Properties

Label 160.4.c.b.129.3
Level $160$
Weight $4$
Character 160.129
Analytic conductor $9.440$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [160,4,Mod(129,160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("160.129");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 160.c (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.44030560092\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{29})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 15x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 129.3
Root \(-2.19258i\) of defining polynomial
Character \(\chi\) \(=\) 160.129
Dual form 160.4.c.b.129.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000i q^{3} +(-3.00000 - 10.7703i) q^{5} +4.00000i q^{7} +11.0000 q^{9} +O(q^{10})\) \(q+4.00000i q^{3} +(-3.00000 - 10.7703i) q^{5} +4.00000i q^{7} +11.0000 q^{9} +43.0813 q^{11} -21.5407i q^{13} +(43.0813 - 12.0000i) q^{15} +43.0813i q^{17} +129.244 q^{19} -16.0000 q^{21} +52.0000i q^{23} +(-107.000 + 64.6220i) q^{25} +152.000i q^{27} +158.000 q^{29} +172.325 q^{31} +172.325i q^{33} +(43.0813 - 12.0000i) q^{35} -280.029i q^{37} +86.1626 q^{39} -170.000 q^{41} -316.000i q^{43} +(-33.0000 - 118.474i) q^{45} +244.000i q^{47} +327.000 q^{49} -172.325 q^{51} -495.435i q^{53} +(-129.244 - 464.000i) q^{55} +516.976i q^{57} -646.220 q^{59} +82.0000 q^{61} +44.0000i q^{63} +(-232.000 + 64.6220i) q^{65} +692.000i q^{67} -208.000 q^{69} -947.789 q^{71} +430.813i q^{73} +(-258.488 - 428.000i) q^{75} +172.325i q^{77} -344.651 q^{79} -311.000 q^{81} -940.000i q^{83} +(464.000 - 129.244i) q^{85} +632.000i q^{87} -6.00000 q^{89} +86.1626 q^{91} +689.301i q^{93} +(-387.732 - 1392.00i) q^{95} -1077.03i q^{97} +473.895 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{5} + 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{5} + 44 q^{9} - 64 q^{21} - 428 q^{25} + 632 q^{29} - 680 q^{41} - 132 q^{45} + 1308 q^{49} + 328 q^{61} - 928 q^{65} - 832 q^{69} - 1244 q^{81} + 1856 q^{85} - 24 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 4.00000i 0.769800i 0.922958 + 0.384900i \(0.125764\pi\)
−0.922958 + 0.384900i \(0.874236\pi\)
\(4\) 0 0
\(5\) −3.00000 10.7703i −0.268328 0.963328i
\(6\) 0 0
\(7\) 4.00000i 0.215980i 0.994152 + 0.107990i \(0.0344414\pi\)
−0.994152 + 0.107990i \(0.965559\pi\)
\(8\) 0 0
\(9\) 11.0000 0.407407
\(10\) 0 0
\(11\) 43.0813 1.18086 0.590432 0.807087i \(-0.298957\pi\)
0.590432 + 0.807087i \(0.298957\pi\)
\(12\) 0 0
\(13\) 21.5407i 0.459562i −0.973242 0.229781i \(-0.926199\pi\)
0.973242 0.229781i \(-0.0738010\pi\)
\(14\) 0 0
\(15\) 43.0813 12.0000i 0.741570 0.206559i
\(16\) 0 0
\(17\) 43.0813i 0.614633i 0.951607 + 0.307316i \(0.0994310\pi\)
−0.951607 + 0.307316i \(0.900569\pi\)
\(18\) 0 0
\(19\) 129.244 1.56056 0.780279 0.625432i \(-0.215077\pi\)
0.780279 + 0.625432i \(0.215077\pi\)
\(20\) 0 0
\(21\) −16.0000 −0.166261
\(22\) 0 0
\(23\) 52.0000i 0.471424i 0.971823 + 0.235712i \(0.0757422\pi\)
−0.971823 + 0.235712i \(0.924258\pi\)
\(24\) 0 0
\(25\) −107.000 + 64.6220i −0.856000 + 0.516976i
\(26\) 0 0
\(27\) 152.000i 1.08342i
\(28\) 0 0
\(29\) 158.000 1.01172 0.505860 0.862616i \(-0.331175\pi\)
0.505860 + 0.862616i \(0.331175\pi\)
\(30\) 0 0
\(31\) 172.325 0.998404 0.499202 0.866486i \(-0.333627\pi\)
0.499202 + 0.866486i \(0.333627\pi\)
\(32\) 0 0
\(33\) 172.325i 0.909030i
\(34\) 0 0
\(35\) 43.0813 12.0000i 0.208059 0.0579534i
\(36\) 0 0
\(37\) 280.029i 1.24423i −0.782927 0.622114i \(-0.786274\pi\)
0.782927 0.622114i \(-0.213726\pi\)
\(38\) 0 0
\(39\) 86.1626 0.353771
\(40\) 0 0
\(41\) −170.000 −0.647550 −0.323775 0.946134i \(-0.604952\pi\)
−0.323775 + 0.946134i \(0.604952\pi\)
\(42\) 0 0
\(43\) 316.000i 1.12069i −0.828260 0.560344i \(-0.810669\pi\)
0.828260 0.560344i \(-0.189331\pi\)
\(44\) 0 0
\(45\) −33.0000 118.474i −0.109319 0.392467i
\(46\) 0 0
\(47\) 244.000i 0.757257i 0.925549 + 0.378628i \(0.123604\pi\)
−0.925549 + 0.378628i \(0.876396\pi\)
\(48\) 0 0
\(49\) 327.000 0.953353
\(50\) 0 0
\(51\) −172.325 −0.473144
\(52\) 0 0
\(53\) 495.435i 1.28402i −0.766695 0.642012i \(-0.778100\pi\)
0.766695 0.642012i \(-0.221900\pi\)
\(54\) 0 0
\(55\) −129.244 464.000i −0.316859 1.13756i
\(56\) 0 0
\(57\) 516.976i 1.20132i
\(58\) 0 0
\(59\) −646.220 −1.42594 −0.712972 0.701193i \(-0.752651\pi\)
−0.712972 + 0.701193i \(0.752651\pi\)
\(60\) 0 0
\(61\) 82.0000 0.172115 0.0860576 0.996290i \(-0.472573\pi\)
0.0860576 + 0.996290i \(0.472573\pi\)
\(62\) 0 0
\(63\) 44.0000i 0.0879917i
\(64\) 0 0
\(65\) −232.000 + 64.6220i −0.442709 + 0.123313i
\(66\) 0 0
\(67\) 692.000i 1.26181i 0.775860 + 0.630905i \(0.217316\pi\)
−0.775860 + 0.630905i \(0.782684\pi\)
