Properties

Label 1520.2.q.m.961.3
Level $1520$
Weight $2$
Character 1520.961
Analytic conductor $12.137$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1520,2,Mod(881,1520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1520, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1520.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.q (of order \(3\), 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: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 9x^{6} + 2x^{5} + 65x^{4} - 20x^{3} + 25x^{2} + 6x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 961.3
Root \(0.354609 - 0.614201i\) of defining polynomial
Character \(\chi\) \(=\) 1520.961
Dual form 1520.2.q.m.881.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.354609 - 0.614201i) q^{3} +(-0.500000 + 0.866025i) q^{5} -3.11079 q^{7} +(1.24850 + 2.16247i) q^{9} +O(q^{10})\) \(q+(0.354609 - 0.614201i) q^{3} +(-0.500000 + 0.866025i) q^{5} -3.11079 q^{7} +(1.24850 + 2.16247i) q^{9} +3.52922 q^{11} +(-0.200785 - 0.347770i) q^{13} +(0.354609 + 0.614201i) q^{15} +(-1.74850 + 3.02850i) q^{17} +(-4.35162 - 0.251824i) q^{19} +(-1.10311 + 1.91065i) q^{21} +(-3.65851 - 6.33672i) q^{23} +(-0.500000 - 0.866025i) q^{25} +3.89858 q^{27} +(3.96540 + 6.86827i) q^{29} -5.73545 q^{31} +(1.25150 - 2.16765i) q^{33} +(1.55539 - 2.69402i) q^{35} -10.5292 q^{37} -0.284801 q^{39} +(0.555394 - 0.961971i) q^{41} +(-4.30390 + 7.45457i) q^{43} -2.49701 q^{45} +(3.76162 + 6.51532i) q^{47} +2.67700 q^{49} +(1.24007 + 2.14787i) q^{51} +(-5.27773 - 9.14130i) q^{53} +(-1.76461 + 3.05640i) q^{55} +(-1.69779 + 2.58347i) q^{57} +(-4.25618 + 7.37192i) q^{59} +(4.61079 + 7.98612i) q^{61} +(-3.88383 - 6.72700i) q^{63} +0.401570 q^{65} +(4.20623 + 7.28540i) q^{67} -5.18936 q^{69} +(-4.31233 + 7.46918i) q^{71} +(-0.870717 + 1.50813i) q^{73} -0.709218 q^{75} -10.9787 q^{77} +(-4.50544 + 7.80366i) q^{79} +(-2.36304 + 4.09291i) q^{81} -4.07857 q^{83} +(-1.74850 - 3.02850i) q^{85} +5.62466 q^{87} +(6.61623 + 11.4596i) q^{89} +(0.624599 + 1.08184i) q^{91} +(-2.03384 + 3.52272i) q^{93} +(2.39390 - 3.64270i) q^{95} +(3.85162 - 6.67120i) q^{97} +(4.40625 + 7.63186i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{3} - 4 q^{5} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{3} - 4 q^{5} - 5 q^{9} - 4 q^{11} + 9 q^{13} + q^{15} + q^{17} - 3 q^{19} + 8 q^{21} - 4 q^{25} - 20 q^{27} + 5 q^{29} + 20 q^{31} + 25 q^{33} - 52 q^{37} + 54 q^{39} - 8 q^{41} - 7 q^{43} + 10 q^{45} - 16 q^{47} + 20 q^{49} - 12 q^{51} + 5 q^{53} + 2 q^{55} + 27 q^{57} - 11 q^{59} + 12 q^{61} + 3 q^{63} - 18 q^{65} + 6 q^{69} - 14 q^{71} - 4 q^{73} - 2 q^{75} - 44 q^{77} - 13 q^{79} - 24 q^{81} - 10 q^{83} + q^{85} + 4 q^{87} + 5 q^{89} + 46 q^{91} - 28 q^{93} + 6 q^{95} - q^{97} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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
\(3\) 0.354609 0.614201i 0.204734 0.354609i −0.745314 0.666713i \(-0.767701\pi\)
0.950048 + 0.312104i \(0.101034\pi\)
\(4\) 0 0
\(5\) −0.500000 + 0.866025i −0.223607 + 0.387298i
\(6\) 0 0
\(7\) −3.11079 −1.17577 −0.587884 0.808945i \(-0.700039\pi\)
−0.587884 + 0.808945i \(0.700039\pi\)
\(8\) 0 0
\(9\) 1.24850 + 2.16247i 0.416168 + 0.720825i
\(10\) 0 0
\(11\) 3.52922 1.06410 0.532051 0.846713i \(-0.321422\pi\)
0.532051 + 0.846713i \(0.321422\pi\)
\(12\) 0 0
\(13\) −0.200785 0.347770i −0.0556877 0.0964540i 0.836838 0.547451i \(-0.184402\pi\)
−0.892525 + 0.450997i \(0.851068\pi\)
\(14\) 0 0
\(15\) 0.354609 + 0.614201i 0.0915597 + 0.158586i
\(16\) 0 0
\(17\) −1.74850 + 3.02850i −0.424075 + 0.734519i −0.996334 0.0855542i \(-0.972734\pi\)
0.572259 + 0.820073i \(0.306067\pi\)
\(18\) 0 0
\(19\) −4.35162 0.251824i −0.998330 0.0577725i
\(20\) 0 0
\(21\) −1.10311 + 1.91065i −0.240719 + 0.416938i
\(22\) 0 0
\(23\) −3.65851 6.33672i −0.762852 1.32130i −0.941375 0.337362i \(-0.890465\pi\)
0.178523 0.983936i \(-0.442868\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.100000 0.173205i
\(26\) 0 0
\(27\) 3.89858 0.750282
\(28\) 0 0
\(29\) 3.96540 + 6.86827i 0.736356 + 1.27541i 0.954126 + 0.299406i \(0.0967884\pi\)
−0.217770 + 0.976000i \(0.569878\pi\)
\(30\) 0 0
\(31\) −5.73545 −1.03012 −0.515059 0.857155i \(-0.672230\pi\)
−0.515059 + 0.857155i \(0.672230\pi\)
\(32\) 0 0
\(33\) 1.25150 2.16765i 0.217857 0.377340i
\(34\) 0 0
\(35\) 1.55539 2.69402i 0.262910 0.455373i
\(36\) 0 0
\(37\) −10.5292 −1.73099 −0.865497 0.500914i \(-0.832997\pi\)
−0.865497 + 0.500914i \(0.832997\pi\)
\(38\) 0 0
\(39\) −0.284801 −0.0456046
\(40\) 0 0
\(41\) 0.555394 0.961971i 0.0867380 0.150235i −0.819392 0.573233i \(-0.805689\pi\)
0.906130 + 0.422998i \(0.139022\pi\)
\(42\) 0 0
\(43\) −4.30390 + 7.45457i −0.656338 + 1.13681i 0.325218 + 0.945639i \(0.394562\pi\)
−0.981556 + 0.191172i \(0.938771\pi\)
\(44\) 0 0
\(45\) −2.49701 −0.372232
\(46\) 0 0
\(47\) 3.76162 + 6.51532i 0.548689 + 0.950357i 0.998365 + 0.0571653i \(0.0182062\pi\)
−0.449676 + 0.893192i \(0.648460\pi\)
\(48\) 0 0
\(49\) 2.67700 0.382429
\(50\) 0 0
\(51\) 1.24007 + 2.14787i 0.173645 + 0.300762i
\(52\) 0 0
\(53\) −5.27773 9.14130i −0.724952 1.25565i −0.958994 0.283427i \(-0.908529\pi\)
0.234042 0.972227i \(-0.424805\pi\)
\(54\) 0 0
\(55\) −1.76461 + 3.05640i −0.237940 + 0.412125i
\(56\) 0 0
\(57\) −1.69779 + 2.58347i −0.224878 + 0.342189i
\(58\) 0 0
