Properties

Label 816.2.bq.f.433.3
Level $816$
Weight $2$
Character 816.433
Analytic conductor $6.516$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [816,2,Mod(49,816)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(816, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("816.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 816.bq (of order \(8\), degree \(4\), 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: \(6.51579280494\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 36x^{14} + 466x^{12} + 2956x^{10} + 10049x^{8} + 18032x^{6} + 14800x^{4} + 3200x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 408)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 433.3
Root \(1.71472i\) of defining polynomial
Character \(\chi\) \(=\) 816.433
Dual form 816.2.bq.f.49.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.382683 - 0.923880i) q^{3} +(-2.50807 - 1.03888i) q^{5} +(-2.22010 + 0.919595i) q^{7} +(-0.707107 - 0.707107i) q^{9} +(1.87280 + 4.52134i) q^{11} +1.07051i q^{13} +(-1.91959 + 1.91959i) q^{15} +(3.25570 - 2.52991i) q^{17} +(-1.61272 + 1.61272i) q^{19} +2.40302i q^{21} +(2.41400 + 5.82790i) q^{23} +(1.67562 + 1.67562i) q^{25} +(-0.923880 + 0.382683i) q^{27} +(3.13555 + 1.29879i) q^{29} +(-2.67656 + 6.46178i) q^{31} +4.89386 q^{33} +6.52351 q^{35} +(-1.63047 + 3.93630i) q^{37} +(0.989020 + 0.409666i) q^{39} +(7.08028 - 2.93275i) q^{41} +(4.37197 + 4.37197i) q^{43} +(1.03888 + 2.50807i) q^{45} -8.19868i q^{47} +(-0.866566 + 0.866566i) q^{49} +(-1.09143 - 3.97603i) q^{51} +(-7.97888 + 7.97888i) q^{53} -13.2855i q^{55} +(0.872800 + 2.10713i) q^{57} +(8.89888 + 8.89888i) q^{59} +(-2.52414 + 1.04553i) q^{61} +(2.22010 + 0.919595i) q^{63} +(1.11213 - 2.68491i) q^{65} -12.8012 q^{67} +6.30808 q^{69} +(1.62627 - 3.92615i) q^{71} +(-9.47623 - 3.92518i) q^{73} +(2.18931 - 0.906841i) q^{75} +(-8.31560 - 8.31560i) q^{77} +(-0.464750 - 1.12200i) q^{79} +1.00000i q^{81} +(-2.92403 + 2.92403i) q^{83} +(-10.7938 + 2.96293i) q^{85} +(2.39984 - 2.39984i) q^{87} -2.96634i q^{89} +(-0.984434 - 2.37663i) q^{91} +(4.94563 + 4.94563i) q^{93} +(5.72025 - 2.36941i) q^{95} +(16.8658 + 6.98605i) q^{97} +(1.87280 - 4.52134i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{11} - 8 q^{15} + 8 q^{19} + 8 q^{23} - 8 q^{25} + 32 q^{29} - 8 q^{31} - 8 q^{33} - 16 q^{35} + 8 q^{37} + 8 q^{39} + 40 q^{41} + 24 q^{43} + 8 q^{45} + 24 q^{49} - 8 q^{51} - 24 q^{53} - 8 q^{57}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/816\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(511\) \(545\) \(613\)
\(\chi(n)\) \(e\left(\frac{5}{8}\right)\) \(1\) \(1\) \(1\)

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
\(3\) 0.382683 0.923880i 0.220942 0.533402i
\(4\) 0 0
\(5\) −2.50807 1.03888i −1.12164 0.464600i −0.256711 0.966488i \(-0.582639\pi\)
−0.864933 + 0.501888i \(0.832639\pi\)
\(6\) 0 0
\(7\) −2.22010 + 0.919595i −0.839118 + 0.347574i −0.760506 0.649331i \(-0.775049\pi\)
−0.0786124 + 0.996905i \(0.525049\pi\)
\(8\) 0 0
\(9\) −0.707107 0.707107i −0.235702 0.235702i
\(10\) 0 0
\(11\) 1.87280 + 4.52134i 0.564671 + 1.36324i 0.905994 + 0.423290i \(0.139125\pi\)
−0.341324 + 0.939946i \(0.610875\pi\)
\(12\) 0 0
\(13\) 1.07051i 0.296906i 0.988920 + 0.148453i \(0.0474293\pi\)
−0.988920 + 0.148453i \(0.952571\pi\)
\(14\) 0 0
\(15\) −1.91959 + 1.91959i −0.495637 + 0.495637i
\(16\) 0 0
\(17\) 3.25570 2.52991i 0.789622 0.613593i
\(18\) 0 0
\(19\) −1.61272 + 1.61272i −0.369984 + 0.369984i −0.867471 0.497487i \(-0.834256\pi\)
0.497487 + 0.867471i \(0.334256\pi\)
\(20\) 0 0
\(21\) 2.40302i 0.524381i
\(22\) 0 0
\(23\) 2.41400 + 5.82790i 0.503353 + 1.21520i 0.947647 + 0.319321i \(0.103455\pi\)
−0.444294 + 0.895881i \(0.646545\pi\)
\(24\) 0 0
\(25\) 1.67562 + 1.67562i 0.335125 + 0.335125i
\(26\) 0 0
\(27\) −0.923880 + 0.382683i −0.177801 + 0.0736475i
\(28\) 0 0
\(29\) 3.13555 + 1.29879i 0.582257 + 0.241179i 0.654315 0.756222i \(-0.272957\pi\)
−0.0720587 + 0.997400i \(0.522957\pi\)
\(30\) 0 0
\(31\) −2.67656 + 6.46178i −0.480724 + 1.16057i 0.478542 + 0.878065i \(0.341166\pi\)
−0.959266 + 0.282506i \(0.908834\pi\)
\(32\) 0 0
\(33\) 4.89386 0.851912
\(34\) 0 0
\(35\) 6.52351 1.10267
\(36\) 0 0
\(37\) −1.63047 + 3.93630i −0.268047 + 0.647123i −0.999391 0.0348855i \(-0.988893\pi\)
0.731344 + 0.682009i \(0.238893\pi\)
\(38\) 0 0
\(39\) 0.989020 + 0.409666i 0.158370 + 0.0655990i
\(40\) 0 0
\(41\) 7.08028 2.93275i 1.10575 0.458018i 0.246280 0.969199i \(-0.420792\pi\)
0.859473 + 0.511180i \(0.170792\pi\)
\(42\) 0 0
\(43\) 4.37197 + 4.37197i 0.666719 + 0.666719i 0.956955 0.290236i \(-0.0937339\pi\)
−0.290236 + 0.956955i \(0.593734\pi\)
\(44\) 0 0
\(45\) 1.03888 + 2.50807i 0.154867 + 0.373881i
\(46\) 0 0
\(47\) 8.19868i 1.19590i −0.801533 0.597950i \(-0.795982\pi\)
0.801533 0.597950i \(-0.204018\pi\)
\(48\) 0 0
\(49\) −0.866566 + 0.866566i −0.123795 + 0.123795i
\(50\) 0 0
\(51\) −1.09143 3.97603i −0.152831 0.556755i
\(52\) 0 0
\(53\) −7.97888 + 7.97888i −1.09598 + 1.09598i −0.101108 + 0.994875i \(0.532239\pi\)
−0.994875 + 0.101108i \(0.967761\pi\)
\(54\) 0 0
\(55\) 13.2855i 1.79141i
\(56\) 0 0
\(57\) 0.872800 + 2.10713i 0.115605 + 0.279096i
\(58\) 0 0
\(59\) 8.89888 + 8.89888i 1.15854 + 1.15854i 0.984791 + 0.173745i \(0.0555869\pi\)
0.173745 + 0.984791i \(0.444413\pi\)
\(60\) 0 0
\(61\) −2.52414 + 1.04553i −0.323183 + 0.133867i −0.538376 0.842705i \(-0.680962\pi\)
