Properties

Label 1600.3.b.v.1151.3
Level $1600$
Weight $3$
Character 1600.1151
Analytic conductor $43.597$
Analytic rank $0$
Dimension $6$
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] = [1600,3,Mod(1151,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.1151");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1600.b (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: \(43.5968422976\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.1827904.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 9x^{4} + 14x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1151.3
Root \(-0.273891i\) of defining polynomial
Character \(\chi\) \(=\) 1600.1151
Dual form 1600.3.b.v.1151.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.547781i q^{3} +10.0566i q^{7} +8.69994 q^{9} +O(q^{10})\) \(q-0.547781i q^{3} +10.0566i q^{7} +8.69994 q^{9} -17.2087i q^{11} -4.41325 q^{13} -27.0176 q^{17} +4.82650i q^{19} +5.50881 q^{21} -15.2653i q^{23} -9.69569i q^{27} -2.38225 q^{29} -38.0352i q^{31} -9.42663 q^{33} +16.5691 q^{37} +2.41749i q^{39} -13.3177 q^{41} -59.7918i q^{43} +62.4388i q^{47} -52.1351 q^{49} +14.7997i q^{51} +71.5952 q^{53} +2.64386 q^{57} -68.8265i q^{59} -40.9439 q^{61} +87.4917i q^{63} -51.0080i q^{67} -8.36206 q^{69} +40.4527i q^{71} -35.8441 q^{73} +173.061 q^{77} -126.800i q^{79} +72.9883 q^{81} +75.1490i q^{83} +1.30495i q^{87} +106.523 q^{89} -44.3822i q^{91} -20.8350 q^{93} -85.4351 q^{97} -149.715i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 18 q^{9} - 80 q^{17} - 8 q^{21} + 44 q^{29} - 144 q^{33} + 208 q^{37} - 68 q^{41} + 62 q^{49} + 64 q^{53} - 400 q^{57} + 100 q^{61} - 184 q^{69} - 80 q^{73} + 400 q^{77} + 238 q^{81} - 76 q^{89} + 320 q^{93} - 208 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) − 0.547781i − 0.182594i −0.995824 0.0912969i \(-0.970899\pi\)
0.995824 0.0912969i \(-0.0291012\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 10.0566i 1.43666i 0.695705 + 0.718328i \(0.255092\pi\)
−0.695705 + 0.718328i \(0.744908\pi\)
\(8\) 0 0
\(9\) 8.69994 0.966660
\(10\) 0 0
\(11\) − 17.2087i − 1.56443i −0.623008 0.782216i \(-0.714089\pi\)
0.623008 0.782216i \(-0.285911\pi\)
\(12\) 0 0
\(13\) −4.41325 −0.339481 −0.169740 0.985489i \(-0.554293\pi\)
−0.169740 + 0.985489i \(0.554293\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −27.0176 −1.58927 −0.794636 0.607086i \(-0.792338\pi\)
−0.794636 + 0.607086i \(0.792338\pi\)
\(18\) 0 0
\(19\) 4.82650i 0.254026i 0.991901 + 0.127013i \(0.0405390\pi\)
−0.991901 + 0.127013i \(0.959461\pi\)
\(20\) 0 0
\(21\) 5.50881 0.262324
\(22\) 0 0
\(23\) − 15.2653i − 0.663710i −0.943330 0.331855i \(-0.892325\pi\)
0.943330 0.331855i \(-0.107675\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 9.69569i − 0.359100i
\(28\) 0 0
\(29\) −2.38225 −0.0821465 −0.0410733 0.999156i \(-0.513078\pi\)
−0.0410733 + 0.999156i \(0.513078\pi\)
\(30\) 0 0
\(31\) − 38.0352i − 1.22694i −0.789717 0.613472i \(-0.789772\pi\)
0.789717 0.613472i \(-0.210228\pi\)
\(32\) 0 0
\(33\) −9.42663 −0.285655
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 16.5691 0.447814 0.223907 0.974610i \(-0.428119\pi\)
0.223907 + 0.974610i \(0.428119\pi\)
\(38\) 0 0
\(39\) 2.41749i 0.0619870i
\(40\) 0 0
\(41\) −13.3177 −0.324822 −0.162411 0.986723i \(-0.551927\pi\)
−0.162411 + 0.986723i \(0.551927\pi\)
\(42\) 0 0
\(43\) − 59.7918i − 1.39051i −0.718765 0.695253i \(-0.755292\pi\)
0.718765 0.695253i \(-0.244708\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 62.4388i 1.32849i 0.747517 + 0.664243i \(0.231246\pi\)
−0.747517 + 0.664243i \(0.768754\pi\)
\(48\) 0 0
\(49\) −52.1351 −1.06398
\(50\) 0 0
\(51\) 14.7997i 0.290191i
\(52\) 0 0
\(53\) 71.5952 1.35085 0.675427 0.737427i \(-0.263959\pi\)
0.675427 + 0.737427i \(0.263959\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.64386 0.0463836
\(58\) 0 0
\(59\) − 68.8265i − 1.16655i −0.812274 0.583275i \(-0.801771\pi\)
0.812274 0.583275i \(-0.198229\pi\)
\(60\) 0 0
\(61\) −40.9439 −0.671212 −0.335606 0.942002i \(-0.608941\pi\)
−0.335606 + 0.942002i \(0.608941\pi\)
\(62\) 0 0
\(63\) 87.4917i 1.38876i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 51.0080i − 0.761313i −0.924717 0.380656i \(-0.875698\pi\)
0.924717 0.380656i \(-0.124302\pi\)
\(68\) 0 0
\(69\) −8.36206 −0.121189
\(70\) 0 0
\(71\) 40.4527i 0.569757i 0.958564 + 0.284878i \(0.0919532\pi\)