\(59\) −4.25618 + 7.37192i −0.554107 + 0.959742i 0.443865 + 0.896094i \(0.353607\pi\)
−0.997972 + 0.0636484i \(0.979726\pi\)
\(60\) 0 0
\(61\) 4.61079 + 7.98612i 0.590351 + 1.02252i 0.994185 + 0.107686i \(0.0343440\pi\)
−0.403834 + 0.914832i \(0.632323\pi\)
\(62\) 0 0
\(63\) −3.88383 6.72700i −0.489317 0.847522i
\(64\) 0 0
\(65\) 0.401570 0.0498086
\(66\) 0 0
\(67\) 4.20623 + 7.28540i 0.513873 + 0.890053i 0.999870 + 0.0160934i \(0.00512290\pi\)
−0.485998 + 0.873960i \(0.661544\pi\)
\(68\) 0 0
\(69\) −5.18936 −0.624726
\(70\) 0 0
\(71\) −4.31233 + 7.46918i −0.511780 + 0.886428i 0.488127 + 0.872773i \(0.337680\pi\)
−0.999907 + 0.0136558i \(0.995653\pi\)
\(72\) 0 0
\(73\) −0.870717 + 1.50813i −0.101910 + 0.176513i −0.912471 0.409140i \(-0.865829\pi\)
0.810562 + 0.585653i \(0.199162\pi\)
\(74\) 0 0
\(75\) −0.709218 −0.0818935
\(76\) 0 0
\(77\) −10.9787 −1.25114
\(78\) 0 0
\(79\) −4.50544 + 7.80366i −0.506902 + 0.877980i 0.493066 + 0.869992i \(0.335876\pi\)
−0.999968 + 0.00798806i \(0.997457\pi\)
\(80\) 0 0
\(81\) −2.36304 + 4.09291i −0.262560 + 0.454768i
\(82\) 0 0
\(83\) −4.07857 −0.447682 −0.223841 0.974626i \(-0.571860\pi\)
−0.223841 + 0.974626i \(0.571860\pi\)
\(84\) 0 0
\(85\) −1.74850 3.02850i −0.189652 0.328487i
\(86\) 0 0
\(87\) 5.62466 0.603027
\(88\) 0 0
\(89\) 6.61623 + 11.4596i 0.701319 + 1.21472i 0.968004 + 0.250936i \(0.0807385\pi\)
−0.266685 + 0.963784i \(0.585928\pi\)
\(90\) 0 0
\(91\) 0.624599 + 1.08184i 0.0654758 + 0.113407i
\(92\) 0 0
\(93\) −2.03384 + 3.52272i −0.210900 + 0.365289i
\(94\) 0 0
\(95\) 2.39390 3.64270i 0.245609 0.373733i
\(96\) 0 0
\(97\) 3.85162 6.67120i 0.391073 0.677358i −0.601519 0.798859i \(-0.705437\pi\)
0.992591 + 0.121501i \(0.0387708\pi\)
\(98\) 0 0
\(99\) 4.40625 + 7.63186i 0.442845 + 0.767030i
\(100\) 0 0
\(101\) −8.68773 15.0476i −0.864462 1.49729i −0.867581 0.497296i \(-0.834326\pi\)
0.00311897 0.999995i \(-0.499007\pi\)
\(102\) 0 0
\(103\) 7.12614 0.702159 0.351080 0.936346i \(-0.385815\pi\)
0.351080 + 0.936346i \(0.385815\pi\)
\(104\) 0 0
\(105\) −1.10311 1.91065i −0.107653 0.186460i
\(106\) 0 0
\(107\) −0.735452 −0.0710989 −0.0355494 0.999368i \(-0.511318\pi\)
−0.0355494 + 0.999368i \(0.511318\pi\)
\(108\) 0 0
\(109\) 8.00468 13.8645i 0.766710 1.32798i −0.172628 0.984987i \(-0.555226\pi\)
0.939338 0.342993i \(-0.111441\pi\)
\(110\) 0 0
\(111\) −3.73376 + 6.46706i −0.354393 + 0.613826i
\(112\) 0 0
\(113\) 4.34923 0.409141 0.204571 0.978852i \(-0.434420\pi\)
0.204571 + 0.978852i \(0.434420\pi\)
\(114\) 0 0
\(115\) 7.31702 0.682315
\(116\) 0 0
\(117\) 0.501362 0.868384i 0.0463509 0.0802822i
\(118\) 0 0
\(119\) 5.43923 9.42102i 0.498613 0.863623i
\(120\) 0 0
\(121\) 1.45543 0.132312
\(122\) 0 0
\(123\) −0.393896 0.682247i −0.0355164 0.0615162i
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −2.32300 4.02355i −0.206133 0.357032i 0.744360 0.667778i \(-0.232755\pi\)
−0.950493 + 0.310746i \(0.899421\pi\)
\(128\) 0 0
\(129\) 3.05240 + 5.28692i 0.268749 + 0.465487i
\(130\) 0 0
\(131\) 5.24775 9.08936i 0.458498 0.794141i −0.540384 0.841418i \(-0.681721\pi\)
0.998882 + 0.0472772i \(0.0150544\pi\)
\(132\) 0 0
\(133\) 13.5370 + 0.783372i 1.17380 + 0.0679270i
\(134\) 0 0
\(135\) −1.94929 + 3.37627i −0.167768 + 0.290583i
\(136\) 0 0
\(137\) 0.696101 + 1.20568i 0.0594719 + 0.103008i 0.894228 0.447611i \(-0.147725\pi\)
−0.834757 + 0.550619i \(0.814392\pi\)
\(138\) 0 0
\(139\) −3.03766 5.26138i −0.257651 0.446264i 0.707961 0.706251i \(-0.249615\pi\)
−0.965612 + 0.259987i \(0.916282\pi\)
\(140\) 0 0
\(141\) 5.33562 0.449340
\(142\) 0 0
\(143\) −0.708615 1.22736i −0.0592574 0.102637i
\(144\) 0 0
\(145\) −7.93079 −0.658617
\(146\) 0 0
\(147\) 0.949290 1.64422i 0.0782961 0.135613i
\(148\) 0 0
\(149\) 0.0422770 0.0732259i 0.00346347 0.00599890i −0.864288 0.502996i \(-0.832231\pi\)
0.867752 + 0.496998i \(0.165564\pi\)
\(150\) 0 0
\(151\) 15.2216 1.23871 0.619357 0.785109i \(-0.287393\pi\)
0.619357 + 0.785109i \(0.287393\pi\)
\(152\) 0 0
\(153\) −8.73207 −0.705946
\(154\) 0 0
\(155\) 2.86773 4.96705i 0.230341 0.398963i
\(156\) 0 0
\(157\) −8.12935 + 14.0804i −0.648793 + 1.12374i 0.334619 + 0.942354i \(0.391392\pi\)
−0.983412 + 0.181388i \(0.941941\pi\)
\(158\) 0 0
\(159\) −7.48612 −0.593688
\(160\) 0 0
\(161\) 11.3808 + 19.7122i 0.896936 + 1.55354i
\(162\) 0 0
\(163\) −14.9461 −1.17067 −0.585336 0.810791i \(-0.699037\pi\)
−0.585336 + 0.810791i \(0.699037\pi\)
\(164\) 0 0
\(165\) 1.25150 + 2.16765i 0.0974288 + 0.168752i
\(166\) 0 0
\(167\) 6.26461 + 10.8506i 0.484770 + 0.839647i 0.999847 0.0174974i \(-0.00556988\pi\)
−0.515077 + 0.857144i \(0.672237\pi\)
\(168\) 0 0
\(169\) 6.41937 11.1187i 0.493798 0.855283i
\(170\) 0 0
\(171\) −4.88845 9.72466i −0.373829 0.743664i
\(172\) 0 0
\(173\) −8.69242 + 15.0557i −0.660872 + 1.14466i 0.319515 + 0.947581i \(0.396480\pi\)
−0.980387 + 0.197083i \(0.936853\pi\)
\(174\) 0 0
\(175\) 1.55539 + 2.69402i 0.117577 + 0.203649i
\(176\) 0 0
\(177\) 3.01856 + 5.22830i 0.226889 + 0.392983i
\(178\) 0 0
\(179\) −20.7830 −1.55340 −0.776698 0.629873i \(-0.783107\pi\)
−0.776698 + 0.629873i \(0.783107\pi\)
\(180\) 0 0
\(181\) −0.814564 1.41087i −0.0605460 0.104869i 0.834164 0.551517i \(-0.185951\pi\)
−0.894710 + 0.446648i \(0.852618\pi\)
\(182\) 0 0
\(183\) 6.54011 0.483459
\(184\) 0 0
\(185\) 5.26461 9.11858i 0.387062 0.670411i
\(186\) 0 0