0.215193 + 0.976572i \(0.430962\pi\)
\(62\) 0 0
\(63\) 2.22010 + 0.919595i 0.279706 + 0.115858i
\(64\) 0 0
\(65\) 1.11213 2.68491i 0.137942 0.333022i
\(66\) 0 0
\(67\) −12.8012 −1.56392 −0.781958 0.623331i \(-0.785779\pi\)
−0.781958 + 0.623331i \(0.785779\pi\)
\(68\) 0 0
\(69\) 6.30808 0.759403
\(70\) 0 0
\(71\) 1.62627 3.92615i 0.193002 0.465948i −0.797521 0.603291i \(-0.793856\pi\)
0.990524 + 0.137342i \(0.0438559\pi\)
\(72\) 0 0
\(73\) −9.47623 3.92518i −1.10911 0.459408i −0.248478 0.968637i \(-0.579931\pi\)
−0.860631 + 0.509229i \(0.829931\pi\)
\(74\) 0 0
\(75\) 2.18931 0.906841i 0.252799 0.104713i
\(76\) 0 0
\(77\) −8.31560 8.31560i −0.947651 0.947651i
\(78\) 0 0
\(79\) −0.464750 1.12200i −0.0522884 0.126235i 0.895577 0.444907i \(-0.146763\pi\)
−0.947865 + 0.318672i \(0.896763\pi\)
\(80\) 0 0
\(81\) 1.00000i 0.111111i
\(82\) 0 0
\(83\) −2.92403 + 2.92403i −0.320954 + 0.320954i −0.849133 0.528179i \(-0.822875\pi\)
0.528179 + 0.849133i \(0.322875\pi\)
\(84\) 0 0
\(85\) −10.7938 + 2.96293i −1.17075 + 0.321374i
\(86\) 0 0
\(87\) 2.39984 2.39984i 0.257290 0.257290i
\(88\) 0 0
\(89\) 2.96634i 0.314431i −0.987564 0.157216i \(-0.949748\pi\)
0.987564 0.157216i \(-0.0502518\pi\)
\(90\) 0 0
\(91\) −0.984434 2.37663i −0.103197 0.249139i
\(92\) 0 0
\(93\) 4.94563 + 4.94563i 0.512838 + 0.512838i
\(94\) 0 0
\(95\) 5.72025 2.36941i 0.586886 0.243096i
\(96\) 0 0
\(97\) 16.8658 + 6.98605i 1.71247 + 0.709326i 0.999971 + 0.00764885i \(0.00243473\pi\)
0.712495 + 0.701678i \(0.247565\pi\)
\(98\) 0 0
\(99\) 1.87280 4.52134i 0.188224 0.454412i
\(100\) 0 0
\(101\) −9.04503 −0.900014 −0.450007 0.893025i \(-0.648579\pi\)
−0.450007 + 0.893025i \(0.648579\pi\)
\(102\) 0 0
\(103\) 2.23568 0.220288 0.110144 0.993916i \(-0.464869\pi\)
0.110144 + 0.993916i \(0.464869\pi\)
\(104\) 0 0
\(105\) 2.49644 6.02694i 0.243628 0.588169i
\(106\) 0 0
\(107\) −5.04118 2.08813i −0.487350 0.201867i 0.125458 0.992099i \(-0.459960\pi\)
−0.612808 + 0.790232i \(0.709960\pi\)
\(108\) 0 0
\(109\) −2.44258 + 1.01175i −0.233956 + 0.0969079i −0.496582 0.867990i \(-0.665412\pi\)
0.262625 + 0.964898i \(0.415412\pi\)
\(110\) 0 0
\(111\) 3.01271 + 3.01271i 0.285954 + 0.285954i
\(112\) 0 0
\(113\) 5.35434 + 12.9265i 0.503694 + 1.21602i 0.947458 + 0.319880i \(0.103643\pi\)
−0.443764 + 0.896144i \(0.646357\pi\)
\(114\) 0 0
\(115\) 17.1246i 1.59688i
\(116\) 0 0
\(117\) 0.756964 0.756964i 0.0699813 0.0699813i
\(118\) 0 0
\(119\) −4.90147 + 8.61057i −0.449317 + 0.789330i
\(120\) 0 0
\(121\) −9.15696 + 9.15696i −0.832451 + 0.832451i
\(122\) 0 0
\(123\) 7.66364i 0.691007i
\(124\) 0 0
\(125\) 2.73257 + 6.59701i 0.244409 + 0.590054i
\(126\) 0 0
\(127\) −10.2828 10.2828i −0.912449 0.912449i 0.0840159 0.996464i \(-0.473225\pi\)
−0.996464 + 0.0840159i \(0.973225\pi\)
\(128\) 0 0
\(129\) 5.71225 2.36609i 0.502936 0.208323i
\(130\) 0 0
\(131\) 2.23079 + 0.924024i 0.194905 + 0.0807323i 0.478001 0.878359i \(-0.341361\pi\)
−0.283096 + 0.959092i \(0.591361\pi\)
\(132\) 0 0
\(133\) 2.09735 5.06346i 0.181864 0.439058i
\(134\) 0 0
\(135\) 2.71472 0.233646
\(136\) 0 0
\(137\) −1.42184 −0.121476 −0.0607382 0.998154i \(-0.519345\pi\)
−0.0607382 + 0.998154i \(0.519345\pi\)
\(138\) 0 0
\(139\) −5.94425 + 14.3507i −0.504184 + 1.21721i 0.443001 + 0.896521i \(0.353914\pi\)
−0.947185 + 0.320687i \(0.896086\pi\)
\(140\) 0 0
\(141\) −7.57460 3.13750i −0.637896 0.264225i
\(142\) 0 0
\(143\) −4.84013 + 2.00485i −0.404752 + 0.167654i
\(144\) 0 0
\(145\) −6.51490 6.51490i −0.541033 0.541033i
\(146\) 0 0
\(147\) 0.468982 + 1.13222i 0.0386810 + 0.0933842i
\(148\) 0 0
\(149\) 16.2390i 1.33035i 0.746687 + 0.665176i \(0.231643\pi\)
−0.746687 + 0.665176i \(0.768357\pi\)
\(150\) 0 0
\(151\) −2.48539 + 2.48539i −0.202258 + 0.202258i −0.800967 0.598709i \(-0.795681\pi\)
0.598709 + 0.800967i \(0.295681\pi\)
\(152\) 0 0
\(153\) −4.09104 0.513208i −0.330741 0.0414904i
\(154\) 0 0
\(155\) 13.4260 13.4260i 1.07840 1.07840i
\(156\) 0 0
\(157\) 21.3904i 1.70714i −0.520975 0.853572i \(-0.674432\pi\)
0.520975 0.853572i \(-0.325568\pi\)
\(158\) 0 0
\(159\) 4.31814 + 10.4249i 0.342451 + 0.826749i
\(160\) 0 0
\(161\) −10.7186 10.7186i −0.844746 0.844746i
\(162\) 0 0
\(163\) −18.3602 + 7.60506i −1.43809 + 0.595675i −0.959333 0.282275i \(-0.908911\pi\)
−0.478752 + 0.877950i \(0.658911\pi\)
\(164\) 0 0
\(165\) −12.2742 5.08412i −0.955542 0.395798i
\(166\) 0 0
\(167\) −3.03768 + 7.33360i −0.235063 + 0.567491i −0.996759 0.0804426i \(-0.974367\pi\)
0.761697 + 0.647934i \(0.224367\pi\)
\(168\) 0 0
\(169\) 11.8540 0.911847
\(170\) 0 0
\(171\) 2.28074 0.174412
\(172\) 0 0
\(173\) 9.18012 22.1628i 0.697951 1.68500i −0.0301604 0.999545i \(-0.509602\pi\)
0.728112 0.685458i \(-0.240398\pi\)
\(174\) 0 0
\(175\) −5.26094 2.17915i −0.397690 0.164728i
\(176\) 0 0
\(177\) 11.6269 4.81604i 0.873935 0.361996i
\(178\) 0 0
\(179\) −17.0828 17.0828i −1.27683 1.27683i −0.942434 0.334392i \(-0.891469\pi\)
−0.334392 0.942434i \(-0.608531\pi\)
\(180\) 0 0
\(181\) 4.65785 + 11.2451i 0.346216 + 0.835838i 0.997060 + 0.0766270i \(0.0244151\pi\)
−0.650844 + 0.759211i \(0.725585\pi\)
\(182\) 0 0
\(183\) 2.73211i 0.201964i
\(184\) 0 0
\(185\) 8.17866 8.17866i 0.601307 0.601307i
\(186\) 0 0
\(187\) 17.5359 + 9.98209i 1.28235 + 0.729963i
\(188\) 0 0
\(189\) 1.69919 1.69919i 0.123598 0.123598i
\(190\) 0 0
\(191\) 3.27804i 0.237191i 0.992943 + 0.118595i \(0.0378392\pi\)