\(68\) 0 0
\(69\) −208.000 −0.362902
\(70\) 0 0
\(71\) −947.789 −1.58425 −0.792126 0.610358i \(-0.791026\pi\)
−0.792126 + 0.610358i \(0.791026\pi\)
\(72\) 0 0
\(73\) 430.813i 0.690724i 0.938470 + 0.345362i \(0.112244\pi\)
−0.938470 + 0.345362i \(0.887756\pi\)
\(74\) 0 0
\(75\) −258.488 428.000i −0.397968 0.658949i
\(76\) 0 0
\(77\) 172.325i 0.255043i
\(78\) 0 0
\(79\) −344.651 −0.490838 −0.245419 0.969417i \(-0.578926\pi\)
−0.245419 + 0.969417i \(0.578926\pi\)
\(80\) 0 0
\(81\) −311.000 −0.426612
\(82\) 0 0
\(83\) 940.000i 1.24311i −0.783369 0.621557i \(-0.786501\pi\)
0.783369 0.621557i \(-0.213499\pi\)
\(84\) 0 0
\(85\) 464.000 129.244i 0.592093 0.164923i
\(86\) 0 0
\(87\) 632.000i 0.778822i
\(88\) 0 0
\(89\) −6.00000 −0.00714605 −0.00357303 0.999994i \(-0.501137\pi\)
−0.00357303 + 0.999994i \(0.501137\pi\)
\(90\) 0 0
\(91\) 86.1626 0.0992560
\(92\) 0 0
\(93\) 689.301i 0.768572i
\(94\) 0 0
\(95\) −387.732 1392.00i −0.418742 1.50333i
\(96\) 0 0
\(97\) 1077.03i 1.12738i −0.825985 0.563691i \(-0.809381\pi\)
0.825985 0.563691i \(-0.190619\pi\)
\(98\) 0 0
\(99\) 473.895 0.481093
\(100\) 0 0
\(101\) 1014.00 0.998978 0.499489 0.866320i \(-0.333521\pi\)
0.499489 + 0.866320i \(0.333521\pi\)
\(102\) 0 0
\(103\) 44.0000i 0.0420917i −0.999779 0.0210459i \(-0.993300\pi\)
0.999779 0.0210459i \(-0.00669960\pi\)
\(104\) 0 0
\(105\) 48.0000 + 172.325i 0.0446126 + 0.160164i
\(106\) 0 0
\(107\) 1812.00i 1.63713i 0.574416 + 0.818564i \(0.305229\pi\)
−0.574416 + 0.818564i \(0.694771\pi\)
\(108\) 0 0
\(109\) −2014.00 −1.76978 −0.884891 0.465798i \(-0.845767\pi\)
−0.884891 + 0.465798i \(0.845767\pi\)
\(110\) 0 0
\(111\) 1120.11 0.957807
\(112\) 0 0
\(113\) 1637.09i 1.36287i −0.731878 0.681436i \(-0.761356\pi\)
0.731878 0.681436i \(-0.238644\pi\)
\(114\) 0 0
\(115\) 560.057 156.000i 0.454136 0.126496i
\(116\) 0 0
\(117\) 236.947i 0.187229i
\(118\) 0 0
\(119\) −172.325 −0.132748
\(120\) 0 0
\(121\) 525.000 0.394440
\(122\) 0 0
\(123\) 680.000i 0.498484i
\(124\) 0 0
\(125\) 1017.00 + 958.559i 0.727706 + 0.685889i
\(126\) 0 0
\(127\) 2788.00i 1.94799i 0.226565 + 0.973996i \(0.427250\pi\)
−0.226565 + 0.973996i \(0.572750\pi\)
\(128\) 0 0
\(129\) 1264.00 0.862705
\(130\) 0 0
\(131\) −43.0813 −0.0287331 −0.0143665 0.999897i \(-0.504573\pi\)
−0.0143665 + 0.999897i \(0.504573\pi\)
\(132\) 0 0
\(133\) 516.976i 0.337049i
\(134\) 0 0
\(135\) 1637.09 456.000i 1.04369 0.290713i
\(136\) 0 0
\(137\) 473.895i 0.295529i 0.989023 + 0.147765i \(0.0472078\pi\)
−0.989023 + 0.147765i \(0.952792\pi\)
\(138\) 0 0
\(139\) 2110.98 1.28814 0.644070 0.764967i \(-0.277245\pi\)
0.644070 + 0.764967i \(0.277245\pi\)
\(140\) 0 0
\(141\) −976.000 −0.582936
\(142\) 0 0
\(143\) 928.000i 0.542680i
\(144\) 0 0
\(145\) −474.000 1701.71i −0.271473 0.974617i
\(146\) 0 0
\(147\) 1308.00i 0.733891i
\(148\) 0 0
\(149\) −2550.00 −1.40204 −0.701021 0.713141i \(-0.747272\pi\)
−0.701021 + 0.713141i \(0.747272\pi\)
\(150\) 0 0
\(151\) 2154.07 1.16090 0.580448 0.814297i \(-0.302877\pi\)
0.580448 + 0.814297i \(0.302877\pi\)
\(152\) 0 0
\(153\) 473.895i 0.250406i
\(154\) 0 0
\(155\) −516.976 1856.00i −0.267900 0.961790i
\(156\) 0 0
\(157\) 3554.21i 1.80673i 0.428872 + 0.903365i \(0.358911\pi\)
−0.428872 + 0.903365i \(0.641089\pi\)
\(158\) 0 0
\(159\) 1981.74 0.988442
\(160\) 0 0
\(161\) −208.000 −0.101818
\(162\) 0 0
\(163\) 228.000i 0.109560i 0.998498 + 0.0547802i \(0.0174458\pi\)
−0.998498 + 0.0547802i \(0.982554\pi\)
\(164\) 0 0
\(165\) 1856.00 516.976i 0.875693 0.243918i
\(166\) 0 0
\(167\) 372.000i 0.172373i 0.996279 + 0.0861863i \(0.0274680\pi\)
−0.996279 + 0.0861863i \(0.972532\pi\)
\(168\) 0 0
\(169\) 1733.00 0.788803
\(170\) 0 0
\(171\) 1421.68 0.635783
\(172\) 0 0
\(173\) 1917.12i 0.842519i 0.906940 + 0.421260i \(0.138412\pi\)
−0.906940 + 0.421260i \(0.861588\pi\)
\(174\) 0 0
\(175\) −258.488 428.000i −0.111656 0.184879i
\(176\) 0 0
\(177\) 2584.88i 1.09769i
\(178\) 0 0
\(179\) −2800.29 −1.16929 −0.584646 0.811289i \(-0.698766\pi\)
−0.584646 + 0.811289i \(0.698766\pi\)
\(180\) 0 0
\(181\) 1542.00 0.633237 0.316619 0.948553i \(-0.397452\pi\)
0.316619 + 0.948553i \(0.397452\pi\)
\(182\) 0 0
\(183\) 328.000i 0.132494i
\(184\) 0 0
\(185\) −3016.00 + 840.086i −1.19860 + 0.333861i
\(186\) 0 0
\(187\) 1856.00i 0.725798i
\(188\) 0 0
\(189\) −608.000 −0.233997
\(190\) 0 0
\(191\) −4480.46 −1.69735 −0.848677 0.528912i \(-0.822600\pi\)
−0.848677 + 0.528912i \(0.822600\pi\)
\(192\) 0 0
\(193\) 2541.80i 0.947993i −0.880527 0.473996i \(-0.842811\pi\)
0.880527 0.473996i \(-0.157189\pi\)
\(194\) 0 0
\(195\) −258.488 928.000i −0.0949267 0.340797i
\(196\) 0 0
