0.302494i
\(220\) 0 0
\(221\) −37421.6 −0.766193
\(222\) 0 0
\(223\) 12067.6 0.242667 0.121334 0.992612i \(-0.461283\pi\)
0.121334 + 0.992612i \(0.461283\pi\)
\(224\) 0 0
\(225\) −11624.4 −0.229619
\(226\) 0 0
\(227\) 38616.4i 0.749411i 0.927144 + 0.374705i \(0.122256\pi\)
−0.927144 + 0.374705i \(0.877744\pi\)
\(228\) 0 0
\(229\) −74080.9 −1.41265 −0.706326 0.707887i \(-0.749649\pi\)
−0.706326 + 0.707887i \(0.749649\pi\)
\(230\) 0 0
\(231\) 46228.6 + 776.408i 0.866337 + 0.0145501i
\(232\) 0 0
\(233\) 51719.5i 0.952670i 0.879264 + 0.476335i \(0.158035\pi\)
−0.879264 + 0.476335i \(0.841965\pi\)
\(234\) 0 0
\(235\) −3410.29 −0.0617527
\(236\) 0 0
\(237\) 16664.4i 0.296683i
\(238\) 0 0
\(239\) 16911.3i 0.296062i 0.988983 + 0.148031i \(0.0472935\pi\)
−0.988983 + 0.148031i \(0.952706\pi\)
\(240\) 0 0
\(241\) 59892.0i 1.03118i 0.856835 + 0.515590i \(0.172427\pi\)
−0.856835 + 0.515590i \(0.827573\pi\)
\(242\) 0 0
\(243\) −59868.5 −1.01388
\(244\) 0 0
\(245\) 24053.2 0.400720
\(246\) 0 0
\(247\) −29747.5 −0.487592
\(248\) 0 0
\(249\) 113979.i 1.83833i
\(250\) 0 0
\(251\) 19186.2 0.304538 0.152269 0.988339i \(-0.451342\pi\)
0.152269 + 0.988339i \(0.451342\pi\)
\(252\) 0 0
\(253\) 1966.15 117068.i 0.0307168 1.82893i
\(254\) 0 0
\(255\) 112278.i 1.72669i
\(256\) 0 0
\(257\) 30625.1 0.463672 0.231836 0.972755i \(-0.425527\pi\)
0.231836 + 0.972755i \(0.425527\pi\)
\(258\) 0 0
\(259\) 41056.4i 0.612042i
\(260\) 0 0
\(261\) 29297.9i 0.430086i
\(262\) 0 0
\(263\) 73815.1i 1.06717i −0.845746 0.533585i \(-0.820844\pi\)
0.845746 0.533585i \(-0.179156\pi\)
\(264\) 0 0
\(265\) 77480.2 1.10331
\(266\) 0 0
\(267\) 57795.3 0.810718
\(268\) 0 0
\(269\) 87292.0 1.20634 0.603170 0.797612i \(-0.293904\pi\)
0.603170 + 0.797612i \(0.293904\pi\)
\(270\) 0 0
\(271\) 4518.78i 0.0615294i −0.999527 0.0307647i \(-0.990206\pi\)
0.999527 0.0307647i \(-0.00979425\pi\)
\(272\) 0 0
\(273\) 27611.2 0.370477
\(274\) 0 0
\(275\) 516.369 30745.4i 0.00682801 0.406551i
\(276\) 0 0
\(277\) 4641.04i 0.0604861i −0.999543 0.0302431i \(-0.990372\pi\)
0.999543 0.0302431i \(-0.00962813\pi\)
\(278\) 0 0
\(279\) 32773.6 0.421032
\(280\) 0 0
\(281\) 58621.1i 0.742406i −0.928552 0.371203i \(-0.878945\pi\)
0.928552 0.371203i \(-0.121055\pi\)
\(282\) 0 0
\(283\) 90326.0i 1.12782i 0.825836 + 0.563910i \(0.190704\pi\)
−0.825836 + 0.563910i \(0.809296\pi\)
\(284\) 0 0
\(285\) 89252.8i 1.09883i
\(286\) 0 0
\(287\) −32813.6 −0.398373
\(288\) 0 0
\(289\) −184671. −2.21108
\(290\) 0 0
\(291\) 25210.4 0.297710
\(292\) 0 0
\(293\) 77687.4i 0.904931i 0.891782 + 0.452465i \(0.149455\pi\)
−0.891782 + 0.452465i \(0.850545\pi\)
\(294\) 0 0
\(295\) 38285.3 0.439935
\(296\) 0 0
\(297\) 806.531 48022.2i 0.00914341 0.544414i
\(298\) 0 0
\(299\) 69922.0i 0.782116i
\(300\) 0 0
\(301\) 92345.3 1.01925
\(302\) 0 0
\(303\) 175399.i 1.91048i
\(304\) 0 0
\(305\) 46155.3i 0.496160i
\(306\) 0 0
\(307\) 60487.2i 0.641781i −0.947116 0.320890i \(-0.896018\pi\)
0.947116 0.320890i \(-0.103982\pi\)
\(308\) 0 0
\(309\) −102667. −1.07526
\(310\) 0 0
\(311\) 32106.2 0.331947 0.165973 0.986130i \(-0.446923\pi\)
0.165973 + 0.986130i \(0.446923\pi\)
\(312\) 0 0
\(313\) 111475. 1.13786 0.568932 0.822384i \(-0.307357\pi\)
0.568932 + 0.822384i \(0.307357\pi\)
\(314\) 0 0