−0.992943 + 0.118595i \(0.962161\pi\)
\(192\) 0 0
\(193\) −7.94091 19.1711i −0.571599 1.37996i −0.900193 0.435492i \(-0.856575\pi\)
0.328593 0.944472i \(-0.393425\pi\)
\(194\) 0 0
\(195\) −2.05494 2.05494i −0.147157 0.147157i
\(196\) 0 0
\(197\) 18.2135 7.54429i 1.29766 0.537508i 0.376401 0.926457i \(-0.377162\pi\)
0.921259 + 0.388949i \(0.127162\pi\)
\(198\) 0 0
\(199\) 14.0934 + 5.83768i 0.999055 + 0.413822i 0.821451 0.570280i \(-0.193165\pi\)
0.177605 + 0.984102i \(0.443165\pi\)
\(200\) 0 0
\(201\) −4.89881 + 11.8268i −0.345535 + 0.834196i
\(202\) 0 0
\(203\) −8.15558 −0.572410
\(204\) 0 0
\(205\) −20.8046 −1.45306
\(206\) 0 0
\(207\) 2.41400 5.82790i 0.167784 0.405067i
\(208\) 0 0
\(209\) −10.3120 4.27137i −0.713295 0.295457i
\(210\) 0 0
\(211\) 11.5631 4.78961i 0.796039 0.329730i 0.0526704 0.998612i \(-0.483227\pi\)
0.743369 + 0.668882i \(0.233227\pi\)
\(212\) 0 0
\(213\) −3.00495 3.00495i −0.205896 0.205896i
\(214\) 0 0
\(215\) −6.42327 15.5071i −0.438063 1.05758i
\(216\) 0 0
\(217\) 16.8071i 1.14094i
\(218\) 0 0
\(219\) −7.25280 + 7.25280i −0.490099 + 0.490099i
\(220\) 0 0
\(221\) 2.70829 + 3.48525i 0.182179 + 0.234443i
\(222\) 0 0
\(223\) 5.32288 5.32288i 0.356446 0.356446i −0.506055 0.862501i \(-0.668897\pi\)
0.862501 + 0.506055i \(0.168897\pi\)
\(224\) 0 0
\(225\) 2.36969i 0.157979i
\(226\) 0 0
\(227\) −1.83963 4.44126i −0.122101 0.294777i 0.850997 0.525171i \(-0.175999\pi\)
−0.973097 + 0.230394i \(0.925999\pi\)
\(228\) 0 0
\(229\) −9.47732 9.47732i −0.626279 0.626279i 0.320851 0.947130i \(-0.396031\pi\)
−0.947130 + 0.320851i \(0.896031\pi\)
\(230\) 0 0
\(231\) −10.8649 + 4.50037i −0.714855 + 0.296103i
\(232\) 0 0
\(233\) 12.7306 + 5.27318i 0.834008 + 0.345457i 0.758488 0.651687i \(-0.225938\pi\)
0.0755198 + 0.997144i \(0.475938\pi\)
\(234\) 0 0
\(235\) −8.51742 + 20.5629i −0.555616 + 1.34137i
\(236\) 0 0
\(237\) −1.21445 −0.0788870
\(238\) 0 0
\(239\) 17.0060 1.10003 0.550014 0.835155i \(-0.314622\pi\)
0.550014 + 0.835155i \(0.314622\pi\)
\(240\) 0 0
\(241\) −7.42077 + 17.9153i −0.478014 + 1.15403i 0.482525 + 0.875882i \(0.339720\pi\)
−0.960539 + 0.278145i \(0.910280\pi\)
\(242\) 0 0
\(243\) 0.923880 + 0.382683i 0.0592669 + 0.0245492i
\(244\) 0 0
\(245\) 3.07366 1.27315i 0.196369 0.0813388i
\(246\) 0 0
\(247\) −1.72644 1.72644i −0.109850 0.109850i
\(248\) 0 0
\(249\) 1.58247 + 3.82042i 0.100285 + 0.242110i
\(250\) 0 0
\(251\) 21.5316i 1.35906i −0.733646 0.679532i \(-0.762183\pi\)
0.733646 0.679532i \(-0.237817\pi\)
\(252\) 0 0
\(253\) −21.8290 + 21.8290i −1.37238 + 1.37238i
\(254\) 0 0
\(255\) −1.39322 + 11.1060i −0.0872466 + 0.695486i
\(256\) 0 0
\(257\) 13.5195 13.5195i 0.843324 0.843324i −0.145966 0.989290i \(-0.546629\pi\)
0.989290 + 0.145966i \(0.0466290\pi\)
\(258\) 0 0
\(259\) 10.2383i 0.636179i
\(260\) 0 0
\(261\) −1.29879 3.13555i −0.0803929 0.194086i
\(262\) 0 0
\(263\) −7.83952 7.83952i −0.483406 0.483406i 0.422812 0.906218i \(-0.361043\pi\)
−0.906218 + 0.422812i \(0.861043\pi\)
\(264\) 0 0
\(265\) 28.3007 11.7225i 1.73850 0.720109i
\(266\) 0 0
\(267\) −2.74054 1.13517i −0.167718 0.0694712i
\(268\) 0 0
\(269\) 1.69352 4.08852i 0.103256 0.249282i −0.863805 0.503826i \(-0.831925\pi\)
0.967061 + 0.254544i \(0.0819254\pi\)
\(270\) 0 0
\(271\) −16.2476 −0.986973 −0.493486 0.869753i \(-0.664278\pi\)
−0.493486 + 0.869753i \(0.664278\pi\)
\(272\) 0 0
\(273\) −2.57245 −0.155692
\(274\) 0 0
\(275\) −4.43795 + 10.7142i −0.267619 + 0.646089i
\(276\) 0 0
\(277\) −0.591916 0.245180i −0.0355648 0.0147314i 0.364830 0.931074i \(-0.381127\pi\)
−0.400395 + 0.916343i \(0.631127\pi\)
\(278\) 0 0
\(279\) 6.46178 2.67656i 0.386857 0.160241i
\(280\) 0 0
\(281\) 0.0521155 + 0.0521155i 0.00310895 + 0.00310895i 0.708660 0.705551i \(-0.249300\pi\)
−0.705551 + 0.708660i \(0.749300\pi\)
\(282\) 0 0
\(283\) 7.82868 + 18.9001i 0.465367 + 1.12350i 0.966164 + 0.257930i \(0.0830404\pi\)
−0.500797 + 0.865565i \(0.666960\pi\)
\(284\) 0 0
\(285\) 6.19156i 0.366756i
\(286\) 0 0
\(287\) −13.0220 + 13.0220i −0.768663 + 0.768663i
\(288\) 0 0
\(289\) 4.19911 16.4732i 0.247007 0.969014i
\(290\) 0 0
\(291\) 12.9085 12.9085i 0.756712 0.756712i
\(292\) 0 0
\(293\) 7.88796i 0.460820i 0.973094 + 0.230410i \(0.0740067\pi\)
−0.973094 + 0.230410i \(0.925993\pi\)
\(294\) 0 0
\(295\) −13.0742 31.5639i −0.761209 1.83772i
\(296\) 0 0
\(297\) −3.46048 3.46048i −0.200798 0.200798i
\(298\) 0 0
\(299\) −6.23882 + 2.58420i −0.360800 + 0.149448i
\(300\) 0 0
\(301\) −13.7266 5.68576i −0.791190 0.327722i
\(302\) 0 0
\(303\) −3.46138 + 8.35652i −0.198851 + 0.480069i
\(304\) 0 0
\(305\) 7.41691 0.424691
\(306\) 0 0
\(307\) 4.63954 0.264793 0.132396 0.991197i \(-0.457733\pi\)
0.132396 + 0.991197i \(0.457733\pi\)
\(308\) 0 0
\(309\) 0.855556 2.06550i 0.0486709 0.117502i
\(310\) 0 0
\(311\) 5.16892 + 2.14104i 0.293103 + 0.121407i 0.524389 0.851479i \(-0.324294\pi\)
−0.231287 + 0.972886i \(0.574294\pi\)
\(312\) 0 0
\(313\) 0.810266 0.335623i 0.0457990 0.0189705i −0.359666 0.933081i \(-0.617109\pi\)
0.405465 + 0.914110i \(0.367109\pi\)
\(314\) 0 0
\(315\) −4.61282 4.61282i −0.259903 0.259903i
\(316\) 0 0
\(317\) 11.3854 + 27.4867i 0.639467 + 1.54381i 0.827392 + 0.561625i \(0.189824\pi\)
−0.187925 + 0.982183i \(0.560176\pi\)
\(318\) 0 0
\(319\) 16.6092i 0.929939i
\(320\) 0 0
\(321\) −3.85836 + 3.85836i −0.215352 + 0.215352i
\(322\) 0 0