\(197\) 2735.66i 0.989381i −0.869069 0.494690i \(-0.835282\pi\)
0.869069 0.494690i \(-0.164718\pi\)
\(198\) 0 0
\(199\) −258.488 −0.0920790 −0.0460395 0.998940i \(-0.514660\pi\)
−0.0460395 + 0.998940i \(0.514660\pi\)
\(200\) 0 0
\(201\) −2768.00 −0.971342
\(202\) 0 0
\(203\) 632.000i 0.218511i
\(204\) 0 0
\(205\) 510.000 + 1830.96i 0.173756 + 0.623803i
\(206\) 0 0
\(207\) 572.000i 0.192062i
\(208\) 0 0
\(209\) 5568.00 1.84281
\(210\) 0 0
\(211\) −904.708 −0.295178 −0.147589 0.989049i \(-0.547151\pi\)
−0.147589 + 0.989049i \(0.547151\pi\)
\(212\) 0 0
\(213\) 3791.16i 1.21956i
\(214\) 0 0
\(215\) −3403.42 + 948.000i −1.07959 + 0.300712i
\(216\) 0 0
\(217\) 689.301i 0.215635i
\(218\) 0 0
\(219\) −1723.25 −0.531720
\(220\) 0 0
\(221\) 928.000 0.282462
\(222\) 0 0
\(223\) 4284.00i 1.28645i −0.765678 0.643224i \(-0.777596\pi\)
0.765678 0.643224i \(-0.222404\pi\)
\(224\) 0 0
\(225\) −1177.00 + 710.842i −0.348741 + 0.210620i
\(226\) 0 0
\(227\) 4956.00i 1.44908i −0.689232 0.724540i \(-0.742052\pi\)
0.689232 0.724540i \(-0.257948\pi\)
\(228\) 0 0
\(229\) −1770.00 −0.510764 −0.255382 0.966840i \(-0.582201\pi\)
−0.255382 + 0.966840i \(0.582201\pi\)
\(230\) 0 0
\(231\) −689.301 −0.196332
\(232\) 0 0
\(233\) 5428.25i 1.52625i 0.646251 + 0.763125i \(0.276336\pi\)
−0.646251 + 0.763125i \(0.723664\pi\)
\(234\) 0 0
\(235\) 2627.96 732.000i 0.729486 0.203193i
\(236\) 0 0
\(237\) 1378.60i 0.377847i
\(238\) 0 0
\(239\) −5514.41 −1.49246 −0.746229 0.665689i \(-0.768138\pi\)
−0.746229 + 0.665689i \(0.768138\pi\)
\(240\) 0 0
\(241\) −1618.00 −0.432467 −0.216233 0.976342i \(-0.569377\pi\)
−0.216233 + 0.976342i \(0.569377\pi\)
\(242\) 0 0
\(243\) 2860.00i 0.755017i
\(244\) 0 0
\(245\) −981.000 3521.90i −0.255811 0.918391i
\(246\) 0 0
\(247\) 2784.00i 0.717173i
\(248\) 0 0
\(249\) 3760.00 0.956949
\(250\) 0 0
\(251\) −4954.35 −1.24588 −0.622940 0.782270i \(-0.714062\pi\)
−0.622940 + 0.782270i \(0.714062\pi\)
\(252\) 0 0
\(253\) 2240.23i 0.556688i
\(254\) 0 0
\(255\) 516.976 + 1856.00i 0.126958 + 0.455793i
\(256\) 0 0
\(257\) 4652.78i 1.12931i 0.825327 + 0.564655i \(0.190991\pi\)
−0.825327 + 0.564655i \(0.809009\pi\)
\(258\) 0 0
\(259\) 1120.11 0.268728
\(260\) 0 0
\(261\) 1738.00 0.412182
\(262\) 0 0
\(263\) 604.000i 0.141613i −0.997490 0.0708065i \(-0.977443\pi\)
0.997490 0.0708065i \(-0.0225573\pi\)
\(264\) 0 0
\(265\) −5336.00 + 1486.31i −1.23694 + 0.344540i
\(266\) 0 0
\(267\) 24.0000i 0.00550103i
\(268\) 0 0
\(269\) −3262.00 −0.739359 −0.369680 0.929159i \(-0.620533\pi\)
−0.369680 + 0.929159i \(0.620533\pi\)
\(270\) 0 0
\(271\) 2067.90 0.463528 0.231764 0.972772i \(-0.425550\pi\)
0.231764 + 0.972772i \(0.425550\pi\)
\(272\) 0 0
\(273\) 344.651i 0.0764073i
\(274\) 0 0
\(275\) −4609.70 + 2784.00i −1.01082 + 0.610478i
\(276\) 0 0
\(277\) 1227.82i 0.266326i −0.991094 0.133163i \(-0.957487\pi\)
0.991094 0.133163i \(-0.0425134\pi\)
\(278\) 0 0
\(279\) 1895.58 0.406757
\(280\) 0 0
\(281\) −1290.00 −0.273861 −0.136931 0.990581i \(-0.543724\pi\)
−0.136931 + 0.990581i \(0.543724\pi\)
\(282\) 0 0
\(283\) 692.000i 0.145354i 0.997356 + 0.0726769i \(0.0231542\pi\)
−0.997356 + 0.0726769i \(0.976846\pi\)
\(284\) 0 0
\(285\) 5568.00 1550.93i 1.15726 0.322347i
\(286\) 0 0
\(287\) 680.000i 0.139858i
\(288\) 0 0
\(289\) 3057.00 0.622227
\(290\) 0 0
\(291\) 4308.13 0.867860
\(292\) 0 0
\(293\) 5535.95i 1.10380i −0.833910 0.551900i \(-0.813903\pi\)
0.833910 0.551900i \(-0.186097\pi\)
\(294\) 0 0
\(295\) 1938.66 + 6960.00i 0.382621 + 1.37365i
\(296\) 0 0
\(297\) 6548.36i 1.27938i
\(298\) 0 0
\(299\) 1120.11 0.216648
\(300\) 0 0
\(301\) 1264.00 0.242046
\(302\) 0 0
\(303\) 4056.00i 0.769014i
\(304\) 0 0
\(305\) −246.000 883.167i −0.0461833 0.165803i
\(306\) 0 0
\(307\) 7436.00i 1.38239i −0.722666 0.691197i \(-0.757084\pi\)
0.722666 0.691197i \(-0.242916\pi\)
\(308\) 0 0
\(309\) 176.000 0.0324022
\(310\) 0 0
\(311\) 2498.72 0.455592 0.227796 0.973709i \(-0.426848\pi\)
0.227796 + 0.973709i \(0.426848\pi\)
\(312\) 0 0
\(313\) 2714.12i 0.490132i −0.969506 0.245066i \(-0.921190\pi\)
0.969506 0.245066i \(-0.0788096\pi\)
\(314\) 0 0
\(315\) 473.895 132.000i 0.0847649 0.0236107i
\(316\) 0 0
\(317\) 3295.72i 0.583931i −0.956429 0.291965i \(-0.905691\pi\)
0.956429 0.291965i \(-0.0943092\pi\)
\(318\) 0 0
\(319\) 6806.85 1.19470
\(320\) 0 0
\(321\) −7248.00 −1.26026
\(322\) 0 0
\(323\) 5568.00i 0.959170i
\(324\) 0 0
\(325\) 1392.00 + 2304.85i 0.237582 + 0.393385i
\(326\) 0 0
\(327\) 8056.00i 1.36238i
\(328\) 0 0
\(329\) −976.000 −0.163552
\(330\) 0 0
\(331\) 2972.61 0.493624 0.246812 0.969063i \(-0.420617\pi\)
0.246812 + 0.969063i \(0.420617\pi\)
\(332\) 0 0
\(333\) 3080.31i 0.506907i
\(334\) 0 0
