Properties

Label 1440.2.bi.d.847.2
Level $1440$
Weight $2$
Character 1440.847
Analytic conductor $11.498$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1440,2,Mod(847,1440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1440, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1440.847");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1440.bi (of order \(4\), degree \(2\), 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: \(11.4984578911\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 5x^{12} + 28x^{8} + 80x^{4} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{49}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: no (minimal twist has level 360)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 847.2
Root \(-1.31813 - 0.512386i\) of defining polynomial
Character \(\chi\) \(=\) 1440.847
Dual form 1440.2.bi.d.1423.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.83051 + 1.28422i) q^{5} +(2.94984 - 2.94984i) q^{7} +O(q^{10})\) \(q+(-1.83051 + 1.28422i) q^{5} +(2.94984 - 2.94984i) q^{7} +1.61148 q^{11} +(2.50967 + 2.50967i) q^{13} +(-4.59398 - 4.59398i) q^{17} -4.00000i q^{19} +(1.09259 + 1.09259i) q^{23} +(1.70156 - 4.70156i) q^{25} -4.75362 q^{29} +5.01934i q^{31} +(-1.61148 + 9.18797i) q^{35} +(-2.50967 + 2.50967i) q^{37} +9.18797 q^{41} +(7.40312 - 7.40312i) q^{43} +(7.32206 - 7.32206i) q^{47} -10.4031i q^{49} +(3.11473 + 3.11473i) q^{53} +(-2.94984 + 2.06950i) q^{55} -1.61148i q^{59} -6.78003i q^{61} +(-7.81695 - 1.37102i) q^{65} +(-7.40312 - 7.40312i) q^{67} +(5.00000 - 5.00000i) q^{73} +(4.75362 - 4.75362i) q^{77} +5.01934 q^{79} +(7.57648 - 7.57648i) q^{83} +(14.3090 + 2.50967i) q^{85} +2.74204i q^{89} +14.8062 q^{91} +(5.13688 + 7.32206i) q^{95} +(2.40312 + 2.40312i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 24 q^{25} + 16 q^{43} - 16 q^{67} + 80 q^{73} + 32 q^{91} - 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(991\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(-1\) \(-1\)

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 0
\(4\) 0 0
\(5\) −1.83051 + 1.28422i −0.818631 + 0.574320i
\(6\) 0 0
\(7\) 2.94984 2.94984i 1.11494 1.11494i 0.122462 0.992473i \(-0.460921\pi\)
0.992473 0.122462i \(-0.0390789\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.61148 0.485880 0.242940 0.970041i \(-0.421888\pi\)
0.242940 + 0.970041i \(0.421888\pi\)
\(12\) 0 0
\(13\) 2.50967 + 2.50967i 0.696057 + 0.696057i 0.963558 0.267501i \(-0.0861978\pi\)
−0.267501 + 0.963558i \(0.586198\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.59398 4.59398i −1.11420 1.11420i −0.992576 0.121629i \(-0.961188\pi\)
−0.121629 0.992576i \(-0.538812\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i −0.888523 0.458831i \(-0.848268\pi\)
0.888523 0.458831i \(-0.151732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.09259 + 1.09259i 0.227821 + 0.227821i 0.811782 0.583961i \(-0.198498\pi\)
−0.583961 + 0.811782i \(0.698498\pi\)
\(24\) 0 0
\(25\) 1.70156 4.70156i 0.340312 0.940312i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.75362 −0.882725 −0.441362 0.897329i \(-0.645505\pi\)
−0.441362 + 0.897329i \(0.645505\pi\)
\(30\) 0 0
\(31\) 5.01934i 0.901500i 0.892650 + 0.450750i \(0.148843\pi\)
−0.892650 + 0.450750i \(0.851157\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.61148 + 9.18797i −0.272390 + 1.55305i
\(36\) 0 0
\(37\) −2.50967 + 2.50967i −0.412587 + 0.412587i −0.882639 0.470052i \(-0.844235\pi\)
0.470052 + 0.882639i \(0.344235\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.18797 1.43492 0.717460 0.696600i \(-0.245305\pi\)
0.717460 + 0.696600i \(0.245305\pi\)
\(42\) 0 0
\(43\) 7.40312 7.40312i 1.12897 1.12897i 0.138620 0.990346i \(-0.455733\pi\)
0.990346 0.138620i \(-0.0442667\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.32206 7.32206i 1.06803 1.06803i 0.0705213 0.997510i \(-0.477534\pi\)
0.997510 0.0705213i \(-0.0224663\pi\)
\(48\) 0 0
\(49\) 10.4031i 1.48616i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.11473 + 3.11473i 0.427841 + 0.427841i 0.887892 0.460051i \(-0.152169\pi\)
−0.460051 + 0.887892i \(0.652169\pi\)
\(54\) 0 0
\(55\) −2.94984 + 2.06950i −0.397756 + 0.279051i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.61148i 0.209797i −0.994483 0.104899i \(-0.966548\pi\)
0.994483 0.104899i \(-0.0334518\pi\)
\(60\) 0 0
\(61\) 6.78003i 0.868093i −0.900890 0.434047i \(-0.857085\pi\)
0.900890 0.434047i \(-0.142915\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −7.81695 1.37102i −0.969573 0.170054i
\(66\) 0 0
\(67\) −7.40312 7.40312i −0.904436 0.904436i 0.0913805 0.995816i \(-0.470872\pi\)