\(323\) −1.17049 + 9.33059i −0.0651280 + 0.519168i
\(324\) 0 0
\(325\) −1.79377 + 1.79377i −0.0995003 + 0.0995003i
\(326\) 0 0
\(327\) 2.64382i 0.146204i
\(328\) 0 0
\(329\) 7.53947 + 18.2019i 0.415664 + 1.00350i
\(330\) 0 0
\(331\) 8.18438 + 8.18438i 0.449854 + 0.449854i 0.895306 0.445452i \(-0.146957\pi\)
−0.445452 + 0.895306i \(0.646957\pi\)
\(332\) 0 0
\(333\) 3.93630 1.63047i 0.215708 0.0893491i
\(334\) 0 0
\(335\) 32.1063 + 13.2989i 1.75416 + 0.726595i
\(336\) 0 0
\(337\) 6.47644 15.6355i 0.352794 0.851720i −0.643479 0.765464i \(-0.722510\pi\)
0.996273 0.0862562i \(-0.0274904\pi\)
\(338\) 0 0
\(339\) 13.9916 0.759917
\(340\) 0 0
\(341\) −34.2286 −1.85358
\(342\) 0 0
\(343\) 7.56414 18.2614i 0.408425 0.986025i
\(344\) 0 0
\(345\) −15.8211 6.55332i −0.851780 0.352819i
\(346\) 0 0
\(347\) 2.02695 0.839592i 0.108813 0.0450716i −0.327613 0.944812i \(-0.606244\pi\)
0.436426 + 0.899740i \(0.356244\pi\)
\(348\) 0 0
\(349\) 18.7192 + 18.7192i 1.00201 + 1.00201i 0.999998 + 0.00201610i \(0.000641744\pi\)
0.00201610 + 0.999998i \(0.499358\pi\)
\(350\) 0 0
\(351\) −0.409666 0.989020i −0.0218663 0.0527900i
\(352\) 0 0
\(353\) 24.9345i 1.32713i 0.748120 + 0.663564i \(0.230957\pi\)
−0.748120 + 0.663564i \(0.769043\pi\)
\(354\) 0 0
\(355\) −8.15758 + 8.15758i −0.432959 + 0.432959i
\(356\) 0 0
\(357\) 6.07942 + 7.82349i 0.321757 + 0.414063i
\(358\) 0 0
\(359\) −2.09699 + 2.09699i −0.110675 + 0.110675i −0.760276 0.649601i \(-0.774936\pi\)
0.649601 + 0.760276i \(0.274936\pi\)
\(360\) 0 0
\(361\) 13.7982i 0.726223i
\(362\) 0 0
\(363\) 4.95571 + 11.9641i 0.260107 + 0.627955i
\(364\) 0 0
\(365\) 19.6893 + 19.6893i 1.03058 + 1.03058i
\(366\) 0 0
\(367\) 21.6341 8.96114i 1.12929 0.467768i 0.261751 0.965135i \(-0.415700\pi\)
0.867540 + 0.497368i \(0.165700\pi\)
\(368\) 0 0
\(369\) −7.08028 2.93275i −0.368585 0.152673i
\(370\) 0 0
\(371\) 10.3766 25.0512i 0.538724 1.30060i
\(372\) 0 0
\(373\) 30.4720 1.57778 0.788890 0.614534i \(-0.210656\pi\)
0.788890 + 0.614534i \(0.210656\pi\)
\(374\) 0 0
\(375\) 7.14055 0.368736
\(376\) 0 0
\(377\) −1.39036 + 3.35663i −0.0716073 + 0.172875i
\(378\) 0 0
\(379\) 2.43151 + 1.00717i 0.124898 + 0.0517346i 0.444257 0.895899i \(-0.353468\pi\)
−0.319359 + 0.947634i \(0.603468\pi\)
\(380\) 0 0
\(381\) −13.4351 + 5.56500i −0.688301 + 0.285103i
\(382\) 0 0
\(383\) −13.8259 13.8259i −0.706473 0.706473i 0.259319 0.965792i \(-0.416502\pi\)
−0.965792 + 0.259319i \(0.916502\pi\)
\(384\) 0 0
\(385\) 12.2172 + 29.4950i 0.622648 + 1.50321i
\(386\) 0 0
\(387\) 6.18290i 0.314294i
\(388\) 0 0
\(389\) −10.3694 + 10.3694i −0.525748 + 0.525748i −0.919302 0.393553i \(-0.871246\pi\)
0.393553 + 0.919302i \(0.371246\pi\)
\(390\) 0 0
\(391\) 22.6033 + 12.8667i 1.14310 + 0.650696i
\(392\) 0 0
\(393\) 1.70737 1.70737i 0.0861256 0.0861256i
\(394\) 0 0
\(395\) 3.29689i 0.165884i
\(396\) 0 0
\(397\) 5.12132 + 12.3639i 0.257031 + 0.620529i 0.998739 0.0501941i \(-0.0159840\pi\)
−0.741708 + 0.670723i \(0.765984\pi\)
\(398\) 0 0
\(399\) −3.87541 3.87541i −0.194013 0.194013i
\(400\) 0 0
\(401\) −6.03313 + 2.49900i −0.301280 + 0.124794i −0.528202 0.849119i \(-0.677134\pi\)
0.226922 + 0.973913i \(0.427134\pi\)
\(402\) 0 0
\(403\) −6.91739 2.86528i −0.344580 0.142730i
\(404\) 0 0
\(405\) 1.03888 2.50807i 0.0516222 0.124627i
\(406\) 0 0
\(407\) −20.8509 −1.03354
\(408\) 0 0
\(409\) −6.45254 −0.319058 −0.159529 0.987193i \(-0.550997\pi\)
−0.159529 + 0.987193i \(0.550997\pi\)
\(410\) 0 0
\(411\) −0.544116 + 1.31361i −0.0268393 + 0.0647958i
\(412\) 0 0
\(413\) −27.9398 11.5730i −1.37483 0.569471i
\(414\) 0 0
\(415\) 10.3714 4.29596i 0.509111 0.210881i
\(416\) 0 0
\(417\) 10.9835 + 10.9835i 0.537866 + 0.537866i
\(418\) 0 0
\(419\) −8.77323 21.1805i −0.428600 1.03473i −0.979732 0.200314i \(-0.935804\pi\)
0.551131 0.834419i \(-0.314196\pi\)
\(420\) 0 0
\(421\) 26.8644i 1.30929i 0.755936 + 0.654645i \(0.227182\pi\)
−0.755936 + 0.654645i \(0.772818\pi\)
\(422\) 0 0
\(423\) −5.79734 + 5.79734i −0.281877 + 0.281877i
\(424\) 0 0
\(425\) 9.69449 + 1.21614i 0.470252 + 0.0589917i
\(426\) 0 0
\(427\) 4.64238 4.64238i 0.224660 0.224660i
\(428\) 0 0
\(429\) 5.23892i 0.252937i
\(430\) 0 0
\(431\) 14.8232 + 35.7865i 0.714010 + 1.72377i 0.689729 + 0.724068i \(0.257730\pi\)
0.0242809 + 0.999705i \(0.492270\pi\)
\(432\) 0 0
\(433\) −21.6205 21.6205i −1.03901 1.03901i −0.999207 0.0398071i \(-0.987326\pi\)
−0.0398071 0.999207i \(-0.512674\pi\)
\(434\) 0 0
\(435\) −8.51213 + 3.52584i −0.408125 + 0.169051i
\(436\) 0 0
\(437\) −13.2919 5.50569i −0.635839 0.263373i
\(438\) 0 0
\(439\) 0.602318 1.45412i 0.0287471 0.0694016i −0.908854 0.417114i \(-0.863042\pi\)
0.937601 + 0.347713i \(0.113042\pi\)
\(440\) 0 0
\(441\) 1.22551 0.0583576
\(442\) 0 0
\(443\) −14.7788 −0.702164 −0.351082 0.936345i \(-0.614186\pi\)
−0.351082 + 0.936345i \(0.614186\pi\)
\(444\) 0 0
\(445\) −3.08166 + 7.43979i −0.146085 + 0.352680i
\(446\) 0 0
\(447\) 15.0029 + 6.21440i 0.709612 + 0.293931i
\(448\) 0 0
\(449\) −9.04315 + 3.74579i −0.426773 + 0.176775i −0.585722 0.810512i \(-0.699189\pi\)
0.158950 + 0.987287i \(0.449189\pi\)
\(450\) 0 0
\(451\) 26.5199 + 26.5199i 1.24877 + 1.24877i
\(452\) 0 0
\(453\) 1.34508 + 3.24731i 0.0631974 + 0.152572i
\(454\) 0 0
\(455\) 6.98347i 0.327390i
\(456\) 0 0
\(457\) −17.3054 + 17.3054i −0.809513 + 0.809513i −0.984560 0.175047i \(-0.943992\pi\)
0.175047 + 0.984560i \(0.443992\pi\)