\(315\) 29898.7i 0.301322i
\(316\) 0 0
\(317\) −139267. −1.38589 −0.692945 0.720990i \(-0.743687\pi\)
−0.692945 + 0.720990i \(0.743687\pi\)
\(318\) 0 0
\(319\) −77490.0 1301.44i −0.761490 0.0127892i
\(320\) 0 0
\(321\) 126145.i 1.22422i
\(322\) 0 0
\(323\) −213194. −2.04347
\(324\) 0 0
\(325\) 18363.5i 0.173856i
\(326\) 0 0
\(327\) 115803.i 1.08299i
\(328\) 0 0
\(329\) 6010.46i 0.0555285i
\(330\) 0 0
\(331\) 60223.2 0.549677 0.274839 0.961490i \(-0.411376\pi\)
0.274839 + 0.961490i \(0.411376\pi\)
\(332\) 0 0
\(333\) −55331.2 −0.498978
\(334\) 0 0
\(335\) 38653.3 0.344427
\(336\) 0 0
\(337\) 85639.9i 0.754078i 0.926197 + 0.377039i \(0.123058\pi\)
−0.926197 + 0.377039i \(0.876942\pi\)
\(338\) 0 0
\(339\) 191688. 1.66800
\(340\) 0 0
\(341\) −1455.83 + 86682.7i −0.0125200 + 0.745459i
\(342\) 0 0
\(343\) 123885.i 1.05301i
\(344\) 0 0
\(345\) −209790. −1.76257
\(346\) 0 0
\(347\) 99499.7i 0.826348i −0.910652 0.413174i \(-0.864420\pi\)
0.910652 0.413174i \(-0.135580\pi\)
\(348\) 0 0
\(349\) 47515.3i 0.390106i 0.980793 + 0.195053i \(0.0624878\pi\)
−0.980793 + 0.195053i \(0.937512\pi\)
\(350\) 0 0
\(351\) 28682.5i 0.232811i
\(352\) 0 0
\(353\) 167482. 1.34406 0.672030 0.740524i \(-0.265423\pi\)
0.672030 + 0.740524i \(0.265423\pi\)
\(354\) 0 0
\(355\) 81943.4 0.650216
\(356\) 0 0
\(357\) 197884. 1.55265
\(358\) 0 0
\(359\) 5063.91i 0.0392914i −0.999807 0.0196457i \(-0.993746\pi\)
0.999807 0.0196457i \(-0.00625382\pi\)
\(360\) 0 0
\(361\) −39152.4 −0.300431
\(362\) 0 0
\(363\) −164735. 5535.00i −1.25018 0.0420053i
\(364\) 0 0
\(365\) 24817.4i 0.186282i
\(366\) 0 0
\(367\) 264021. 1.96023 0.980115 0.198429i \(-0.0635841\pi\)
0.980115 + 0.198429i \(0.0635841\pi\)
\(368\) 0 0
\(369\) 44222.5i 0.324781i
\(370\) 0 0
\(371\) 136555.i 0.992106i
\(372\) 0 0
\(373\) 270510.i 1.94431i −0.234339 0.972155i \(-0.575293\pi\)
0.234339 0.972155i \(-0.424707\pi\)
\(374\) 0 0
\(375\) −190601. −1.35538
\(376\) 0 0
\(377\) −46282.9 −0.325640
\(378\) 0 0
\(379\) 38850.0 0.270466 0.135233 0.990814i \(-0.456822\pi\)
0.135233 + 0.990814i \(0.456822\pi\)
\(380\) 0 0
\(381\) 313118.i 2.15704i
\(382\) 0 0
\(383\) −243070. −1.65704 −0.828522 0.559956i \(-0.810818\pi\)
−0.828522 + 0.559956i \(0.810818\pi\)
\(384\) 0 0
\(385\) 79079.0 + 1328.13i 0.533506 + 0.00896022i
\(386\) 0 0
\(387\) 124453.i 0.830964i
\(388\) 0 0
\(389\) 142544. 0.941995 0.470998 0.882135i \(-0.343894\pi\)
0.470998 + 0.882135i \(0.343894\pi\)
\(390\) 0 0
\(391\) 501115.i 3.27781i
\(392\) 0 0
\(393\) 71046.2i 0.459998i
\(394\) 0 0
\(395\) 28506.2i 0.182703i
\(396\) 0 0
\(397\) 73860.0 0.468628 0.234314 0.972161i \(-0.424716\pi\)
0.234314 + 0.972161i \(0.424716\pi\)
\(398\) 0 0
\(399\) 157303. 0.988079
\(400\) 0 0
\(401\) 225776. 1.40407 0.702034 0.712143i \(-0.252275\pi\)
0.702034 + 0.712143i \(0.252275\pi\)
\(402\) 0 0
\(403\) 51773.5i 0.318785i
\(404\) 0 0
\(405\) −157410. −0.959673
\(406\) 0 0
\(407\) 2457.86 146345.i 0.0148378 0.883466i
\(408\) 0 0
\(409\) 97972.9i 0.585679i 0.956162 + 0.292839i \(0.0946001\pi\)
−0.956162 + 0.292839i \(0.905400\pi\)
\(410\) 0 0
\(411\) −159943. −0.946851
\(412\) 0 0
\(413\) 67475.8i 0.395592i
\(414\) 0 0
\(415\) 194973.i 1.13208i
\(416\) 0 0
\(417\) 312663.i 1.79806i
\(418\) 0 0