−0.958564 + 0.284878i \(0.908047\pi\)
\(72\) 0 0
\(73\) −35.8441 −0.491015 −0.245508 0.969395i \(-0.578955\pi\)
−0.245508 + 0.969395i \(0.578955\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 173.061 2.24755
\(78\) 0 0
\(79\) − 126.800i − 1.60506i −0.596612 0.802530i \(-0.703487\pi\)
0.596612 0.802530i \(-0.296513\pi\)
\(80\) 0 0
\(81\) 72.9883 0.901090
\(82\) 0 0
\(83\) 75.1490i 0.905409i 0.891661 + 0.452705i \(0.149541\pi\)
−0.891661 + 0.452705i \(0.850459\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.30495i 0.0149994i
\(88\) 0 0
\(89\) 106.523 1.19689 0.598445 0.801164i \(-0.295785\pi\)
0.598445 + 0.801164i \(0.295785\pi\)
\(90\) 0 0
\(91\) − 44.3822i − 0.487717i
\(92\) 0 0
\(93\) −20.8350 −0.224032
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −85.4351 −0.880774 −0.440387 0.897808i \(-0.645159\pi\)
−0.440387 + 0.897808i \(0.645159\pi\)
\(98\) 0 0
\(99\) − 149.715i − 1.51227i
\(100\) 0 0
\(101\) 83.2877 0.824631 0.412315 0.911041i \(-0.364720\pi\)
0.412315 + 0.911041i \(0.364720\pi\)
\(102\) 0 0
\(103\) − 47.2653i − 0.458887i −0.973322 0.229443i \(-0.926309\pi\)
0.973322 0.229443i \(-0.0736906\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 189.628i − 1.77223i −0.463467 0.886114i \(-0.653395\pi\)
0.463467 0.886114i \(-0.346605\pi\)
\(108\) 0 0
\(109\) 84.5617 0.775795 0.387898 0.921702i \(-0.373201\pi\)
0.387898 + 0.921702i \(0.373201\pi\)
\(110\) 0 0
\(111\) − 9.07626i − 0.0817681i
\(112\) 0 0
\(113\) 114.558 1.01379 0.506896 0.862007i \(-0.330793\pi\)
0.506896 + 0.862007i \(0.330793\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −38.3950 −0.328162
\(118\) 0 0
\(119\) − 271.705i − 2.28324i
\(120\) 0 0
\(121\) −175.141 −1.44745
\(122\) 0 0
\(123\) 7.29518i 0.0593104i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 44.4121i 0.349701i 0.984595 + 0.174851i \(0.0559443\pi\)
−0.984595 + 0.174851i \(0.944056\pi\)
\(128\) 0 0
\(129\) −32.7528 −0.253898
\(130\) 0 0
\(131\) − 47.1200i − 0.359695i −0.983695 0.179847i \(-0.942440\pi\)
0.983695 0.179847i \(-0.0575604\pi\)
\(132\) 0 0
\(133\) −48.5381 −0.364948
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −62.8617 −0.458845 −0.229422 0.973327i \(-0.573684\pi\)
−0.229422 + 0.973327i \(0.573684\pi\)
\(138\) 0 0
\(139\) − 128.320i − 0.923167i −0.887097 0.461584i \(-0.847281\pi\)
0.887097 0.461584i \(-0.152719\pi\)
\(140\) 0 0
\(141\) 34.2028 0.242573
\(142\) 0 0
\(143\) 75.9465i 0.531094i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 28.5586i 0.194276i
\(148\) 0 0
\(149\) −165.561 −1.11115 −0.555574 0.831467i \(-0.687501\pi\)
−0.555574 + 0.831467i \(0.687501\pi\)
\(150\) 0 0
\(151\) − 218.558i − 1.44741i −0.690111 0.723704i \(-0.742438\pi\)
0.690111 0.723704i \(-0.257562\pi\)
\(152\) 0 0
\(153\) −235.052 −1.53628
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −174.293 −1.11014 −0.555072 0.831802i \(-0.687309\pi\)
−0.555072 + 0.831802i \(0.687309\pi\)
\(158\) 0 0
\(159\) − 39.2185i − 0.246657i
\(160\) 0 0
\(161\) 153.517 0.953524
\(162\) 0 0
\(163\) − 52.6353i − 0.322916i −0.986880 0.161458i \(-0.948380\pi\)
0.986880 0.161458i \(-0.0516196\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 76.6812i − 0.459169i −0.973289 0.229584i \(-0.926263\pi\)
0.973289 0.229584i \(-0.0737367\pi\)
\(168\) 0 0
\(169\) −149.523 −0.884753
\(170\) 0 0
\(171\) 41.9902i 0.245557i
\(172\) 0 0
\(173\) −9.72437 −0.0562102 −0.0281051 0.999605i \(-0.508947\pi\)
−0.0281051 + 0.999605i \(0.508947\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −37.7019 −0.213005
\(178\) 0 0
\(179\) − 155.502i − 0.868728i −0.900738 0.434364i \(-0.856973\pi\)
0.900738 0.434364i \(-0.143027\pi\)
\(180\) 0 0
\(181\) −250.346 −1.38313 −0.691563 0.722316i \(-0.743077\pi\)
−0.691563 + 0.722316i \(0.743077\pi\)
\(182\) 0 0
\(183\) 22.4283i 0.122559i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 464.939i 2.48631i
\(188\) 0 0
\(189\) 97.5056 0.515903
\(190\) 0 0
\(191\) 111.128i 0.581825i 0.956750 + 0.290912i \(0.0939588\pi\)
−0.956750 + 0.290912i \(0.906041\pi\)
\(192\) 0 0
\(193\) −10.2701 −0.0532130 −0.0266065 0.999646i \(-0.508470\pi\)
−0.0266065 + 0.999646i \(0.508470\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −163.213 −0.828492 −0.414246 0.910165i \(-0.635955\pi\)
−0.414246 + 0.910165i \(0.635955\pi\)
\(198\) 0 0
\(199\) 195.028i 0.980041i 0.871711 + 0.490021i \(0.163011\pi\)
−0.871711 + 0.490021i \(0.836989\pi\)