\(458\) 0 0
\(459\) −2.03972 + 3.58323i −0.0952058 + 0.167251i
\(460\) 0 0
\(461\) 9.62092 9.62092i 0.448091 0.448091i −0.446629 0.894719i \(-0.647375\pi\)
0.894719 + 0.446629i \(0.147375\pi\)
\(462\) 0 0
\(463\) 10.8498i 0.504233i −0.967697 0.252117i \(-0.918873\pi\)
0.967697 0.252117i \(-0.0811267\pi\)
\(464\) 0 0
\(465\) −7.26610 17.5419i −0.336957 0.813487i
\(466\) 0 0
\(467\) −10.9692 10.9692i −0.507594 0.507594i 0.406193 0.913787i \(-0.366856\pi\)
−0.913787 + 0.406193i \(0.866856\pi\)
\(468\) 0 0
\(469\) 28.4199 11.7719i 1.31231 0.543577i
\(470\) 0 0
\(471\) −19.7622 8.18577i −0.910594 0.377180i
\(472\) 0 0
\(473\) −11.5793 + 27.9550i −0.532418 + 1.28537i
\(474\) 0 0
\(475\) −5.40464 −0.247982
\(476\) 0 0
\(477\) 11.2838 0.516652
\(478\) 0 0
\(479\) −16.4855 + 39.7996i −0.753244 + 1.81849i −0.212836 + 0.977088i \(0.568270\pi\)
−0.540407 + 0.841403i \(0.681730\pi\)
\(480\) 0 0
\(481\) −4.21384 1.74543i −0.192134 0.0795847i
\(482\) 0 0
\(483\) −14.0046 + 5.80087i −0.637229 + 0.263949i
\(484\) 0 0
\(485\) −35.0430 35.0430i −1.59122 1.59122i
\(486\) 0 0
\(487\) −16.2097 39.1337i −0.734531 1.77331i −0.626865 0.779128i \(-0.715662\pi\)
−0.107666 0.994187i \(-0.534338\pi\)
\(488\) 0 0
\(489\) 19.8730i 0.898688i
\(490\) 0 0
\(491\) −3.65738 + 3.65738i −0.165055 + 0.165055i −0.784802 0.619747i \(-0.787235\pi\)
0.619747 + 0.784802i \(0.287235\pi\)
\(492\) 0 0
\(493\) 13.4942 3.70420i 0.607748 0.166829i
\(494\) 0 0
\(495\) −9.39423 + 9.39423i −0.422239 + 0.422239i
\(496\) 0 0
\(497\) 10.2119i 0.458068i
\(498\) 0 0
\(499\) 8.48908 + 20.4945i 0.380023 + 0.917458i 0.991960 + 0.126551i \(0.0403907\pi\)
−0.611937 + 0.790907i \(0.709609\pi\)
\(500\) 0 0
\(501\) 5.61290 + 5.61290i 0.250766 + 0.250766i
\(502\) 0 0
\(503\) 25.0563 10.3786i 1.11720 0.462761i 0.253791 0.967259i \(-0.418322\pi\)
0.863413 + 0.504498i \(0.168322\pi\)
\(504\) 0 0
\(505\) 22.6856 + 9.39667i 1.00950 + 0.418147i
\(506\) 0 0
\(507\) 4.53633 10.9517i 0.201466 0.486381i
\(508\) 0 0
\(509\) 25.9336 1.14949 0.574743 0.818334i \(-0.305102\pi\)
0.574743 + 0.818334i \(0.305102\pi\)
\(510\) 0 0
\(511\) 24.6478 1.09035
\(512\) 0 0
\(513\) 0.872800 2.10713i 0.0385351 0.0930319i
\(514\) 0 0
\(515\) −5.60724 2.32259i −0.247084 0.102346i
\(516\) 0 0
\(517\) 37.0690 15.3545i 1.63029 0.675290i
\(518\) 0 0
\(519\) −16.9626 16.9626i −0.744577 0.744577i
\(520\) 0 0
\(521\) −14.8942 35.9578i −0.652527 1.57534i −0.809099 0.587672i \(-0.800045\pi\)
0.156573 0.987666i \(-0.449955\pi\)
\(522\) 0 0
\(523\) 29.2087i 1.27721i 0.769536 + 0.638603i \(0.220488\pi\)
−0.769536 + 0.638603i \(0.779512\pi\)
\(524\) 0 0
\(525\) −4.02655 + 4.02655i −0.175733 + 0.175733i
\(526\) 0 0
\(527\) 7.63367 + 27.8091i 0.332528 + 1.21138i
\(528\) 0 0
\(529\) −11.8736 + 11.8736i −0.516244 + 0.516244i
\(530\) 0 0
\(531\) 12.5849i 0.546139i
\(532\) 0 0
\(533\) 3.13953 + 7.57950i 0.135988 + 0.328304i
\(534\) 0 0
\(535\) 10.4743 + 10.4743i 0.452845 + 0.452845i
\(536\) 0 0
\(537\) −22.3197 + 9.24513i −0.963166 + 0.398957i
\(538\) 0 0
\(539\) −5.54094 2.29513i −0.238665 0.0988584i
\(540\) 0 0
\(541\) 5.71438 13.7957i 0.245680 0.593125i −0.752148 0.658994i \(-0.770982\pi\)
0.997828 + 0.0658697i \(0.0209822\pi\)
\(542\) 0 0
\(543\) 12.1716 0.522332
\(544\) 0 0
\(545\) 7.17724 0.307439
\(546\) 0 0
\(547\) −8.00943 + 19.3365i −0.342458 + 0.826767i 0.655008 + 0.755622i \(0.272665\pi\)
−0.997466 + 0.0711452i \(0.977335\pi\)
\(548\) 0 0
\(549\) 2.52414 + 1.04553i 0.107728 + 0.0446223i
\(550\) 0 0
\(551\) −7.15136 + 2.96219i −0.304658 + 0.126194i
\(552\) 0 0
\(553\) 2.06358 + 2.06358i 0.0877523 + 0.0877523i
\(554\) 0 0
\(555\) −4.42626 10.6859i −0.187884 0.453592i
\(556\) 0 0
\(557\) 11.4512i 0.485203i −0.970126 0.242602i \(-0.921999\pi\)
0.970126 0.242602i \(-0.0780008\pi\)
\(558\) 0 0
\(559\) −4.68023 + 4.68023i −0.197952 + 0.197952i
\(560\) 0 0
\(561\) 15.9329 12.3810i 0.672689 0.522728i
\(562\) 0 0
\(563\) 16.1023 16.1023i 0.678629 0.678629i −0.281061 0.959690i \(-0.590686\pi\)
0.959690 + 0.281061i \(0.0906862\pi\)
\(564\) 0 0
\(565\) 37.9831i 1.59796i
\(566\) 0 0
\(567\) −0.919595 2.22010i −0.0386194 0.0932354i
\(568\) 0 0
\(569\) −22.4856 22.4856i −0.942644 0.942644i 0.0557985 0.998442i \(-0.482230\pi\)
−0.998442 + 0.0557985i \(0.982230\pi\)
\(570\) 0 0
\(571\) 36.9087 15.2881i 1.54458 0.639787i 0.562256 0.826963i \(-0.309933\pi\)
0.982326 + 0.187176i \(0.0599335\pi\)
\(572\) 0 0
\(573\) 3.02852 + 1.25445i 0.126518 + 0.0524055i
\(574\) 0 0
\(575\) −5.72042 + 13.8103i −0.238558 + 0.575930i
\(576\) 0 0
\(577\) 28.3973 1.18219 0.591097 0.806600i \(-0.298695\pi\)
0.591097 + 0.806600i \(0.298695\pi\)
\(578\) 0 0
\(579\) −20.7506 −0.862366
\(580\) 0 0
\(581\) 3.80271 9.18055i 0.157763 0.380873i
\(582\) 0 0
\(583\) −51.0181 21.1324i −2.11295 0.875214i
\(584\) 0 0
\(585\) −2.68491 + 1.11213i −0.111007 + 0.0459808i
\(586\) 0 0
\(587\) 7.76184 + 7.76184i 0.320365 + 0.320365i 0.848907 0.528542i \(-0.177261\pi\)
−0.528542 + 0.848907i \(0.677261\pi\)
\(588\) 0 0
\(589\) −6.10453 14.7376i −0.251533 0.607254i
\(590\) 0 0
\(591\) 19.7142i 0.810933i
\(592\) 0 0
\(593\) −2.22864 + 2.22864i −0.0915194 + 0.0915194i −0.751384 0.659865i \(-0.770613\pi\)
0.659865 + 0.751384i \(0.270613\pi\)
\(594\) 0 0
\(595\) 21.2386 16.5039i 0.870696 0.676594i
\(596\) 0 0
\(597\) 10.7866 10.7866i 0.441467 0.441467i
\(598\) 0 0