−0.995816 + 0.0913805i \(0.970872\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 5.00000 5.00000i 0.585206 0.585206i −0.351123 0.936329i \(-0.614200\pi\)
0.936329 + 0.351123i \(0.114200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.75362 4.75362i 0.541725 0.541725i
\(78\) 0 0
\(79\) 5.01934 0.564720 0.282360 0.959309i \(-0.408883\pi\)
0.282360 + 0.959309i \(0.408883\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.57648 7.57648i 0.831627 0.831627i −0.156112 0.987739i \(-0.549896\pi\)
0.987739 + 0.156112i \(0.0498961\pi\)
\(84\) 0 0
\(85\) 14.3090 + 2.50967i 1.55203 + 0.272212i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.74204i 0.290655i 0.989384 + 0.145328i \(0.0464236\pi\)
−0.989384 + 0.145328i \(0.953576\pi\)
\(90\) 0 0
\(91\) 14.8062 1.55212
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.13688 + 7.32206i 0.527032 + 0.751227i
\(96\) 0 0
\(97\) 2.40312 + 2.40312i 0.244000 + 0.244000i 0.818503 0.574503i \(-0.194804\pi\)
−0.574503 + 0.818503i \(0.694804\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 8.79790i 0.875424i −0.899115 0.437712i \(-0.855789\pi\)
0.899115 0.437712i \(-0.144211\pi\)
\(102\) 0 0
\(103\) 7.96918 + 7.96918i 0.785227 + 0.785227i 0.980707 0.195481i \(-0.0626268\pi\)
−0.195481 + 0.980707i \(0.562627\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.61148 + 1.61148i 0.155788 + 0.155788i 0.780697 0.624909i \(-0.214864\pi\)
−0.624909 + 0.780697i \(0.714864\pi\)
\(108\) 0 0
\(109\) 11.7994 1.13017 0.565087 0.825031i \(-0.308843\pi\)
0.565087 + 0.825031i \(0.308843\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −4.59398 + 4.59398i −0.432166 + 0.432166i −0.889364 0.457199i \(-0.848853\pi\)
0.457199 + 0.889364i \(0.348853\pi\)
\(114\) 0 0
\(115\) −3.40312 0.596876i −0.317343 0.0556590i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −27.1030 −2.48453
\(120\) 0 0
\(121\) −8.40312 −0.763920
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 2.92310 + 10.7915i 0.261450 + 0.965217i
\(126\) 0 0
\(127\) 3.83019 3.83019i 0.339874 0.339874i −0.516446 0.856320i \(-0.672745\pi\)
0.856320 + 0.516446i \(0.172745\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −13.5415 −1.18313 −0.591563 0.806259i \(-0.701489\pi\)
−0.591563 + 0.806259i \(0.701489\pi\)
\(132\) 0 0
\(133\) −11.7994 11.7994i −1.02313 1.02313i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 11.0399 + 11.0399i 0.943203 + 0.943203i 0.998472 0.0552681i \(-0.0176013\pi\)
−0.0552681 + 0.998472i \(0.517601\pi\)
\(138\) 0 0
\(139\) 4.00000i 0.339276i 0.985506 + 0.169638i \(0.0542598\pi\)
−0.985506 + 0.169638i \(0.945740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.04429 + 4.04429i 0.338200 + 0.338200i
\(144\) 0 0
\(145\) 8.70156 6.10469i 0.722625 0.506967i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −8.79790 −0.720752 −0.360376 0.932807i \(-0.617352\pi\)
−0.360376 + 0.932807i \(0.617352\pi\)
\(150\) 0 0
\(151\) 0.880344i 0.0716414i 0.999358 + 0.0358207i \(0.0114045\pi\)
−0.999358 + 0.0358207i \(0.988595\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −6.44593 9.18797i −0.517750 0.737995i
\(156\) 0 0
\(157\) −9.28970 + 9.28970i −0.741398 + 0.741398i −0.972847 0.231449i \(-0.925653\pi\)
0.231449 + 0.972847i \(0.425653\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 6.44593 0.508010
\(162\) 0 0
\(163\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −7.32206 + 7.32206i −0.566598 + 0.566598i −0.931174 0.364576i \(-0.881214\pi\)
0.364576 + 0.931174i \(0.381214\pi\)
\(168\) 0 0
\(169\) 0.403124i 0.0310096i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 7.86835 + 7.86835i 0.598220 + 0.598220i 0.939839 0.341619i \(-0.110975\pi\)
−0.341619 + 0.939839i \(0.610975\pi\)
\(174\) 0 0
\(175\) −8.84952 18.8882i −0.668961 1.42781i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 26.4333i 1.97572i −0.155343 0.987861i \(-0.549648\pi\)
0.155343 0.987861i \(-0.450352\pi\)
\(180\) 0 0
\(181\) 11.7994i 0.877040i 0.898721 + 0.438520i \(0.144497\pi\)
−0.898721 + 0.438520i \(0.855503\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.37102 7.81695i 0.100799 0.574713i
\(186\) 0 0
\(187\) −7.40312 7.40312i −0.541370 0.541370i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 19.0145i 1.37584i −0.725787 0.687919i \(-0.758524\pi\)
0.725787 0.687919i \(-0.241476\pi\)
\(192\) 0 0
\(193\) −12.4031 + 12.4031i −0.892796 + 0.892796i −0.994786 0.101989i \(-0.967479\pi\)
0.101989 + 0.994786i \(0.467479\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 9.34420 9.34420i 0.665747 0.665747i −0.290982 0.956729i \(-0.593982\pi\)