\(187\) −6.17087 + 10.6883i −0.451258 + 0.781603i
\(188\) 0 0
\(189\) −12.1277 −0.882157
\(190\) 0 0
\(191\) 13.4016 0.969704 0.484852 0.874596i \(-0.338874\pi\)
0.484852 + 0.874596i \(0.338874\pi\)
\(192\) 0 0
\(193\) 5.34693 9.26116i 0.384881 0.666633i −0.606872 0.794800i \(-0.707576\pi\)
0.991753 + 0.128167i \(0.0409092\pi\)
\(194\) 0 0
\(195\) 0.142400 0.246645i 0.0101975 0.0176626i
\(196\) 0 0
\(197\) −7.00611 −0.499165 −0.249582 0.968354i \(-0.580293\pi\)
−0.249582 + 0.968354i \(0.580293\pi\)
\(198\) 0 0
\(199\) −10.3909 17.9976i −0.736592 1.27581i −0.954021 0.299738i \(-0.903101\pi\)
0.217430 0.976076i \(-0.430233\pi\)
\(200\) 0 0
\(201\) 5.96627 0.420828
\(202\) 0 0
\(203\) −12.3355 21.3657i −0.865783 1.49958i
\(204\) 0 0
\(205\) 0.555394 + 0.961971i 0.0387904 + 0.0671870i
\(206\) 0 0
\(207\) 9.13533 15.8229i 0.634949 1.09976i
\(208\) 0 0
\(209\) −15.3578 0.888745i −1.06232 0.0614758i
\(210\) 0 0
\(211\) −5.61623 + 9.72760i −0.386637 + 0.669675i −0.991995 0.126278i \(-0.959697\pi\)
0.605358 + 0.795954i \(0.293030\pi\)
\(212\) 0 0
\(213\) 3.05838 + 5.29728i 0.209557 + 0.362963i
\(214\) 0 0
\(215\) −4.30390 7.45457i −0.293523 0.508398i
\(216\) 0 0
\(217\) 17.8418 1.21118
\(218\) 0 0
\(219\) 0.617528 + 1.06959i 0.0417287 + 0.0722762i
\(220\) 0 0
\(221\) 1.40429 0.0944630
\(222\) 0 0
\(223\) −5.45005 + 9.43976i −0.364962 + 0.632133i −0.988770 0.149445i \(-0.952251\pi\)
0.623808 + 0.781578i \(0.285585\pi\)
\(224\) 0 0
\(225\) 1.24850 2.16247i 0.0832336 0.144165i
\(226\) 0 0
\(227\) −3.96627 −0.263250 −0.131625 0.991300i \(-0.542020\pi\)
−0.131625 + 0.991300i \(0.542020\pi\)
\(228\) 0 0
\(229\) −10.9139 −0.721213 −0.360606 0.932718i \(-0.617430\pi\)
−0.360606 + 0.932718i \(0.617430\pi\)
\(230\) 0 0
\(231\) −3.89314 + 6.74311i −0.256150 + 0.443664i
\(232\) 0 0
\(233\) 7.79547 13.5021i 0.510698 0.884555i −0.489225 0.872157i \(-0.662720\pi\)
0.999923 0.0123973i \(-0.00394628\pi\)
\(234\) 0 0
\(235\) −7.52324 −0.490762
\(236\) 0 0
\(237\) 3.19534 + 5.53450i 0.207560 + 0.359504i
\(238\) 0 0
\(239\) 8.09544 0.523650 0.261825 0.965115i \(-0.415676\pi\)
0.261825 + 0.965115i \(0.415676\pi\)
\(240\) 0 0
\(241\) 12.6425 + 21.8974i 0.814373 + 1.41054i 0.909777 + 0.415096i \(0.136252\pi\)
−0.0954047 + 0.995439i \(0.530415\pi\)
\(242\) 0 0
\(243\) 7.52378 + 13.0316i 0.482651 + 0.835976i
\(244\) 0 0
\(245\) −1.33850 + 2.31835i −0.0855137 + 0.148114i
\(246\) 0 0
\(247\) 0.786163 + 1.56392i 0.0500223 + 0.0995101i
\(248\) 0 0
\(249\) −1.44630 + 2.50506i −0.0916555 + 0.158752i
\(250\) 0 0
\(251\) 3.94461 + 6.83226i 0.248981 + 0.431248i 0.963243 0.268630i \(-0.0865709\pi\)
−0.714262 + 0.699878i \(0.753238\pi\)
\(252\) 0 0
\(253\) −12.9117 22.3637i −0.811751 1.40599i
\(254\) 0 0
\(255\) −2.48014 −0.155313
\(256\) 0 0
\(257\) 4.59305 + 7.95540i 0.286507 + 0.496244i 0.972973 0.230917i \(-0.0741726\pi\)
−0.686467 + 0.727161i \(0.740839\pi\)
\(258\) 0 0
\(259\) 32.7542 2.03525
\(260\) 0 0
\(261\) −9.90163 + 17.1501i −0.612896 + 1.06157i
\(262\) 0 0
\(263\) 12.1217 20.9954i 0.747454 1.29463i −0.201585 0.979471i \(-0.564609\pi\)
0.949039 0.315158i \(-0.102058\pi\)
\(264\) 0 0
\(265\) 10.5555 0.648417
\(266\) 0 0
\(267\) 9.38470 0.574335
\(268\) 0 0
\(269\) −0.712209 + 1.23358i −0.0434241 + 0.0752128i −0.886921 0.461922i \(-0.847160\pi\)
0.843496 + 0.537135i \(0.180493\pi\)
\(270\) 0 0
\(271\) −5.19617 + 9.00002i −0.315645 + 0.546712i −0.979574 0.201083i \(-0.935554\pi\)
0.663930 + 0.747795i \(0.268887\pi\)
\(272\) 0 0
\(273\) 0.885955 0.0536204
\(274\) 0 0
\(275\) −1.76461 3.05640i −0.106410 0.184308i
\(276\) 0 0
\(277\) 20.3078 1.22018 0.610088 0.792334i \(-0.291134\pi\)
0.610088 + 0.792334i \(0.291134\pi\)
\(278\) 0 0
\(279\) −7.16074 12.4028i −0.428702 0.742534i
\(280\) 0 0
\(281\) −7.79547 13.5021i −0.465038 0.805470i 0.534165 0.845380i \(-0.320626\pi\)
−0.999203 + 0.0399101i \(0.987293\pi\)
\(282\) 0 0
\(283\) 13.6323 23.6119i 0.810358 1.40358i −0.102255 0.994758i \(-0.532606\pi\)
0.912613 0.408824i \(-0.134061\pi\)
\(284\) 0 0
\(285\) −1.38845 2.76207i −0.0822448 0.163611i
\(286\) 0 0
\(287\) −1.72771 + 2.99249i −0.101984 + 0.176641i
\(288\) 0 0
\(289\) 2.38546 + 4.13174i 0.140321 + 0.243044i
\(290\) 0 0
\(291\) −2.73164 4.73134i −0.160131 0.277356i
\(292\) 0 0
\(293\) −12.6710 −0.740249 −0.370125 0.928982i \(-0.620685\pi\)
−0.370125 + 0.928982i \(0.620685\pi\)
\(294\) 0 0
\(295\) −4.25618 7.37192i −0.247804 0.429210i
\(296\) 0 0
\(297\) 13.7590 0.798376
\(298\) 0 0
\(299\) −1.46915 + 2.54464i −0.0849630 + 0.147160i
\(300\) 0 0
\(301\) 13.3885 23.1896i 0.771701 1.33663i
\(302\) 0 0
\(303\) −12.3230 −0.707938
\(304\) 0 0
\(305\) −9.22158 −0.528026
\(306\) 0 0
\(307\) 1.28540 2.22638i 0.0733619 0.127066i −0.827011 0.562186i \(-0.809961\pi\)
0.900373 + 0.435119i \(0.143294\pi\)
\(308\) 0 0
\(309\) 2.52699 4.37688i 0.143756 0.248992i
\(310\) 0 0
\(311\) −19.7568 −1.12030 −0.560152 0.828390i \(-0.689257\pi\)
−0.560152 + 0.828390i \(0.689257\pi\)
\(312\) 0 0
\(313\) −11.1877 19.3777i −0.632368 1.09529i −0.987066 0.160313i \(-0.948750\pi\)
0.354698 0.934981i \(-0.384584\pi\)
\(314\) 0 0
\(315\) 7.76767 0.437658
\(316\) 0 0
\(317\) −6.65313 11.5236i −0.373677 0.647228i 0.616451 0.787393i \(-0.288570\pi\)
−0.990128 + 0.140166i \(0.955236\pi\)
\(318\) 0 0
\(319\) 13.9948 + 24.2397i 0.783557 + 1.35716i
\(320\) 0 0