\(599\) 15.4176i 0.629945i −0.949101 0.314972i \(-0.898005\pi\)
0.949101 0.314972i \(-0.101995\pi\)
\(600\) 0 0
\(601\) 8.10132 + 19.5583i 0.330460 + 0.797801i 0.998556 + 0.0537256i \(0.0171096\pi\)
−0.668096 + 0.744075i \(0.732890\pi\)
\(602\) 0 0
\(603\) 9.05182 + 9.05182i 0.368618 + 0.368618i
\(604\) 0 0
\(605\) 32.4793 13.4534i 1.32047 0.546957i
\(606\) 0 0
\(607\) 14.0086 + 5.80253i 0.568590 + 0.235518i 0.648409 0.761292i \(-0.275435\pi\)
−0.0798199 + 0.996809i \(0.525435\pi\)
\(608\) 0 0
\(609\) −3.12101 + 7.53478i −0.126470 + 0.305325i
\(610\) 0 0
\(611\) 8.77676 0.355070
\(612\) 0 0
\(613\) 43.4543 1.75510 0.877551 0.479484i \(-0.159176\pi\)
0.877551 + 0.479484i \(0.159176\pi\)
\(614\) 0 0
\(615\) −7.96158 + 19.2210i −0.321042 + 0.775064i
\(616\) 0 0
\(617\) −36.3420 15.0534i −1.46307 0.606025i −0.497807 0.867288i \(-0.665861\pi\)
−0.965268 + 0.261263i \(0.915861\pi\)
\(618\) 0 0
\(619\) −9.69556 + 4.01603i −0.389697 + 0.161418i −0.568924 0.822390i \(-0.692640\pi\)
0.179227 + 0.983808i \(0.442640\pi\)
\(620\) 0 0
\(621\) −4.46048 4.46048i −0.178993 0.178993i
\(622\) 0 0
\(623\) 2.72783 + 6.58557i 0.109288 + 0.263845i
\(624\) 0 0
\(625\) 31.2330i 1.24932i
\(626\) 0 0
\(627\) −7.89246 + 7.89246i −0.315194 + 0.315194i
\(628\) 0 0
\(629\) 4.65017 + 16.9403i 0.185414 + 0.675455i
\(630\) 0 0
\(631\) 32.2236 32.2236i 1.28280 1.28280i 0.343734 0.939067i \(-0.388308\pi\)
0.939067 0.343734i \(-0.111692\pi\)
\(632\) 0 0
\(633\) 12.5159i 0.497460i
\(634\) 0 0
\(635\) 15.1074 + 36.4725i 0.599519 + 1.44737i
\(636\) 0 0
\(637\) −0.927666 0.927666i −0.0367555 0.0367555i
\(638\) 0 0
\(639\) −3.92615 + 1.62627i −0.155316 + 0.0643341i
\(640\) 0 0
\(641\) 13.8538 + 5.73842i 0.547191 + 0.226654i 0.639114 0.769112i \(-0.279301\pi\)
−0.0919230 + 0.995766i \(0.529301\pi\)
\(642\) 0 0
\(643\) −7.89107 + 19.0507i −0.311193 + 0.751287i 0.688468 + 0.725267i \(0.258284\pi\)
−0.999661 + 0.0260206i \(0.991716\pi\)
\(644\) 0 0
\(645\) −16.7848 −0.660901
\(646\) 0 0
\(647\) −26.8160 −1.05424 −0.527122 0.849790i \(-0.676729\pi\)
−0.527122 + 0.849790i \(0.676729\pi\)
\(648\) 0 0
\(649\) −23.5690 + 56.9007i −0.925166 + 2.23355i
\(650\) 0 0
\(651\) −15.5278 6.43182i −0.608582 0.252083i
\(652\) 0 0
\(653\) −12.6601 + 5.24399i −0.495428 + 0.205213i −0.616385 0.787445i \(-0.711404\pi\)
0.120957 + 0.992658i \(0.461404\pi\)
\(654\) 0 0
\(655\) −4.63504 4.63504i −0.181106 0.181106i
\(656\) 0 0
\(657\) 3.92518 + 9.47623i 0.153136 + 0.369703i
\(658\) 0 0
\(659\) 20.2130i 0.787388i 0.919241 + 0.393694i \(0.128803\pi\)
−0.919241 + 0.393694i \(0.871197\pi\)
\(660\) 0 0
\(661\) 5.04254 5.04254i 0.196132 0.196132i −0.602208 0.798340i \(-0.705712\pi\)
0.798340 + 0.602208i \(0.205712\pi\)
\(662\) 0 0
\(663\) 4.25637 1.16839i 0.165304 0.0453763i
\(664\) 0 0
\(665\) −10.5206 + 10.5206i −0.407972 + 0.407972i
\(666\) 0 0
\(667\) 21.4089i 0.828957i
\(668\) 0 0
\(669\) −2.88072 6.95468i −0.111375 0.268883i
\(670\) 0 0
\(671\) −9.45443 9.45443i −0.364984 0.364984i
\(672\) 0 0
\(673\) −34.6300 + 14.3442i −1.33489 + 0.552929i −0.932046 0.362339i \(-0.881978\pi\)
−0.402844 + 0.915269i \(0.631978\pi\)
\(674\) 0 0
\(675\) −2.18931 0.906841i −0.0842665 0.0349043i
\(676\) 0 0
\(677\) 14.0912 34.0193i 0.541571 1.30747i −0.382044 0.924144i \(-0.624780\pi\)
0.923614 0.383323i \(-0.125220\pi\)
\(678\) 0 0
\(679\) −43.8681 −1.68350
\(680\) 0 0
\(681\) −4.80718 −0.184212
\(682\) 0 0
\(683\) −6.42199 + 15.5041i −0.245731 + 0.593246i −0.997833 0.0658003i \(-0.979040\pi\)
0.752102 + 0.659047i \(0.229040\pi\)
\(684\) 0 0
\(685\) 3.56609 + 1.47712i 0.136253 + 0.0564379i
\(686\) 0 0
\(687\) −12.3827 + 5.12909i −0.472430 + 0.195687i
\(688\) 0 0
\(689\) −8.54146 8.54146i −0.325404 0.325404i
\(690\) 0 0
\(691\) −5.24785 12.6694i −0.199638 0.481968i 0.792078 0.610420i \(-0.208999\pi\)
−0.991716 + 0.128452i \(0.958999\pi\)
\(692\) 0 0
\(693\) 11.7600i 0.446727i
\(694\) 0 0
\(695\) 29.8172 29.8172i 1.13103 1.13103i
\(696\) 0 0
\(697\) 15.6316 27.4606i 0.592091 1.04014i
\(698\) 0 0
\(699\) 9.74356 9.74356i 0.368535 0.368535i
\(700\) 0 0
\(701\) 11.5629i 0.436724i −0.975868 0.218362i \(-0.929929\pi\)
0.975868 0.218362i \(-0.0700713\pi\)
\(702\) 0 0
\(703\) −3.71867 8.97766i −0.140252 0.338599i
\(704\) 0 0
\(705\) 15.7381 + 15.7381i 0.592733 + 0.592733i
\(706\) 0 0
\(707\) 20.0809 8.31776i 0.755218 0.312822i
\(708\) 0 0
\(709\) −13.9142 5.76346i −0.522560 0.216451i 0.105781 0.994389i \(-0.466266\pi\)
−0.628341 + 0.777938i \(0.716266\pi\)
\(710\) 0 0
\(711\) −0.464750 + 1.12200i −0.0174295 + 0.0420785i
\(712\) 0 0
\(713\) −44.1199 −1.65230
\(714\) 0 0
\(715\) 14.2222 0.531880
\(716\) 0 0
\(717\) 6.50793 15.7115i 0.243043 0.586758i
\(718\) 0 0
\(719\) −1.76251 0.730055i −0.0657305 0.0272264i 0.349576 0.936908i \(-0.386326\pi\)
−0.415306 + 0.909682i \(0.636326\pi\)
\(720\) 0 0
\(721\) −4.96342 + 2.05592i −0.184847 + 0.0765663i
\(722\) 0 0
\(723\) 13.7118 + 13.7118i 0.509947 + 0.509947i
\(724\) 0 0
\(725\) 3.07772 + 7.43027i 0.114304 + 0.275953i
\(726\) 0 0
\(727\) 4.61649i 0.171216i −0.996329 0.0856082i \(-0.972717\pi\)
0.996329 0.0856082i \(-0.0272833\pi\)
\(728\) 0 0
\(729\) 0.707107 0.707107i 0.0261891 0.0261891i
\(730\) 0 0
\(731\) 25.2945 + 3.17311i 0.935550 + 0.117362i
\(732\) 0 0
\(733\) 4.76113 4.76113i 0.175856 0.175856i −0.613691 0.789547i \(-0.710316\pi\)
0.789547 + 0.613691i \(0.210316\pi\)
\(734\) 0 0