\(335\) 7453.07 2076.00i 1.21554 0.338579i
\(336\) 0 0
\(337\) 9133.24i 1.47632i −0.674627 0.738159i \(-0.735695\pi\)
0.674627 0.738159i \(-0.264305\pi\)
\(338\) 0 0
\(339\) 6548.36 1.04914
\(340\) 0 0
\(341\) 7424.00 1.17898
\(342\) 0 0
\(343\) 2680.00i 0.421885i
\(344\) 0 0
\(345\) 624.000 + 2240.23i 0.0973769 + 0.349594i
\(346\) 0 0
\(347\) 5916.00i 0.915238i −0.889148 0.457619i \(-0.848702\pi\)
0.889148 0.457619i \(-0.151298\pi\)
\(348\) 0 0
\(349\) −7522.00 −1.15371 −0.576853 0.816848i \(-0.695719\pi\)
−0.576853 + 0.816848i \(0.695719\pi\)
\(350\) 0 0
\(351\) 3274.18 0.497900
\(352\) 0 0
\(353\) 8185.45i 1.23419i 0.786890 + 0.617093i \(0.211690\pi\)
−0.786890 + 0.617093i \(0.788310\pi\)
\(354\) 0 0
\(355\) 2843.37 + 10208.0i 0.425099 + 1.52615i
\(356\) 0 0
\(357\) 689.301i 0.102190i
\(358\) 0 0
\(359\) −1464.76 −0.215341 −0.107670 0.994187i \(-0.534339\pi\)
−0.107670 + 0.994187i \(0.534339\pi\)
\(360\) 0 0
\(361\) 9845.00 1.43534
\(362\) 0 0
\(363\) 2100.00i 0.303640i
\(364\) 0 0
\(365\) 4640.00 1292.44i 0.665394 0.185341i
\(366\) 0 0
\(367\) 516.000i 0.0733923i 0.999326 + 0.0366962i \(0.0116834\pi\)
−0.999326 + 0.0366962i \(0.988317\pi\)
\(368\) 0 0
\(369\) −1870.00 −0.263817
\(370\) 0 0
\(371\) 1981.74 0.277323
\(372\) 0 0
\(373\) 8077.75i 1.12131i 0.828048 + 0.560657i \(0.189451\pi\)
−0.828048 + 0.560657i \(0.810549\pi\)
\(374\) 0 0
\(375\) −3834.24 + 4068.00i −0.527998 + 0.560188i
\(376\) 0 0
\(377\) 3403.42i 0.464948i
\(378\) 0 0
\(379\) −2197.15 −0.297783 −0.148892 0.988854i \(-0.547571\pi\)
−0.148892 + 0.988854i \(0.547571\pi\)
\(380\) 0 0
\(381\) −11152.0 −1.49957
\(382\) 0 0
\(383\) 5988.00i 0.798884i 0.916759 + 0.399442i \(0.130796\pi\)
−0.916759 + 0.399442i \(0.869204\pi\)
\(384\) 0 0
\(385\) 1856.00 516.976i 0.245690 0.0684351i
\(386\) 0 0
\(387\) 3476.00i 0.456576i
\(388\) 0 0
\(389\) −7974.00 −1.03933 −0.519663 0.854371i \(-0.673943\pi\)
−0.519663 + 0.854371i \(0.673943\pi\)
\(390\) 0 0
\(391\) −2240.23 −0.289753
\(392\) 0 0
\(393\) 172.325i 0.0221187i
\(394\) 0 0
\(395\) 1033.95 + 3712.00i 0.131706 + 0.472838i
\(396\) 0 0
\(397\) 7991.58i 1.01029i 0.863034 + 0.505146i \(0.168561\pi\)
−0.863034 + 0.505146i \(0.831439\pi\)
\(398\) 0 0
\(399\) −2067.90 −0.259460
\(400\) 0 0
\(401\) 418.000 0.0520547 0.0260273 0.999661i \(-0.491714\pi\)
0.0260273 + 0.999661i \(0.491714\pi\)
\(402\) 0 0
\(403\) 3712.00i 0.458829i
\(404\) 0 0
\(405\) 933.000 + 3349.57i 0.114472 + 0.410967i
\(406\) 0 0
\(407\) 12064.0i 1.46926i
\(408\) 0 0
\(409\) 1414.00 0.170948 0.0854741 0.996340i \(-0.472760\pi\)
0.0854741 + 0.996340i \(0.472760\pi\)
\(410\) 0 0
\(411\) −1895.58 −0.227499
\(412\) 0 0
\(413\) 2584.88i 0.307975i
\(414\) 0 0
\(415\) −10124.1 + 2820.00i −1.19753 + 0.333562i
\(416\) 0 0
\(417\) 8443.94i 0.991610i
\(418\) 0 0
\(419\) −3661.91 −0.426960 −0.213480 0.976947i \(-0.568480\pi\)
−0.213480 + 0.976947i \(0.568480\pi\)
\(420\) 0 0
\(421\) 3290.00 0.380866 0.190433 0.981700i \(-0.439011\pi\)
0.190433 + 0.981700i \(0.439011\pi\)
\(422\) 0 0
\(423\) 2684.00i 0.308512i
\(424\) 0 0
\(425\) −2784.00 4609.70i −0.317750 0.526126i
\(426\) 0 0
\(427\) 328.000i 0.0371734i
\(428\) 0 0
\(429\) 3712.00 0.417755
\(430\) 0 0
\(431\) −1723.25 −0.192590 −0.0962949 0.995353i \(-0.530699\pi\)
−0.0962949 + 0.995353i \(0.530699\pi\)
\(432\) 0 0
\(433\) 16414.0i 1.82172i −0.412713 0.910861i \(-0.635419\pi\)
0.412713 0.910861i \(-0.364581\pi\)
\(434\) 0 0
\(435\) 6806.85 1896.00i 0.750261 0.208980i
\(436\) 0 0
\(437\) 6720.69i 0.735684i
\(438\) 0 0
\(439\) −3704.99 −0.402801 −0.201401 0.979509i \(-0.564549\pi\)
−0.201401 + 0.979509i \(0.564549\pi\)
\(440\) 0 0
\(441\) 3597.00 0.388403
\(442\) 0 0
\(443\) 5484.00i 0.588155i −0.955782 0.294078i \(-0.904988\pi\)
0.955782 0.294078i \(-0.0950124\pi\)
\(444\) 0 0
\(445\) 18.0000 + 64.6220i 0.00191749 + 0.00688399i
\(446\) 0 0
\(447\) 10200.0i 1.07929i
\(448\) 0 0
\(449\) −3458.00 −0.363459 −0.181730 0.983349i \(-0.558170\pi\)
−0.181730 + 0.983349i \(0.558170\pi\)
\(450\) 0 0
\(451\) −7323.82 −0.764668
\(452\) 0 0
\(453\) 8616.26i 0.893659i
\(454\) 0 0
\(455\) −258.488 928.000i −0.0266332 0.0956161i
\(456\) 0 0
\(457\) 10468.8i 1.07157i −0.844354 0.535786i \(-0.820016\pi\)
0.844354 0.535786i \(-0.179984\pi\)
\(458\) 0 0
\(459\) −6548.36 −0.665907
\(460\) 0 0
\(461\) 5806.00 0.586578 0.293289 0.956024i \(-0.405250\pi\)
0.293289 + 0.956024i \(0.405250\pi\)
\(462\) 0 0
\(463\) 5004.00i 0.502280i −0.967951 0.251140i \(-0.919195\pi\)
0.967951 0.251140i \(-0.0808054\pi\)
\(464\) 0 0
\(465\) 7424.00 2067.90i 0.740387 0.206230i
\(466\) 0 0
\(467\) 5148.00i 0.510109i −0.966927 0.255055i \(-0.917907\pi\)