\(321\) −0.260798 + 0.451716i −0.0145563 + 0.0252123i
\(322\) 0 0
\(323\) 8.37148 12.7386i 0.465801 0.708792i
\(324\) 0 0
\(325\) −0.200785 + 0.347770i −0.0111375 + 0.0192908i
\(326\) 0 0
\(327\) −5.67707 9.83297i −0.313943 0.543764i
\(328\) 0 0
\(329\) −11.7016 20.2678i −0.645131 1.11740i
\(330\) 0 0
\(331\) 19.6173 1.07826 0.539132 0.842221i \(-0.318752\pi\)
0.539132 + 0.842221i \(0.318752\pi\)
\(332\) 0 0
\(333\) −13.1458 22.7692i −0.720385 1.24774i
\(334\) 0 0
\(335\) −8.41246 −0.459622
\(336\) 0 0
\(337\) 14.5909 25.2722i 0.794819 1.37667i −0.128136 0.991757i \(-0.540899\pi\)
0.922954 0.384910i \(-0.125767\pi\)
\(338\) 0 0
\(339\) 1.54228 2.67130i 0.0837650 0.145085i
\(340\) 0 0
\(341\) −20.2417 −1.09615
\(342\) 0 0
\(343\) 13.4479 0.726120
\(344\) 0 0
\(345\) 2.59468 4.49412i 0.139693 0.241955i
\(346\) 0 0
\(347\) −6.03935 + 10.4605i −0.324209 + 0.561547i −0.981352 0.192219i \(-0.938432\pi\)
0.657143 + 0.753766i \(0.271765\pi\)
\(348\) 0 0
\(349\) 20.2894 1.08607 0.543033 0.839711i \(-0.317276\pi\)
0.543033 + 0.839711i \(0.317276\pi\)
\(350\) 0 0
\(351\) −0.782776 1.35581i −0.0417815 0.0723677i
\(352\) 0 0
\(353\) 6.69387 0.356279 0.178139 0.984005i \(-0.442992\pi\)
0.178139 + 0.984005i \(0.442992\pi\)
\(354\) 0 0
\(355\) −4.31233 7.46918i −0.228875 0.396423i
\(356\) 0 0
\(357\) −3.85760 6.68156i −0.204166 0.353626i
\(358\) 0 0
\(359\) 4.94536 8.56562i 0.261006 0.452076i −0.705503 0.708707i \(-0.749279\pi\)
0.966510 + 0.256630i \(0.0826123\pi\)
\(360\) 0 0
\(361\) 18.8732 + 2.19169i 0.993325 + 0.115352i
\(362\) 0 0
\(363\) 0.516108 0.893925i 0.0270886 0.0469189i
\(364\) 0 0
\(365\) −0.870717 1.50813i −0.0455754 0.0789389i
\(366\) 0 0
\(367\) 8.09936 + 14.0285i 0.422783 + 0.732282i 0.996211 0.0869742i \(-0.0277198\pi\)
−0.573427 + 0.819257i \(0.694386\pi\)
\(368\) 0 0
\(369\) 2.77365 0.144390
\(370\) 0 0
\(371\) 16.4179 + 28.4366i 0.852375 + 1.47636i
\(372\) 0 0
\(373\) 10.3003 0.533328 0.266664 0.963790i \(-0.414079\pi\)
0.266664 + 0.963790i \(0.414079\pi\)
\(374\) 0 0
\(375\) 0.354609 0.614201i 0.0183119 0.0317172i
\(376\) 0 0
\(377\) 1.59238 2.75809i 0.0820120 0.142049i
\(378\) 0 0
\(379\) 21.2587 1.09199 0.545993 0.837790i \(-0.316153\pi\)
0.545993 + 0.837790i \(0.316153\pi\)
\(380\) 0 0
\(381\) −3.29502 −0.168809
\(382\) 0 0
\(383\) −6.79002 + 11.7607i −0.346954 + 0.600942i −0.985707 0.168470i \(-0.946117\pi\)
0.638753 + 0.769412i \(0.279451\pi\)
\(384\) 0 0
\(385\) 5.48934 9.50781i 0.279762 0.484563i
\(386\) 0 0
\(387\) −21.4938 −1.09259
\(388\) 0 0
\(389\) −8.98771 15.5672i −0.455695 0.789287i 0.543033 0.839711i \(-0.317276\pi\)
−0.998728 + 0.0504248i \(0.983942\pi\)
\(390\) 0 0
\(391\) 25.5877 1.29402
\(392\) 0 0
\(393\) −3.72180 6.44634i −0.187740 0.325175i
\(394\) 0 0
\(395\) −4.50544 7.80366i −0.226693 0.392645i
\(396\) 0 0
\(397\) −12.0169 + 20.8139i −0.603112 + 1.04462i 0.389234 + 0.921139i \(0.372740\pi\)
−0.992347 + 0.123483i \(0.960594\pi\)
\(398\) 0 0
\(399\) 5.28148 8.03663i 0.264405 0.402335i
\(400\) 0 0
\(401\) 5.91076 10.2377i 0.295169 0.511248i −0.679855 0.733347i \(-0.737957\pi\)
0.975024 + 0.222098i \(0.0712906\pi\)
\(402\) 0 0
\(403\) 1.15159 + 1.99462i 0.0573649 + 0.0993589i
\(404\) 0 0
\(405\) −2.36304 4.09291i −0.117421 0.203378i
\(406\) 0 0
\(407\) −37.1600 −1.84195
\(408\) 0 0
\(409\) 15.6300 + 27.0719i 0.772851 + 1.33862i 0.935994 + 0.352015i \(0.114504\pi\)
−0.163143 + 0.986602i \(0.552163\pi\)
\(410\) 0 0
\(411\) 0.987375 0.0487036
\(412\) 0 0
\(413\) 13.2401 22.9325i 0.651501 1.12843i
\(414\) 0 0
\(415\) 2.03929 3.53215i 0.100105 0.173386i
\(416\) 0 0
\(417\) −4.30872 −0.210999
\(418\) 0 0
\(419\) 22.4217 1.09537 0.547686 0.836684i \(-0.315509\pi\)
0.547686 + 0.836684i \(0.315509\pi\)
\(420\) 0 0
\(421\) −13.6431 + 23.6305i −0.664922 + 1.15168i 0.314384 + 0.949296i \(0.398202\pi\)
−0.979306 + 0.202384i \(0.935131\pi\)
\(422\) 0 0
\(423\) −9.39281 + 16.2688i −0.456694 + 0.791017i
\(424\) 0 0
\(425\) 3.49701 0.169630
\(426\) 0 0
\(427\) −14.3432 24.8431i −0.694115 1.20224i
\(428\) 0 0
\(429\) −1.00513 −0.0485279
\(430\) 0 0
\(431\) 13.6686 + 23.6748i 0.658395 + 1.14037i 0.981031 + 0.193850i \(0.0620977\pi\)
−0.322636 + 0.946523i \(0.604569\pi\)
\(432\) 0 0
\(433\) 3.67761 + 6.36980i 0.176734 + 0.306113i 0.940760 0.339073i \(-0.110113\pi\)
−0.764026 + 0.645186i \(0.776780\pi\)
\(434\) 0 0
\(435\) −2.81233 + 4.87110i −0.134841 + 0.233551i
\(436\) 0 0
\(437\) 14.3247 + 28.4963i 0.685243 + 1.36316i
\(438\) 0 0
\(439\) 2.11862 3.66955i 0.101116 0.175138i −0.811029 0.585006i \(-0.801092\pi\)
0.912145 + 0.409868i \(0.134425\pi\)
\(440\) 0 0
\(441\) 3.34225 + 5.78895i 0.159155 + 0.275664i
\(442\) 0 0
\(443\) −11.1310 19.2794i −0.528849 0.915993i −0.999434 0.0336383i \(-0.989291\pi\)
0.470585 0.882354i \(-0.344043\pi\)
\(444\) 0 0
\(445\) −13.2325 −0.627279
\(446\) 0 0
\(447\) −0.0299836 0.0519332i −0.00141818 0.00245635i
\(448\) 0 0
\(449\) 4.46328 0.210635 0.105318 0.994439i \(-0.466414\pi\)
0.105318 + 0.994439i \(0.466414\pi\)
\(450\) 0 0
\(451\) 1.96011 3.39501i 0.0922980 0.159865i
\(452\) 0 0
\(453\) 5.39771 9.34911i 0.253607 0.439259i
\(454\) 0 0
\(455\) −1.24920 −0.0585634
\(456\) 0 0
\(457\) 10.7463 0.502692 0.251346 0.967897i \(-0.419127\pi\)
0.251346 + 0.967897i \(0.419127\pi\)
\(458\) 0 0
\(459\) −6.81668 + 11.8068i −0.318176 + 0.551096i
\(460\) 0 0