\(200\) 0 0
\(201\) −27.9412 −0.139011
\(202\) 0 0
\(203\) − 23.9573i − 0.118016i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 132.807i − 0.641582i
\(208\) 0 0
\(209\) 83.0580 0.397407
\(210\) 0 0
\(211\) − 297.038i − 1.40776i −0.710317 0.703881i \(-0.751449\pi\)
0.710317 0.703881i \(-0.248551\pi\)
\(212\) 0 0
\(213\) 22.1592 0.104034
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 382.505 1.76270
\(218\) 0 0
\(219\) 19.6347i 0.0896563i
\(220\) 0 0
\(221\) 119.236 0.539527
\(222\) 0 0
\(223\) − 405.335i − 1.81764i −0.417184 0.908822i \(-0.636983\pi\)
0.417184 0.908822i \(-0.363017\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 36.6013i 0.161239i 0.996745 + 0.0806196i \(0.0256899\pi\)
−0.996745 + 0.0806196i \(0.974310\pi\)
\(228\) 0 0
\(229\) −91.1467 −0.398021 −0.199010 0.979997i \(-0.563773\pi\)
−0.199010 + 0.979997i \(0.563773\pi\)
\(230\) 0 0
\(231\) − 94.7997i − 0.410388i
\(232\) 0 0
\(233\) 338.802 1.45409 0.727043 0.686592i \(-0.240894\pi\)
0.727043 + 0.686592i \(0.240894\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −69.4585 −0.293074
\(238\) 0 0
\(239\) − 30.7997i − 0.128869i −0.997922 0.0644346i \(-0.979476\pi\)
0.997922 0.0644346i \(-0.0205244\pi\)
\(240\) 0 0
\(241\) 240.282 0.997020 0.498510 0.866884i \(-0.333881\pi\)
0.498510 + 0.866884i \(0.333881\pi\)
\(242\) 0 0
\(243\) − 127.243i − 0.523633i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 21.3005i − 0.0862370i
\(248\) 0 0
\(249\) 41.1652 0.165322
\(250\) 0 0
\(251\) 86.0268i 0.342736i 0.985207 + 0.171368i \(0.0548187\pi\)
−0.985207 + 0.171368i \(0.945181\pi\)
\(252\) 0 0
\(253\) −262.697 −1.03833
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 355.963 1.38507 0.692536 0.721383i \(-0.256493\pi\)
0.692536 + 0.721383i \(0.256493\pi\)
\(258\) 0 0
\(259\) 166.629i 0.643355i
\(260\) 0 0
\(261\) −20.7254 −0.0794077
\(262\) 0 0
\(263\) 73.1254i 0.278043i 0.990289 + 0.139022i \(0.0443958\pi\)
−0.990289 + 0.139022i \(0.955604\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 58.3514i − 0.218545i
\(268\) 0 0
\(269\) 268.002 0.996290 0.498145 0.867094i \(-0.334015\pi\)
0.498145 + 0.867094i \(0.334015\pi\)
\(270\) 0 0
\(271\) − 229.412i − 0.846538i −0.906004 0.423269i \(-0.860883\pi\)
0.906004 0.423269i \(-0.139117\pi\)
\(272\) 0 0
\(273\) −24.3118 −0.0890541
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 126.327 0.456053 0.228026 0.973655i \(-0.426773\pi\)
0.228026 + 0.973655i \(0.426773\pi\)
\(278\) 0 0
\(279\) − 330.904i − 1.18604i
\(280\) 0 0
\(281\) −458.746 −1.63255 −0.816275 0.577664i \(-0.803964\pi\)
−0.816275 + 0.577664i \(0.803964\pi\)
\(282\) 0 0
\(283\) 465.558i 1.64508i 0.568706 + 0.822541i \(0.307444\pi\)
−0.568706 + 0.822541i \(0.692556\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 133.931i − 0.466657i
\(288\) 0 0
\(289\) 440.952 1.52579
\(290\) 0 0
\(291\) 46.7997i 0.160824i
\(292\) 0 0
\(293\) 165.218 0.563884 0.281942 0.959431i \(-0.409021\pi\)
0.281942 + 0.959431i \(0.409021\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −166.851 −0.561787
\(298\) 0 0
\(299\) 67.3697i 0.225317i
\(300\) 0 0
\(301\) 601.302 1.99768
\(302\) 0 0
\(303\) − 45.6234i − 0.150572i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 235.339i 0.766577i 0.923629 + 0.383288i \(0.125208\pi\)
−0.923629 + 0.383288i \(0.874792\pi\)
\(308\) 0 0
\(309\) −25.8911 −0.0837898
\(310\) 0 0
\(311\) 210.665i 0.677381i 0.940898 + 0.338691i \(0.109984\pi\)
−0.940898 + 0.338691i \(0.890016\pi\)
\(312\) 0 0
\(313\) 318.738 1.01833 0.509166 0.860668i \(-0.329954\pi\)
0.509166 + 0.860668i \(0.329954\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 70.3950 0.222066 0.111033 0.993817i \(-0.464584\pi\)
0.111033 + 0.993817i \(0.464584\pi\)
\(318\) 0 0
\(319\) 40.9955i 0.128513i
\(320\) 0 0
\(321\) −103.875 −0.323598
\(322\) 0 0
\(323\) − 130.401i − 0.403717i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 46.3213i − 0.141655i
\(328\) 0 0
\(329\) −627.922 −1.90858
\(330\) 0 0
\(331\) − 280.807i − 0.848359i −0.905578 0.424180i \(-0.860562\pi\)
0.905578 0.424180i \(-0.139438\pi\)
\(332\) 0 0
\(333\) 144.150 0.432884
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 120.969 0.358959 0.179480 0.983762i \(-0.442559\pi\)
0.179480 + 0.983762i \(0.442559\pi\)
\(338\) 0 0
\(339\) − 62.7530i − 0.185112i
\(340\) 0 0
\(341\) −654.539 −1.91947
\(342\) 0 0