0.966927 0.255055i \(-0.0820935\pi\)
\(468\) 0 0
\(469\) −2768.00 −0.272525
\(470\) 0 0
\(471\) −14216.8 −1.39082
\(472\) 0 0
\(473\) 13613.7i 1.32338i
\(474\) 0 0
\(475\) −13829.1 + 8352.00i −1.33584 + 0.806771i
\(476\) 0 0
\(477\) 5449.79i 0.523121i
\(478\) 0 0
\(479\) 4480.46 0.427385 0.213692 0.976901i \(-0.431451\pi\)
0.213692 + 0.976901i \(0.431451\pi\)
\(480\) 0 0
\(481\) −6032.00 −0.571799
\(482\) 0 0
\(483\) 832.000i 0.0783795i
\(484\) 0 0
\(485\) −11600.0 + 3231.10i −1.08604 + 0.302509i
\(486\) 0 0
\(487\) 3044.00i 0.283238i 0.989921 + 0.141619i \(0.0452308\pi\)
−0.989921 + 0.141619i \(0.954769\pi\)
\(488\) 0 0
\(489\) −912.000 −0.0843396
\(490\) 0 0
\(491\) 215.407 0.0197987 0.00989935 0.999951i \(-0.496849\pi\)
0.00989935 + 0.999951i \(0.496849\pi\)
\(492\) 0 0
\(493\) 6806.85i 0.621836i
\(494\) 0 0
\(495\) −1421.68 5104.00i −0.129091 0.463450i
\(496\) 0 0
\(497\) 3791.16i 0.342166i
\(498\) 0 0
\(499\) −11416.5 −1.02420 −0.512099 0.858926i \(-0.671132\pi\)
−0.512099 + 0.858926i \(0.671132\pi\)
\(500\) 0 0
\(501\) −1488.00 −0.132692
\(502\) 0 0
\(503\) 15348.0i 1.36050i 0.732978 + 0.680252i \(0.238130\pi\)
−0.732978 + 0.680252i \(0.761870\pi\)
\(504\) 0 0
\(505\) −3042.00 10921.1i −0.268054 0.962343i
\(506\) 0 0
\(507\) 6932.00i 0.607221i
\(508\) 0 0
\(509\) 17502.0 1.52409 0.762046 0.647523i \(-0.224195\pi\)
0.762046 + 0.647523i \(0.224195\pi\)
\(510\) 0 0
\(511\) −1723.25 −0.149182
\(512\) 0 0
\(513\) 19645.1i 1.69074i
\(514\) 0 0
\(515\) −473.895 + 132.000i −0.0405481 + 0.0112944i
\(516\) 0 0
\(517\) 10511.8i 0.894217i
\(518\) 0 0
\(519\) −7668.47 −0.648572
\(520\) 0 0
\(521\) 2874.00 0.241674 0.120837 0.992672i \(-0.461442\pi\)
0.120837 + 0.992672i \(0.461442\pi\)
\(522\) 0 0
\(523\) 14604.0i 1.22101i −0.792012 0.610505i \(-0.790966\pi\)
0.792012 0.610505i \(-0.209034\pi\)
\(524\) 0 0
\(525\) 1712.00 1033.95i 0.142320 0.0859530i
\(526\) 0 0
\(527\) 7424.00i 0.613652i
\(528\) 0 0
\(529\) 9463.00 0.777760
\(530\) 0 0
\(531\) −7108.42 −0.580940
\(532\) 0 0
\(533\) 3661.91i 0.297589i
\(534\) 0 0
\(535\) 19515.8 5436.00i 1.57709 0.439287i
\(536\) 0 0
\(537\) 11201.1i 0.900121i
\(538\) 0 0
\(539\) 14087.6 1.12578
\(540\) 0 0
\(541\) −19330.0 −1.53616 −0.768079 0.640355i \(-0.778787\pi\)
−0.768079 + 0.640355i \(0.778787\pi\)
\(542\) 0 0
\(543\) 6168.00i 0.487466i
\(544\) 0 0
\(545\) 6042.00 + 21691.4i 0.474882 + 1.70488i
\(546\) 0 0
\(547\) 708.000i 0.0553417i 0.999617 + 0.0276708i \(0.00880902\pi\)
−0.999617 + 0.0276708i \(0.991191\pi\)
\(548\) 0 0
\(549\) 902.000 0.0701210
\(550\) 0 0
\(551\) 20420.5 1.57885
\(552\) 0 0
\(553\) 1378.60i 0.106011i
\(554\) 0 0
\(555\) −3360.34 12064.0i −0.257007 0.922682i
\(556\) 0 0
\(557\) 9284.02i 0.706242i −0.935578 0.353121i \(-0.885120\pi\)
0.935578 0.353121i \(-0.114880\pi\)
\(558\) 0 0
\(559\) −6806.85 −0.515025
\(560\) 0 0
\(561\) −7424.00 −0.558719
\(562\) 0 0
\(563\) 9220.00i 0.690189i 0.938568 + 0.345095i \(0.112153\pi\)
−0.938568 + 0.345095i \(0.887847\pi\)
\(564\) 0 0
\(565\) −17632.0 + 4911.27i −1.31289 + 0.365697i
\(566\) 0 0
\(567\) 1244.00i 0.0921395i
\(568\) 0 0
\(569\) −10458.0 −0.770513 −0.385257 0.922809i \(-0.625887\pi\)
−0.385257 + 0.922809i \(0.625887\pi\)
\(570\) 0 0
\(571\) 13312.1 0.975648 0.487824 0.872942i \(-0.337791\pi\)
0.487824 + 0.872942i \(0.337791\pi\)
\(572\) 0 0
\(573\) 17921.8i 1.30662i
\(574\) 0 0
\(575\) −3360.34 5564.00i −0.243715 0.403539i
\(576\) 0 0
\(577\) 17491.0i 1.26198i 0.775792 + 0.630988i \(0.217350\pi\)
−0.775792 + 0.630988i \(0.782650\pi\)
\(578\) 0 0
\(579\) 10167.2 0.729765
\(580\) 0 0
\(581\) 3760.00 0.268487
\(582\) 0 0
\(583\) 21344.0i 1.51626i
\(584\) 0 0
\(585\) −2552.00 + 710.842i −0.180363 + 0.0502388i
\(586\) 0 0
\(587\) 11460.0i 0.805800i 0.915244 + 0.402900i \(0.131998\pi\)
−0.915244 + 0.402900i \(0.868002\pi\)
\(588\) 0 0
\(589\) 22272.0 1.55807
\(590\) 0 0
\(591\) 10942.7 0.761626
\(592\) 0 0
\(593\) 2326.39i 0.161102i 0.996750 + 0.0805510i \(0.0256680\pi\)
−0.996750 + 0.0805510i \(0.974332\pi\)
\(594\) 0 0
\(595\) 516.976 + 1856.00i 0.0356201 + 0.127880i
\(596\) 0 0
\(597\) 1033.95i 0.0708825i
\(598\) 0 0
\(599\) 24384.0 1.66328 0.831640 0.555316i \(-0.187403\pi\)
0.831640 + 0.555316i \(0.187403\pi\)
\(600\) 0 0
\(601\) 24054.0 1.63258 0.816292 0.577639i \(-0.196026\pi\)
0.816292 + 0.577639i \(0.196026\pi\)
\(602\) 0 0
\(603\) 7612.00i 0.514071i
\(604\) 0 0
\(605\) −1575.00 5654.42i −0.105839 0.379975i
\(606\) 0 0
\(607\) 17364.0i 1.16109i 0.814227 + 0.580546i \(0.197161\pi\)
−0.814227 + 0.580546i \(0.802839\pi\)
\(608\) 0 0
\(609\) −2528.00 −0.168210
\(610\) 0 0
\(611\) 5255.92 0.348006
\(612\) 0 0