0.956729 + 0.290982i \(0.0939820\pi\)
\(198\) 0 0
\(199\) −20.9577 −1.48565 −0.742826 0.669485i \(-0.766515\pi\)
−0.742826 + 0.669485i \(0.766515\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −14.0224 + 14.0224i −0.984181 + 0.984181i
\(204\) 0 0
\(205\) −16.8187 + 11.7994i −1.17467 + 0.824103i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 6.44593i 0.445874i
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −4.04429 + 23.0588i −0.275818 + 1.57259i
\(216\) 0 0
\(217\) 14.8062 + 14.8062i 1.00511 + 1.00511i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 23.0588i 1.55110i
\(222\) 0 0
\(223\) 3.83019 + 3.83019i 0.256488 + 0.256488i 0.823624 0.567136i \(-0.191949\pi\)
−0.567136 + 0.823624i \(0.691949\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 15.6339 + 15.6339i 1.03766 + 1.03766i 0.999263 + 0.0383956i \(0.0122247\pi\)
0.0383956 + 0.999263i \(0.487775\pi\)
\(228\) 0 0
\(229\) −6.78003 −0.448037 −0.224018 0.974585i \(-0.571918\pi\)
−0.224018 + 0.974585i \(0.571918\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −11.0399 + 11.0399i −0.723249 + 0.723249i −0.969266 0.246017i \(-0.920878\pi\)
0.246017 + 0.969266i \(0.420878\pi\)
\(234\) 0 0
\(235\) −4.00000 + 22.8062i −0.260931 + 1.48772i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 8.08857 0.523206 0.261603 0.965176i \(-0.415749\pi\)
0.261603 + 0.965176i \(0.415749\pi\)
\(240\) 0 0
\(241\) −6.80625 −0.438429 −0.219215 0.975677i \(-0.570349\pi\)
−0.219215 + 0.975677i \(0.570349\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 13.3599 + 19.0431i 0.853532 + 1.21662i
\(246\) 0 0
\(247\) 10.0387 10.0387i 0.638746 0.638746i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 19.9874 1.26159 0.630797 0.775948i \(-0.282728\pi\)
0.630797 + 0.775948i \(0.282728\pi\)
\(252\) 0 0
\(253\) 1.76069 + 1.76069i 0.110694 + 0.110694i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 13.7820 + 13.7820i 0.859694 + 0.859694i 0.991302 0.131607i \(-0.0420138\pi\)
−0.131607 + 0.991302i \(0.542014\pi\)
\(258\) 0 0
\(259\) 14.8062i 0.920016i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −2.95170 2.95170i −0.182009 0.182009i 0.610221 0.792231i \(-0.291080\pi\)
−0.792231 + 0.610221i \(0.791080\pi\)
\(264\) 0 0
\(265\) −9.70156 1.70156i −0.595962 0.104526i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −18.3051 −1.11608 −0.558042 0.829813i \(-0.688447\pi\)
−0.558042 + 0.829813i \(0.688447\pi\)
\(270\) 0 0
\(271\) 12.6797i 0.770237i 0.922867 + 0.385119i \(0.125840\pi\)
−0.922867 + 0.385119i \(0.874160\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.74204 7.57648i 0.165351 0.456879i
\(276\) 0 0
\(277\) 7.52901 7.52901i 0.452374 0.452374i −0.443768 0.896142i \(-0.646358\pi\)
0.896142 + 0.443768i \(0.146358\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −27.5639 −1.64432 −0.822162 0.569253i \(-0.807232\pi\)
−0.822162 + 0.569253i \(0.807232\pi\)
\(282\) 0 0
\(283\) −7.40312 + 7.40312i −0.440070 + 0.440070i −0.892035 0.451965i \(-0.850723\pi\)
0.451965 + 0.892035i \(0.350723\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 27.1030 27.1030i 1.59984 1.59984i
\(288\) 0 0
\(289\) 25.2094i 1.48290i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −3.11473 3.11473i −0.181965 0.181965i 0.610247 0.792211i \(-0.291070\pi\)
−0.792211 + 0.610247i \(0.791070\pi\)
\(294\) 0 0
\(295\) 2.06950 + 2.94984i 0.120491 + 0.171746i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 5.48408i 0.317152i
\(300\) 0 0
\(301\) 43.6761i 2.51745i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 8.70704 + 12.4109i 0.498564 + 0.710648i
\(306\) 0 0
\(307\) 20.0000 + 20.0000i 1.14146 + 1.14146i 0.988183 + 0.153277i \(0.0489827\pi\)
0.153277 + 0.988183i \(0.451017\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 8.08857i 0.458661i −0.973349 0.229330i \(-0.926346\pi\)
0.973349 0.229330i \(-0.0736537\pi\)
\(312\) 0 0
\(313\) −19.8062 + 19.8062i −1.11952 + 1.11952i −0.127703 + 0.991812i \(0.540760\pi\)
−0.991812 + 0.127703i \(0.959240\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4.59058 4.59058i 0.257833 0.257833i −0.566339 0.824172i \(-0.691641\pi\)
0.824172 + 0.566339i \(0.191641\pi\)
\(318\) 0 0
\(319\) −7.66037 −0.428898
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −18.3759 + 18.3759i −1.02246 + 1.02246i
\(324\) 0 0
\(325\) 16.0697 7.52901i 0.891388 0.417634i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 43.1978i 2.38157i
\(330\) 0 0
\(331\) −2.80625 −0.154245 −0.0771227 0.997022i \(-0.524573\pi\)
−0.0771227 + 0.997022i \(0.524573\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 23.0588 + 4.04429i 1.25983 + 0.220963i