\(461\) −11.9163 + 20.6397i −0.554998 + 0.961285i 0.442906 + 0.896568i \(0.353948\pi\)
−0.997904 + 0.0647167i \(0.979386\pi\)
\(462\) 0 0
\(463\) 25.3695 1.17902 0.589510 0.807761i \(-0.299321\pi\)
0.589510 + 0.807761i \(0.299321\pi\)
\(464\) 0 0
\(465\) −2.03384 3.52272i −0.0943172 0.163362i
\(466\) 0 0
\(467\) −5.62020 −0.260072 −0.130036 0.991509i \(-0.541509\pi\)
−0.130036 + 0.991509i \(0.541509\pi\)
\(468\) 0 0
\(469\) −13.0847 22.6633i −0.604195 1.04650i
\(470\) 0 0
\(471\) 5.76548 + 9.98611i 0.265659 + 0.460136i
\(472\) 0 0
\(473\) −15.1894 + 26.3089i −0.698411 + 1.20968i
\(474\) 0 0
\(475\) 1.95772 + 3.89452i 0.0898265 + 0.178693i
\(476\) 0 0
\(477\) 13.1785 22.8259i 0.603404 1.04513i
\(478\) 0 0
\(479\) 6.83611 + 11.8405i 0.312350 + 0.541006i 0.978871 0.204480i \(-0.0655505\pi\)
−0.666521 + 0.745487i \(0.732217\pi\)
\(480\) 0 0
\(481\) 2.11411 + 3.66175i 0.0963951 + 0.166961i
\(482\) 0 0
\(483\) 16.1430 0.734532
\(484\) 0 0
\(485\) 3.85162 + 6.67120i 0.174893 + 0.302924i
\(486\) 0 0
\(487\) 7.39233 0.334979 0.167489 0.985874i \(-0.446434\pi\)
0.167489 + 0.985874i \(0.446434\pi\)
\(488\) 0 0
\(489\) −5.30004 + 9.17994i −0.239676 + 0.415131i
\(490\) 0 0
\(491\) 0.979053 1.69577i 0.0441840 0.0765290i −0.843088 0.537776i \(-0.819265\pi\)
0.887272 + 0.461247i \(0.152598\pi\)
\(492\) 0 0
\(493\) −27.7341 −1.24908
\(494\) 0 0
\(495\) −8.81251 −0.396093
\(496\) 0 0
\(497\) 13.4148 23.2350i 0.601734 1.04223i
\(498\) 0 0
\(499\) 2.72234 4.71522i 0.121868 0.211082i −0.798636 0.601814i \(-0.794445\pi\)
0.920505 + 0.390732i \(0.127778\pi\)
\(500\) 0 0
\(501\) 8.88595 0.396995
\(502\) 0 0
\(503\) −8.29629 14.3696i −0.369913 0.640709i 0.619638 0.784887i \(-0.287279\pi\)
−0.989552 + 0.144179i \(0.953946\pi\)
\(504\) 0 0
\(505\) 17.3755 0.773198
\(506\) 0 0
\(507\) −4.55273 7.88557i −0.202194 0.350210i
\(508\) 0 0
\(509\) −6.06552 10.5058i −0.268849 0.465661i 0.699716 0.714422i \(-0.253310\pi\)
−0.968565 + 0.248761i \(0.919977\pi\)
\(510\) 0 0
\(511\) 2.70862 4.69146i 0.119822 0.207538i
\(512\) 0 0
\(513\) −16.9651 0.981757i −0.749029 0.0433456i
\(514\) 0 0
\(515\) −3.56307 + 6.17142i −0.157008 + 0.271945i
\(516\) 0 0
\(517\) 13.2756 + 22.9940i 0.583861 + 1.01128i
\(518\) 0 0
\(519\) 6.16482 + 10.6778i 0.270606 + 0.468703i
\(520\) 0 0
\(521\) 14.1249 0.618824 0.309412 0.950928i \(-0.399868\pi\)
0.309412 + 0.950928i \(0.399868\pi\)
\(522\) 0 0
\(523\) −13.1840 22.8353i −0.576495 0.998519i −0.995877 0.0907094i \(-0.971087\pi\)
0.419382 0.907810i \(-0.362247\pi\)
\(524\) 0 0
\(525\) 2.20623 0.0962877
\(526\) 0 0
\(527\) 10.0285 17.3698i 0.436847 0.756641i
\(528\) 0 0
\(529\) −15.2694 + 26.4473i −0.663885 + 1.14988i
\(530\) 0 0
\(531\) −21.2554 −0.922407
\(532\) 0 0
\(533\) −0.446059 −0.0193210
\(534\) 0 0
\(535\) 0.367726 0.636920i 0.0158982 0.0275365i
\(536\) 0 0
\(537\) −7.36985 + 12.7649i −0.318032 + 0.550848i
\(538\) 0 0
\(539\) 9.44775 0.406943
\(540\) 0 0
\(541\) 14.6986 + 25.4586i 0.631940 + 1.09455i 0.987155 + 0.159768i \(0.0510745\pi\)
−0.355214 + 0.934785i \(0.615592\pi\)
\(542\) 0 0
\(543\) −1.15541 −0.0495833
\(544\) 0 0
\(545\) 8.00468 + 13.8645i 0.342883 + 0.593891i
\(546\) 0 0
\(547\) 10.8762 + 18.8381i 0.465031 + 0.805457i 0.999203 0.0399185i \(-0.0127098\pi\)
−0.534172 + 0.845376i \(0.679377\pi\)
\(548\) 0 0
\(549\) −11.5132 + 19.9414i −0.491371 + 0.851079i
\(550\) 0 0
\(551\) −15.5263 30.8867i −0.661443 1.31582i
\(552\) 0 0
\(553\) 14.0155 24.2755i 0.595999 1.03230i
\(554\) 0 0
\(555\) −3.73376 6.46706i −0.158489 0.274511i
\(556\) 0 0
\(557\) 12.0716 + 20.9086i 0.511489 + 0.885924i 0.999911 + 0.0133172i \(0.00423912\pi\)
−0.488423 + 0.872607i \(0.662428\pi\)
\(558\) 0 0
\(559\) 3.45663 0.146200
\(560\) 0 0
\(561\) 4.37649 + 7.58030i 0.184776 + 0.320041i
\(562\) 0 0
\(563\) 15.0693 0.635097 0.317548 0.948242i \(-0.397140\pi\)
0.317548 + 0.948242i \(0.397140\pi\)
\(564\) 0 0
\(565\) −2.17462 + 3.76654i −0.0914868 + 0.158460i
\(566\) 0 0
\(567\) 7.35092 12.7322i 0.308710 0.534701i
\(568\) 0 0
\(569\) 0.302873 0.0126971 0.00634855 0.999980i \(-0.497979\pi\)
0.00634855 + 0.999980i \(0.497979\pi\)
\(570\) 0 0
\(571\) −30.3107 −1.26846 −0.634232 0.773143i \(-0.718684\pi\)
−0.634232 + 0.773143i \(0.718684\pi\)
\(572\) 0 0
\(573\) 4.75232 8.23126i 0.198531 0.343866i
\(574\) 0 0
\(575\) −3.65851 + 6.33672i −0.152570 + 0.264260i
\(576\) 0 0
\(577\) 18.6601 0.776832 0.388416 0.921484i \(-0.373022\pi\)
0.388416 + 0.921484i \(0.373022\pi\)
\(578\) 0 0
\(579\) −3.79214 6.56819i −0.157596 0.272964i
\(580\) 0 0
\(581\) 12.6876 0.526369
\(582\) 0 0
\(583\) −18.6263 32.2617i −0.771422 1.33614i
\(584\) 0 0
\(585\) 0.501362 + 0.868384i 0.0207288 + 0.0359033i
\(586\) 0 0
\(587\) −13.5986 + 23.5535i −0.561275 + 0.972156i 0.436111 + 0.899893i \(0.356355\pi\)
−0.997386 + 0.0722631i \(0.976978\pi\)
\(588\) 0 0
\(589\) 24.9585 + 1.44433i 1.02840 + 0.0595124i
\(590\) 0 0
\(591\) −2.48443 + 4.30316i −0.102196 + 0.177008i
\(592\) 0 0
\(593\) 14.7987 + 25.6321i 0.607709 + 1.05258i 0.991617 + 0.129211i \(0.0412446\pi\)
−0.383908 + 0.923371i \(0.625422\pi\)
\(594\) 0 0
\(595\) 5.43923 + 9.42102i 0.222987 + 0.386224i
\(596\) 0 0
\(597\) −14.7388 −0.603221
\(598\) 0 0
\(599\) 7.54404 + 13.0667i 0.308241 + 0.533889i 0.977978 0.208710i \(-0.0669264\pi\)
−0.669737 + 0.742599i \(0.733593\pi\)
\(600\) 0 0