\(613\) 2434.09i 0.160379i 0.996780 + 0.0801894i \(0.0255525\pi\)
−0.996780 + 0.0801894i \(0.974447\pi\)
\(614\) 0 0
\(615\) −7323.82 + 2040.00i −0.480203 + 0.133757i
\(616\) 0 0
\(617\) 8616.26i 0.562201i 0.959678 + 0.281100i \(0.0906993\pi\)
−0.959678 + 0.281100i \(0.909301\pi\)
\(618\) 0 0
\(619\) 9176.32 0.595844 0.297922 0.954590i \(-0.403706\pi\)
0.297922 + 0.954590i \(0.403706\pi\)
\(620\) 0 0
\(621\) −7904.00 −0.510751
\(622\) 0 0
\(623\) 24.0000i 0.00154340i
\(624\) 0 0
\(625\) 7273.00 13829.1i 0.465472 0.885063i
\(626\) 0 0
\(627\) 22272.0i 1.41859i
\(628\) 0 0
\(629\) 12064.0 0.764743
\(630\) 0 0
\(631\) −9219.40 −0.581646 −0.290823 0.956777i \(-0.593929\pi\)
−0.290823 + 0.956777i \(0.593929\pi\)
\(632\) 0 0
\(633\) 3618.83i 0.227228i
\(634\) 0 0
\(635\) 30027.7 8364.00i 1.87655 0.522701i
\(636\) 0 0
\(637\) 7043.80i 0.438125i
\(638\) 0 0
\(639\) −10425.7 −0.645436
\(640\) 0 0
\(641\) −25538.0 −1.57362 −0.786810 0.617195i \(-0.788269\pi\)
−0.786810 + 0.617195i \(0.788269\pi\)
\(642\) 0 0
\(643\) 22060.0i 1.35297i −0.736455 0.676486i \(-0.763502\pi\)
0.736455 0.676486i \(-0.236498\pi\)
\(644\) 0 0
\(645\) −3792.00 13613.7i −0.231488 0.831068i
\(646\) 0 0
\(647\) 29444.0i 1.78912i 0.446944 + 0.894562i \(0.352512\pi\)
−0.446944 + 0.894562i \(0.647488\pi\)
\(648\) 0 0
\(649\) −27840.0 −1.68385
\(650\) 0 0
\(651\) −2757.20 −0.165996
\(652\) 0 0
\(653\) 20786.7i 1.24571i −0.782338 0.622854i \(-0.785973\pi\)
0.782338 0.622854i \(-0.214027\pi\)
\(654\) 0 0
\(655\) 129.244 + 464.000i 0.00770989 + 0.0276794i
\(656\) 0 0
\(657\) 4738.95i 0.281406i
\(658\) 0 0
\(659\) 13742.9 0.812366 0.406183 0.913792i \(-0.366860\pi\)
0.406183 + 0.913792i \(0.366860\pi\)
\(660\) 0 0
\(661\) 11530.0 0.678464 0.339232 0.940703i \(-0.389833\pi\)
0.339232 + 0.940703i \(0.389833\pi\)
\(662\) 0 0
\(663\) 3712.00i 0.217439i
\(664\) 0 0
\(665\) 5568.00 1550.93i 0.324688 0.0904397i
\(666\) 0 0
\(667\) 8216.00i 0.476949i
\(668\) 0 0
\(669\) 17136.0 0.990308
\(670\) 0 0
\(671\) 3532.67 0.203245
\(672\) 0 0
\(673\) 23910.1i 1.36949i 0.728782 + 0.684746i \(0.240087\pi\)
−0.728782 + 0.684746i \(0.759913\pi\)
\(674\) 0 0
\(675\) −9822.54 16264.0i −0.560103 0.927410i
\(676\) 0 0
\(677\) 2175.61i 0.123509i 0.998091 + 0.0617543i \(0.0196695\pi\)
−0.998091 + 0.0617543i \(0.980330\pi\)
\(678\) 0 0
\(679\) 4308.13 0.243492
\(680\) 0 0
\(681\) 19824.0 1.11550
\(682\) 0 0
\(683\) 20708.0i 1.16013i 0.814570 + 0.580066i \(0.196973\pi\)
−0.814570 + 0.580066i \(0.803027\pi\)
\(684\) 0 0
\(685\) 5104.00 1421.68i 0.284692 0.0792989i
\(686\) 0 0
\(687\) 7080.00i 0.393186i
\(688\) 0 0
\(689\) −10672.0 −0.590088
\(690\) 0 0
\(691\) 4609.70 0.253779 0.126890 0.991917i \(-0.459501\pi\)
0.126890 + 0.991917i \(0.459501\pi\)
\(692\) 0 0
\(693\) 1895.58i 0.103906i
\(694\) 0 0
\(695\) −6332.95 22736.0i −0.345644 1.24090i
\(696\) 0 0
\(697\) 7323.82i 0.398005i
\(698\) 0 0
\(699\) −21713.0 −1.17491
\(700\) 0 0
\(701\) −8942.00 −0.481790 −0.240895 0.970551i \(-0.577441\pi\)
−0.240895 + 0.970551i \(0.577441\pi\)
\(702\) 0 0
\(703\) 36192.0i 1.94169i
\(704\) 0 0
\(705\) 2928.00 + 10511.8i 0.156418 + 0.561559i
\(706\) 0 0
\(707\) 4056.00i 0.215759i
\(708\) 0 0
\(709\) 3670.00 0.194400 0.0972001 0.995265i \(-0.469011\pi\)
0.0972001 + 0.995265i \(0.469011\pi\)
\(710\) 0 0
\(711\) −3791.16 −0.199971
\(712\) 0 0
\(713\) 8960.91i 0.470672i
\(714\) 0 0
\(715\) −9994.87 + 2784.00i −0.522779 + 0.145616i
\(716\) 0 0
\(717\) 22057.6i 1.14889i
\(718\) 0 0
\(719\) 8616.26 0.446916 0.223458 0.974714i \(-0.428265\pi\)
0.223458 + 0.974714i \(0.428265\pi\)
\(720\) 0 0
\(721\) 176.000 0.00909096
\(722\) 0 0
\(723\) 6472.00i 0.332913i
\(724\) 0 0
\(725\) −16906.0 + 10210.3i −0.866032 + 0.523034i
\(726\) 0 0
\(727\) 32004.0i 1.63269i 0.577567 + 0.816343i \(0.304002\pi\)
−0.577567 + 0.816343i \(0.695998\pi\)
\(728\) 0 0
\(729\) −19837.0 −1.00782
\(730\) 0 0
\(731\) 13613.7 0.688811
\(732\) 0 0
\(733\) 5363.62i 0.270273i 0.990827 + 0.135136i \(0.0431473\pi\)
−0.990827 + 0.135136i \(0.956853\pi\)
\(734\) 0 0
\(735\) 14087.6 3924.00i 0.706978 0.196924i
\(736\) 0 0
\(737\) 29812.3i 1.49003i
\(738\) 0 0
\(739\) −25374.9 −1.26310 −0.631550 0.775335i \(-0.717581\pi\)
−0.631550 + 0.775335i \(0.717581\pi\)
\(740\) 0 0
\(741\) 11136.0 0.552080
\(742\) 0 0
\(743\) 17404.0i 0.859342i −0.902986 0.429671i \(-0.858630\pi\)
0.902986 0.429671i \(-0.141370\pi\)
\(744\) 0 0
\(745\) 7650.00 + 27464.3i 0.376207 + 1.35062i
\(746\) 0 0
\(747\) 10340.0i 0.506454i
\(748\) 0 0
\(749\) −7248.00 −0.353586
\(750\) 0 0
\(751\) −32569.5 −1.58253 −0.791263 0.611476i \(-0.790576\pi\)
−0.791263 + 0.611476i \(0.790576\pi\)