\(336\) 0 0
\(337\) −2.40312 2.40312i −0.130907 0.130907i 0.638618 0.769524i \(-0.279507\pi\)
−0.769524 + 0.638618i \(0.779507\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 8.08857i 0.438021i
\(342\) 0 0
\(343\) −10.0387 10.0387i −0.542038 0.542038i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2.74204 2.74204i −0.147200 0.147200i 0.629666 0.776866i \(-0.283192\pi\)
−0.776866 + 0.629666i \(0.783192\pi\)
\(348\) 0 0
\(349\) 6.78003 0.362926 0.181463 0.983398i \(-0.441917\pi\)
0.181463 + 0.983398i \(0.441917\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 13.7820 13.7820i 0.733539 0.733539i −0.237780 0.971319i \(-0.576420\pi\)
0.971319 + 0.237780i \(0.0764197\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −35.1916 −1.85734 −0.928671 0.370904i \(-0.879048\pi\)
−0.928671 + 0.370904i \(0.879048\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −2.73147 + 15.5737i −0.142972 + 0.815163i
\(366\) 0 0
\(367\) −14.7492 + 14.7492i −0.769902 + 0.769902i −0.978089 0.208187i \(-0.933244\pi\)
0.208187 + 0.978089i \(0.433244\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 18.3759 0.954031
\(372\) 0 0
\(373\) −14.3090 14.3090i −0.740894 0.740894i 0.231856 0.972750i \(-0.425520\pi\)
−0.972750 + 0.231856i \(0.925520\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −11.9300 11.9300i −0.614427 0.614427i
\(378\) 0 0
\(379\) 18.8062i 0.966012i 0.875617 + 0.483006i \(0.160455\pi\)
−0.875617 + 0.483006i \(0.839545\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 26.0105 + 26.0105i 1.32907 + 1.32907i 0.906180 + 0.422892i \(0.138985\pi\)
0.422892 + 0.906180i \(0.361015\pi\)
\(384\) 0 0
\(385\) −2.59688 + 14.8062i −0.132349 + 0.754596i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −37.3196 −1.89218 −0.946090 0.323905i \(-0.895004\pi\)
−0.946090 + 0.323905i \(0.895004\pi\)
\(390\) 0 0
\(391\) 10.0387i 0.507678i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −9.18797 + 6.44593i −0.462297 + 0.324330i
\(396\) 0 0
\(397\) −19.3284 + 19.3284i −0.970063 + 0.970063i −0.999565 0.0295016i \(-0.990608\pi\)
0.0295016 + 0.999565i \(0.490608\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 12.8919 0.643789 0.321894 0.946776i \(-0.395680\pi\)
0.321894 + 0.946776i \(0.395680\pi\)
\(402\) 0 0
\(403\) −12.5969 + 12.5969i −0.627495 + 0.627495i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −4.04429 + 4.04429i −0.200468 + 0.200468i
\(408\) 0 0
\(409\) 3.40312i 0.168274i 0.996454 + 0.0841368i \(0.0268133\pi\)
−0.996454 + 0.0841368i \(0.973187\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −4.75362 4.75362i −0.233910 0.233910i
\(414\) 0 0
\(415\) −4.13899 + 23.5987i −0.203175 + 1.15842i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 31.9174i 1.55927i 0.626235 + 0.779634i \(0.284595\pi\)
−0.626235 + 0.779634i \(0.715405\pi\)
\(420\) 0 0
\(421\) 30.3788i 1.48057i 0.672293 + 0.740285i \(0.265309\pi\)
−0.672293 + 0.740285i \(0.734691\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −29.4158 + 13.7820i −1.42688 + 0.668523i
\(426\) 0 0
\(427\) −20.0000 20.0000i −0.967868 0.967868i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 27.1030i 1.30551i −0.757570 0.652754i \(-0.773614\pi\)
0.757570 0.652754i \(-0.226386\pi\)
\(432\) 0 0
\(433\) 12.4031 12.4031i 0.596056 0.596056i −0.343205 0.939261i \(-0.611512\pi\)
0.939261 + 0.343205i \(0.111512\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.37036 4.37036i 0.209063 0.209063i
\(438\) 0 0
\(439\) 10.9190 0.521136 0.260568 0.965455i \(-0.416090\pi\)
0.260568 + 0.965455i \(0.416090\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 15.6339 15.6339i 0.742789 0.742789i −0.230325 0.973114i \(-0.573979\pi\)
0.973114 + 0.230325i \(0.0739789\pi\)
\(444\) 0 0
\(445\) −3.52138 5.01934i −0.166929 0.237939i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 6.44593i 0.304202i −0.988365 0.152101i \(-0.951396\pi\)
0.988365 0.152101i \(-0.0486039\pi\)
\(450\) 0 0
\(451\) 14.8062 0.697199
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −27.1030 + 19.0145i −1.27061 + 0.891412i
\(456\) 0 0
\(457\) −9.80625 9.80625i −0.458717 0.458717i 0.439517 0.898234i \(-0.355150\pi\)
−0.898234 + 0.439517i \(0.855150\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 18.3051i 0.852555i 0.904592 + 0.426278i \(0.140175\pi\)
−0.904592 + 0.426278i \(0.859825\pi\)
\(462\) 0 0
\(463\) −14.7492 14.7492i −0.685454 0.685454i 0.275770 0.961224i \(-0.411067\pi\)
−0.961224 + 0.275770i \(0.911067\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −10.7994 10.7994i −0.499739 0.499739i 0.411618 0.911357i \(-0.364964\pi\)