\(735\) 3.32691i 0.122715i
\(736\) 0 0
\(737\) −23.9741 57.8786i −0.883097 2.13199i
\(738\) 0 0
\(739\) −6.49918 6.49918i −0.239076 0.239076i 0.577391 0.816468i \(-0.304071\pi\)
−0.816468 + 0.577391i \(0.804071\pi\)
\(740\) 0 0
\(741\) −2.25570 + 0.934340i −0.0828651 + 0.0343238i
\(742\) 0 0
\(743\) −7.90655 3.27500i −0.290063 0.120148i 0.232907 0.972499i \(-0.425176\pi\)
−0.522970 + 0.852351i \(0.675176\pi\)
\(744\) 0 0
\(745\) 16.8703 40.7286i 0.618081 1.49218i
\(746\) 0 0
\(747\) 4.13520 0.151299
\(748\) 0 0
\(749\) 13.1122 0.479108
\(750\) 0 0
\(751\) −3.59617 + 8.68192i −0.131226 + 0.316808i −0.975812 0.218613i \(-0.929847\pi\)
0.844586 + 0.535420i \(0.179847\pi\)
\(752\) 0 0
\(753\) −19.8926 8.23980i −0.724928 0.300275i
\(754\) 0 0
\(755\) 8.81554 3.65151i 0.320830 0.132892i
\(756\) 0 0
\(757\) −30.9701 30.9701i −1.12563 1.12563i −0.990880 0.134750i \(-0.956977\pi\)
−0.134750 0.990880i \(-0.543023\pi\)
\(758\) 0 0
\(759\) 11.8138 + 28.5210i 0.428813 + 1.03525i
\(760\) 0 0
\(761\) 7.85599i 0.284779i 0.989811 + 0.142390i \(0.0454786\pi\)
−0.989811 + 0.142390i \(0.954521\pi\)
\(762\) 0 0
\(763\) 4.49236 4.49236i 0.162634 0.162634i
\(764\) 0 0
\(765\) 9.72746 + 5.53725i 0.351697 + 0.200200i
\(766\) 0 0
\(767\) −9.52632 + 9.52632i −0.343976 + 0.343976i
\(768\) 0 0
\(769\) 33.7233i 1.21609i 0.793901 + 0.608047i \(0.208047\pi\)
−0.793901 + 0.608047i \(0.791953\pi\)
\(770\) 0 0
\(771\) −7.31670 17.6641i −0.263505 0.636156i
\(772\) 0 0
\(773\) 19.9064 + 19.9064i 0.715983 + 0.715983i 0.967780 0.251797i \(-0.0810215\pi\)
−0.251797 + 0.967780i \(0.581022\pi\)
\(774\) 0 0
\(775\) −15.3124 + 6.34261i −0.550038 + 0.227833i
\(776\) 0 0
\(777\) −9.45899 3.91804i −0.339339 0.140559i
\(778\) 0 0
\(779\) −6.68883 + 16.1483i −0.239652 + 0.578571i
\(780\) 0 0
\(781\) 20.7971 0.744180
\(782\) 0 0
\(783\) −3.39389 −0.121288
\(784\) 0 0
\(785\) −22.2220 + 53.6488i −0.793139 + 1.91481i
\(786\) 0 0
\(787\) −31.9518 13.2349i −1.13896 0.471772i −0.268141 0.963380i \(-0.586409\pi\)
−0.870817 + 0.491608i \(0.836409\pi\)
\(788\) 0 0
\(789\) −10.2428 + 4.24272i −0.364654 + 0.151045i
\(790\) 0 0
\(791\) −23.7743 23.7743i −0.845317 0.845317i
\(792\) 0 0
\(793\) −1.11925 2.70211i −0.0397458 0.0959549i
\(794\) 0 0
\(795\) 30.6324i 1.08642i
\(796\) 0 0
\(797\) 31.8112 31.8112i 1.12681 1.12681i 0.136119 0.990692i \(-0.456537\pi\)
0.990692 0.136119i \(-0.0434630\pi\)
\(798\) 0 0
\(799\) −20.7419 26.6924i −0.733797 0.944310i
\(800\) 0 0
\(801\) −2.09752 + 2.09752i −0.0741122 + 0.0741122i
\(802\) 0 0
\(803\) 50.1964i 1.77139i
\(804\) 0 0
\(805\) 15.7477 + 38.0184i 0.555035 + 1.33997i
\(806\) 0 0
\(807\) −3.12922 3.12922i −0.110154 0.110154i
\(808\) 0 0
\(809\) 28.7452 11.9066i 1.01063 0.418615i 0.184943 0.982749i \(-0.440790\pi\)
0.825683 + 0.564134i \(0.190790\pi\)
\(810\) 0 0
\(811\) −44.6958 18.5136i −1.56948 0.650101i −0.582779 0.812631i \(-0.698035\pi\)
−0.986704 + 0.162529i \(0.948035\pi\)
\(812\) 0 0
\(813\) −6.21770 + 15.0108i −0.218064 + 0.526453i
\(814\) 0 0
\(815\) 53.9495 1.88977
\(816\) 0 0
\(817\) −14.1016 −0.493351
\(818\) 0 0
\(819\) −0.984434 + 2.37663i −0.0343989 + 0.0830463i
\(820\) 0 0
\(821\) −21.2625 8.80722i −0.742067 0.307374i −0.0205669 0.999788i \(-0.506547\pi\)
−0.721500 + 0.692414i \(0.756547\pi\)
\(822\) 0 0
\(823\) −10.0413 + 4.15925i −0.350018 + 0.144982i −0.550764 0.834661i \(-0.685664\pi\)
0.200746 + 0.979643i \(0.435664\pi\)
\(824\) 0 0
\(825\) 8.20027 + 8.20027i 0.285497 + 0.285497i
\(826\) 0 0
\(827\) −12.2934 29.6790i −0.427485 1.03204i −0.980082 0.198591i \(-0.936363\pi\)
0.552598 0.833448i \(-0.313637\pi\)
\(828\) 0 0
\(829\) 21.0657i 0.731642i −0.930685 0.365821i \(-0.880788\pi\)
0.930685 0.365821i \(-0.119212\pi\)
\(830\) 0 0
\(831\) −0.453033 + 0.453033i −0.0157155 + 0.0157155i
\(832\) 0 0
\(833\) −0.628942 + 5.01361i −0.0217915 + 0.173711i
\(834\) 0 0
\(835\) 15.2374 15.2374i 0.527313 0.527313i
\(836\) 0 0
\(837\) 6.99418i 0.241754i
\(838\) 0 0
\(839\) 16.3781 + 39.5402i 0.565435 + 1.36508i 0.905367 + 0.424631i \(0.139596\pi\)
−0.339932 + 0.940450i \(0.610404\pi\)
\(840\) 0 0
\(841\) −12.3613 12.3613i −0.426251 0.426251i
\(842\) 0 0
\(843\) 0.0680922 0.0282047i 0.00234522 0.000971422i
\(844\) 0 0
\(845\) −29.7307 12.3149i −1.02277 0.423644i
\(846\) 0 0
\(847\) 11.9087 28.7500i 0.409186 0.987863i
\(848\) 0 0
\(849\) 20.4573 0.702094
\(850\) 0 0
\(851\) −26.8763 −0.921307
\(852\) 0 0
\(853\) 5.75449 13.8926i 0.197030 0.475673i −0.794226 0.607622i \(-0.792124\pi\)
0.991256 + 0.131949i \(0.0421236\pi\)
\(854\) 0 0
\(855\) −5.72025 2.36941i −0.195629 0.0810320i
\(856\) 0 0
\(857\) 11.2462 4.65833i 0.384163 0.159126i −0.182240 0.983254i \(-0.558335\pi\)
0.566403 + 0.824129i \(0.308335\pi\)
\(858\) 0 0
\(859\) −9.12775 9.12775i −0.311435 0.311435i 0.534030 0.845465i \(-0.320677\pi\)
−0.845465 + 0.534030i \(0.820677\pi\)
\(860\) 0 0
\(861\) 7.04744 + 17.0140i 0.240176 + 0.579837i
\(862\) 0 0
\(863\) 21.4811i 0.731226i −0.930767 0.365613i \(-0.880859\pi\)
0.930767 0.365613i \(-0.119141\pi\)
\(864\) 0 0
\(865\) −46.0488 + 46.0488i −1.56571 + 1.56571i
\(866\) 0 0
\(867\) −13.6124 10.1835i −0.462300 0.345850i
\(868\) 0 0
\(869\) 4.20258 4.20258i 0.142563 0.142563i
\(870\) 0 0
\(871\) 13.7038i 0.464335i
\(872\) 0 0
\(873\) −6.98605 16.8658i −0.236442 0.570822i
\(874\) 0 0
\(875\) −12.1332 12.1332i −0.410175 0.410175i
\(876\) 0 0