\(752\) 0 0
\(753\) 19817.4i 0.959079i
\(754\) 0 0
\(755\) −6462.20 23200.0i −0.311501 1.11832i
\(756\) 0 0
\(757\) 6139.09i 0.294754i −0.989080 0.147377i \(-0.952917\pi\)
0.989080 0.147377i \(-0.0470831\pi\)
\(758\) 0 0
\(759\) −8960.91 −0.428538
\(760\) 0 0
\(761\) 27850.0 1.32663 0.663313 0.748342i \(-0.269150\pi\)
0.663313 + 0.748342i \(0.269150\pi\)
\(762\) 0 0
\(763\) 8056.00i 0.382237i
\(764\) 0 0
\(765\) 5104.00 1421.68i 0.241223 0.0671909i
\(766\) 0 0
\(767\) 13920.0i 0.655309i
\(768\) 0 0
\(769\) −33550.0 −1.57327 −0.786635 0.617419i \(-0.788178\pi\)
−0.786635 + 0.617419i \(0.788178\pi\)
\(770\) 0 0
\(771\) −18611.1 −0.869343
\(772\) 0 0
\(773\) 1658.63i 0.0771757i 0.999255 + 0.0385878i \(0.0122859\pi\)
−0.999255 + 0.0385878i \(0.987714\pi\)
\(774\) 0 0
\(775\) −18438.8 + 11136.0i −0.854634 + 0.516151i
\(776\) 0 0
\(777\) 4480.46i 0.206867i
\(778\) 0 0
\(779\) −21971.5 −1.01054
\(780\) 0 0
\(781\) −40832.0 −1.87079
\(782\) 0 0
\(783\) 24016.0i 1.09612i
\(784\) 0 0
\(785\) 38280.0 10662.6i 1.74047 0.484797i
\(786\) 0 0
\(787\) 27028.0i 1.22420i 0.790781 + 0.612099i \(0.209675\pi\)
−0.790781 + 0.612099i \(0.790325\pi\)
\(788\) 0 0
\(789\) 2416.00 0.109014
\(790\) 0 0
\(791\) 6548.36 0.294353
\(792\) 0 0
\(793\) 1766.33i 0.0790976i
\(794\) 0 0
\(795\) −5945.22 21344.0i −0.265227 0.952193i
\(796\) 0 0
\(797\) 2692.58i 0.119669i −0.998208 0.0598345i \(-0.980943\pi\)
0.998208 0.0598345i \(-0.0190573\pi\)
\(798\) 0 0
\(799\) −10511.8 −0.465435
\(800\) 0 0
\(801\) −66.0000 −0.00291135
\(802\) 0 0
\(803\) 18560.0i 0.815652i
\(804\) 0 0
\(805\) 624.000 + 2240.23i 0.0273206 + 0.0980841i
\(806\) 0 0
\(807\) 13048.0i 0.569159i
\(808\) 0 0
\(809\) 1434.00 0.0623198 0.0311599 0.999514i \(-0.490080\pi\)
0.0311599 + 0.999514i \(0.490080\pi\)
\(810\) 0 0
\(811\) −42004.3 −1.81871 −0.909353 0.416026i \(-0.863422\pi\)
−0.909353 + 0.416026i \(0.863422\pi\)
\(812\) 0 0
\(813\) 8271.61i 0.356824i
\(814\) 0 0
\(815\) 2455.64 684.000i 0.105543 0.0293981i
\(816\) 0 0
\(817\) 40841.1i 1.74890i
\(818\) 0 0
\(819\) 947.789 0.0404376
\(820\) 0 0
\(821\) 21322.0 0.906386 0.453193 0.891412i \(-0.350285\pi\)
0.453193 + 0.891412i \(0.350285\pi\)
\(822\) 0 0
\(823\) 21548.0i 0.912656i −0.889812 0.456328i \(-0.849164\pi\)
0.889812 0.456328i \(-0.150836\pi\)
\(824\) 0 0
\(825\) −11136.0 18438.8i −0.469946 0.778129i
\(826\) 0 0
\(827\) 9660.00i 0.406180i −0.979160 0.203090i \(-0.934902\pi\)
0.979160 0.203090i \(-0.0650985\pi\)
\(828\) 0 0
\(829\) 24082.0 1.00893 0.504465 0.863432i \(-0.331690\pi\)
0.504465 + 0.863432i \(0.331690\pi\)
\(830\) 0 0
\(831\) 4911.27 0.205018
\(832\) 0 0
\(833\) 14087.6i 0.585962i
\(834\) 0 0
\(835\) 4006.56 1116.00i 0.166051 0.0462524i
\(836\) 0 0
\(837\) 26193.4i 1.08169i
\(838\) 0 0
\(839\) −14906.1 −0.613369 −0.306685 0.951811i \(-0.599220\pi\)
−0.306685 + 0.951811i \(0.599220\pi\)
\(840\) 0 0
\(841\) 575.000 0.0235762
\(842\) 0 0
\(843\) 5160.00i 0.210818i
\(844\) 0 0
\(845\) −5199.00 18665.0i −0.211658 0.759876i
\(846\) 0 0
\(847\) 2100.00i 0.0851911i
\(848\) 0 0
\(849\) −2768.00 −0.111893
\(850\) 0 0
\(851\) 14561.5 0.586559
\(852\) 0 0
\(853\) 21002.1i 0.843024i −0.906823 0.421512i \(-0.861499\pi\)
0.906823 0.421512i \(-0.138501\pi\)
\(854\) 0 0
\(855\) −4265.05 15312.0i −0.170598 0.612467i
\(856\) 0 0
\(857\) 19860.5i 0.791624i −0.918332 0.395812i \(-0.870463\pi\)
0.918332 0.395812i \(-0.129537\pi\)
\(858\) 0 0
\(859\) 34852.8 1.38436 0.692178 0.721727i \(-0.256651\pi\)
0.692178 + 0.721727i \(0.256651\pi\)
\(860\) 0 0
\(861\) 2720.00 0.107662
\(862\) 0 0
\(863\) 11468.0i 0.452347i −0.974087 0.226173i \(-0.927378\pi\)
0.974087 0.226173i \(-0.0726216\pi\)
\(864\) 0 0
\(865\) 20648.0 5751.36i 0.811622 0.226072i
\(866\) 0 0
\(867\) 12228.0i 0.478990i
\(868\) 0 0
\(869\) −14848.0 −0.579613
\(870\) 0 0
\(871\) 14906.1 0.579880
\(872\) 0 0
\(873\) 11847.4i 0.459304i
\(874\) 0 0
\(875\) −3834.24 + 4068.00i −0.148138 + 0.157170i
\(876\) 0 0
\(877\) 34572.8i 1.33117i 0.746321 + 0.665587i \(0.231819\pi\)
−0.746321 + 0.665587i \(0.768181\pi\)
\(878\) 0 0
\(879\) 22143.8 0.849706
\(880\) 0 0
\(881\) −33186.0 −1.26909 −0.634543 0.772888i \(-0.718812\pi\)
−0.634543 + 0.772888i \(0.718812\pi\)
\(882\) 0 0
\(883\) 15196.0i 0.579146i −0.957156 0.289573i \(-0.906487\pi\)
0.957156 0.289573i \(-0.0935133\pi\)
\(884\) 0 0
\(885\) −27840.0 + 7754.64i −1.05744 + 0.294542i
\(886\) 0 0
\(887\) 3036.00i 0.114925i −0.998348 0.0574627i \(-0.981699\pi\)
0.998348 0.0574627i \(-0.0183010\pi\)
\(888\) 0 0
\(889\) −11152.0 −0.420727
\(890\) 0 0
\(891\) −13398.3 −0.503771
\(892\) 0 0
\(893\) 31535.5i 1.18174i
\(894\) 0 0