\(601\) −44.9249 −1.83253 −0.916263 0.400576i \(-0.868810\pi\)
−0.916263 + 0.400576i \(0.868810\pi\)
\(602\) 0 0
\(603\) −10.5030 + 18.1917i −0.427715 + 0.740824i
\(604\) 0 0
\(605\) −0.727713 + 1.26044i −0.0295858 + 0.0512440i
\(606\) 0 0
\(607\) 24.9570 1.01297 0.506487 0.862247i \(-0.330944\pi\)
0.506487 + 0.862247i \(0.330944\pi\)
\(608\) 0 0
\(609\) −17.4971 −0.709020
\(610\) 0 0
\(611\) 1.51055 2.61636i 0.0611105 0.105846i
\(612\) 0 0
\(613\) 9.64316 16.7024i 0.389484 0.674605i −0.602897 0.797819i \(-0.705987\pi\)
0.992380 + 0.123214i \(0.0393202\pi\)
\(614\) 0 0
\(615\) 0.787791 0.0317668
\(616\) 0 0
\(617\) −9.72294 16.8406i −0.391431 0.677978i 0.601208 0.799093i \(-0.294686\pi\)
−0.992639 + 0.121115i \(0.961353\pi\)
\(618\) 0 0
\(619\) 1.70324 0.0684589 0.0342294 0.999414i \(-0.489102\pi\)
0.0342294 + 0.999414i \(0.489102\pi\)
\(620\) 0 0
\(621\) −14.2630 24.7042i −0.572354 0.991346i
\(622\) 0 0
\(623\) −20.5817 35.6485i −0.824588 1.42823i
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) −5.99190 + 9.11764i −0.239293 + 0.364124i
\(628\) 0 0
\(629\) 18.4104 31.8877i 0.734071 1.27145i
\(630\) 0 0
\(631\) −4.39314 7.60914i −0.174888 0.302915i 0.765235 0.643752i \(-0.222623\pi\)
−0.940123 + 0.340837i \(0.889290\pi\)
\(632\) 0 0
\(633\) 3.98313 + 6.89899i 0.158315 + 0.274210i
\(634\) 0 0
\(635\) 4.64599 0.184371
\(636\) 0 0
\(637\) −0.537502 0.930981i −0.0212966 0.0368868i
\(638\) 0 0
\(639\) −21.5359 −0.851946
\(640\) 0 0
\(641\) 17.9033 31.0095i 0.707139 1.22480i −0.258775 0.965938i \(-0.583319\pi\)
0.965914 0.258863i \(-0.0833478\pi\)
\(642\) 0 0
\(643\) 1.30758 2.26480i 0.0515661 0.0893150i −0.839090 0.543992i \(-0.816912\pi\)
0.890656 + 0.454677i \(0.150245\pi\)
\(644\) 0 0
\(645\) −6.10481 −0.240377
\(646\) 0 0
\(647\) −30.8849 −1.21421 −0.607105 0.794622i \(-0.707669\pi\)
−0.607105 + 0.794622i \(0.707669\pi\)
\(648\) 0 0
\(649\) −15.0210 + 26.0172i −0.589626 + 1.02126i
\(650\) 0 0
\(651\) 6.32686 10.9584i 0.247969 0.429495i
\(652\) 0 0
\(653\) −17.2198 −0.673864 −0.336932 0.941529i \(-0.609389\pi\)
−0.336932 + 0.941529i \(0.609389\pi\)
\(654\) 0 0
\(655\) 5.24775 + 9.08936i 0.205046 + 0.355151i
\(656\) 0 0
\(657\) −4.34838 −0.169646
\(658\) 0 0
\(659\) −0.0769444 0.133272i −0.00299733 0.00519153i 0.864523 0.502594i \(-0.167621\pi\)
−0.867520 + 0.497402i \(0.834287\pi\)
\(660\) 0 0
\(661\) 13.8595 + 24.0054i 0.539073 + 0.933701i 0.998954 + 0.0457209i \(0.0145585\pi\)
−0.459882 + 0.887980i \(0.652108\pi\)
\(662\) 0 0
\(663\) 0.497975 0.862519i 0.0193398 0.0334975i
\(664\) 0 0
\(665\) −7.44690 + 11.3317i −0.288778 + 0.439423i
\(666\) 0 0
\(667\) 29.0149 50.2552i 1.12346 1.94589i
\(668\) 0 0
\(669\) 3.86527 + 6.69485i 0.149440 + 0.258838i
\(670\) 0 0
\(671\) 16.2725 + 28.1848i 0.628193 + 1.08806i
\(672\) 0 0
\(673\) −48.0820 −1.85342 −0.926712 0.375773i \(-0.877377\pi\)
−0.926712 + 0.375773i \(0.877377\pi\)
\(674\) 0 0
\(675\) −1.94929 3.37627i −0.0750282 0.129953i
\(676\) 0 0
\(677\) −42.8392 −1.64644 −0.823222 0.567720i \(-0.807826\pi\)
−0.823222 + 0.567720i \(0.807826\pi\)
\(678\) 0 0
\(679\) −11.9816 + 20.7527i −0.459810 + 0.796415i
\(680\) 0 0
\(681\) −1.40647 + 2.43609i −0.0538962 + 0.0933510i
\(682\) 0 0
\(683\) −41.4022 −1.58421 −0.792106 0.610383i \(-0.791015\pi\)
−0.792106 + 0.610383i \(0.791015\pi\)
\(684\) 0 0
\(685\) −1.39220 −0.0531933
\(686\) 0 0
\(687\) −3.87018 + 6.70335i −0.147657 + 0.255749i
\(688\) 0 0
\(689\) −2.11938 + 3.67087i −0.0807418 + 0.139849i
\(690\) 0 0
\(691\) −16.9248 −0.643850 −0.321925 0.946765i \(-0.604330\pi\)
−0.321925 + 0.946765i \(0.604330\pi\)
\(692\) 0 0
\(693\) −13.7069 23.7411i −0.520683 0.901849i
\(694\) 0 0
\(695\) 6.07532 0.230450
\(696\) 0 0
\(697\) 1.94222 + 3.36402i 0.0735668 + 0.127421i
\(698\) 0 0
\(699\) −5.52869 9.57597i −0.209114 0.362196i
\(700\) 0 0
\(701\) 18.7780 32.5244i 0.709233 1.22843i −0.255908 0.966701i \(-0.582375\pi\)
0.965142 0.261727i \(-0.0842921\pi\)
\(702\) 0 0
\(703\) 45.8192 + 2.65152i 1.72810 + 0.100004i
\(704\) 0 0
\(705\) −2.66781 + 4.62078i −0.100476 + 0.174029i
\(706\) 0 0
\(707\) 27.0257 + 46.8099i 1.01641 + 1.76047i
\(708\) 0 0
\(709\) 15.3872 + 26.6514i 0.577879 + 1.00092i 0.995722 + 0.0923969i \(0.0294529\pi\)
−0.417843 + 0.908519i \(0.637214\pi\)
\(710\) 0 0
\(711\) −22.5003 −0.843826
\(712\) 0 0
\(713\) 20.9832 + 36.3440i 0.785827 + 1.36109i
\(714\) 0 0
\(715\) 1.41723 0.0530014
\(716\) 0 0
\(717\) 2.87072 4.97223i 0.107209 0.185691i
\(718\) 0 0
\(719\) −23.1509 + 40.0985i −0.863383 + 1.49542i 0.00526119 + 0.999986i \(0.498325\pi\)
−0.868644 + 0.495437i \(0.835008\pi\)
\(720\) 0 0
\(721\) −22.1679 −0.825576
\(722\) 0 0
\(723\) 17.9325 0.666918
\(724\) 0 0
\(725\) 3.96540 6.86827i 0.147271 0.255081i
\(726\) 0 0
\(727\) −2.93454 + 5.08278i −0.108836 + 0.188510i −0.915299 0.402775i \(-0.868046\pi\)
0.806463 + 0.591285i \(0.201379\pi\)
\(728\) 0 0
\(729\) −3.50625 −0.129861
\(730\) 0 0
\(731\) −15.0508 26.0687i −0.556673 0.964186i
\(732\) 0 0
\(733\) 11.3955 0.420901 0.210450 0.977605i \(-0.432507\pi\)
0.210450 + 0.977605i \(0.432507\pi\)
\(734\) 0 0
\(735\) 0.949290 + 1.64422i 0.0350151 + 0.0606479i
\(736\) 0 0
\(737\) 14.8447 + 25.7118i 0.546812 + 0.947107i
\(738\) 0 0
\(739\) 6.86621 11.8926i 0.252578 0.437477i −0.711657 0.702527i \(-0.752055\pi\)
0.964235 + 0.265050i \(0.0853884\pi\)
\(740\) 0 0