−0.911357 + 0.411618i \(0.864964\pi\)
\(468\) 0 0
\(469\) −43.6761 −2.01677
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 11.9300 11.9300i 0.548542 0.548542i
\(474\) 0 0
\(475\) −18.8062 6.80625i −0.862890 0.312292i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 8.08857 0.369576 0.184788 0.982778i \(-0.440840\pi\)
0.184788 + 0.982778i \(0.440840\pi\)
\(480\) 0 0
\(481\) −12.5969 −0.574368
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −7.48509 1.31281i −0.339880 0.0596118i
\(486\) 0 0
\(487\) −2.06950 + 2.06950i −0.0937778 + 0.0937778i −0.752439 0.658662i \(-0.771123\pi\)
0.658662 + 0.752439i \(0.271123\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 11.2804 0.509076 0.254538 0.967063i \(-0.418077\pi\)
0.254538 + 0.967063i \(0.418077\pi\)
\(492\) 0 0
\(493\) 21.8380 + 21.8380i 0.983536 + 0.983536i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 18.8062i 0.841883i −0.907088 0.420942i \(-0.861700\pi\)
0.907088 0.420942i \(-0.138300\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −5.13688 5.13688i −0.229042 0.229042i 0.583250 0.812292i \(-0.301781\pi\)
−0.812292 + 0.583250i \(0.801781\pi\)
\(504\) 0 0
\(505\) 11.2984 + 16.1047i 0.502774 + 0.716649i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −12.8422 −0.569220 −0.284610 0.958643i \(-0.591864\pi\)
−0.284610 + 0.958643i \(0.591864\pi\)
\(510\) 0 0
\(511\) 29.4984i 1.30493i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −24.8219 4.35352i −1.09378 0.191839i
\(516\) 0 0
\(517\) 11.7994 11.7994i 0.518935 0.518935i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 6.44593 0.282401 0.141201 0.989981i \(-0.454904\pi\)
0.141201 + 0.989981i \(0.454904\pi\)
\(522\) 0 0
\(523\) 5.19375 5.19375i 0.227107 0.227107i −0.584376 0.811483i \(-0.698661\pi\)
0.811483 + 0.584376i \(0.198661\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 23.0588 23.0588i 1.00446 1.00446i
\(528\) 0 0
\(529\) 20.6125i 0.896196i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 23.0588 + 23.0588i 0.998786 + 0.998786i
\(534\) 0 0
\(535\) −5.01934 0.880344i −0.217005 0.0380606i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 16.7645i 0.722096i
\(540\) 0 0
\(541\) 8.27799i 0.355898i 0.984040 + 0.177949i \(0.0569463\pi\)
−0.984040 + 0.177949i \(0.943054\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −21.5989 + 15.1530i −0.925195 + 0.649082i
\(546\) 0 0
\(547\) 22.2094 + 22.2094i 0.949604 + 0.949604i 0.998790 0.0491855i \(-0.0156625\pi\)
−0.0491855 + 0.998790i \(0.515663\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 19.0145i 0.810044i
\(552\) 0 0
\(553\) 14.8062 14.8062i 0.629626 0.629626i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 13.7146 13.7146i 0.581104 0.581104i −0.354102 0.935207i \(-0.615214\pi\)
0.935207 + 0.354102i \(0.115214\pi\)
\(558\) 0 0
\(559\) 37.1588 1.57165
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 2.74204 2.74204i 0.115563 0.115563i −0.646960 0.762524i \(-0.723960\pi\)
0.762524 + 0.646960i \(0.223960\pi\)
\(564\) 0 0
\(565\) 2.50967 14.3090i 0.105583 0.601986i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 36.7519i 1.54072i 0.637610 + 0.770359i \(0.279923\pi\)
−0.637610 + 0.770359i \(0.720077\pi\)
\(570\) 0 0
\(571\) −17.6125 −0.737060 −0.368530 0.929616i \(-0.620139\pi\)
−0.368530 + 0.929616i \(0.620139\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 6.99599 3.27777i 0.291753 0.136692i
\(576\) 0 0
\(577\) 12.4031 + 12.4031i 0.516349 + 0.516349i 0.916465 0.400116i \(-0.131030\pi\)
−0.400116 + 0.916465i \(0.631030\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 44.6989i 1.85442i
\(582\) 0 0
\(583\) 5.01934 + 5.01934i 0.207880 + 0.207880i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.61148 1.61148i −0.0665130 0.0665130i 0.673068 0.739581i \(-0.264976\pi\)
−0.739581 + 0.673068i \(0.764976\pi\)
\(588\) 0 0
\(589\) 20.0774 0.827273
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 4.59398 4.59398i 0.188652 0.188652i −0.606461 0.795113i \(-0.707411\pi\)
0.795113 + 0.606461i \(0.207411\pi\)
\(594\) 0 0
\(595\) 49.6125 34.8062i 2.03391 1.42692i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −27.1030 −1.10740 −0.553700 0.832716i \(-0.686785\pi\)
−0.553700 + 0.832716i \(0.686785\pi\)
\(600\) 0 0
\(601\) 20.2094 0.824358 0.412179 0.911103i \(-0.364768\pi\)
0.412179 + 0.911103i \(0.364768\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 15.3820 10.7915i 0.625369 0.438735i
\(606\) 0 0
\(607\) 18.8882 18.8882i 0.766648 0.766648i −0.210867 0.977515i \(-0.567629\pi\)