\(343\) − 31.5280i − 0.0919182i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 214.333i 0.617675i 0.951115 + 0.308838i \(0.0999399\pi\)
−0.951115 + 0.308838i \(0.900060\pi\)
\(348\) 0 0
\(349\) −418.041 −1.19782 −0.598912 0.800815i \(-0.704400\pi\)
−0.598912 + 0.800815i \(0.704400\pi\)
\(350\) 0 0
\(351\) 42.7895i 0.121907i
\(352\) 0 0
\(353\) −364.929 −1.03379 −0.516897 0.856048i \(-0.672913\pi\)
−0.516897 + 0.856048i \(0.672913\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −148.835 −0.416905
\(358\) 0 0
\(359\) 242.169i 0.674567i 0.941403 + 0.337283i \(0.109508\pi\)
−0.941403 + 0.337283i \(0.890492\pi\)
\(360\) 0 0
\(361\) 337.705 0.935471
\(362\) 0 0
\(363\) 95.9389i 0.264295i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 224.564i − 0.611891i −0.952049 0.305945i \(-0.901028\pi\)
0.952049 0.305945i \(-0.0989725\pi\)
\(368\) 0 0
\(369\) −115.863 −0.313992
\(370\) 0 0
\(371\) 720.004i 1.94071i
\(372\) 0 0
\(373\) −572.174 −1.53398 −0.766989 0.641660i \(-0.778246\pi\)
−0.766989 + 0.641660i \(0.778246\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 10.5135 0.0278872
\(378\) 0 0
\(379\) − 122.896i − 0.324263i −0.986769 0.162132i \(-0.948163\pi\)
0.986769 0.162132i \(-0.0518369\pi\)
\(380\) 0 0
\(381\) 24.3281 0.0638533
\(382\) 0 0
\(383\) 289.717i 0.756442i 0.925715 + 0.378221i \(0.123464\pi\)
−0.925715 + 0.378221i \(0.876536\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 520.185i − 1.34415i
\(388\) 0 0
\(389\) 111.590 0.286863 0.143432 0.989660i \(-0.454186\pi\)
0.143432 + 0.989660i \(0.454186\pi\)
\(390\) 0 0
\(391\) 412.433i 1.05482i
\(392\) 0 0
\(393\) −25.8114 −0.0656780
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −705.314 −1.77661 −0.888305 0.459253i \(-0.848117\pi\)
−0.888305 + 0.459253i \(0.848117\pi\)
\(398\) 0 0
\(399\) 26.5883i 0.0666373i
\(400\) 0 0
\(401\) 432.052 1.07744 0.538718 0.842486i \(-0.318909\pi\)
0.538718 + 0.842486i \(0.318909\pi\)
\(402\) 0 0
\(403\) 167.859i 0.416524i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 285.134i − 0.700575i
\(408\) 0 0
\(409\) 98.8102 0.241590 0.120795 0.992677i \(-0.461456\pi\)
0.120795 + 0.992677i \(0.461456\pi\)
\(410\) 0 0
\(411\) 34.4345i 0.0837822i
\(412\) 0 0
\(413\) 692.160 1.67593
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −70.2914 −0.168565
\(418\) 0 0
\(419\) 204.980i 0.489213i 0.969622 + 0.244607i \(0.0786588\pi\)
−0.969622 + 0.244607i \(0.921341\pi\)
\(420\) 0 0
\(421\) −449.956 −1.06878 −0.534390 0.845238i \(-0.679459\pi\)
−0.534390 + 0.845238i \(0.679459\pi\)
\(422\) 0 0
\(423\) 543.214i 1.28419i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 411.756i − 0.964301i
\(428\) 0 0
\(429\) 41.6021 0.0969745
\(430\) 0 0
\(431\) 314.588i 0.729903i 0.931026 + 0.364952i \(0.118914\pi\)
−0.931026 + 0.364952i \(0.881086\pi\)
\(432\) 0 0
\(433\) 330.441 0.763143 0.381571 0.924339i \(-0.375383\pi\)
0.381571 + 0.924339i \(0.375383\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 73.6781 0.168600
\(438\) 0 0
\(439\) 664.815i 1.51439i 0.653191 + 0.757193i \(0.273430\pi\)
−0.653191 + 0.757193i \(0.726570\pi\)
\(440\) 0 0
\(441\) −453.572 −1.02851
\(442\) 0 0
\(443\) − 312.860i − 0.706229i −0.935580 0.353115i \(-0.885123\pi\)
0.935580 0.353115i \(-0.114877\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 90.6912i 0.202889i
\(448\) 0 0
\(449\) 246.330 0.548620 0.274310 0.961641i \(-0.411551\pi\)
0.274310 + 0.961641i \(0.411551\pi\)
\(450\) 0 0
\(451\) 229.181i 0.508161i
\(452\) 0 0
\(453\) −119.722 −0.264287
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −707.094 −1.54725 −0.773626 0.633642i \(-0.781559\pi\)
−0.773626 + 0.633642i \(0.781559\pi\)
\(458\) 0 0
\(459\) 261.955i 0.570707i
\(460\) 0 0
\(461\) 611.288 1.32600 0.663002 0.748618i \(-0.269282\pi\)
0.663002 + 0.748618i \(0.269282\pi\)
\(462\) 0 0
\(463\) − 334.063i − 0.721519i −0.932659 0.360760i \(-0.882518\pi\)
0.932659 0.360760i \(-0.117482\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 128.864i 0.275939i 0.990436 + 0.137970i \(0.0440577\pi\)
−0.990436 + 0.137970i \(0.955942\pi\)
\(468\) 0 0
\(469\) 512.966 1.09374
\(470\) 0 0
\(471\) 95.4742i 0.202705i
\(472\) 0 0
\(473\) −1028.94 −2.17535
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 622.874 1.30582
\(478\) 0 0
\(479\) 510.257i 1.06525i 0.846350 + 0.532627i \(0.178795\pi\)
−0.846350 + 0.532627i \(0.821205\pi\)
\(480\) 0 0