\(419\) 64572.5 0.367807 0.183903 0.982944i \(-0.441127\pi\)
0.183903 + 0.982944i \(0.441127\pi\)
\(420\) 0 0
\(421\) −203272. −1.14687 −0.573434 0.819252i \(-0.694389\pi\)
−0.573434 + 0.819252i \(0.694389\pi\)
\(422\) 0 0
\(423\) −8100.21 −0.0452706
\(424\) 0 0
\(425\) 131607.i 0.728621i
\(426\) 0 0
\(427\) −81346.2 −0.446151
\(428\) 0 0
\(429\) −98420.1 1652.96i −0.534773 0.00898149i
\(430\) 0 0
\(431\) 11113.4i 0.0598266i 0.999552 + 0.0299133i \(0.00952311\pi\)
−0.999552 + 0.0299133i \(0.990477\pi\)
\(432\) 0 0
\(433\) −288512. −1.53882 −0.769409 0.638756i \(-0.779449\pi\)
−0.769409 + 0.638756i \(0.779449\pi\)
\(434\) 0 0
\(435\) 138865.i 0.733861i
\(436\) 0 0
\(437\) 398350.i 2.08594i
\(438\) 0 0
\(439\) 107467.i 0.557630i 0.960345 + 0.278815i \(0.0899417\pi\)
−0.960345 + 0.278815i \(0.910058\pi\)
\(440\) 0 0
\(441\) 57131.8 0.293765
\(442\) 0 0
\(443\) 130812. 0.666563 0.333281 0.942827i \(-0.391844\pi\)
0.333281 + 0.942827i \(0.391844\pi\)
\(444\) 0 0
\(445\) 98865.0 0.499255
\(446\) 0 0
\(447\) 263851.i 1.32052i
\(448\) 0 0
\(449\) −158898. −0.788181 −0.394090 0.919072i \(-0.628940\pi\)
−0.394090 + 0.919072i \(0.628940\pi\)
\(450\) 0 0
\(451\) 116964. + 1964.41i 0.575041 + 0.00965780i
\(452\) 0 0
\(453\) 73808.9i 0.359677i
\(454\) 0 0
\(455\) 47232.0 0.228146
\(456\) 0 0
\(457\) 291561.i 1.39604i −0.716079 0.698019i \(-0.754065\pi\)
0.716079 0.698019i \(-0.245935\pi\)
\(458\) 0 0
\(459\) 205561.i 0.975698i
\(460\) 0 0
\(461\) 62731.9i 0.295180i 0.989049 + 0.147590i \(0.0471515\pi\)
−0.989049 + 0.147590i \(0.952848\pi\)
\(462\) 0 0
\(463\) −170883. −0.797146 −0.398573 0.917137i \(-0.630494\pi\)
−0.398573 + 0.917137i \(0.630494\pi\)
\(464\) 0 0
\(465\) 155339. 0.718411
\(466\) 0 0
\(467\) −94559.0 −0.433580 −0.216790 0.976218i \(-0.569559\pi\)
−0.216790 + 0.976218i \(0.569559\pi\)
\(468\) 0 0
\(469\) 68124.3i 0.309711i
\(470\) 0 0
\(471\) −26064.1 −0.117490
\(472\) 0 0
\(473\) −329164. 5528.30i −1.47126 0.0247098i
\(474\) 0 0
\(475\) 104618.i 0.463682i
\(476\) 0 0
\(477\) 184033. 0.808832
\(478\) 0 0
\(479\) 154781.i 0.674600i 0.941397 + 0.337300i \(0.109514\pi\)
−0.941397 + 0.337300i \(0.890486\pi\)
\(480\) 0 0
\(481\) 87408.6i 0.377802i
\(482\) 0 0
\(483\) 369744.i 1.58492i
\(484\) 0 0
\(485\) 43125.1 0.183336
\(486\) 0 0
\(487\) 205330. 0.865754 0.432877 0.901453i \(-0.357498\pi\)
0.432877 + 0.901453i \(0.357498\pi\)
\(488\) 0 0
\(489\) 244614. 1.02297
\(490\) 0 0
\(491\) 180799.i 0.749950i −0.927035 0.374975i \(-0.877651\pi\)
0.927035 0.374975i \(-0.122349\pi\)
\(492\) 0 0
\(493\) −331699. −1.36474
\(494\) 0 0
\(495\) −1789.90 + 106574.i −0.00730498 + 0.434950i
\(496\) 0 0
\(497\) 144421.i 0.584678i
\(498\) 0 0
\(499\) −198403. −0.796797 −0.398399 0.917212i \(-0.630434\pi\)
−0.398399 + 0.917212i \(0.630434\pi\)
\(500\) 0 0
\(501\) 298006.i 1.18727i
\(502\) 0 0
\(503\) 278604.i 1.10116i −0.834781 0.550582i \(-0.814406\pi\)
0.834781 0.550582i \(-0.185594\pi\)
\(504\) 0 0
\(505\) 300040.i 1.17651i
\(506\) 0 0
\(507\) 262755. 1.02220
\(508\) 0 0
\(509\) 195175. 0.753336 0.376668 0.926348i \(-0.377070\pi\)
0.376668 + 0.926348i \(0.377070\pi\)
\(510\) 0 0
\(511\) −43739.2 −0.167506
\(512\) 0 0
\(513\) 163406.i 0.620917i
\(514\) 0 0
\(515\) −175622. −0.662164
\(516\) 0 0