\(877\) −47.3960 + 19.6321i −1.60045 + 0.662928i −0.991479 0.130264i \(-0.958417\pi\)
−0.608971 + 0.793192i \(0.708417\pi\)
\(878\) 0 0
\(879\) 7.28753 + 3.01859i 0.245802 + 0.101815i
\(880\) 0 0
\(881\) −15.5491 + 37.5388i −0.523861 + 1.26471i 0.411626 + 0.911353i \(0.364961\pi\)
−0.935487 + 0.353361i \(0.885039\pi\)
\(882\) 0 0
\(883\) −3.59541 −0.120995 −0.0604975 0.998168i \(-0.519269\pi\)
−0.0604975 + 0.998168i \(0.519269\pi\)
\(884\) 0 0
\(885\) −34.1645 −1.14843
\(886\) 0 0
\(887\) 15.3637 37.0913i 0.515864 1.24540i −0.424560 0.905400i \(-0.639571\pi\)
0.940424 0.340005i \(-0.110429\pi\)
\(888\) 0 0
\(889\) 32.2848 + 13.3728i 1.08280 + 0.448509i
\(890\) 0 0
\(891\) −4.52134 + 1.87280i −0.151471 + 0.0627412i
\(892\) 0 0
\(893\) 13.2222 + 13.2222i 0.442465 + 0.442465i
\(894\) 0 0
\(895\) 25.0979 + 60.5917i 0.838930 + 2.02536i
\(896\) 0 0
\(897\) 6.75285i 0.225471i
\(898\) 0 0
\(899\) −16.7850 + 16.7850i −0.559810 + 0.559810i
\(900\) 0 0
\(901\) −5.79096 + 46.1627i −0.192925 + 1.53790i
\(902\) 0 0
\(903\) −10.5059 + 10.5059i −0.349615 + 0.349615i
\(904\) 0 0
\(905\) 33.0423i 1.09836i
\(906\) 0 0
\(907\) −16.9424 40.9026i −0.562563 1.35815i −0.907710 0.419599i \(-0.862171\pi\)
0.345147 0.938549i \(-0.387829\pi\)
\(908\) 0 0
\(909\) 6.39580 + 6.39580i 0.212135 + 0.212135i
\(910\) 0 0
\(911\) −8.08964 + 3.35084i −0.268022 + 0.111018i −0.512647 0.858599i \(-0.671335\pi\)
0.244625 + 0.969618i \(0.421335\pi\)
\(912\) 0 0
\(913\) −18.6966 7.74440i −0.618768 0.256302i
\(914\) 0 0
\(915\) 2.83833 6.85233i 0.0938323 0.226531i
\(916\) 0 0
\(917\) −5.80230 −0.191609
\(918\) 0 0
\(919\) 32.6562 1.07723 0.538615 0.842552i \(-0.318948\pi\)
0.538615 + 0.842552i \(0.318948\pi\)
\(920\) 0 0
\(921\) 1.77548 4.28638i 0.0585039 0.141241i
\(922\) 0 0
\(923\) 4.20298 + 1.74093i 0.138343 + 0.0573034i
\(924\) 0 0
\(925\) −9.32780 + 3.86370i −0.306696 + 0.127038i
\(926\) 0 0
\(927\) −1.58086 1.58086i −0.0519223 0.0519223i
\(928\) 0 0
\(929\) −18.8006 45.3888i −0.616829 1.48916i −0.855366 0.518024i \(-0.826668\pi\)
0.238537 0.971133i \(-0.423332\pi\)
\(930\) 0 0
\(931\) 2.79506i 0.0916045i
\(932\) 0 0
\(933\) 3.95612 3.95612i 0.129518 0.129518i
\(934\) 0 0
\(935\) −33.6110 43.2534i −1.09920 1.41454i
\(936\) 0 0
\(937\) 28.6009 28.6009i 0.934352 0.934352i −0.0636224 0.997974i \(-0.520265\pi\)
0.997974 + 0.0636224i \(0.0202653\pi\)
\(938\) 0 0
\(939\) 0.877026i 0.0286207i
\(940\) 0 0
\(941\) 1.10983 + 2.67938i 0.0361796 + 0.0873452i 0.940937 0.338583i \(-0.109948\pi\)
−0.904757 + 0.425928i \(0.859948\pi\)
\(942\) 0 0
\(943\) 34.1835 + 34.1835i 1.11317 + 1.11317i
\(944\) 0 0
\(945\) −6.02694 + 2.49644i −0.196056 + 0.0812092i
\(946\) 0 0
\(947\) 41.7742 + 17.3034i 1.35748 + 0.562286i 0.938364 0.345648i \(-0.112341\pi\)
0.419114 + 0.907934i \(0.362341\pi\)
\(948\) 0 0
\(949\) 4.20194 10.1444i 0.136401 0.329301i
\(950\) 0 0
\(951\) 29.7514 0.964756
\(952\) 0 0
\(953\) 17.8145 0.577069 0.288534 0.957470i \(-0.406832\pi\)
0.288534 + 0.957470i \(0.406832\pi\)
\(954\) 0 0
\(955\) 3.40549 8.22157i 0.110199 0.266044i
\(956\) 0 0
\(957\) 15.3449 + 6.35608i 0.496032 + 0.205463i
\(958\) 0 0
\(959\) 3.15663 1.30752i 0.101933 0.0422221i
\(960\) 0 0
\(961\) −12.6704 12.6704i −0.408722 0.408722i
\(962\) 0 0
\(963\) 2.08813 + 5.04118i 0.0672889 + 0.162450i
\(964\) 0 0
\(965\) 56.3320i 1.81339i
\(966\) 0 0
\(967\) −19.1664 + 19.1664i −0.616349 + 0.616349i −0.944593 0.328244i \(-0.893543\pi\)
0.328244 + 0.944593i \(0.393543\pi\)
\(968\) 0 0
\(969\) 8.17241 + 4.65206i 0.262536 + 0.149446i
\(970\) 0 0
\(971\) −1.02700 + 1.02700i −0.0329582 + 0.0329582i −0.723394 0.690436i \(-0.757419\pi\)
0.690436 + 0.723394i \(0.257419\pi\)
\(972\) 0 0
\(973\) 37.3262i 1.19662i
\(974\) 0 0
\(975\) 0.970780 + 2.34367i 0.0310898 + 0.0750575i
\(976\) 0 0
\(977\) 34.2951 + 34.2951i 1.09720 + 1.09720i 0.994737 + 0.102462i \(0.0326720\pi\)
0.102462 + 0.994737i \(0.467328\pi\)
\(978\) 0 0
\(979\) 13.4118 5.55536i 0.428644 0.177550i
\(980\) 0 0
\(981\) 2.44258 + 1.01175i 0.0779854 + 0.0323026i
\(982\) 0 0
\(983\) −6.76821 + 16.3399i −0.215872 + 0.521162i −0.994306 0.106565i \(-0.966015\pi\)
0.778433 + 0.627727i \(0.216015\pi\)
\(984\) 0 0
\(985\) −53.5184 −1.70524
\(986\) 0 0
\(987\) 19.7016 0.627108
\(988\) 0 0
\(989\) −14.9255 + 36.0333i −0.474603 + 1.14579i
\(990\) 0 0
\(991\) 26.3051 + 10.8959i 0.835610 + 0.346121i 0.759121 0.650950i \(-0.225629\pi\)
0.0764888 + 0.997070i \(0.475629\pi\)
\(992\) 0 0
\(993\) 10.6934 4.42935i 0.339345 0.140561i
\(994\) 0 0
\(995\) −29.2826 29.2826i −0.928322 0.928322i
\(996\) 0 0
\(997\) −4.92753 11.8961i −0.156056 0.376753i 0.826443 0.563021i \(-0.190361\pi\)
−0.982499 + 0.186267i \(0.940361\pi\)
\(998\) 0 0
\(999\) 4.26062i 0.134800i
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 816.2.bq.f.433.3 16
4.3 odd 2 408.2.ba.a.25.1 16
12.11 even 2 1224.2.bq.e.433.4 16
17.15 even 8 inner 816.2.bq.f.49.3 16
68.7 even 16 6936.2.a.bo.1.7 8
68.15 odd 8 408.2.ba.a.49.1 yes 16
68.27 even 16 6936.2.a.bl.1.2 8
204.83 even 8 1224.2.bq.e.865.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
408.2.ba.a.25.1 16 4.3 odd 2
408.2.ba.a.49.1 yes 16 68.15 odd 8
816.2.bq.f.49.3 16 17.15 even 8 inner
816.2.bq.f.433.3 16 1.1 even 1 trivial
1224.2.bq.e.433.4 16 12.11 even 2
1224.2.bq.e.865.4 16 204.83 even 8
6936.2.a.bl.1.2 8 68.27 even 16
6936.2.a.bo.1.7 8 68.7 even 16