\(481\) −73.1237 −0.152024
\(482\) 0 0
\(483\) − 84.0939i − 0.174107i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 616.472i 1.26586i 0.774211 + 0.632928i \(0.218147\pi\)
−0.774211 + 0.632928i \(0.781853\pi\)
\(488\) 0 0
\(489\) −28.8326 −0.0589624
\(490\) 0 0
\(491\) 603.190i 1.22849i 0.789114 + 0.614247i \(0.210540\pi\)
−0.789114 + 0.614247i \(0.789460\pi\)
\(492\) 0 0
\(493\) 64.3627 0.130553
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −406.817 −0.818545
\(498\) 0 0
\(499\) 867.976i 1.73943i 0.493553 + 0.869716i \(0.335698\pi\)
−0.493553 + 0.869716i \(0.664302\pi\)
\(500\) 0 0
\(501\) −42.0045 −0.0838413
\(502\) 0 0
\(503\) − 57.8408i − 0.114992i −0.998346 0.0574958i \(-0.981688\pi\)
0.998346 0.0574958i \(-0.0183116\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 81.9060i 0.161550i
\(508\) 0 0
\(509\) −168.498 −0.331038 −0.165519 0.986207i \(-0.552930\pi\)
−0.165519 + 0.986207i \(0.552930\pi\)
\(510\) 0 0
\(511\) − 360.470i − 0.705420i
\(512\) 0 0
\(513\) 46.7962 0.0912207
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 1074.49 2.07833
\(518\) 0 0
\(519\) 5.32682i 0.0102636i
\(520\) 0 0
\(521\) −87.1060 −0.167190 −0.0835951 0.996500i \(-0.526640\pi\)
−0.0835951 + 0.996500i \(0.526640\pi\)
\(522\) 0 0
\(523\) − 243.469i − 0.465524i −0.972534 0.232762i \(-0.925224\pi\)
0.972534 0.232762i \(-0.0747763\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1027.62i 1.94995i
\(528\) 0 0
\(529\) 295.969 0.559488
\(530\) 0 0
\(531\) − 598.786i − 1.12766i
\(532\) 0 0
\(533\) 58.7743 0.110271
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −85.1812 −0.158624
\(538\) 0 0
\(539\) 897.179i 1.66452i
\(540\) 0 0
\(541\) 812.616 1.50206 0.751032 0.660266i \(-0.229557\pi\)
0.751032 + 0.660266i \(0.229557\pi\)
\(542\) 0 0
\(543\) 137.135i 0.252550i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 104.713i − 0.191431i −0.995409 0.0957155i \(-0.969486\pi\)
0.995409 0.0957155i \(-0.0305139\pi\)
\(548\) 0 0
\(549\) −356.210 −0.648833
\(550\) 0 0
\(551\) − 11.4979i − 0.0208674i
\(552\) 0 0
\(553\) 1275.17 2.30592
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 260.018 0.466818 0.233409 0.972379i \(-0.425012\pi\)
0.233409 + 0.972379i \(0.425012\pi\)
\(558\) 0 0
\(559\) 263.876i 0.472050i
\(560\) 0 0
\(561\) 254.685 0.453984
\(562\) 0 0
\(563\) − 39.9073i − 0.0708833i −0.999372 0.0354416i \(-0.988716\pi\)
0.999372 0.0354416i \(-0.0112838\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 734.014i 1.29456i
\(568\) 0 0
\(569\) 211.588 0.371860 0.185930 0.982563i \(-0.440470\pi\)
0.185930 + 0.982563i \(0.440470\pi\)
\(570\) 0 0
\(571\) − 556.938i − 0.975372i −0.873019 0.487686i \(-0.837841\pi\)
0.873019 0.487686i \(-0.162159\pi\)
\(572\) 0 0
\(573\) 60.8741 0.106237
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 516.233 0.894684 0.447342 0.894363i \(-0.352371\pi\)
0.447342 + 0.894363i \(0.352371\pi\)
\(578\) 0 0
\(579\) 5.62577i 0.00971636i
\(580\) 0 0
\(581\) −755.742 −1.30076
\(582\) 0 0
\(583\) − 1232.06i − 2.11332i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 934.677i − 1.59229i −0.605103 0.796147i \(-0.706868\pi\)
0.605103 0.796147i \(-0.293132\pi\)
\(588\) 0 0
\(589\) 183.577 0.311676
\(590\) 0 0
\(591\) 89.4050i 0.151277i
\(592\) 0 0
\(593\) −634.665 −1.07026 −0.535131 0.844769i \(-0.679738\pi\)
−0.535131 + 0.844769i \(0.679738\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 106.833 0.178949
\(598\) 0 0
\(599\) 914.029i 1.52593i 0.646443 + 0.762963i \(0.276256\pi\)
−0.646443 + 0.762963i \(0.723744\pi\)
\(600\) 0 0
\(601\) −598.569 −0.995955 −0.497977 0.867190i \(-0.665924\pi\)
−0.497977 + 0.867190i \(0.665924\pi\)
\(602\) 0 0
\(603\) − 443.766i − 0.735930i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 201.830i − 0.332504i −0.986083 0.166252i \(-0.946834\pi\)
0.986083 0.166252i \(-0.0531664\pi\)
\(608\) 0 0
\(609\) −13.1234 −0.0215490
\(610\) 0 0
\(611\) − 275.558i − 0.450995i
\(612\) 0 0
\(613\) 71.9205 0.117325 0.0586627 0.998278i \(-0.481316\pi\)
0.0586627 + 0.998278i \(0.481316\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −204.485 −0.331418 −0.165709 0.986175i \(-0.552991\pi\)
−0.165709 + 0.986175i \(0.552991\pi\)
\(618\) 0 0
\(619\) 287.512i 0.464479i 0.972659 + 0.232239i \(0.0746053\pi\)
−0.972659 + 0.232239i \(0.925395\pi\)
\(620\) 0 0
\(621\) −148.008 −0.238338
\(622\) 0 0