0.977515 + 0.210867i \(0.0676285\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 36.7519 1.48682
\(612\) 0 0
\(613\) 19.3284 + 19.3284i 0.780666 + 0.780666i 0.979943 0.199278i \(-0.0638595\pi\)
−0.199278 + 0.979943i \(0.563860\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 7.33602 + 7.33602i 0.295337 + 0.295337i 0.839184 0.543847i \(-0.183033\pi\)
−0.543847 + 0.839184i \(0.683033\pi\)
\(618\) 0 0
\(619\) 10.8062i 0.434340i −0.976134 0.217170i \(-0.930317\pi\)
0.976134 0.217170i \(-0.0696826\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 8.08857 + 8.08857i 0.324062 + 0.324062i
\(624\) 0 0
\(625\) −19.2094 16.0000i −0.768375 0.640000i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 23.0588 0.919413
\(630\) 0 0
\(631\) 25.0967i 0.999083i 0.866290 + 0.499542i \(0.166498\pi\)
−0.866290 + 0.499542i \(0.833502\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −2.09241 + 11.9300i −0.0830348 + 0.473428i
\(636\) 0 0
\(637\) 26.1084 26.1084i 1.03445 1.03445i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 2.74204 0.108304 0.0541520 0.998533i \(-0.482754\pi\)
0.0541520 + 0.998533i \(0.482754\pi\)
\(642\) 0 0
\(643\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −19.7810 + 19.7810i −0.777671 + 0.777671i −0.979434 0.201763i \(-0.935333\pi\)
0.201763 + 0.979434i \(0.435333\pi\)
\(648\) 0 0
\(649\) 2.59688i 0.101936i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 15.1904 + 15.1904i 0.594447 + 0.594447i 0.938829 0.344383i \(-0.111912\pi\)
−0.344383 + 0.938829i \(0.611912\pi\)
\(654\) 0 0
\(655\) 24.7879 17.3902i 0.968543 0.679493i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 4.83445i 0.188323i −0.995557 0.0941617i \(-0.969983\pi\)
0.995557 0.0941617i \(-0.0300171\pi\)
\(660\) 0 0
\(661\) 26.8574i 1.04463i 0.852752 + 0.522315i \(0.174932\pi\)
−0.852752 + 0.522315i \(0.825068\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 36.7519 + 6.44593i 1.42518 + 0.249962i
\(666\) 0 0
\(667\) −5.19375 5.19375i −0.201103 0.201103i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 10.9259i 0.421789i
\(672\) 0 0
\(673\) 15.0000 15.0000i 0.578208 0.578208i −0.356202 0.934409i \(-0.615928\pi\)
0.934409 + 0.356202i \(0.115928\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 18.1421 18.1421i 0.697258 0.697258i −0.266560 0.963818i \(-0.585887\pi\)
0.963818 + 0.266560i \(0.0858872\pi\)
\(678\) 0 0
\(679\) 14.1777 0.544089
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.13056 1.13056i 0.0432595 0.0432595i −0.685146 0.728406i \(-0.740262\pi\)
0.728406 + 0.685146i \(0.240262\pi\)
\(684\) 0 0
\(685\) −34.3864 6.03105i −1.31384 0.230434i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 15.6339i 0.595604i
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −5.13688 7.32206i −0.194853 0.277741i
\(696\) 0 0
\(697\) −42.2094 42.2094i −1.59879 1.59879i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 3.33496i 0.125960i −0.998015 0.0629798i \(-0.979940\pi\)
0.998015 0.0629798i \(-0.0200604\pi\)
\(702\) 0 0
\(703\) 10.0387 + 10.0387i 0.378616 + 0.378616i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −25.9524 25.9524i −0.976041 0.976041i
\(708\) 0 0
\(709\) 26.8574 1.00865 0.504325 0.863514i \(-0.331741\pi\)
0.504325 + 0.863514i \(0.331741\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −5.48408 + 5.48408i −0.205380 + 0.205380i
\(714\) 0 0
\(715\) −12.5969 2.20937i −0.471096 0.0826259i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 8.08857 0.301653 0.150826 0.988560i \(-0.451807\pi\)
0.150826 + 0.988560i \(0.451807\pi\)
\(720\) 0 0
\(721\) 47.0156 1.75095
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −8.08857 + 22.3494i −0.300402 + 0.830037i
\(726\) 0 0
\(727\) −35.7069 + 35.7069i −1.32430 + 1.32430i −0.414034 + 0.910261i \(0.635881\pi\)
−0.910261 + 0.414034i \(0.864119\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −68.0197 −2.51580
\(732\) 0 0
\(733\) 2.50967 + 2.50967i 0.0926967 + 0.0926967i 0.751935 0.659238i \(-0.229121\pi\)
−0.659238 + 0.751935i \(0.729121\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −11.9300 11.9300i −0.439447 0.439447i
\(738\) 0 0
\(739\) 48.4187i 1.78111i −0.454873 0.890556i \(-0.650315\pi\)
0.454873 0.890556i \(-0.349685\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −9.18116 9.18116i −0.336824 0.336824i 0.518346 0.855171i \(-0.326548\pi\)
−0.855171 + 0.518346i \(0.826548\pi\)
\(744\) 0 0
\(745\) 16.1047 11.2984i 0.590030 0.413943i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 9.50723 0.347387
\(750\) 0 0
\(751\) 15.0580i 0.549475i 0.961519 + 0.274737i \(0.0885909\pi\)