\(517\) 359.819 21424.2i 0.00134618 0.0801538i
\(518\) 0 0
\(519\) 69089.0i 0.256492i
\(520\) 0 0
\(521\) −2617.52 −0.00964304 −0.00482152 0.999988i \(-0.501535\pi\)
−0.00482152 + 0.999988i \(0.501535\pi\)
\(522\) 0 0
\(523\) 309803.i 1.13261i −0.824194 0.566307i \(-0.808372\pi\)
0.824194 0.566307i \(-0.191628\pi\)
\(524\) 0 0
\(525\) 97105.3i 0.352310i
\(526\) 0 0
\(527\) 371049.i 1.33601i
\(528\) 0 0
\(529\) 656488. 2.34593
\(530\) 0 0
\(531\) 90936.2 0.322513
\(532\) 0 0
\(533\) 69859.8 0.245908
\(534\) 0 0
\(535\) 215785.i 0.753899i
\(536\) 0 0
\(537\) 91913.7 0.318736
\(538\) 0 0
\(539\) −2537.85 + 151108.i −0.00873551 + 0.520127i
\(540\) 0 0
\(541\) 197545.i 0.674948i 0.941335 + 0.337474i \(0.109573\pi\)
−0.941335 + 0.337474i \(0.890427\pi\)
\(542\) 0 0
\(543\) 114050. 0.386809
\(544\) 0 0
\(545\) 198093.i 0.666923i
\(546\) 0 0
\(547\) 440541.i 1.47235i 0.676790 + 0.736176i \(0.263370\pi\)
−0.676790 + 0.736176i \(0.736630\pi\)
\(548\) 0 0
\(549\) 109629.i 0.363732i
\(550\) 0 0
\(551\) −263677. −0.868498
\(552\) 0 0
\(553\) −50240.6 −0.164288
\(554\) 0 0
\(555\) −262256. −0.851411
\(556\) 0 0
\(557\) 422793.i 1.36275i 0.731933 + 0.681376i \(0.238618\pi\)
−0.731933 + 0.681376i \(0.761382\pi\)
\(558\) 0 0
\(559\) −196602. −0.629165
\(560\) 0 0
\(561\) −705355. 11846.4i −2.24121 0.0376410i
\(562\) 0 0
\(563\) 52520.6i 0.165696i 0.996562 + 0.0828482i \(0.0264017\pi\)
−0.996562 + 0.0828482i \(0.973598\pi\)
\(564\) 0 0
\(565\) 327903. 1.02718
\(566\) 0 0
\(567\) 277427.i 0.862944i
\(568\) 0 0
\(569\) 140197.i 0.433025i −0.976280 0.216513i \(-0.930532\pi\)
0.976280 0.216513i \(-0.0694683\pi\)
\(570\) 0 0
\(571\) 207548.i 0.636569i −0.947995 0.318284i \(-0.896893\pi\)
0.947995 0.318284i \(-0.103107\pi\)
\(572\) 0 0
\(573\) 155964. 0.475023
\(574\) 0 0
\(575\) 245907. 0.743764
\(576\) 0 0
\(577\) −273572. −0.821712 −0.410856 0.911700i \(-0.634770\pi\)
−0.410856 + 0.911700i \(0.634770\pi\)
\(578\) 0 0
\(579\) 627664.i 1.87228i
\(580\) 0 0
\(581\) −343628. −1.01797
\(582\) 0 0
\(583\) −8174.91 + 486748.i −0.0240517 + 1.43208i
\(584\) 0 0
\(585\) 63654.0i 0.186000i
\(586\) 0 0
\(587\) 576701. 1.67369 0.836843 0.547442i \(-0.184398\pi\)
0.836843 + 0.547442i \(0.184398\pi\)
\(588\) 0 0
\(589\) 294957.i 0.850214i
\(590\) 0 0
\(591\) 434309.i 1.24344i
\(592\) 0 0
\(593\) 671956.i 1.91087i −0.295202 0.955435i \(-0.595387\pi\)
0.295202 0.955435i \(-0.404613\pi\)
\(594\) 0 0
\(595\) 338501. 0.956150
\(596\) 0 0
\(597\) −492438. −1.38167
\(598\) 0 0
\(599\) 582984. 1.62481 0.812406 0.583092i \(-0.198157\pi\)
0.812406 + 0.583092i \(0.198157\pi\)
\(600\) 0 0
\(601\) 377092.i 1.04399i 0.852947 + 0.521997i \(0.174813\pi\)
−0.852947 + 0.521997i \(0.825187\pi\)
\(602\) 0 0
\(603\) 91810.2 0.252497
\(604\) 0 0
\(605\) −281797. 9468.21i −0.769885 0.0258677i
\(606\) 0 0
\(607\) 465535.i 1.26350i 0.775173 + 0.631750i \(0.217663\pi\)
−0.775173 + 0.631750i \(0.782337\pi\)
\(608\) 0 0
\(609\) 244742. 0.659893
\(610\) 0 0
\(611\) 12796.2i 0.0342766i
\(612\) 0 0
\(613\) 623300.i 1.65873i 0.558705 + 0.829366i \(0.311298\pi\)
−0.558705 + 0.829366i \(0.688702\pi\)
\(614\) 0 0
\(615\) 209604.i 0.554177i
\(616\) 0 0
\(617\) 197189. 0.517980 0.258990 0.965880i \(-0.416610\pi\)
0.258990 + 0.965880i \(0.416610\pi\)
\(618\) 0 0