\(623\) 1071.26i 1.71952i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 45.4976i − 0.0725640i
\(628\) 0 0
\(629\) −447.658 −0.711699
\(630\) 0 0
\(631\) − 274.203i − 0.434554i −0.976110 0.217277i \(-0.930283\pi\)
0.976110 0.217277i \(-0.0697175\pi\)
\(632\) 0 0
\(633\) −162.712 −0.257049
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 230.085 0.361201
\(638\) 0 0
\(639\) 351.936i 0.550761i
\(640\) 0 0
\(641\) 681.328 1.06291 0.531457 0.847085i \(-0.321645\pi\)
0.531457 + 0.847085i \(0.321645\pi\)
\(642\) 0 0
\(643\) 527.270i 0.820016i 0.912082 + 0.410008i \(0.134474\pi\)
−0.912082 + 0.410008i \(0.865526\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 551.627i − 0.852592i −0.904584 0.426296i \(-0.859818\pi\)
0.904584 0.426296i \(-0.140182\pi\)
\(648\) 0 0
\(649\) −1184.42 −1.82499
\(650\) 0 0
\(651\) − 209.529i − 0.321857i
\(652\) 0 0
\(653\) −699.422 −1.07109 −0.535545 0.844507i \(-0.679894\pi\)
−0.535545 + 0.844507i \(0.679894\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −311.842 −0.474645
\(658\) 0 0
\(659\) − 38.3257i − 0.0581574i −0.999577 0.0290787i \(-0.990743\pi\)
0.999577 0.0290787i \(-0.00925734\pi\)
\(660\) 0 0
\(661\) −392.364 −0.593591 −0.296796 0.954941i \(-0.595918\pi\)
−0.296796 + 0.954941i \(0.595918\pi\)
\(662\) 0 0
\(663\) − 65.3150i − 0.0985143i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 36.3658i 0.0545215i
\(668\) 0 0
\(669\) −222.035 −0.331890
\(670\) 0 0
\(671\) 704.594i 1.05007i
\(672\) 0 0
\(673\) −427.290 −0.634904 −0.317452 0.948274i \(-0.602827\pi\)
−0.317452 + 0.948274i \(0.602827\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −131.234 −0.193847 −0.0969233 0.995292i \(-0.530900\pi\)
−0.0969233 + 0.995292i \(0.530900\pi\)
\(678\) 0 0
\(679\) − 859.186i − 1.26537i
\(680\) 0 0
\(681\) 20.0495 0.0294413
\(682\) 0 0
\(683\) − 896.229i − 1.31219i −0.754676 0.656097i \(-0.772206\pi\)
0.754676 0.656097i \(-0.227794\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 49.9285i 0.0726761i
\(688\) 0 0
\(689\) −315.968 −0.458589
\(690\) 0 0
\(691\) 520.465i 0.753206i 0.926375 + 0.376603i \(0.122908\pi\)
−0.926375 + 0.376603i \(0.877092\pi\)
\(692\) 0 0
\(693\) 1505.62 2.17262
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 359.812 0.516230
\(698\) 0 0
\(699\) − 185.589i − 0.265507i
\(700\) 0 0
\(701\) −80.7527 −0.115196 −0.0575982 0.998340i \(-0.518344\pi\)
−0.0575982 + 0.998340i \(0.518344\pi\)
\(702\) 0 0
\(703\) 79.9709i 0.113757i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 837.591i 1.18471i
\(708\) 0 0
\(709\) −174.828 −0.246584 −0.123292 0.992370i \(-0.539345\pi\)
−0.123292 + 0.992370i \(0.539345\pi\)
\(710\) 0 0
\(711\) − 1103.15i − 1.55155i
\(712\) 0 0
\(713\) −580.621 −0.814335
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −16.8715 −0.0235307
\(718\) 0 0
\(719\) 889.905i 1.23770i 0.785510 + 0.618849i \(0.212401\pi\)
−0.785510 + 0.618849i \(0.787599\pi\)
\(720\) 0 0
\(721\) 475.328 0.659263
\(722\) 0 0
\(723\) − 131.622i − 0.182050i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 408.818i − 0.562335i −0.959659 0.281168i \(-0.909278\pi\)
0.959659 0.281168i \(-0.0907217\pi\)
\(728\) 0 0
\(729\) 587.194 0.805478
\(730\) 0 0
\(731\) 1615.43i 2.20989i
\(732\) 0 0
\(733\) −388.369 −0.529835 −0.264917 0.964271i \(-0.585345\pi\)
−0.264917 + 0.964271i \(0.585345\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −877.783 −1.19102
\(738\) 0 0
\(739\) − 109.561i − 0.148256i −0.997249 0.0741280i \(-0.976383\pi\)
0.997249 0.0741280i \(-0.0236173\pi\)
\(740\) 0 0
\(741\) −11.6680 −0.0157463
\(742\) 0 0
\(743\) 732.717i 0.986160i 0.869984 + 0.493080i \(0.164129\pi\)
−0.869984 + 0.493080i \(0.835871\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 653.791i 0.875222i
\(748\) 0 0
\(749\) 1907.02 2.54608
\(750\) 0 0
\(751\) 420.809i 0.560332i 0.959952 + 0.280166i \(0.0903895\pi\)
−0.959952 + 0.280166i \(0.909610\pi\)
\(752\) 0 0
\(753\) 47.1238 0.0625814
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1306.99 −1.72654 −0.863269 0.504744i \(-0.831587\pi\)
−0.863269 + 0.504744i \(0.831587\pi\)
\(758\) 0 0
\(759\) 143.901i 0.189592i
\(760\) 0 0
\(761\) −1402.25 −1.84264 −0.921320 0.388804i \(-0.872888\pi\)
−0.921320 + 0.388804i \(0.872888\pi\)
\(762\) 0 0
\(763\) 850.402i 1.11455i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 303.748i 0.396021i
\(768\) 0 0
\(769\) 359.952 0.468078 0.234039 0.972227i \(-0.424806\pi\)