\(895\) 8400.86 + 30160.0i 0.313754 + 1.12641i
\(896\) 0 0
\(897\) 4480.46i 0.166776i
\(898\) 0 0
\(899\) 27227.4 1.01011
\(900\) 0 0
\(901\) 21344.0 0.789203
\(902\) 0 0
\(903\) 5056.00i 0.186327i
\(904\) 0 0
\(905\) −4626.00 16607.8i −0.169915 0.610015i
\(906\) 0 0
\(907\) 7404.00i 0.271054i −0.990774 0.135527i \(-0.956727\pi\)
0.990774 0.135527i \(-0.0432727\pi\)
\(908\) 0 0
\(909\) 11154.0 0.406991
\(910\) 0 0
\(911\) 6548.36 0.238152 0.119076 0.992885i \(-0.462007\pi\)
0.119076 + 0.992885i \(0.462007\pi\)
\(912\) 0 0
\(913\) 40496.4i 1.46795i
\(914\) 0 0
\(915\) 3532.67 984.000i 0.127635 0.0355520i
\(916\) 0 0
\(917\) 172.325i 0.00620576i
\(918\) 0 0
\(919\) −34206.6 −1.22782 −0.613912 0.789374i \(-0.710405\pi\)
−0.613912 + 0.789374i \(0.710405\pi\)
\(920\) 0 0
\(921\) 29744.0 1.06417
\(922\) 0 0
\(923\) 20416.0i 0.728062i
\(924\) 0 0
\(925\) 18096.0 + 29963.1i 0.643235 + 1.06506i
\(926\) 0 0
\(927\) 484.000i 0.0171485i
\(928\) 0 0
\(929\) 24286.0 0.857694 0.428847 0.903377i \(-0.358920\pi\)
0.428847 + 0.903377i \(0.358920\pi\)
\(930\) 0 0
\(931\) 42262.8 1.48776
\(932\) 0 0
\(933\) 9994.87i 0.350715i
\(934\) 0 0
\(935\) 19989.7 5568.00i 0.699181 0.194752i
\(936\) 0 0
\(937\) 37997.7i 1.32479i 0.749153 + 0.662397i \(0.230461\pi\)
−0.749153 + 0.662397i \(0.769539\pi\)
\(938\) 0 0
\(939\) 10856.5 0.377304
\(940\) 0 0
\(941\) −36722.0 −1.27216 −0.636080 0.771623i \(-0.719445\pi\)
−0.636080 + 0.771623i \(0.719445\pi\)
\(942\) 0 0
\(943\) 8840.00i 0.305270i
\(944\) 0 0
\(945\) 1824.00 + 6548.36i 0.0627881 + 0.225416i
\(946\) 0 0
\(947\) 16364.0i 0.561519i −0.959778 0.280760i \(-0.909414\pi\)
0.959778 0.280760i \(-0.0905863\pi\)
\(948\) 0 0
\(949\) 9280.00 0.317431
\(950\) 0 0
\(951\) 13182.9 0.449510
\(952\) 0 0
\(953\) 7797.72i 0.265050i −0.991180 0.132525i \(-0.957691\pi\)
0.991180 0.132525i \(-0.0423085\pi\)
\(954\) 0 0
\(955\) 13441.4 + 48256.0i 0.455448 + 1.63511i
\(956\) 0 0
\(957\) 27227.4i 0.919683i
\(958\) 0 0
\(959\) −1895.58 −0.0638284
\(960\) 0 0
\(961\) −95.0000 −0.00318888
\(962\) 0 0
\(963\) 19932.0i 0.666978i
\(964\) 0 0
\(965\) −27376.0 + 7625.39i −0.913228 + 0.254373i
\(966\) 0 0
\(967\) 16756.0i 0.557225i 0.960404 + 0.278613i \(0.0898745\pi\)
−0.960404 + 0.278613i \(0.910125\pi\)
\(968\) 0 0
\(969\) −22272.0 −0.738369
\(970\) 0 0
\(971\) 16414.0 0.542482 0.271241 0.962512i \(-0.412566\pi\)
0.271241 + 0.962512i \(0.412566\pi\)
\(972\) 0 0
\(973\) 8443.94i 0.278212i
\(974\) 0 0
\(975\) −9219.40 + 5568.00i −0.302828 + 0.182891i
\(976\) 0 0
\(977\) 50362.1i 1.64916i −0.565749 0.824578i \(-0.691413\pi\)
0.565749 0.824578i \(-0.308587\pi\)
\(978\) 0 0
\(979\) −258.488 −0.00843852
\(980\) 0 0
\(981\) −22154.0 −0.721022
\(982\) 0 0
\(983\) 60612.0i 1.96666i 0.181841 + 0.983328i \(0.441794\pi\)
−0.181841 + 0.983328i \(0.558206\pi\)
\(984\) 0 0
\(985\) −29464.0 + 8206.99i −0.953098 + 0.265479i
\(986\) 0 0
\(987\) 3904.00i 0.125902i
\(988\) 0 0
\(989\) 16432.0 0.528319
\(990\) 0 0
\(991\) 16198.6 0.519238 0.259619 0.965711i \(-0.416403\pi\)
0.259619 + 0.965711i \(0.416403\pi\)
\(992\) 0 0
\(993\) 11890.4i 0.379992i
\(994\) 0 0
\(995\) 775.464 + 2784.00i 0.0247074 + 0.0887022i
\(996\) 0 0
\(997\) 54002.4i 1.71542i −0.514133 0.857710i \(-0.671886\pi\)
0.514133 0.857710i \(-0.328114\pi\)
\(998\) 0 0
\(999\) 42564.3 1.34802
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 160.4.c.b.129.3 yes 4
3.2 odd 2 1440.4.f.h.289.4 4
4.3 odd 2 inner 160.4.c.b.129.1 4
5.2 odd 4 800.4.a.r.1.2 2
5.3 odd 4 800.4.a.n.1.2 2
5.4 even 2 inner 160.4.c.b.129.2 yes 4
8.3 odd 2 320.4.c.i.129.4 4
8.5 even 2 320.4.c.i.129.2 4
12.11 even 2 1440.4.f.h.289.3 4
15.14 odd 2 1440.4.f.h.289.1 4
20.3 even 4 800.4.a.r.1.1 2
20.7 even 4 800.4.a.n.1.1 2
20.19 odd 2 inner 160.4.c.b.129.4 yes 4
40.3 even 4 1600.4.a.cc.1.2 2
40.13 odd 4 1600.4.a.co.1.1 2
40.19 odd 2 320.4.c.i.129.1 4
40.27 even 4 1600.4.a.co.1.2 2
40.29 even 2 320.4.c.i.129.3 4
40.37 odd 4 1600.4.a.cc.1.1 2
60.59 even 2 1440.4.f.h.289.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.4.c.b.129.1 4 4.3 odd 2 inner
160.4.c.b.129.2 yes 4 5.4 even 2 inner
160.4.c.b.129.3 yes 4 1.1 even 1 trivial
160.4.c.b.129.4 yes 4 20.19 odd 2 inner
320.4.c.i.129.1 4 40.19 odd 2
320.4.c.i.129.2 4 8.5 even 2
320.4.c.i.129.3 4 40.29 even 2
320.4.c.i.129.4 4 8.3 odd 2
800.4.a.n.1.1 2 20.7 even 4
800.4.a.n.1.2 2 5.3 odd 4
800.4.a.r.1.1 2 20.3 even 4
800.4.a.r.1.2 2 5.2 odd 4
1440.4.f.h.289.1 4 15.14 odd 2
1440.4.f.h.289.2 4 60.59 even 2
1440.4.f.h.289.3 4 12.11 even 2
1440.4.f.h.289.4 4 3.2 odd 2
1600.4.a.cc.1.1 2 40.37 odd 4
1600.4.a.cc.1.2 2 40.3 even 4
1600.4.a.co.1.1 2 40.13 odd 4
1600.4.a.co.1.2 2 40.27 even 4