\(619\) −214900. −0.560861 −0.280431 0.959874i \(-0.590477\pi\)
−0.280431 + 0.959874i \(0.590477\pi\)
\(620\) 0 0
\(621\) 384089. 0.995976
\(622\) 0 0
\(623\) 174244.i 0.448934i
\(624\) 0 0
\(625\) −167211. −0.428061
\(626\) 0 0
\(627\) −560706. 9417.04i −1.42626 0.0239541i
\(628\) 0 0
\(629\) 626437.i 1.58335i
\(630\) 0 0
\(631\) −40334.6 −0.101302 −0.0506511 0.998716i \(-0.516130\pi\)
−0.0506511 + 0.998716i \(0.516130\pi\)
\(632\) 0 0
\(633\) 455507.i 1.13681i
\(634\) 0 0
\(635\) 535623.i 1.32835i
\(636\) 0 0
\(637\) 90253.1i 0.222425i
\(638\) 0 0
\(639\) 194634. 0.476669
\(640\) 0 0
\(641\) −19965.1 −0.0485909 −0.0242954 0.999705i \(-0.507734\pi\)
−0.0242954 + 0.999705i \(0.507734\pi\)
\(642\) 0 0
\(643\) −21885.5 −0.0529340 −0.0264670 0.999650i \(-0.508426\pi\)
−0.0264670 + 0.999650i \(0.508426\pi\)
\(644\) 0 0
\(645\) 589875.i 1.41788i
\(646\) 0 0
\(647\) −345969. −0.826472 −0.413236 0.910624i \(-0.635602\pi\)
−0.413236 + 0.910624i \(0.635602\pi\)
\(648\) 0 0
\(649\) −4039.47 + 240517.i −0.00959037 + 0.571027i
\(650\) 0 0
\(651\) 273776.i 0.646000i
\(652\) 0 0
\(653\) −380939. −0.893366 −0.446683 0.894692i \(-0.647395\pi\)
−0.446683 + 0.894692i \(0.647395\pi\)
\(654\) 0 0
\(655\) 121532.i 0.283275i
\(656\) 0 0
\(657\) 58946.8i 0.136562i
\(658\) 0 0
\(659\) 149577.i 0.344424i 0.985060 + 0.172212i \(0.0550914\pi\)
−0.985060 + 0.172212i \(0.944909\pi\)
\(660\) 0 0
\(661\) 637662. 1.45945 0.729723 0.683743i \(-0.239649\pi\)
0.729723 + 0.683743i \(0.239649\pi\)
\(662\) 0 0
\(663\) −421292. −0.958420
\(664\) 0 0
\(665\) 269084. 0.608477
\(666\) 0 0
\(667\) 619777.i 1.39311i
\(668\) 0 0
\(669\) 135857. 0.303549
\(670\) 0 0
\(671\) 289958. + 4869.84i 0.644006 + 0.0108161i
\(672\) 0 0
\(673\) 165463.i 0.365318i −0.983176 0.182659i \(-0.941530\pi\)
0.983176 0.182659i \(-0.0584704\pi\)
\(674\) 0 0
\(675\) 100873. 0.221394
\(676\) 0 0
\(677\) 420292.i 0.917009i 0.888692 + 0.458504i \(0.151615\pi\)
−0.888692 + 0.458504i \(0.848385\pi\)
\(678\) 0 0
\(679\) 76005.7i 0.164857i
\(680\) 0 0
\(681\) 434742.i 0.937428i
\(682\) 0 0
\(683\) −265281. −0.568677 −0.284338 0.958724i \(-0.591774\pi\)
−0.284338 + 0.958724i \(0.591774\pi\)
\(684\) 0 0
\(685\) −273600. −0.583089
\(686\) 0 0
\(687\) −834001. −1.76707
\(688\) 0 0
\(689\) 290723.i 0.612408i
\(690\) 0 0
\(691\) 72314.2 0.151449 0.0757247 0.997129i \(-0.475873\pi\)
0.0757247 + 0.997129i \(0.475873\pi\)
\(692\) 0 0
\(693\) 187830. + 3154.60i 0.391110 + 0.00656868i
\(694\) 0 0
\(695\) 534843.i 1.10728i
\(696\) 0 0
\(697\) 500669. 1.03059
\(698\) 0 0
\(699\) 582257.i 1.19168i
\(700\) 0 0
\(701\) 142549.i 0.290088i −0.989425 0.145044i \(-0.953668\pi\)
0.989425 0.145044i \(-0.0463324\pi\)
\(702\) 0 0
\(703\) 497972.i 1.00761i
\(704\) 0 0
\(705\) −38393.0 −0.0772456
\(706\) 0 0
\(707\) 528803. 1.05793
\(708\) 0 0
\(709\) 588664. 1.17105 0.585525 0.810654i \(-0.300888\pi\)
0.585525 + 0.810654i \(0.300888\pi\)
\(710\) 0 0
\(711\) 67708.7i 0.133938i
\(712\) 0 0
\(713\) −693302. −1.36378
\(714\) 0 0
\(715\) −168358. 2827.57i −0.329323 0.00553097i
\(716\) 0 0
\(717\) 190387.i 0.370340i
\(718\) 0 0
\(719\) −133520. −0.258280 −0.129140 0.991626i \(-0.541222\pi\)
−0.129140 + 0.991626i \(0.541222\pi\)
\(720\) 0 0
\(721\) 309525.i 0.595422i