\(741\) 1.23934 + 0.0717198i 0.0455284 + 0.00263469i
\(742\) 0 0
\(743\) −4.33763 + 7.51300i −0.159132 + 0.275625i −0.934556 0.355816i \(-0.884203\pi\)
0.775424 + 0.631441i \(0.217536\pi\)
\(744\) 0 0
\(745\) 0.0422770 + 0.0732259i 0.00154891 + 0.00268279i
\(746\) 0 0
\(747\) −5.09212 8.81981i −0.186311 0.322700i
\(748\) 0 0
\(749\) 2.28784 0.0835957
\(750\) 0 0
\(751\) 3.35314 + 5.80780i 0.122358 + 0.211930i 0.920697 0.390278i \(-0.127621\pi\)
−0.798339 + 0.602208i \(0.794288\pi\)
\(752\) 0 0
\(753\) 5.59517 0.203899
\(754\) 0 0
\(755\) −7.61079 + 13.1823i −0.276985 + 0.479752i
\(756\) 0 0
\(757\) −7.19480 + 12.4618i −0.261500 + 0.452931i −0.966641 0.256136i \(-0.917550\pi\)
0.705141 + 0.709067i \(0.250884\pi\)
\(758\) 0 0
\(759\) −18.3144 −0.664771
\(760\) 0 0
\(761\) −50.9650 −1.84748 −0.923740 0.383020i \(-0.874884\pi\)
−0.923740 + 0.383020i \(0.874884\pi\)
\(762\) 0 0
\(763\) −24.9009 + 43.1296i −0.901472 + 1.56140i
\(764\) 0 0
\(765\) 4.36603 7.56219i 0.157854 0.273412i
\(766\) 0 0
\(767\) 3.41831 0.123428
\(768\) 0 0
\(769\) −11.9988 20.7825i −0.432687 0.749435i 0.564417 0.825490i \(-0.309101\pi\)
−0.997104 + 0.0760546i \(0.975768\pi\)
\(770\) 0 0
\(771\) 6.51495 0.234630
\(772\) 0 0
\(773\) 21.9925 + 38.0920i 0.791014 + 1.37008i 0.925340 + 0.379139i \(0.123780\pi\)
−0.134326 + 0.990937i \(0.542887\pi\)
\(774\) 0 0
\(775\) 2.86773 + 4.96705i 0.103012 + 0.178422i
\(776\) 0 0
\(777\) 11.6149 20.1177i 0.416683 0.721717i
\(778\) 0 0
\(779\) −2.65911 + 4.04627i −0.0952725 + 0.144973i
\(780\) 0 0
\(781\) −15.2192 + 26.3604i −0.544585 + 0.943250i
\(782\) 0 0
\(783\) 15.4594 + 26.7765i 0.552474 + 0.956914i
\(784\) 0 0
\(785\) −8.12935 14.0804i −0.290149 0.502553i
\(786\) 0 0
\(787\) −16.4815 −0.587501 −0.293751 0.955882i \(-0.594904\pi\)
−0.293751 + 0.955882i \(0.594904\pi\)
\(788\) 0 0
\(789\) −8.59691 14.8903i −0.306058 0.530108i
\(790\) 0 0
\(791\) −13.5295 −0.481055
\(792\) 0 0
\(793\) 1.85155 3.20699i 0.0657506 0.113883i
\(794\) 0 0
\(795\) 3.74306 6.48317i 0.132753 0.229934i
\(796\) 0 0
\(797\) −47.9346 −1.69793 −0.848966 0.528448i \(-0.822774\pi\)
−0.848966 + 0.528448i \(0.822774\pi\)
\(798\) 0 0
\(799\) −26.3089 −0.930740
\(800\) 0 0
\(801\) −16.5208 + 28.6149i −0.583733 + 1.01106i
\(802\) 0 0
\(803\) −3.07295 + 5.32251i −0.108442 + 0.187827i
\(804\) 0 0
\(805\) −22.7617 −0.802244
\(806\) 0 0
\(807\) 0.505111 + 0.874879i 0.0177808 + 0.0307972i
\(808\) 0 0
\(809\) 11.7879 0.414441 0.207221 0.978294i \(-0.433558\pi\)
0.207221 + 0.978294i \(0.433558\pi\)
\(810\) 0 0
\(811\) 15.9432 + 27.6144i 0.559840 + 0.969671i 0.997509 + 0.0705351i \(0.0224707\pi\)
−0.437669 + 0.899136i \(0.644196\pi\)
\(812\) 0 0
\(813\) 3.68522 + 6.38298i 0.129246 + 0.223861i
\(814\) 0 0
\(815\) 7.47307 12.9437i 0.261770 0.453399i
\(816\) 0 0
\(817\) 20.6062 31.3556i 0.720919 1.09699i
\(818\) 0 0
\(819\) −1.55963 + 2.70136i −0.0544979 + 0.0943932i
\(820\) 0 0
\(821\) −16.2579 28.1596i −0.567406 0.982776i −0.996821 0.0796688i \(-0.974614\pi\)
0.429415 0.903107i \(-0.358720\pi\)
\(822\) 0 0
\(823\) 24.6985 + 42.7790i 0.860934 + 1.49118i 0.871028 + 0.491233i \(0.163454\pi\)
−0.0100940 + 0.999949i \(0.503213\pi\)
\(824\) 0 0
\(825\) −2.50299 −0.0871429
\(826\) 0 0
\(827\) 12.0147 + 20.8101i 0.417794 + 0.723640i 0.995717 0.0924511i \(-0.0294702\pi\)
−0.577924 + 0.816091i \(0.696137\pi\)
\(828\) 0 0
\(829\) 8.42673 0.292672 0.146336 0.989235i \(-0.453252\pi\)
0.146336 + 0.989235i \(0.453252\pi\)
\(830\) 0 0
\(831\) 7.20132 12.4731i 0.249811 0.432686i
\(832\) 0 0
\(833\) −4.68075 + 8.10730i −0.162178 + 0.280901i
\(834\) 0 0
\(835\) −12.5292 −0.433592
\(836\) 0 0
\(837\) −22.3601 −0.772879
\(838\) 0 0
\(839\) 11.0809 19.1926i 0.382554 0.662603i −0.608872 0.793268i \(-0.708378\pi\)
0.991427 + 0.130665i \(0.0417112\pi\)
\(840\) 0 0
\(841\) −16.9488 + 29.3561i −0.584440 + 1.01228i
\(842\) 0 0
\(843\) −11.0574 −0.380836
\(844\) 0 0
\(845\) 6.41937 + 11.1187i 0.220833 + 0.382494i
\(846\) 0 0
\(847\) −4.52752 −0.155568
\(848\) 0 0
\(849\) −9.66830 16.7460i −0.331815 0.574721i
\(850\) 0 0
\(851\) 38.5213 + 66.7208i 1.32049 + 2.28716i
\(852\) 0 0
\(853\) 19.4153 33.6283i 0.664767 1.15141i −0.314582 0.949230i \(-0.601864\pi\)
0.979348 0.202180i \(-0.0648025\pi\)
\(854\) 0 0
\(855\) 10.8660 + 0.628808i 0.371610 + 0.0215048i
\(856\) 0 0
\(857\) 17.9215 31.0409i 0.612186 1.06034i −0.378685 0.925526i \(-0.623624\pi\)
0.990871 0.134812i \(-0.0430432\pi\)
\(858\) 0 0
\(859\) −9.06313 15.6978i −0.309230 0.535602i 0.668964 0.743295i \(-0.266738\pi\)
−0.978194 + 0.207692i \(0.933405\pi\)
\(860\) 0 0
\(861\) 1.22533 + 2.12233i 0.0417590 + 0.0723287i
\(862\) 0 0
\(863\) −48.5099 −1.65130 −0.825648 0.564186i \(-0.809190\pi\)
−0.825648 + 0.564186i \(0.809190\pi\)
\(864\) 0 0
\(865\) −8.69242 15.0557i −0.295551 0.511909i
\(866\) 0 0
\(867\) 3.38363 0.114914
\(868\) 0 0
\(869\) −15.9007 + 27.5409i −0.539395 + 0.934259i
\(870\) 0 0
\(871\) 1.68909 2.92560i 0.0572328 0.0991301i
\(872\) 0 0
\(873\) 19.2351 0.651008
\(874\) 0 0
\(875\) −3.11079 −0.105164
\(876\) 0 0
\(877\) −13.4708 + 23.3322i −0.454878 + 0.787872i −0.998681 0.0513410i \(-0.983650\pi\)
0.543803 + 0.839213i \(0.316984\pi\)
\(878\) 0 0
\(879\) −4.49326 + 7.78255i −0.151554 + 0.262499i
\(880\) 0 0
\(881\) 27.0603 0.911685 0.455843 0.890060i \(-0.349338\pi\)
0.455843 + 0.890060i \(0.349338\pi\)