0.234039 + 0.972227i \(0.424806\pi\)
\(770\) 0 0
\(771\) − 194.990i − 0.252905i
\(772\) 0 0
\(773\) −437.539 −0.566027 −0.283013 0.959116i \(-0.591334\pi\)
−0.283013 + 0.959116i \(0.591334\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 91.2762 0.117473
\(778\) 0 0
\(779\) − 64.2778i − 0.0825132i
\(780\) 0 0
\(781\) 696.141 0.891346
\(782\) 0 0
\(783\) 23.0975i 0.0294988i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 438.946i − 0.557746i −0.960328 0.278873i \(-0.910039\pi\)
0.960328 0.278873i \(-0.0899608\pi\)
\(788\) 0 0
\(789\) 40.0567 0.0507690
\(790\) 0 0
\(791\) 1152.07i 1.45647i
\(792\) 0 0
\(793\) 180.696 0.227864
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −982.414 −1.23264 −0.616320 0.787496i \(-0.711377\pi\)
−0.616320 + 0.787496i \(0.711377\pi\)
\(798\) 0 0
\(799\) − 1686.95i − 2.11133i
\(800\) 0 0
\(801\) 926.745 1.15699
\(802\) 0 0
\(803\) 616.832i 0.768160i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 146.806i − 0.181916i
\(808\) 0 0
\(809\) 449.408 0.555510 0.277755 0.960652i \(-0.410410\pi\)
0.277755 + 0.960652i \(0.410410\pi\)
\(810\) 0 0
\(811\) 610.106i 0.752288i 0.926561 + 0.376144i \(0.122750\pi\)
−0.926561 + 0.376144i \(0.877250\pi\)
\(812\) 0 0
\(813\) −125.667 −0.154572
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 288.585 0.353225
\(818\) 0 0
\(819\) − 386.123i − 0.471456i
\(820\) 0 0
\(821\) 1099.76 1.33954 0.669770 0.742568i \(-0.266393\pi\)
0.669770 + 0.742568i \(0.266393\pi\)
\(822\) 0 0
\(823\) 236.092i 0.286867i 0.989660 + 0.143434i \(0.0458143\pi\)
−0.989660 + 0.143434i \(0.954186\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 1266.81i − 1.53182i −0.642950 0.765908i \(-0.722290\pi\)
0.642950 0.765908i \(-0.277710\pi\)
\(828\) 0 0
\(829\) −41.7642 −0.0503790 −0.0251895 0.999683i \(-0.508019\pi\)
−0.0251895 + 0.999683i \(0.508019\pi\)
\(830\) 0 0
\(831\) − 69.1993i − 0.0832723i
\(832\) 0 0
\(833\) 1408.57 1.69095
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −368.778 −0.440595
\(838\) 0 0
\(839\) − 816.273i − 0.972912i −0.873705 0.486456i \(-0.838289\pi\)
0.873705 0.486456i \(-0.161711\pi\)
\(840\) 0 0
\(841\) −835.325 −0.993252
\(842\) 0 0
\(843\) 251.293i 0.298093i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 1761.32i − 2.07948i
\(848\) 0 0
\(849\) 255.024 0.300381
\(850\) 0 0
\(851\) − 252.933i − 0.297219i
\(852\) 0 0
\(853\) −1215.58 −1.42507 −0.712533 0.701639i \(-0.752452\pi\)
−0.712533 + 0.701639i \(0.752452\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 997.998 1.16452 0.582262 0.813001i \(-0.302168\pi\)
0.582262 + 0.813001i \(0.302168\pi\)
\(858\) 0 0
\(859\) − 1110.42i − 1.29269i −0.763044 0.646347i \(-0.776296\pi\)
0.763044 0.646347i \(-0.223704\pi\)
\(860\) 0 0
\(861\) −73.3646 −0.0852086
\(862\) 0 0
\(863\) − 1372.33i − 1.59019i −0.606486 0.795094i \(-0.707422\pi\)
0.606486 0.795094i \(-0.292578\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 241.545i − 0.278599i
\(868\) 0 0
\(869\) −2182.06 −2.51101
\(870\) 0 0
\(871\) 225.111i 0.258451i
\(872\) 0 0
\(873\) −743.280 −0.851409
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 776.337 0.885219 0.442609 0.896714i \(-0.354053\pi\)
0.442609 + 0.896714i \(0.354053\pi\)
\(878\) 0 0
\(879\) − 90.5034i − 0.102962i
\(880\) 0 0
\(881\) 1047.38 1.18885 0.594427 0.804150i \(-0.297379\pi\)
0.594427 + 0.804150i \(0.297379\pi\)
\(882\) 0 0
\(883\) 1175.41i 1.33116i 0.746327 + 0.665579i \(0.231815\pi\)
−0.746327 + 0.665579i \(0.768185\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 109.768i − 0.123752i −0.998084 0.0618762i \(-0.980292\pi\)
0.998084 0.0618762i \(-0.0197084\pi\)
\(888\) 0 0
\(889\) −446.634 −0.502401
\(890\) 0 0
\(891\) − 1256.04i − 1.40969i
\(892\) 0 0
\(893\) −301.361 −0.337470
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 36.9039 0.0411414
\(898\) 0 0
\(899\) 90.6094i 0.100789i
\(900\) 0 0
\(901\) −1934.33 −2.14687
\(902\) 0 0
\(903\) − 329.382i − 0.364764i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1639.15i 1.80722i 0.428358 + 0.903609i \(0.359092\pi\)
−0.428358 + 0.903609i \(0.640908\pi\)
\(908\) 0 0
\(909\) 724.598 0.797137
\(910\) 0 0
\(911\) − 6.82023i − 0.00748654i −0.999993 0.00374327i \(-0.998808\pi\)
0.999993 0.00374327i \(-0.00119152\pi\)
\(912\) 0 0
\(913\) 1293.22 1.41645
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 473.867 0.516757