\(722\) 0 0
\(723\) 674263.i 1.28989i
\(724\) 0 0
\(725\) 162771.i 0.309672i
\(726\) 0 0
\(727\) 224313. 0.424410 0.212205 0.977225i \(-0.431936\pi\)
0.212205 + 0.977225i \(0.431936\pi\)
\(728\) 0 0
\(729\) −11922.8 −0.0224348
\(730\) 0 0
\(731\) −1.40900e6 −2.63680
\(732\) 0 0
\(733\) 297252.i 0.553243i −0.960979 0.276622i \(-0.910785\pi\)
0.960979 0.276622i \(-0.0892149\pi\)
\(734\) 0 0
\(735\) 270790. 0.501255
\(736\) 0 0
\(737\) −4078.30 + 242829.i −0.00750834 + 0.447059i
\(738\) 0 0
\(739\) 268988.i 0.492543i 0.969201 + 0.246272i \(0.0792056\pi\)
−0.969201 + 0.246272i \(0.920794\pi\)
\(740\) 0 0
\(741\) −334896. −0.609922
\(742\) 0 0
\(743\) 133653.i 0.242104i −0.992646 0.121052i \(-0.961373\pi\)
0.992646 0.121052i \(-0.0386267\pi\)
\(744\) 0 0
\(745\) 451346.i 0.813200i
\(746\) 0 0
\(747\) 463104.i 0.829921i
\(748\) 0 0
\(749\) −380309. −0.677911
\(750\) 0 0
\(751\) 201267. 0.356856 0.178428 0.983953i \(-0.442899\pi\)
0.178428 + 0.983953i \(0.442899\pi\)
\(752\) 0 0
\(753\) 215998. 0.380942
\(754\) 0 0
\(755\) 126258.i 0.221496i
\(756\) 0 0
\(757\) −835429. −1.45787 −0.728934 0.684584i \(-0.759984\pi\)
−0.728934 + 0.684584i \(0.759984\pi\)
\(758\) 0 0
\(759\) 22134.9 1.31795e6i 0.0384233 2.28778i
\(760\) 0 0
\(761\) 345963.i 0.597393i −0.954348 0.298696i \(-0.903448\pi\)
0.954348 0.298696i \(-0.0965518\pi\)
\(762\) 0 0
\(763\) 349128. 0.599702
\(764\) 0 0
\(765\) 456194.i 0.779518i
\(766\) 0 0
\(767\) 143655.i 0.244191i
\(768\) 0 0
\(769\) 1.05669e6i 1.78688i 0.449183 + 0.893440i \(0.351715\pi\)
−0.449183 + 0.893440i \(0.648285\pi\)
\(770\) 0 0
\(771\) 344776. 0.580001
\(772\) 0 0
\(773\) 224559. 0.375813 0.187907 0.982187i \(-0.439830\pi\)
0.187907 + 0.982187i \(0.439830\pi\)
\(774\) 0 0
\(775\) −182081. −0.303153
\(776\) 0 0
\(777\) 462212.i 0.765595i
\(778\) 0 0
\(779\) 397996. 0.655849
\(780\) 0 0
\(781\) −8645.83 + 514787.i −0.0141744 + 0.843967i
\(782\) 0 0
\(783\) 254237.i 0.414682i
\(784\) 0 0
\(785\) −44585.4 −0.0723525
\(786\) 0 0
\(787\) 227376.i 0.367109i −0.983010 0.183554i \(-0.941240\pi\)
0.983010 0.183554i \(-0.0587603\pi\)
\(788\) 0 0
\(789\) 831008.i 1.33491i
\(790\) 0 0
\(791\) 577911.i 0.923651i
\(792\) 0 0
\(793\) 173185. 0.275400
\(794\) 0 0
\(795\) 872270. 1.38012
\(796\) 0 0
\(797\) 972019. 1.53023 0.765117 0.643891i \(-0.222681\pi\)
0.765117 + 0.643891i \(0.222681\pi\)
\(798\) 0 0
\(799\) 91707.4i 0.143652i
\(800\) 0 0
\(801\) 234827. 0.366001
\(802\) 0 0
\(803\) 155908. + 2618.47i 0.241790 + 0.00406085i
\(804\) 0 0
\(805\) 632486.i 0.976021i
\(806\) 0 0
\(807\) 982732. 1.50900
\(808\) 0 0
\(809\) 56475.6i 0.0862906i 0.999069 + 0.0431453i \(0.0137378\pi\)
−0.999069 + 0.0431453i \(0.986262\pi\)
\(810\) 0 0
\(811\) 224590.i 0.341467i −0.985317 0.170733i \(-0.945386\pi\)
0.985317 0.170733i \(-0.0546137\pi\)
\(812\) 0 0
\(813\) 50872.3i 0.0769663i
\(814\) 0 0
\(815\) 418438. 0.629964
\(816\) 0 0
\(817\) −1.12006e6 −1.67801
\(818\) 0 0
\(819\) 112187. 0.167253
\(820\) 0 0
\(821\) 719097.i 1.06684i −0.845849 0.533422i \(-0.820906\pi\)
0.845849 0.533422i \(-0.179094\pi\)
\(822\) 0 0
\(823\) 52447.6 0.0774330 0.0387165 0.999250i \(-0.487673\pi\)
0.0387165 + 0.999250i \(0.487673\pi\)
\(824\) 0 0
\(825\) 5813.26 346131.i 0.00854107 0.508549i
\(826\) 0 0