\(882\) 0 0
\(883\) 18.3088 + 31.7118i 0.616140 + 1.06719i 0.990183 + 0.139775i \(0.0446379\pi\)
−0.374043 + 0.927411i \(0.622029\pi\)
\(884\) 0 0
\(885\) −6.03712 −0.202936
\(886\) 0 0
\(887\) 21.2342 + 36.7786i 0.712973 + 1.23491i 0.963736 + 0.266857i \(0.0859852\pi\)
−0.250763 + 0.968049i \(0.580681\pi\)
\(888\) 0 0
\(889\) 7.22635 + 12.5164i 0.242364 + 0.419787i
\(890\) 0 0
\(891\) −8.33971 + 14.4448i −0.279391 + 0.483919i
\(892\) 0 0
\(893\) −14.7284 29.2995i −0.492868 0.980469i
\(894\) 0 0
\(895\) 10.3915 17.9986i 0.347350 0.601628i
\(896\) 0 0
\(897\) 1.04195 + 1.80470i 0.0347896 + 0.0602573i
\(898\) 0 0
\(899\) −22.7433 39.3926i −0.758533 1.31382i
\(900\) 0 0
\(901\) 36.9125 1.22973
\(902\) 0 0
\(903\) −9.49538 16.4465i −0.315986 0.547305i
\(904\) 0 0
\(905\) 1.62913 0.0541540
\(906\) 0 0
\(907\) 26.8140 46.4433i 0.890345 1.54212i 0.0508831 0.998705i \(-0.483796\pi\)
0.839462 0.543418i \(-0.182870\pi\)
\(908\) 0 0
\(909\) 21.6934 37.5740i 0.719523 1.24625i
\(910\) 0 0
\(911\) 10.1631 0.336719 0.168360 0.985726i \(-0.446153\pi\)
0.168360 + 0.985726i \(0.446153\pi\)
\(912\) 0 0
\(913\) −14.3942 −0.476379
\(914\) 0 0
\(915\) −3.27006 + 5.66390i −0.108105 + 0.187243i
\(916\) 0 0
\(917\) −16.3246 + 28.2751i −0.539087 + 0.933725i
\(918\) 0 0
\(919\) 16.6831 0.550325 0.275163 0.961398i \(-0.411268\pi\)
0.275163 + 0.961398i \(0.411268\pi\)
\(920\) 0 0
\(921\) −0.911632 1.57899i −0.0300393 0.0520296i
\(922\) 0 0
\(923\) 3.46341 0.113999
\(924\) 0 0
\(925\) 5.26461 + 9.11858i 0.173099 + 0.299817i
\(926\) 0 0
\(927\) 8.89702 + 15.4101i 0.292216 + 0.506134i
\(928\) 0 0
\(929\) −4.49844 + 7.79152i −0.147589 + 0.255631i −0.930336 0.366709i \(-0.880485\pi\)
0.782747 + 0.622340i \(0.213818\pi\)
\(930\) 0 0
\(931\) −11.6493 0.674135i −0.381790 0.0220939i
\(932\) 0 0
\(933\) −7.00594 + 12.1346i −0.229364 + 0.397270i
\(934\) 0 0
\(935\) −6.17087 10.6883i −0.201809 0.349543i
\(936\) 0 0
\(937\) −19.9187 34.5001i −0.650714 1.12707i −0.982950 0.183874i \(-0.941136\pi\)
0.332236 0.943196i \(-0.392197\pi\)
\(938\) 0 0
\(939\) −15.8691 −0.517868
\(940\) 0 0
\(941\) −5.92236 10.2578i −0.193063 0.334396i 0.753201 0.657791i \(-0.228509\pi\)
−0.946264 + 0.323395i \(0.895176\pi\)
\(942\) 0 0
\(943\) −8.12765 −0.264673
\(944\) 0 0
\(945\) 6.06383 10.5029i 0.197256 0.341658i
\(946\) 0 0
\(947\) −6.69404 + 11.5944i −0.217527 + 0.376768i −0.954051 0.299643i \(-0.903132\pi\)
0.736524 + 0.676411i \(0.236466\pi\)
\(948\) 0 0
\(949\) 0.699307 0.0227005
\(950\) 0 0
\(951\) −9.43704 −0.306017
\(952\) 0 0
\(953\) −23.0994 + 40.0094i −0.748264 + 1.29603i 0.200390 + 0.979716i \(0.435779\pi\)
−0.948654 + 0.316315i \(0.897554\pi\)
\(954\) 0 0
\(955\) −6.70078 + 11.6061i −0.216832 + 0.375565i
\(956\) 0 0
\(957\) 19.8507 0.641682
\(958\) 0 0
\(959\) −2.16542 3.75062i −0.0699252 0.121114i
\(960\) 0 0
\(961\) 1.89541 0.0611424
\(962\) 0 0
\(963\) −0.918216 1.59040i −0.0295891 0.0512498i
\(964\) 0 0
\(965\) 5.34693 + 9.26116i 0.172124 + 0.298127i
\(966\) 0 0
\(967\) 11.8556 20.5345i 0.381251 0.660346i −0.609991 0.792409i \(-0.708827\pi\)
0.991241 + 0.132063i \(0.0421602\pi\)
\(968\) 0 0
\(969\) −4.85543 9.65898i −0.155979 0.310291i
\(970\) 0 0
\(971\) −17.2724 + 29.9166i −0.554296 + 0.960069i 0.443662 + 0.896194i \(0.353679\pi\)
−0.997958 + 0.0638748i \(0.979654\pi\)
\(972\) 0 0
\(973\) 9.44951 + 16.3670i 0.302937 + 0.524703i
\(974\) 0 0
\(975\) 0.142400 + 0.246645i 0.00456046 + 0.00789895i
\(976\) 0 0
\(977\) −52.4467 −1.67792 −0.838959 0.544195i \(-0.816835\pi\)
−0.838959 + 0.544195i \(0.816835\pi\)
\(978\) 0 0
\(979\) 23.3502 + 40.4437i 0.746275 + 1.29259i
\(980\) 0 0
\(981\) 39.9755 1.27632
\(982\) 0 0
\(983\) 7.01464 12.1497i 0.223732 0.387515i −0.732206 0.681083i \(-0.761509\pi\)
0.955938 + 0.293568i \(0.0948426\pi\)
\(984\) 0 0
\(985\) 3.50306 6.06747i 0.111617 0.193326i
\(986\) 0 0
\(987\) −16.5980 −0.528320
\(988\) 0 0
\(989\) 62.9834 2.00276
\(990\) 0 0
\(991\) 4.94168 8.55924i 0.156978 0.271893i −0.776800 0.629748i \(-0.783158\pi\)
0.933777 + 0.357854i \(0.116492\pi\)
\(992\) 0 0
\(993\) 6.95647 12.0490i 0.220757 0.382362i
\(994\) 0 0
\(995\) 20.7818 0.658828
\(996\) 0 0
\(997\) 21.2210 + 36.7559i 0.672077 + 1.16407i 0.977314 + 0.211795i \(0.0679310\pi\)
−0.305237 + 0.952276i \(0.598736\pi\)
\(998\) 0 0
\(999\) −41.0490 −1.29873
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 1520.2.q.m.961.3 8
4.3 odd 2 380.2.i.c.201.2 yes 8
12.11 even 2 3420.2.t.w.3241.4 8
19.7 even 3 inner 1520.2.q.m.881.3 8
20.3 even 4 1900.2.s.d.49.3 16
20.7 even 4 1900.2.s.d.49.6 16
20.19 odd 2 1900.2.i.d.201.3 8
76.7 odd 6 380.2.i.c.121.2 8
76.11 odd 6 7220.2.a.r.1.3 4
76.27 even 6 7220.2.a.p.1.2 4
228.83 even 6 3420.2.t.w.1261.4 8
380.7 even 12 1900.2.s.d.349.3 16
380.83 even 12 1900.2.s.d.349.6 16
380.159 odd 6 1900.2.i.d.501.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.i.c.121.2 8 76.7 odd 6
380.2.i.c.201.2 yes 8 4.3 odd 2
1520.2.q.m.881.3 8 19.7 even 3 inner
1520.2.q.m.961.3 8 1.1 even 1 trivial
1900.2.i.d.201.3 8 20.19 odd 2
1900.2.i.d.501.3 8 380.159 odd 6
1900.2.s.d.49.3 16 20.3 even 4
1900.2.s.d.49.6 16 20.7 even 4
1900.2.s.d.349.3 16 380.7 even 12
1900.2.s.d.349.6 16 380.83 even 12
3420.2.t.w.1261.4 8 228.83 even 6
3420.2.t.w.3241.4 8 12.11 even 2
7220.2.a.p.1.2 4 76.27 even 6
7220.2.a.r.1.3 4 76.11 odd 6