\(918\) 0 0
\(919\) − 72.4188i − 0.0788017i −0.999223 0.0394009i \(-0.987455\pi\)
0.999223 0.0394009i \(-0.0125449\pi\)
\(920\) 0 0
\(921\) 128.914 0.139972
\(922\) 0 0
\(923\) − 178.528i − 0.193421i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 411.205i − 0.443587i
\(928\) 0 0
\(929\) 368.537 0.396703 0.198352 0.980131i \(-0.436441\pi\)
0.198352 + 0.980131i \(0.436441\pi\)
\(930\) 0 0
\(931\) − 251.630i − 0.270279i
\(932\) 0 0
\(933\) 115.399 0.123686
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 590.172 0.629852 0.314926 0.949116i \(-0.398020\pi\)
0.314926 + 0.949116i \(0.398020\pi\)
\(938\) 0 0
\(939\) − 174.599i − 0.185941i
\(940\) 0 0
\(941\) 189.017 0.200869 0.100434 0.994944i \(-0.467977\pi\)
0.100434 + 0.994944i \(0.467977\pi\)
\(942\) 0 0
\(943\) 203.299i 0.215588i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 664.319i 0.701498i 0.936470 + 0.350749i \(0.114073\pi\)
−0.936470 + 0.350749i \(0.885927\pi\)
\(948\) 0 0
\(949\) 158.189 0.166690
\(950\) 0 0
\(951\) − 38.5610i − 0.0405479i
\(952\) 0 0
\(953\) −173.943 −0.182522 −0.0912610 0.995827i \(-0.529090\pi\)
−0.0912610 + 0.995827i \(0.529090\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 22.4566 0.0234656
\(958\) 0 0
\(959\) − 632.175i − 0.659202i
\(960\) 0 0
\(961\) −485.680 −0.505390
\(962\) 0 0
\(963\) − 1649.76i − 1.71314i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 255.364i − 0.264078i −0.991245 0.132039i \(-0.957848\pi\)
0.991245 0.132039i \(-0.0421524\pi\)
\(968\) 0 0
\(969\) −71.4309 −0.0737161
\(970\) 0 0
\(971\) − 1623.60i − 1.67209i −0.548659 0.836046i \(-0.684861\pi\)
0.548659 0.836046i \(-0.315139\pi\)
\(972\) 0 0
\(973\) 1290.46 1.32627
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −104.946 −0.107416 −0.0537082 0.998557i \(-0.517104\pi\)
−0.0537082 + 0.998557i \(0.517104\pi\)
\(978\) 0 0
\(979\) − 1833.13i − 1.87245i
\(980\) 0 0
\(981\) 735.681 0.749930
\(982\) 0 0
\(983\) − 889.933i − 0.905323i −0.891682 0.452662i \(-0.850475\pi\)
0.891682 0.452662i \(-0.149525\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 343.964i 0.348494i
\(988\) 0 0
\(989\) −912.742 −0.922894
\(990\) 0 0
\(991\) − 1509.57i − 1.52328i −0.648001 0.761639i \(-0.724395\pi\)
0.648001 0.761639i \(-0.275605\pi\)
\(992\) 0 0
\(993\) −153.821 −0.154905
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1599.93 −1.60474 −0.802371 0.596825i \(-0.796429\pi\)
−0.802371 + 0.596825i \(0.796429\pi\)
\(998\) 0 0
\(999\) − 160.649i − 0.160810i
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 1600.3.b.v.1151.3 6
4.3 odd 2 inner 1600.3.b.v.1151.4 6
5.2 odd 4 320.3.h.g.319.4 6
5.3 odd 4 320.3.h.f.319.3 6
5.4 even 2 1600.3.b.w.1151.4 6
8.3 odd 2 800.3.b.h.351.3 6
8.5 even 2 800.3.b.h.351.4 6
20.3 even 4 320.3.h.g.319.3 6
20.7 even 4 320.3.h.f.319.4 6
20.19 odd 2 1600.3.b.w.1151.3 6
40.3 even 4 160.3.h.a.159.4 yes 6
40.13 odd 4 160.3.h.b.159.4 yes 6
40.19 odd 2 800.3.b.i.351.4 6
40.27 even 4 160.3.h.b.159.3 yes 6
40.29 even 2 800.3.b.i.351.3 6
40.37 odd 4 160.3.h.a.159.3 6
80.3 even 4 1280.3.e.i.639.3 6
80.13 odd 4 1280.3.e.h.639.4 6
80.27 even 4 1280.3.e.h.639.3 6
80.37 odd 4 1280.3.e.i.639.4 6
80.43 even 4 1280.3.e.g.639.4 6
80.53 odd 4 1280.3.e.f.639.3 6
80.67 even 4 1280.3.e.f.639.4 6
80.77 odd 4 1280.3.e.g.639.3 6
120.53 even 4 1440.3.j.b.1279.5 6
120.77 even 4 1440.3.j.a.1279.6 6
120.83 odd 4 1440.3.j.a.1279.5 6
120.107 odd 4 1440.3.j.b.1279.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.3.h.a.159.3 6 40.37 odd 4
160.3.h.a.159.4 yes 6 40.3 even 4
160.3.h.b.159.3 yes 6 40.27 even 4
160.3.h.b.159.4 yes 6 40.13 odd 4
320.3.h.f.319.3 6 5.3 odd 4
320.3.h.f.319.4 6 20.7 even 4
320.3.h.g.319.3 6 20.3 even 4
320.3.h.g.319.4 6 5.2 odd 4
800.3.b.h.351.3 6 8.3 odd 2
800.3.b.h.351.4 6 8.5 even 2
800.3.b.i.351.3 6 40.29 even 2
800.3.b.i.351.4 6 40.19 odd 2
1280.3.e.f.639.3 6 80.53 odd 4
1280.3.e.f.639.4 6 80.67 even 4
1280.3.e.g.639.3 6 80.77 odd 4
1280.3.e.g.639.4 6 80.43 even 4
1280.3.e.h.639.3 6 80.27 even 4
1280.3.e.h.639.4 6 80.13 odd 4
1280.3.e.i.639.3 6 80.3 even 4
1280.3.e.i.639.4 6 80.37 odd 4
1440.3.j.a.1279.5 6 120.83 odd 4
1440.3.j.a.1279.6 6 120.77 even 4
1440.3.j.b.1279.5 6 120.53 even 4
1440.3.j.b.1279.6 6 120.107 odd 4
1600.3.b.v.1151.3 6 1.1 even 1 trivial
1600.3.b.v.1151.4 6 4.3 odd 2 inner
1600.3.b.w.1151.3 6 20.19 odd 2
1600.3.b.w.1151.4 6 5.4 even 2