−0.961519 + 0.274737i \(0.911409\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1.13056 1.61148i −0.0411451 0.0586479i
\(756\) 0 0
\(757\) 11.0504 11.0504i 0.401633 0.401633i −0.477175 0.878808i \(-0.658339\pi\)
0.878808 + 0.477175i \(0.158339\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 43.1978 1.56592 0.782960 0.622073i \(-0.213709\pi\)
0.782960 + 0.622073i \(0.213709\pi\)
\(762\) 0 0
\(763\) 34.8062 34.8062i 1.26007 1.26007i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.04429 4.04429i 0.146031 0.146031i
\(768\) 0 0
\(769\) 1.40312i 0.0505980i 0.999680 + 0.0252990i \(0.00805377\pi\)
−0.999680 + 0.0252990i \(0.991946\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −25.4642 25.4642i −0.915882 0.915882i 0.0808446 0.996727i \(-0.474238\pi\)
−0.996727 + 0.0808446i \(0.974238\pi\)
\(774\) 0 0
\(775\) 23.5987 + 8.54071i 0.847691 + 0.306792i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 36.7519i 1.31677i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 5.07491 28.9349i 0.181131 1.03273i
\(786\) 0 0
\(787\) 20.0000 + 20.0000i 0.712923 + 0.712923i 0.967146 0.254223i \(-0.0818196\pi\)
−0.254223 + 0.967146i \(0.581820\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 27.1030i 0.963673i
\(792\) 0 0
\(793\) 17.0156 17.0156i 0.604242 0.604242i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −6.00924 + 6.00924i −0.212858 + 0.212858i −0.805481 0.592622i \(-0.798093\pi\)
0.592622 + 0.805481i \(0.298093\pi\)
\(798\) 0 0
\(799\) −67.2748 −2.38001
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 8.05741 8.05741i 0.284340 0.284340i
\(804\) 0 0
\(805\) −11.7994 + 8.27799i −0.415873 + 0.291761i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 2.74204i 0.0964049i 0.998838 + 0.0482025i \(0.0153493\pi\)
−0.998838 + 0.0482025i \(0.984651\pi\)
\(810\) 0 0
\(811\) 42.8062 1.50313 0.751565 0.659659i \(-0.229299\pi\)
0.751565 + 0.659659i \(0.229299\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −29.6125 29.6125i −1.03601 1.03601i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 39.9452i 1.39410i 0.717023 + 0.697049i \(0.245504\pi\)
−0.717023 + 0.697049i \(0.754496\pi\)
\(822\) 0 0
\(823\) −5.59087 5.59087i −0.194886 0.194886i 0.602918 0.797803i \(-0.294005\pi\)
−0.797803 + 0.602918i \(0.794005\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 36.7519 + 36.7519i 1.27799 + 1.27799i 0.941792 + 0.336196i \(0.109140\pi\)
0.336196 + 0.941792i \(0.390860\pi\)
\(828\) 0 0
\(829\) −8.27799 −0.287506 −0.143753 0.989614i \(-0.545917\pi\)
−0.143753 + 0.989614i \(0.545917\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −47.7918 + 47.7918i −1.65589 + 1.65589i
\(834\) 0 0
\(835\) 4.00000 22.8062i 0.138426 0.789243i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −27.1030 −0.935701 −0.467850 0.883808i \(-0.654971\pi\)
−0.467850 + 0.883808i \(0.654971\pi\)
\(840\) 0 0
\(841\) −6.40312 −0.220797
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0.517700 + 0.737925i 0.0178094 + 0.0253854i
\(846\) 0 0
\(847\) −24.7879 + 24.7879i −0.851722 + 0.851722i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −5.48408 −0.187992
\(852\) 0 0
\(853\) −14.3090 14.3090i −0.489932 0.489932i 0.418353 0.908285i \(-0.362608\pi\)
−0.908285 + 0.418353i \(0.862608\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 13.7820 + 13.7820i 0.470782 + 0.470782i 0.902168 0.431385i \(-0.141975\pi\)
−0.431385 + 0.902168i \(0.641975\pi\)
\(858\) 0 0
\(859\) 25.6125i 0.873887i −0.899489 0.436944i \(-0.856061\pi\)
0.899489 0.436944i \(-0.143939\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −21.9662 21.9662i −0.747737 0.747737i 0.226317 0.974054i \(-0.427332\pi\)
−0.974054 + 0.226317i \(0.927332\pi\)
\(864\) 0 0
\(865\) −24.5078 4.29844i −0.833291 0.146151i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 8.08857 0.274386
\(870\) 0 0
\(871\) 37.1588i 1.25908i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 40.4558 + 23.2104i 1.36765 + 0.784654i
\(876\) 0 0
\(877\) 14.3090 14.3090i 0.483182 0.483182i −0.422965 0.906146i \(-0.639011\pi\)
0.906146 + 0.422965i \(0.139011\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −46.9017 −1.58016 −0.790079 0.613005i \(-0.789961\pi\)
−0.790079 + 0.613005i \(0.789961\pi\)
\(882\) 0 0
\(883\) −14.8062 + 14.8062i −0.498270 + 0.498270i −0.910899 0.412629i \(-0.864611\pi\)
0.412629 + 0.910899i \(0.364611\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −26.3365 + 26.3365i −0.884294 + 0.884294i −0.993968 0.109674i \(-0.965019\pi\)
0.109674 + 0.993968i \(0.465019\pi\)
\(888\) 0 0
\(889\) 22.5969i 0.757875i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −29.2882 29.2882i −0.980093 0.980093i