\(827\) 121409.i 0.177517i 0.996053 + 0.0887586i \(0.0282900\pi\)
−0.996053 + 0.0887586i \(0.971710\pi\)
\(828\) 0 0
\(829\) −555136. −0.807774 −0.403887 0.914809i \(-0.632341\pi\)
−0.403887 + 0.914809i \(0.632341\pi\)
\(830\) 0 0
\(831\) 52248.7i 0.0756612i
\(832\) 0 0
\(833\) 646823.i 0.932171i
\(834\) 0 0
\(835\) 509771.i 0.731143i
\(836\) 0 0
\(837\) −284398. −0.405952
\(838\) 0 0
\(839\) −728123. −1.03438 −0.517191 0.855870i \(-0.673022\pi\)
−0.517191 + 0.855870i \(0.673022\pi\)
\(840\) 0 0
\(841\) 297036. 0.419969
\(842\) 0 0
\(843\) 659955.i 0.928665i
\(844\) 0 0
\(845\) 449471. 0.629489
\(846\) 0 0
\(847\) −16687.2 + 496652.i −0.0232604 + 0.692285i
\(848\) 0 0
\(849\) 1.01689e6i 1.41078i
\(850\) 0 0
\(851\) 1.17049e6 1.61625
\(852\) 0 0
\(853\) 560810.i 0.770757i 0.922759 + 0.385379i \(0.125929\pi\)
−0.922759 + 0.385379i \(0.874071\pi\)
\(854\) 0 0
\(855\) 362641.i 0.496072i
\(856\) 0 0
\(857\) 1.02077e6i 1.38984i −0.719086 0.694921i \(-0.755439\pi\)
0.719086 0.694921i \(-0.244561\pi\)
\(858\) 0 0
\(859\) −350246. −0.474665 −0.237332 0.971428i \(-0.576273\pi\)
−0.237332 + 0.971428i \(0.576273\pi\)
\(860\) 0 0
\(861\) −369415. −0.498320
\(862\) 0 0
\(863\) −516997. −0.694170 −0.347085 0.937834i \(-0.612829\pi\)
−0.347085 + 0.937834i \(0.612829\pi\)
\(864\) 0 0
\(865\) 118184.i 0.157953i
\(866\) 0 0
\(867\) −2.07903e6 −2.76581
\(868\) 0 0
\(869\) 179082. + 3007.68i 0.237145 + 0.00398284i
\(870\) 0 0
\(871\) 145036.i 0.191178i
\(872\) 0 0
\(873\) 102432. 0.134402
\(874\) 0 0
\(875\) 574632.i 0.750540i
\(876\) 0 0
\(877\) 657406.i 0.854741i 0.904077 + 0.427371i \(0.140560\pi\)
−0.904077 + 0.427371i \(0.859440\pi\)
\(878\) 0 0
\(879\) 874603.i 1.13197i
\(880\) 0 0
\(881\) 869395. 1.12012 0.560061 0.828452i \(-0.310778\pi\)
0.560061 + 0.828452i \(0.310778\pi\)
\(882\) 0 0
\(883\) −955758. −1.22582 −0.612910 0.790153i \(-0.710001\pi\)
−0.612910 + 0.790153i \(0.710001\pi\)
\(884\) 0 0
\(885\) 431015. 0.550308
\(886\) 0 0
\(887\) 436265.i 0.554503i 0.960797 + 0.277251i \(0.0894235\pi\)
−0.960797 + 0.277251i \(0.910577\pi\)
\(888\) 0 0
\(889\) −944006. −1.19446
\(890\) 0 0
\(891\) 16608.3 988887.i 0.0209204 1.24564i
\(892\) 0 0
\(893\) 72900.7i 0.0914174i
\(894\) 0 0
\(895\) 157228. 0.196284
\(896\) 0 0
\(897\) 787180.i 0.978338i
\(898\) 0 0
\(899\) 458912.i 0.567819i
\(900\) 0 0
\(901\) 2.08355e6i 2.56657i
\(902\) 0 0
\(903\) 1.03962e6 1.27497
\(904\) 0 0
\(905\) 195095. 0.238204
\(906\) 0 0
\(907\) 707203. 0.859666 0.429833 0.902909i \(-0.358573\pi\)
0.429833 + 0.902909i \(0.358573\pi\)
\(908\) 0 0
\(909\) 712661.i 0.862493i
\(910\) 0 0
\(911\) 132785. 0.159998 0.0799989 0.996795i \(-0.474508\pi\)
0.0799989 + 0.996795i \(0.474508\pi\)
\(912\) 0 0
\(913\) 1.22486e6 + 20571.5i 1.46942 + 0.0246788i
\(914\) 0 0
\(915\) 519615.i 0.620640i
\(916\) 0 0
\(917\) 214194. 0.254723
\(918\) 0 0
\(919\) 964589.i 1.14212i −0.820909 0.571059i \(-0.806533\pi\)
0.820909 0.571059i \(-0.193467\pi\)
\(920\) 0 0
\(921\) 680963.i 0.802795i
\(922\) 0 0
\(923\) 307470.i 0.360911i
\(924\) 0 0
\(925\) 307405. 0.359275
\(926\) 0 0
\(927\) −417143. −0.485429
\(928\) 0 0
\(929\) 634849. 0.735596 0.367798 0.929906i \(-0.380112\pi\)
0.367798 + 0.929906i \(0.380112\pi\)
\(930\) 0 0
\(931\) 514178.i 0.593217i