\(894\) 0 0
\(895\) 33.9462 + 48.3866i 1.13470 + 1.61739i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 23.8600i 0.795776i
\(900\) 0 0
\(901\) 28.6181i 0.953406i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −15.1530 21.5989i −0.503702 0.717972i
\(906\) 0 0
\(907\) −12.5969 12.5969i −0.418272 0.418272i 0.466336 0.884608i \(-0.345574\pi\)
−0.884608 + 0.466336i \(0.845574\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 27.1030i 0.897964i 0.893541 + 0.448982i \(0.148213\pi\)
−0.893541 + 0.448982i \(0.851787\pi\)
\(912\) 0 0
\(913\) 12.2094 12.2094i 0.404071 0.404071i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −39.9452 + 39.9452i −1.31911 + 1.31911i
\(918\) 0 0
\(919\) 52.2168 1.72247 0.861237 0.508204i \(-0.169691\pi\)
0.861237 + 0.508204i \(0.169691\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 7.52901 + 16.0697i 0.247552 + 0.528369i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 43.1978i 1.41727i −0.705573 0.708637i \(-0.749310\pi\)
0.705573 0.708637i \(-0.250690\pi\)
\(930\) 0 0
\(931\) −41.6125 −1.36379
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 23.0588 + 4.04429i 0.754102 + 0.132262i
\(936\) 0 0
\(937\) 22.4031 + 22.4031i 0.731878 + 0.731878i 0.970992 0.239113i \(-0.0768568\pi\)
−0.239113 + 0.970992i \(0.576857\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 10.2166i 0.333050i −0.986037 0.166525i \(-0.946745\pi\)
0.986037 0.166525i \(-0.0532547\pi\)
\(942\) 0 0
\(943\) 10.0387 + 10.0387i 0.326904 + 0.326904i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 18.3759 + 18.3759i 0.597138 + 0.597138i 0.939550 0.342412i \(-0.111244\pi\)
−0.342412 + 0.939550i \(0.611244\pi\)
\(948\) 0 0
\(949\) 25.0967 0.814673
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.85195 1.85195i 0.0599904 0.0599904i −0.676475 0.736466i \(-0.736493\pi\)
0.736466 + 0.676475i \(0.236493\pi\)
\(954\) 0 0
\(955\) 24.4187 + 34.8062i 0.790172 + 1.12630i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 65.1320 2.10322
\(960\) 0 0
\(961\) 5.80625 0.187298
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 6.77576 38.6324i 0.218119 1.24362i
\(966\) 0 0
\(967\) −2.06950 + 2.06950i −0.0665505 + 0.0665505i −0.739599 0.673048i \(-0.764985\pi\)
0.673048 + 0.739599i \(0.264985\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −38.3634 −1.23114 −0.615569 0.788083i \(-0.711074\pi\)
−0.615569 + 0.788083i \(0.711074\pi\)
\(972\) 0 0
\(973\) 11.7994 + 11.7994i 0.378270 + 0.378270i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −7.33602 7.33602i −0.234700 0.234700i 0.579951 0.814651i \(-0.303072\pi\)
−0.814651 + 0.579951i \(0.803072\pi\)
\(978\) 0 0
\(979\) 4.41875i 0.141224i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −9.18116 9.18116i −0.292834 0.292834i 0.545365 0.838199i \(-0.316391\pi\)
−0.838199 + 0.545365i \(0.816391\pi\)
\(984\) 0 0
\(985\) −5.10469 + 29.1047i −0.162649 + 0.927352i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 16.1771 0.514403
\(990\) 0 0
\(991\) 42.7957i 1.35945i −0.733466 0.679726i \(-0.762099\pi\)
0.733466 0.679726i \(-0.237901\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 38.3634 26.9143i 1.21620 0.853240i
\(996\) 0 0
\(997\) −2.50967 + 2.50967i −0.0794820 + 0.0794820i −0.745730 0.666248i \(-0.767899\pi\)
0.666248 + 0.745730i \(0.267899\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1440.2.bi.d.847.2 16
3.2 odd 2 inner 1440.2.bi.d.847.8 16
4.3 odd 2 360.2.w.d.307.5 yes 16
5.3 odd 4 inner 1440.2.bi.d.1423.7 16
8.3 odd 2 inner 1440.2.bi.d.847.7 16
8.5 even 2 360.2.w.d.307.1 yes 16
12.11 even 2 360.2.w.d.307.4 yes 16
15.8 even 4 inner 1440.2.bi.d.1423.1 16
20.3 even 4 360.2.w.d.163.1 16
24.5 odd 2 360.2.w.d.307.8 yes 16
24.11 even 2 inner 1440.2.bi.d.847.1 16
40.3 even 4 inner 1440.2.bi.d.1423.2 16
40.13 odd 4 360.2.w.d.163.5 yes 16
60.23 odd 4 360.2.w.d.163.8 yes 16
120.53 even 4 360.2.w.d.163.4 yes 16
120.83 odd 4 inner 1440.2.bi.d.1423.8 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.2.w.d.163.1 16 20.3 even 4
360.2.w.d.163.4 yes 16 120.53 even 4
360.2.w.d.163.5 yes 16 40.13 odd 4
360.2.w.d.163.8 yes 16 60.23 odd 4
360.2.w.d.307.1 yes 16 8.5 even 2
360.2.w.d.307.4 yes 16 12.11 even 2
360.2.w.d.307.5 yes 16 4.3 odd 2
360.2.w.d.307.8 yes 16 24.5 odd 2
1440.2.bi.d.847.1 16 24.11 even 2 inner
1440.2.bi.d.847.2 16 1.1 even 1 trivial
1440.2.bi.d.847.7 16 8.3 odd 2 inner
1440.2.bi.d.847.8 16 3.2 odd 2 inner
1440.2.bi.d.1423.1 16 15.8 even 4 inner
1440.2.bi.d.1423.2 16 40.3 even 4 inner
1440.2.bi.d.1423.7 16 5.3 odd 4 inner
1440.2.bi.d.1423.8 16 120.83 odd 4 inner