\(932\) 0 0
\(933\) 361451. 0.415228
\(934\) 0 0
\(935\) −1.20659e6 20264.6i −1.38018 0.0231800i
\(936\) 0 0
\(937\) 3331.90i 0.00379501i 0.999998 + 0.00189750i \(0.000603995\pi\)
−0.999998 + 0.00189750i \(0.999396\pi\)
\(938\) 0 0
\(939\) 1.25499e6 1.42334
\(940\) 0 0
\(941\) 1.13619e6i 1.28313i −0.767069 0.641564i \(-0.778286\pi\)
0.767069 0.641564i \(-0.221714\pi\)
\(942\) 0 0
\(943\) 935496.i 1.05201i
\(944\) 0 0
\(945\) 259451.i 0.290530i
\(946\) 0 0
\(947\) −156679. −0.174707 −0.0873536 0.996177i \(-0.527841\pi\)
−0.0873536 + 0.996177i \(0.527841\pi\)
\(948\) 0 0
\(949\) 93120.3 0.103398
\(950\) 0 0
\(951\) −1.56786e6 −1.73359
\(952\) 0 0
\(953\) 343607.i 0.378334i 0.981945 + 0.189167i \(0.0605788\pi\)
−0.981945 + 0.189167i \(0.939421\pi\)
\(954\) 0 0
\(955\) 266793. 0.292528
\(956\) 0 0
\(957\) −872380. 14651.6i −0.952537 0.0159978i
\(958\) 0 0
\(959\) 482205.i 0.524317i
\(960\) 0 0
\(961\) −410168. −0.444135
\(962\) 0 0
\(963\) 512537.i 0.552679i
\(964\) 0 0
\(965\) 1.07369e6i 1.15298i
\(966\) 0 0
\(967\) 395609.i 0.423071i 0.977370 + 0.211536i \(0.0678464\pi\)
−0.977370 + 0.211536i \(0.932154\pi\)
\(968\) 0 0
\(969\) −2.40013e6 −2.55615
\(970\) 0 0
\(971\) −1.28188e6 −1.35960 −0.679799 0.733398i \(-0.737933\pi\)
−0.679799 + 0.733398i \(0.737933\pi\)
\(972\) 0 0
\(973\) −942632. −0.995672
\(974\) 0 0
\(975\) 206736.i 0.217474i
\(976\) 0 0
\(977\) −1.02959e6 −1.07863 −0.539316 0.842103i \(-0.681317\pi\)
−0.539316 + 0.842103i \(0.681317\pi\)
\(978\) 0 0
\(979\) −10431.2 + 621092.i −0.0108835 + 0.648024i
\(980\) 0 0
\(981\) 470515.i 0.488918i
\(982\) 0 0
\(983\) 1.35529e6 1.40257 0.701284 0.712882i \(-0.252610\pi\)
0.701284 + 0.712882i \(0.252610\pi\)
\(984\) 0 0
\(985\) 742933.i 0.765732i
\(986\) 0 0
\(987\) 67665.6i 0.0694598i
\(988\) 0 0
\(989\) 2.63271e6i 2.69160i
\(990\) 0 0
\(991\) −47967.0 −0.0488422 −0.0244211 0.999702i \(-0.507774\pi\)
−0.0244211 + 0.999702i \(0.507774\pi\)
\(992\) 0 0
\(993\) 677991. 0.687584
\(994\) 0 0
\(995\) −842368. −0.850855
\(996\) 0 0
\(997\) 1.71845e6i 1.72881i 0.502798 + 0.864404i \(0.332304\pi\)
−0.502798 + 0.864404i \(0.667696\pi\)
\(998\) 0 0
\(999\) 480145. 0.481106
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 704.5.h.i.65.3 4
4.3 odd 2 704.5.h.j.65.2 4
8.3 odd 2 176.5.h.e.65.4 4
8.5 even 2 22.5.b.a.21.1 4
11.10 odd 2 inner 704.5.h.i.65.4 4
24.5 odd 2 198.5.d.a.109.4 4
40.13 odd 4 550.5.c.a.549.2 8
40.29 even 2 550.5.d.a.351.4 4
40.37 odd 4 550.5.c.a.549.7 8
44.43 even 2 704.5.h.j.65.1 4
88.21 odd 2 22.5.b.a.21.3 yes 4
88.43 even 2 176.5.h.e.65.3 4
264.197 even 2 198.5.d.a.109.2 4
440.109 odd 2 550.5.d.a.351.2 4
440.197 even 4 550.5.c.a.549.3 8
440.373 even 4 550.5.c.a.549.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
22.5.b.a.21.1 4 8.5 even 2
22.5.b.a.21.3 yes 4 88.21 odd 2
176.5.h.e.65.3 4 88.43 even 2
176.5.h.e.65.4 4 8.3 odd 2
198.5.d.a.109.2 4 264.197 even 2
198.5.d.a.109.4 4 24.5 odd 2
550.5.c.a.549.2 8 40.13 odd 4
550.5.c.a.549.3 8 440.197 even 4
550.5.c.a.549.6 8 440.373 even 4
550.5.c.a.549.7 8 40.37 odd 4
550.5.d.a.351.2 4 440.109 odd 2
550.5.d.a.351.4 4 40.29 even 2
704.5.h.i.65.3 4 1.1 even 1 trivial
704.5.h.i.65.4 4 11.10 odd 2 inner
704.5.h.j.65.1 4 44.43 even 2
704.5.h.j.65.2 4 4.3 odd 2